mm-1070
 
===
Subject: Re: Turing Machines and Physical Computation
> The following are taken from a series of messages posted internally
> within a closed network back in 1990. I thought they might be worth
> citing once again (although without their original context for reasons
> which some of the more thoughtful here will understand). 
I doubt that anyone can guess why you posted this here again.
> A lot has been
> said here since. Most folk here won't see a fraction of what it's all
> about for many reasons that I've already covered. Some might be wise to
> take their emotional responses as something to further act upon.
Well i went through it again in detail.  I understand most of the 
pieces.  I  have commented on what i got from each in this context.  In 
one case i have expressed my own emotion as i felt it was an appropriate 
  dialectic to Quine's (and your) own emotional judgment.  However, why 
you have concatenated these particular pieces together and included your 
own piece about naloxone, i cannot guess.
> ON THE COMPUTER METAPHOR
> 'It   has  always  bothered  me  that  models  of   psychological
> processing  were thought to be inspired by our  understanding  of
> the  computer. The statement has always been false.  Indeed,  the
> architecture  of the modern digital computer - the so-called  Von
> Neumann architecture - was heavily influenced by people's (naive)
> view  of  how the mind operated. Perhaps I had better document
> this.  Simply  read the work on cybernetics and  thought in the
> 1940's  and  1950's  prior  to the  development of the digital
> computer.   The  group  of  workers  included people from all
> disciplines:  See  the Macy Conferences on Cybernetics,  or Her
> Majesty's  Conference on Thought processes. Read the preface  to
> Wiener's book on cybernetics. Everyone who was working together -
> engineers,     physicists,     mathematicians,     psychologists,
> neuroscientists  (not yet named) - consciously  and  deliberately
> claimed to be modelling brain processes.'
> Reflections on Cognition and Parallel Distributed Processing
> D.A. Norman
> (Ch 26, p534, Parallel Distributed Processing Volume 2)
> McClelland J and Rumelhart D 1986
I get that computer behavior and human behavior are essentially 
different.  Why they are so different is certainly what we came here to 
discuss.
>     'A  trait  is  EFFECTIVE if there is  a  hard  and  fast
>     routine by which we can check for it, without  guesswork
>     or imagination...It came to be appreciated, in the  mid-
>     thirties, that recursiveness affords a sharp explication
>     of  effectiveness. This has come to be  called  Church's
>     Thesis....By its nature, Church's Thesis was not open to
>     formal proof; for the thesis equated a precise property,
>     recursiveness,  with a property - effectiveness  -  that
>     was  to be rendered precise only by the  thesis  itself.
>     But the thesis was supported by such instances as  could
>     be  mustered,  and soon it was pretty well  clinched  by
>     Alan  Turing's  pioneer work in the abstract  theory  of
>     computing   machines.  His  formulation  of   mechanical
>     computability,  in terms of ideal mechanization,  turned
>     out  to  be  equivalent  to  recursiveness.   Mechanical
>     computability,  surely, is very much what our  intuitive
>     talk  of  effectiveness  was aiming  at  all  along;  so
>     Church's Thesis is well sustained.'
>     -
>     Quine (1987)
>     Recursion
?
>     'The  first  three  chapters actually grew  out  of  two
>     earlier  papers.  Those papers were, in  part,  polemics
>     against  the views of my good friend and  student  Jerry
>     Fodor. Fodor I hasten to say, is not the main target  of
>     this  book;  but  I have retained  some  of  my  polemic
>     against what I call MIT mentalism...  The main  target
>     of  the present book is one H Putnam (one of  my  former
>     selves) and those who have adopted his views. Or perhaps
>     it  would be more accurate to say that the present  book
>     doesn't have a main target; for its aim is not so much
>     to  refute one particular view as to establish the  need
>     for a different way of looking at problems about mental
>     states. At any rate, the intended contribution of these
>     three  chapters to that end is to do two things: (1)  to
>     establish a close connection (discovered and  emphasised
>     throughout  his  career by W V Quine)  between  problems
>     about  meaning  and problems about belief  fixation,  by
>     showing  that the holistic character of belief  fixation
>     in science bears deeply on the issue of individuation of
>     meanings  (or contents or intentions, as they  are
>     called  by various philosophers; and (2) to argue  that,
>     in  fact,  thinking  of meanings  (or  contents)  as
>     theoretical entities - as scientific objects,  objects
>     which can be isolated and which can play an  explanatory
>     role in scientific theory - is a mistake. In the  course
>     of  the  argument  I defend the view that  there  is  no
>     criterion   for  sameness  of  meaning   except   actual
>     interpretative  practice - a view made famous  by  Quine
>     and Davidson'
>     H Putnam (1988)
>     Representation and Reality
for sameness of meaning except actual interpretative practice, which 
works fine for me.  I then go a bit beyond that quote to observe that 
interpretive practice *establishes* meaning [*].  I then fail to see 
how meaning would not be relative to the agent practicing the 
interpretation.  Yet, i believe, you have denied that implication.   Now 
i do see how my quote [*] would be considered nonsense, were it our 
agenda to excise the very concept called meaning from our ontology. 
Sans that agenda, however, it makes total sense.  In fact we could 
define meaning as that which interpretive practice establishes, or 
perhaps more precisely:  meaning is that which changes as a result of 
interpretive practice.  One, of course, hopes that the agenda is not 
going to be used to justify the agenda.  Is it?
Incidentally that dispute is the only connection that i can find between 
your collage here and recent context in c.a.p.
>     'We  cannot  individuate concepts  and  beliefs  without
>     reference   to the ENVIRONMENT. Meanings aren't in  the
>     head.
>     The upshot of our discussion for the philosophy of  mind
>     is that propositional  attitudes,  as philosophers  call
>     them -  that  is, such  things  as  'believing that snow
>     is   white'  and  'feeling certain  that  the cat is on
>     the  mat' - are not states of the human brain and
>     nervous system considered in isolation from  the  social
>     and   nonhuman   environment.  A fortiori they are
>     not  functional  states  -  that is, states  definable
>     in   terms   of  parameters  which would  enter  into  a
>     software description of  the organism.  FUNCTIONALISM,
>     CONSTRUED AS THE THESIS THAT PROPOSITIONAL
>     ATTITUDES ARE JUST  COMPUTATIONAL STATES OF
>     THE BRAIN,  CANNOT BE CORRECT'.
>     The   arguments  I  just summarised were,  it  might  be
>     pointed  out  in  this  connection,  arguments   against
>     methodological solipsism.
>     H. Putnam (1988)
>     'Representation and Reality'
>     (Professor of Mathematical Logic Harvard)
Again we get it that meanings are not in the head;  rather they are 
established by interpretive practice.  The practice is public behavior 
which is not in the head.
>     -
>     'I subscribe entirely to these sentences of Count Verri:
>     On  the  Nature of Pleasure and Pain:  The  only  moving
>     principle of man is pain. Pain precedes every  pleasure.
>     Pleasure is not a positive state.'
>     -
>     Immanuel Kant 1781
>     -
>     -
>     'Meanwhile  our eager-beaver researcher,  undismayed  by
>     logic-of-science  considerations and relying  blissfully
>     on  the exactitude of modern  statistical  hypothesis-
>     testing,  has produced a long publication list and  been
>     promoted  to  a  full professorship.  In  terms  of  his
>     contribution  to  the  enduring  body  of  psychological
>     knowledge,  he  has  done  hardly  anything.  His   true
>     position  is that of a  potent-but-sterile  intellectual
>     rake,  who  leaves  in his merry path a  long  train  of
>     ravished maidens but no scientific offspring.'
>     -
>     P. E. Meehl (1967)
>     Theory Testing in Psychology and Physics
>     Philosophy of Science pp 103-115
I wonder why this quote was included here or what its context is.
> C56.
> Naloxone enhances neophobia
> J.F.W. DEAKIN & D.C. LONGLEY*
> (introduced by T.J. Crow)
> National Institute for Medical Research,
> Mill Hill, London, NW7 1AA
> Several   studies  report  that  naloxone,  an  opiate   receptor
> antagonist,  reduces  deprivation induced  eating  and  drinking.
> However,  in  the present study, naloxone (5mg/kg,i.p.)  did  not
> reduce food intake of rats maintained on a 22 h deprivation - 2 h
> feeding   schedule.   In  contrast,   naloxone   (5   mg/kg,i.p.)
> progressively reduced water intake in deprived animals to 46%  of
> saline  treated controls. No effects of naloxone (1, 5 mg/kg)  on
> established  bar  pressing for food or water were  observed  with
> either  continuous  or fixed ratio  schedules  of  reinforcement.
> However,  naloxone (5mg/kg) accelerated extinction of  responding
> when food and water were no longer available.
> Animals  treated  with naloxone (5mg/kg) during training  of  the
> bar-pressing  ate  only  26% of  the  pellets  delivered  whereas
> controls  ate  all pellets delivered. Since the animals  had  not
> previously experienced the pellets or the operant apparatus,  the
> possibilities  arose that naloxone effects were due  to  enhanced
> neophobic  effects of the novel food pellets or  novel  apparatus
> cues, or were due to conditioned taste aversion. Therefore,  food
> novelty,   apparatus  novelty  and  timing  of  injections   were
> independently  varied  in different groups of 8-10  rats  treated
> with saline or naloxone. Rats were maintained at 85% body  weight
> with 12g lab chow per day. On experimental days 46 small  pellets
> (Cambden  instruments) were placed on a small petri dish  in  the
> home  cage of some groups or released from a pellet dispenser  in
> an  operant box for other groups. The dependent variable was  the
> number of pellets eaten over 15 minutes.
> Naloxone (1,5 mg/kg i.p.) injected 5 or 20 min before test almost
> completely  suppressed pellet eating if the animals had not  been
> previously  exposed  to the pellets (p<0.01 't'  test  vs  saline
> groups).  This  occurred  independently  of  whether  tests  were
> carried  out  in  the home cage or novel  operant  box.  Naloxone
> induced  suppression  of  pellet  eating  was  almost  completely
> abolished  in either environment if animals had been  exposed  to
> the pellets for the five preceding days in the same or  different
> environment.  Naloxone  (5mg/kg, i.p.)  administered  immediately
> after  pellet eating tests failed to suppress  subsequent  pellet
> eating.
> Thus, naloxone suppressed pellet eating if the pellets were novel
> and if naloxone was administered before eating tests. The results
> suggest  naloxone enhances neophobic effects of novel  foods  and
> that  suppression of novel pellet eating is not due  to  enhanced
> effects  of  novelty of apparatus cues or  to  conditioned  taste
> aversion.
> Reference
> FRENK, H & ROGERS G.H. (1979) The suppressant effects of naloxone  on
> food and water intake in the rat.
> Behav. Neural. Biol, 26, 23-40.
Hmmmm ... why was this included ?
> PROCEEDINGS OF THE BRITISH PHARMACOLOGICAL SOCIETY (BPS) 1-3 April 1981
> (Also British J Pharmacology 1981)
>     'A  word  now about similarity:  subjective  similarity,
>     which is crucial to the learning process. I don't  think
>     anyone has an innate notion of similarity. What one  has
>     incontestably   is  an  innate   subjective   behavioral
>     standard  of similarity. It can be tested in people  and
>     other animals by conditioning.
>     -
>     It   is   unfortunate  that  my  phrase   'standard   of
>     similarity'  suggests judgment or deliberate  comparison
>     on  the subject's part, but I am at a loss  for  another
>     word.    What   is   afoot   is    just    conditioning,
>     discrimination,  reinforcement, extinction. It  is  what
>     Stemmer  and  some other psychologists treat  under  the
>     head  of 'generalization class', but I prefer  to  allow
>     differences  of degree. If a subject is rewarded  for  a
>     response  to one stimulation and penalized for the  same
>     response  to  another,  then  a  third  stimulation   is
>     subjectively  more  similar  to the first  than  to  the
>     second, for him, if it elicits the response.
... or he is being deceitful, or is no longer interested in responding
... or has interpreted the third stimulus in a different context
... or in other words examining external behavior alone is not a 
scientific method for determining the subjective similarity of the 
stimulus ... hmmm, you go into that (p, p*, p**) in detail below ?
>     -
>     Since it is basic to the mechanism of any learning,  any
>     conditioning,  a  similarity standard must be  there  to
>     start  the  first  learning. That is why  I  say  it  is
>     innate.   But  our  similarity  standards  evolve   very
>     decidedly  as we learn, and that is part of what  I  was
>     dealing with in 'Natural Kinds'......
>     .....
>     I first took up subjective similarity under the head  of
>     'quality space' on page 85 of WORD AND OBJECT. In  ROOTS
>     OF REFERENCE I went into it more fully under the head of
>     'perceptual similarity'. I was concerned with it for its
>     role in tying stimulations to observation sentences,  as
>     a basis for a theory of evidence...'
>     -
>     Quine
>     'Comment on Hacking'
>     In Perspectives on Quine (1990)
>     -
Hmmmm ... why was this included ?
> been  a lot of interest in the foundations of mathematical  logic
> and  the concepts of 'identity',  'analyticity',  'synonymy'  and
> 'similarity'.
> 'The  Morning Star' and 'Venus', 'the victor at  Jena'  and  'the
> vanquished  at  Waterloo', and even  'consciousness'  and  'brain
> processes'  have all  occupied philosophers  concerned  with  the
> relationship between Sense and  Meaning, the  'is' of definition'
> and the 'is' of  composition, 'Use  & Mention'  over the past 100
> years.
> Oedipus believed that Jocasta was fair game, but not his  mother.
> Ryle's foreigner saw all the buildings and grounds of Oxford, but
> hadn't 'seen' the university. Jenny accepts that James has seen a
> good range of her behaviour but doesn't believe he knows the real
> 'her'.
> Ryle offered a solution to part of this problem in 1949 with  his
> example  of  the category mistake, ie  that  'mind'  like  'the
> university'  is a concept or category which includes a number  of
> members,  distributed  elements, or dispositions to  behave.  (In
> Quine we see these being noted as observation sentences, or  more
> specifically, as occasion sentences, dated and timed).
> But there are a few problems here. If we say that these  elements
> fall  under  a particular class, the elements of that  class  are
> themselves not identical, nor even similar in sense. It may  well
> be that amongst Jenny's dispositions we include 'cake maker'  and
> 'garden  tender'  as well as 'teacher', but we would  rarely  say
> that these are identical or even similar behaviours any more than
> we  would say that the library is the same as or similar  to  the
> refectory.  They are similar or associated only in that they  are
> elements  of a  class, either 'Jenny'  or 'university' -  just as
> different  people can be said to be similar in terms of a  higher
> type,  such  as  female, or human, or position  in  an  actuarial
> table.  However, we would acknowledge that 'rabbit',  'undetached
> part  of  rabbit',  and  'stage of  a  rabbits  development'  are
> interchangeable   in  identifying  the  same   referent,   'salve
> veritate'. This is the thesis of indeterminacy of translation,  a
> thesis   which  leads  into  another  Quinean  thesis,  that   of
> Ontological Relativity - which dispenses with the intensional.
>     -
What Quine actually says (see below) was each of us is free to 
internalize in his peculiar neural way ... apparently this ends up in 
your mind as dispenses with the intensional.  I don't take dispenses 
with as being the same as free to ... for me there is a grand 
distinction.
>     'The view that I have come to, regarding intersubjective
>     likeness of stimulation, is rather that we can simply do
>     without  it. The observation sentence 'Rabbit'  has  its
>     stimulus  meaning for the linguist, and the  observation
>     sentence  'Gavagai'  has its stimulus  meaning  for  the
>     informant.  The linguist, observes natives assenting  to
>     'Gavagai'  when  he,  in  their  position,  would   have
>     assented to 'Rabbit'. So he tries assigning HIS stimulus
>     meaning of 'Rabbit' to 'Gavagai' on subsequent occasions
>     for his informant's approval. Encouraged, he tentatively
>     adopts 'Rabbit' as translation.'.........
>     -
>     Discussion  with  Dreben  helped  me  to  clarify  these
>     consequences of my new stance. In WORD AND OBJECT I  had
>     already  pointed out that communication  presupposes  no
>     similarity in nerve nets; verbal behavior is  inculcated
>     on the strength only of surface stimulation. Such was my
>     parable  of the trimmed bushes (p.8), alike  in  outward
>     form  but  wildly  unlike  in  their  inward  twigs  and
>     branches.  Save the surface, in the paintmaker's  words,
>     and you save all. But now, leaving the surface itself to
>     Sherwin-Williams' tender mercies, I give the  individual
>     yet wider berth. His privacy widens apace.
>     -
>     Unlike  Davidson,  I  leave  the  stimulations  at   the
>     subject's  surface,  and private stimulus  meaning  with
>     them. But they may be as idiosyncratic, for all I  care,
>     as the subject's internal wiring itself. What floats  in
>     the open air is our common language, which each of us is
>     free to internalize in his peculiar neural way. Language
>     is  where  intersubjectivity sets in.  Communication  is
>     well named.'
>     -
>     Quine (1990)
>     Three Indeterminacies
In other words (see comment after Putman above) ... Meanings are not in 
the head;  rather they are established by interpretive practice.  The 
practice is public behavior which is not in the head.
> There  are pervasive problems with 'properties',  'essences',  or
> 'propositions'  and their intensional kin. Nowhere is  this  more
> apparent  than with the 'propositional attitudes'. The  linch-pin
> is the resistance of these to the principle of 'substitutivity of
> identity'  'salve  veritate', and in anything scientific  we  are
> exclusively    interested   in   truth   functions   where    the
> substitutivity of identity is guaranteed. Failure to respect this
> principle  results  in invalid reasoning through the  fallacy  of
> equivocation.
> In the case of propositional attitudes, we can not quote  someone
> indirectly  without thereby uttering an untruth, and we  can  not
> make  statements  about  what someone  believes,  thinks,  hopes,
> fears,  or understands except in the actual context in which  the
> propositional  attitudes expressing such intensional idioms  take
> place (which amounts to quoting them directly and  contextually).
> If  one reports to someone else 'what someone said', one  has  to
> either directly quote (verbatim) or else acknowledge that one  is
> not  making  a  report  at all. Such  paraphrase  amounts  to  an
> 'interpretation' which is an imputation or inference, a  creative
> act
.. or the acknowledgment of a successful communication ...
> .  Propositional  attitudes are not  projectible outside the
> immediate context of their utterance, so: -
>     'Reports' are not 'bona fide' reports at all unless they
>     are  reports verbatim (ie records of behaviour)  -  they
>     become  interpretations through a process of  imputation
>     at best, and in almost all cases, they are creative acts
>     which  do not permit substitutivity of  identity  'salve
>     veritate'.   They   are  non-truth   functional   unless
>     expressed  actuarially  via  relation  to  a  population
>     distribution (normative psychometric measure).
Incidentally i agree with Quine that for a journalistic report, direct 
quotes in context are strongly indicated.  Paraphrase, however is useful 
as an acknowledgment of a successful communication and as a curative 
means to add additional interpretive practice.  These become the very 
acts which create our language distinguishing it from bird calls! 
Emotionally i love it when someone *correctly* paraphrase me, for it is 
proof that i have communicated to them ... i no longer need to merely 
guess ... i have finally cast my bottle into the ocean of doubt and 
ambiguity and some other creature has discovered it :)  When they can 
add a new idea not contained or imagined in my original, then i must 
leap for joy as the very purpose of my struggle with these 
multi-interpretive marks has finally become a step (however minor) in 
the cultural dance.  So i must find Quine's judgmental comment, at best 
they are creative acts, to be sourly lacking in insight.
Unfortunately i don't have time to guess what connects all of the above 
to what you say below.
> This  is, I argue, why interviews, exams and similar devices  for
> measuring  behaviour (e.g. minute taking in meetings) are  widely
> subjected  to the  criticism that they are not reliable  measures
> of  behaviour,  particularly when such measures are  supposed  to
> comprise important elements in a human regression process (review
> board) which  seeks to make inferences or projections about other
> behaviour. It is also why continuous recording of what happens is
> more  promising, since such measures provide more  representative
> samples  under naturalistic conditions (ie the settings are  more
> prone to generalise) - Such a record is a profile of behaviour.
> These  principles are observable in the processes of operant  and
> classical   conditioning.  In  the  latter  paradigms,   one   is
> interested  in  the basic processes  of  association  ascertained
> after  the acquisition process through observation  of  behaviour
> following  the  presentation  of  a  CS  (the  whole  process  is
> therefore one of recognition or recall). The touchstone of  rival
> models  is the 'blocking effect' (Kamin 1968)  which  illustrates
> that  elements  of a stimulus compound may come  not  to  control
> behaviour  unless  those  elements  have  come  to  provide   new
> information, ie change the conditional probability of the  event-
> event  relationship.  In operant acquisition, one  tends  not  to
> focus  on  the  response topography, but  records  the  class  of
> behaviours comprising lever pressing as the operant. But when one
> looks  at the extinction of this class of behaviours why does  it
> take  so  long  to  reach  criterion? May  it  not  be  that  the
> acquisition  of  the  operant is in fact no  different  from  the
> configuring process in classical conditioning, ie the acquisition
> process  is  the acquisition of multiple R-S  contingencies,  and
> extinction involves a testing of all the elements:
> In its most basic form each R-S* during acquisition differs, each
> is  a  slightly different 'perspective'. This  might  comprise  a
> slight  motor  variation  on  a CRF task,  such  as  a   slightly
> different pressure on the lever above threshold, or angle of  the
> body, in other cases it may be the ratio of presses on a FR or VR
> schedule  (which  will increase resistance to  extinction).  Note
> that  the element of behaviour here is lever press (p or p*)  and
> the outcome is pellet delivery (q) ie
>                    if 'p'ress then 'q'ualia.
> Each CRF trial  RECORDS either 'p then q', 'p* then q', 'p** then
> q'  and  so  on.  In fact  every  supra-criterion  R-S  variation
> possible,  leads to a testing of almost all of the  contingencies
> during extinction :
>                   not 'q'ualia then not 'p'ress
> but...perhaps  it will be  p*.... or perhaps p**, so each has  to
> be tested/subjected to falsification (negation). After all,  each
> variation or 'perspective' was reinforced during acquisition.  On
> partial reinforcement schedules we interleave a little extinction
> or impossible discrimination training.
> Do each of these behaviours, dispositions, or properties, learned
> only  in their particular context, occupy their own  location  or
> node in the operant repertoire, only becoming extensional through
> the  eyes of the observer? Or are they really configured  into  a
> class   of  such  behaviours?.  Surely  all  trainers  have   had
> experience  of  this  process whereby the  basic  elements  of  a
> complex skill may be taught but the trainee just fails to pull it
> all  together  coherently, and surely all assessors have had  the
> problem of deciding whether the student 'really knows'.
> The  same processes seem to be operating in 'group work'  of  all
> sorts.  Here,  as  with  performance on  a  test,  the  behaviour
> occurring in such settings, (which is predominantly verbal),  may
> not  generalise  to  other situations  or  be  representative  of
> general  performance  (other than to other similar  behaviour  in
> groups or interviews!). Such context effects have been central to
> Cognitive  Psychology for decades e.g. 'the encoding  specificity
> principle'  of  Tulving and Thompson. Such processes seem  to  be
> quite  basic to the very nature of habit formation (cf. notes  on
> novelty,  opioids & habit formation). Similarly, if 'role  plays'
> had  any substantial (generalisable) effects on  behaviour,  then
> many  of our oscar winning actors would surely be more likely  to
> be  roaming  cities killing and maiming, (or  living  in  idyllic
> relationships). Acting is no less real or true behaviour than  is
> any other.
> David Longley
> July 7, 1991
patty
===
Subject: Re: Turing Machines and Physical Computation
Selected message from thread 
 
===
Subject: Re: Turing Machines and Physical Computation 
 
  
 
[...]
>>  ON THE COMPUTER METAPHOR
>>  'It   has  always  bothered  me  that  models  of   psychological
>> processing  were thought to be inspired by our  understanding  of
>> the  computer. The statement has always been false.  Indeed,  the
>> architecture  of the modern digital computer - the so-called  Von
>> Neumann architecture - was heavily influenced by people's (naive)
>> view  of  how the mind operated. Perhaps I had better document
>> this.  Simply  read the work on cybernetics and  thought in the
>> 1940's  and  1950's  prior  to the  development of the digital
>> computer.   The  group  of  workers  included people from all
>> disciplines:  See  the Macy Conferences on Cybernetics,  or Her
>> Majesty's  Conference on Thought processes. Read the preface  to
>> Wiener's book on cybernetics. Everyone who was working together -
>> engineers,     physicists,     mathematicians,     psychologists,
>> neuroscientists  (not yet named) - consciously  and  deliberately
>> claimed to be modelling brain processes.'
>>  Reflections on Cognition and Parallel Distributed Processing
>>  D.A. Norman
>> (Ch 26, p534, Parallel Distributed Processing Volume 2)
>> McClelland J and Rumelhart D 1986
>I get that computer behavior and human behavior are essentially 
>different.  Why they are so different is certainly what we came
> here to discuss.
Computers don't have behaviours only the programs running on them.
These models of the brain (programs) behave differently because
they are incomplete emulations of how brains process sensory data.
John
===
Subject: Re: Turing Machines and Physical Computation
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Another one from April 1991 (10 years after the naloxone paper to the 
British J. Pharmacology). Let's see if anyone can make the connection. 
Hint: what might this have to do with 'predicting stimulation of our 
sensory surfaces'? Clue: what do we find in all our primary sensory 
nuclei? What has this got to do with physical computation and the 
extensional stance?
'INTUITION', METAPHYSICS AND PSYCHOLOGY
'There is  another  difficulty with intuitions,  too,  much  more
serious than the suspicion that the universe is having a colossal
joke at our expense. That difficulty is that intuition  justifies
too  much. It justifies, for example, a naive set theory  riddled
with paradox. It justifies, apparently, Euclidean geometry at the
expense  of  consistent  and  useful  alternatives.  And  because
intuitions  notoriously differ from person to person, it can  and
does  justify  theories that conflict with one another.  It  even
justifies  logics that conflict with one another. It is far  from
obvious  that an appeal to intuition is better than just  leaving
the axioms wholly without justification. Indeed, our epistemology
would  be  equally good if, instead of relying on  intuition,  we
relied frankly and forthrightly on prejudice.
At  any rate, shortly before the turn of this  century  there
arose  the  beginning of an alternative.  Poincare  and  Hilbert,
perhaps  the most prominent mathematicians of their  day,  argued
that  the axioms of geometry and mathematics can be construed  as
implicit definitions of the terms they contain.'
Creath (1990)
Carnap, Quine and the Rejection of Intuition
Perspectives on Quine (1990)
-- 
David Longley
http://www.longley.demon.co.uk/Frag.htm
===
Subject: Re: Turing Machines and Physical Computation
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The following are taken from a series of messages posted internally
>> within a closed network back in 1990. I thought they might be worth
>> citing once again (although without their original context for reasons
>> which some of the more thoughtful here will understand).
>I doubt that anyone can guess why you posted this here again.
I know. That's why I posted it. You're supposed to do some work to make 
the connections.
>> A lot has been
>> said here since. Most folk here won't see a fraction of what it's all
>> about for many reasons that I've already covered. Some might be wise to
>> take their emotional responses as something to further act upon.
>Well i went through it again in detail.  I understand most of the 
>pieces.  I  have commented on what i got from each in this context. In 
>one case i have expressed my own emotion as i felt it was an 
>appropriate  dialectic to Quine's (and your) own emotional judgment.
Your affective behaviour is not the issue. You will easily be misled if 
you just assess what you read in terms of how it makes you feel.
>However, why you have concatenated these particular pieces together and 
>included your own piece about naloxone, i cannot guess.
The piece about naloxone is actually implicitly about the endogenous 
opiates, their receptors, and habit formation (also known as 
learning, although I don't use that term much. Instead I tend to talk 
of changing operant levels through reinforcement of behaviours). If you 
want some hints on where this goes you would need to look into the 
monoamines like dopamine, activity, the nucleus accumbens and the the 
mesolimbic system/ventral striatum. I said not long ago that there were 
two classes of responding (behaviour) which I found very dramatic a) 
ICSS and b) neophobia. Once you have looked into some of that, you might 
like to look into (primarily) young males and especially adolescents, 
testosterone, alcohol, and the above system, especially viz a viz the 
field I am always implicitly referring to (cf. PROBE).
>>  ON THE COMPUTER METAPHOR
>>  'It   has  always  bothered  me  that  models  of   psychological
>> processing  were thought to be inspired by our  understanding  of
>> the  computer. The statement has always been false.  Indeed,  the
>> architecture  of the modern digital computer - the so-called  Von
>> Neumann architecture - was heavily influenced by people's (naive)
>> view  of  how the mind operated. Perhaps I had better document
>> this.  Simply  read the work on cybernetics and  thought in the
>> 1940's  and  1950's  prior  to the  development of the digital
>> computer.   The  group  of  workers  included people from all
>> disciplines:  See  the Macy Conferences on Cybernetics,  or Her
>> Majesty's  Conference on Thought processes. Read the preface  to
>> Wiener's book on cybernetics. Everyone who was working together -
>> engineers,     physicists,     mathematicians,     psychologists,
>> neuroscientists  (not yet named) - consciously  and  deliberately
>> claimed to be modelling brain processes.'
>>  Reflections on Cognition and Parallel Distributed Processing
>>  D.A. Norman
>> (Ch 26, p534, Parallel Distributed Processing Volume 2)
>> McClelland J and Rumelhart D 1986
>I get that computer behavior and human behavior are essentially 
>different.  Why they are so different is certainly what we came here to 
>discuss.
And that's what Fragments is largely about as I have said many times.
>>       'A  trait  is  EFFECTIVE if there is  a  hard  and  fast
>>     routine by which we can check for it, without  guesswork
>>     or imagination...It came to be appreciated, in the  mid-
>>     thirties, that recursiveness affords a sharp explication
>>     of  effectiveness. This has come to be  called  Church's
>>     Thesis....By its nature, Church's Thesis was not open to
>>     formal proof; for the thesis equated a precise property,
>>     recursiveness,  with a property - effectiveness  -  that
>>     was  to be rendered precise only by the  thesis  itself.
>>     But the thesis was supported by such instances as  could
>>     be  mustered,  and soon it was pretty well  clinched  by
>>     Alan  Turing's  pioneer work in the abstract  theory  of
>>     computing   machines.  His  formulation  of   mechanical
>>     computability,  in terms of ideal mechanization,  turned
>>     out  to  be  equivalent  to  recursiveness.   Mechanical
>>     computability,  surely, is very much what our  intuitive
>>     talk  of  effectiveness  was aiming  at  all  along;  so
>>     Church's Thesis is well sustained.'
>>     -
>>     Quine (1987)
>>     Recursion
Thread title - but we call it rule governed behaviour - do you not get 
it yet? What is rule governed behaviour Patty? How does it relate to 
PROBE?
>>      'The  first  three  chapters actually grew  out  of  two
>>     earlier  papers.  Those papers were, in  part,  polemics
>>     against  the views of my good friend and  student  Jerry
>>     Fodor. Fodor I hasten to say, is not the main target  of
>>     this  book;  but  I have retained  some  of  my  polemic
>>     against what I call MIT mentalism...  The main  target
>>     of  the present book is one H Putnam (one of  my  former
>>     selves) and those who have adopted his views. Or perhaps
>>     it  would be more accurate to say that the present  book
>>     doesn't have a main target; for its aim is not so much
>>     to  refute one particular view as to establish the  need
>>     for a different way of looking at problems about mental
>>     states. At any rate, the intended contribution of these
>>     three  chapters to that end is to do two things: (1)  to
>>     establish a close connection (discovered and  emphasised
>>     throughout  his  career by W V Quine)  between  problems
>>     about  meaning  and problems about belief  fixation,  by
>>     showing  that the holistic character of belief  fixation
>>     in science bears deeply on the issue of individuation of
>>     meanings  (or contents or intentions, as they  are
>>     called  by various philosophers; and (2) to argue  that,
>>     in  fact,  thinking  of meanings  (or  contents)  as
>>     theoretical entities - as scientific objects,  objects
>>     which can be isolated and which can play an  explanatory
>>     role in scientific theory - is a mistake. In the  course
>>     of  the  argument  I defend the view that  there  is  no
>>     criterion   for  sameness  of  meaning   except   actual
>>     interpretative  practice - a view made famous  by  Quine
>>     and Davidson'
>>      H Putnam (1988)
>>     Representation and Reality
>for sameness of meaning except actual interpretative practice, which 
>works fine for me.  I then go a bit beyond that quote to observe that 
>interpretive practice *establishes* meaning [*].  I then fail to see 
>how meaning would not be relative to the agent practicing the 
>interpretation.  Yet, i believe, you have denied that implication. Now 
>i do see how my quote [*] would be considered nonsense, were it our 
>agenda to excise the very concept called meaning from our ontology. 
>Sans that agenda, however, it makes total sense.  In fact we could 
>define meaning as that which interpretive practice establishes, or 
>perhaps more precisely:  meaning is that which changes as a result of 
>interpretive practice.  One, of course, hopes that the agenda is not 
>going to be used to justify the agenda. Is it?
The term meaning is scientifically (and technologically) useless. It 
is mentalistic (intensional). That's why Quine talks of exorcising it. 
What do we deal with instead? We have been through this at great length, 
and you will see Quine say it very explicitly in his comment on Hacking. 
What work do you think he is referring to there?
>Incidentally that dispute is the only connection that i can find 
>between your collage here and recent context in c.a.p.
But all you are doing there is stating what you don't know. I know that. 
I've also suggested what you must do to change that (or are you now 
doing a Verhey etc on me?). It just doesn't do to tell people you can't 
see something. You are the one who has to change that not me.
>>      'We  cannot  individuate concepts  and  beliefs  without
>>     reference   to the ENVIRONMENT. Meanings aren't in  the
>>     head.
>>      The upshot of our discussion for the philosophy of  mind
>>     is that propositional  attitudes,  as philosophers  call
>>     them -  that  is, such  things  as  'believing that snow
>>     is   white'  and  'feeling certain  that  the cat is on
>>     the  mat' - are not states of the human brain and
>>     nervous system considered in isolation from  the  social
>>     and   nonhuman   environment.  A fortiori they are
>>     not  functional  states  -  that is, states  definable
>>     in   terms   of  parameters  which would  enter  into  a
>>     software description of  the organism.  FUNCTIONALISM,
>>     CONSTRUED AS THE THESIS THAT PROPOSITIONAL
>>     ATTITUDES ARE JUST  COMPUTATIONAL STATES OF
>>     THE BRAIN,  CANNOT BE CORRECT'.
>>      The   arguments  I  just summarised were,  it  might  be
>>     pointed  out  in  this  connection,  arguments   against
>>     methodological solipsism.
>>      H. Putnam (1988)
>>     'Representation and Reality'
>>     (Professor of Mathematical Logic Harvard)
>Again we get it that meanings are not in the head;  rather they are 
>established by interpretive practice.  The practice is public behavior 
>which is not in the head.
But people don't get it do they. They spend an awful lot of time in 
c.a.p and elsewhere showing that they don't get it. So do you, you just 
don't see it. Nearly all the posts to c.a.p tacitly assume it. When it's 
pointed out, those doing so don't see what they are doing and just deny 
that they are doing it! It's called lack of insight. This is why this 
discipline (behavioural science) is so difficult. Most of the 
mathematicians and computer scientist folk here and elsewhere are 
metaphysical never mind, methodological solipsists. I can see that, so 
could Glen. So could Skinner and Quine. You can't.
>>      -
>>     'I subscribe entirely to these sentences of Count Verri:
>>     On  the  Nature of Pleasure and Pain:  The  only  moving
>>     principle of man is pain. Pain precedes every  pleasure.
>>     Pleasure is not a positive state.'
>>     -
>>     Immanuel Kant 1781
>>     -
>>     -
>>     'Meanwhile  our eager-beaver researcher,  undismayed  by
>>     logic-of-science  considerations and relying  blissfully
>>     on  the exactitude of modern  statistical  hypothesis-
>>     testing,  has produced a long publication list and  been
>>     promoted  to  a  full professorship.  In  terms  of  his
>>     contribution  to  the  enduring  body  of  psychological
>>     knowledge,  he  has  done  hardly  anything.  His   true
>>     position  is that of a  potent-but-sterile  intellectual
>>     rake,  who  leaves  in his merry path a  long  train  of
>>     ravished maidens but no scientific offspring.'
>>     -
>>     P. E. Meehl (1967)
>>     Theory Testing in Psychology and Physics
>>     Philosophy of Science pp 103-115
>I wonder why this quote was included here or what its context is.
The Kant quote leads onto the neophobia abstract. The Meehl paper is a 
pointer to one of the segments of Fragments which covers the point in 
detail - it's a damning critique of mainstream psychology as well as a 
pointer to the actuarial vs. clinical section of Fragments
>> -
>> -
>> C56.
>> -
>> Naloxone enhances neophobia
>> -
>> J.F.W. DEAKIN & D.C. LONGLEY*
>> (introduced by T.J. Crow)
>> -
>> National Institute for Medical Research,
>> Mill Hill, London, NW7 1AA
>> -
>> Several   studies  report  that  naloxone,  an  opiate   receptor
>> antagonist,  reduces  deprivation induced  eating  and  drinking.
>> However,  in  the present study, naloxone (5mg/kg,i.p.)  did  not
>> reduce food intake of rats maintained on a 22 h deprivation - 2 h
>> feeding   schedule.   In  contrast,   naloxone   (5   mg/kg,i.p.)
>> progressively reduced water intake in deprived animals to 46%  of
>> saline  treated controls. No effects of naloxone (1, 5 mg/kg)  on
>> established  bar  pressing for food or water were  observed  with
>> either  continuous  or fixed ratio  schedules  of  reinforcement.
>> However,  naloxone (5mg/kg) accelerated extinction of  responding
>> when food and water were no longer available.
>> -
>> Animals  treated  with naloxone (5mg/kg) during training  of  the
>> bar-pressing  ate  only  26% of  the  pellets  delivered  whereas
>> controls  ate  all pellets delivered. Since the animals  had  not
>> previously experienced the pellets or the operant apparatus,  the
>> possibilities  arose that naloxone effects were due  to  enhanced
>> neophobic  effects of the novel food pellets or  novel  apparatus
>> cues, or were due to conditioned taste aversion. Therefore,  food
>> novelty,   apparatus  novelty  and  timing  of  injections   were
>> independently  varied  in different groups of 8-10  rats  treated
>> with saline or naloxone. Rats were maintained at 85% body  weight
>> with 12g lab chow per day. On experimental days 46 small  pellets
>> (Cambden  instruments) were placed on a small petri dish  in  the
>> home  cage of some groups or released from a pellet dispenser  in
>> an  operant box for other groups. The dependent variable was  the
>> number of pellets eaten over 15 minutes.
>> -
>> Naloxone (1,5 mg/kg i.p.) injected 5 or 20 min before test almost
>> completely  suppressed pellet eating if the animals had not  been
>> previously  exposed  to the pellets (p<0.01 't'  test  vs  saline
>> groups).  This  occurred  independently  of  whether  tests  were
>> carried  out  in  the home cage or novel  operant  box.  Naloxone
>> induced  suppression  of  pellet  eating  was  almost  completely
>> abolished  in either environment if animals had been  exposed  to
>> the pellets for the five preceding days in the same or  different
>> environment.  Naloxone  (5mg/kg, i.p.)  administered  immediately
>> after  pellet eating tests failed to suppress  subsequent  pellet
>> eating.
>> -
>> Thus, naloxone suppressed pellet eating if the pellets were novel
>> and if naloxone was administered before eating tests. The results
>> suggest  naloxone enhances neophobic effects of novel  foods  and
>> that  suppression of novel pellet eating is not due  to  enhanced
>> effects  of  novelty of apparatus cues or  to  conditioned  taste
>> aversion.
>> -
>> Reference
>> -
>> FRENK, H & ROGERS G.H. (1979) The suppressant effects of naloxone  on
>> food and water intake in the rat.
>> Behav. Neural. Biol, 26, 23-40.
>Hmmmm ... why was this included ?
As I said, habit formation. Those in the EAB consider the analysis of 
the control of operant behaviour to be the analysis of intelligent 
behaviour. We see your computer rule governed behaviour (programming) 
as only a part of this. We see computer scientists as naive and 
misguided technicians in this respect when they speak of AI as they 
get behaviour wrong. They tend to be pre 1929 Carnapian as I have said 
before, and they won't be told that they have simply got their facts 
wrong (I have explained this before - it's a factual error as clear as 
pointing out to someone that they are wrong when they say that snow is 
black).
>> -
>> PROCEEDINGS OF THE BRITISH PHARMACOLOGICAL SOCIETY (BPS) 1-3 April 1981
>> (Also British J Pharmacology 1981)
>>      'A  word  now about similarity:  subjective  similarity,
>>     which is crucial to the learning process. I don't  think
>>     anyone has an innate notion of similarity. What one  has
>>     incontestably   is  an  innate   subjective   behavioral
>>     standard  of similarity. It can be tested in people  and
>>     other animals by conditioning.
>>     -
>>     It   is   unfortunate  that  my  phrase   'standard   of
>>     similarity'  suggests judgment or deliberate  comparison
>>     on  the subject's part, but I am at a loss  for  another
>>     word.    What   is   afoot   is    just    conditioning,
>>     discrimination,  reinforcement, extinction. It  is  what
>>     Stemmer  and  some other psychologists treat  under  the
>>     head  of 'generalization class', but I prefer  to  allow
>>     differences  of degree. If a subject is rewarded  for  a
>>     response  to one stimulation and penalized for the  same
>>     response  to  another,  then  a  third  stimulation   is
>>     subjectively  more  similar  to the first  than  to  the
>>     second, for him, if it elicits the response.
>... or he is being deceitful, or is no longer interested in responding
>... or has interpreted the third stimulus in a different context
>... or in other words examining external behavior alone is not a 
>scientific method for determining the subjective similarity of the 
>stimulus ... hmmm, you go into that (p, p*, p**) in detail below ?
The above makes no sense at all as you mix languages.
>>     -
>>     Since it is basic to the mechanism of any learning,  any
>>     conditioning,  a  similarity standard must be  there  to
>>     start  the  first  learning. That is why  I  say  it  is
>>     innate.   But  our  similarity  standards  evolve   very
>>     decidedly  as we learn, and that is part of what  I  was
>>     dealing with in 'Natural Kinds'......
>>     .....
>>     I first took up subjective similarity under the head  of
>>     'quality space' on page 85 of WORD AND OBJECT. In  ROOTS
>>     OF REFERENCE I went into it more fully under the head of
>>     'perceptual similarity'. I was concerned with it for its
>>     role in tying stimulations to observation sentences,  as
>>     a basis for a theory of evidence...'
>>     -
>>     Quine
>>     'Comment on Hacking'
>>     In Perspectives on Quine (1990)
>>     -
>Hmmmm ... why was this included ?
I've already explained. I've also explained before that the fact that 
what goes on inside the head can vary dramatically between individuals 
should be taken as a clear indication that it doesn't matter! The point 
here is that hoards of people are discussing things that really don't 
make any difference. We are talking about whole professions here, not 
a few posters to USENET.
>> been  a lot of interest in the foundations of mathematical  logic
>> and  the concepts of 'identity',  'analyticity',  'synonymy'  and
>> 'similarity'.
>> -
>> 'The  Morning Star' and 'Venus', 'the victor at  Jena'  and  'the
>> vanquished  at  Waterloo', and even  'consciousness'  and  'brain
>> processes'  have all  occupied philosophers  concerned  with  the
>> relationship between Sense and  Meaning, the  'is' of definition'
>> and the 'is' of  composition, 'Use  & Mention'  over the past 100
>> years.
>> -
>> Oedipus believed that Jocasta was fair game, but not his  mother.
>> Ryle's foreigner saw all the buildings and grounds of Oxford, but
>> hadn't 'seen' the university. Jenny accepts that James has seen a
>> good range of her behaviour but doesn't believe he knows the real
>> 'her'.
>> -
>> Ryle offered a solution to part of this problem in 1949 with  his
>> example  of  the category mistake, ie  that  'mind'  like  'the
>> university'  is a concept or category which includes a number  of
>> members,  distributed  elements, or dispositions to  behave.  (In
>> Quine we see these being noted as observation sentences, or  more
>> specifically, as occasion sentences, dated and timed).
>> -
>> But there are a few problems here. If we say that these  elements
>> fall  under  a particular class, the elements of that  class  are
>> themselves not identical, nor even similar in sense. It may  well
>> be that amongst Jenny's dispositions we include 'cake maker'  and
>> 'garden  tender'  as well as 'teacher', but we would  rarely  say
>> that these are identical or even similar behaviours any more than
>> we  would say that the library is the same as or similar  to  the
>> refectory.  They are similar or associated only in that they  are
>> elements  of a  class, either 'Jenny'  or 'university' -  just as
>> different  people can be said to be similar in terms of a  higher
>> type,  such  as  female, or human, or position  in  an  actuarial
>> table.  However, we would acknowledge that 'rabbit',  'undetached
>> part  of  rabbit',  and  'stage of  a  rabbits  development'  are
>> interchangeable   in  identifying  the  same   referent,   'salve
>> veritate'. This is the thesis of indeterminacy of translation,  a
>> thesis   which  leads  into  another  Quinean  thesis,  that   of
>> Ontological Relativity - which dispenses with the intensional.
>>     -
>What Quine actually says (see below) was each of us is free to 
>internalize in his peculiar neural way ... apparently this ends up in 
>your mind as dispenses with the intensional.  I don't take dispenses 
>with as being the same as free to ... for me there is a grand 
>distinction.
You've missed the point, along with most folk here. This is a radical 
failing on all of your part. It prevents you from grasping just how 
profound the EAB revolution was. Skinner is up there with Darwin.
>>     'The view that I have come to, regarding intersubjective
>>     likeness of stimulation, is rather that we can simply do
>>     without  it. The observation sentence 'Rabbit'  has  its
>>     stimulus  meaning for the linguist, and the  observation
>>     sentence  'Gavagai'  has its stimulus  meaning  for  the
>>     informant.  The linguist, observes natives assenting  to
>>     'Gavagai'  when  he,  in  their  position,  would   have
>>     assented to 'Rabbit'. So he tries assigning HIS stimulus
>>     meaning of 'Rabbit' to 'Gavagai' on subsequent occasions
>>     for his informant's approval. Encouraged, he tentatively
>>     adopts 'Rabbit' as translation.'.........
>>     -
>>     Discussion  with  Dreben  helped  me  to  clarify  these
>>     consequences of my new stance. In WORD AND OBJECT I  had
>>     already  pointed out that communication  presupposes  no
>>     similarity in nerve nets; verbal behavior is  inculcated
>>     on the strength only of surface stimulation. Such was my
>>     parable  of the trimmed bushes (p.8), alike  in  outward
>>     form  but  wildly  unlike  in  their  inward  twigs  and
>>     branches.  Save the surface, in the paintmaker's  words,
>>     and you save all. But now, leaving the surface itself to
>>     Sherwin-Williams' tender mercies, I give the  individual
>>     yet wider berth. His privacy widens apace.
>>     -
>>     Unlike  Davidson,  I  leave  the  stimulations  at   the
>>     subject's  surface,  and private stimulus  meaning  with
>>     them. But they may be as idiosyncratic, for all I  care,
>>     as the subject's internal wiring itself. What floats  in
>>     the open air is our common language, which each of us is
>>     free to internalize in his peculiar neural way. Language
>>     is  where  intersubjectivity sets in.  Communication  is
>>     well named.'
>>     -
>>     Quine (1990)
>>     Three Indeterminacies
>> -
>In other words (see comment after Putman above) ... Meanings are not in 
>the head;  rather they are established by interpretive practice. The 
>practice is public behavior which is not in the head.
Yes. I've covered this from a number of perspectives over the years.
>> There  are pervasive problems with 'properties',  'essences',  or
>> 'propositions'  and their intensional kin. Nowhere is  this  more
>> apparent  than with the 'propositional attitudes'. The  linch-pin
>> is the resistance of these to the principle of 'substitutivity of
>> identity'  'salve  veritate', and in anything scientific  we  are
>> exclusively    interested   in   truth   functions   where    the
>> substitutivity of identity is guaranteed. Failure to respect this
>> principle  results  in invalid reasoning through the  fallacy  of
>> equivocation.
>> -
>> In the case of propositional attitudes, we can not quote  someone
>> indirectly  without thereby uttering an untruth, and we  can  not
>> make  statements  about  what someone  believes,  thinks,  hopes,
>> fears,  or understands except in the actual context in which  the
>> propositional  attitudes expressing such intensional idioms  take
>> place (which amounts to quoting them directly and  contextually).
>> If  one reports to someone else 'what someone said', one  has  to
>> either directly quote (verbatim) or else acknowledge that one  is
>> not  making  a  report  at all. Such  paraphrase  amounts  to  an
>> 'interpretation' which is an imputation or inference, a  creative
>> act
>.. or the acknowledgment of a successful communication ...
>> .  Propositional  attitudes are not  projectible outside the
>> immediate context of their utterance, so: -
>> -
>>     'Reports' are not 'bona fide' reports at all unless they
>>     are  reports verbatim (ie records of behaviour)  -  they
>>     become  interpretations through a process of  imputation
>>     at best, and in almost all cases, they are creative acts
>>     which  do not permit substitutivity of  identity  'salve
>>     veritate'.   They   are  non-truth   functional   unless
>>     expressed  actuarially  via  relation  to  a  population
>>     distribution (normative psychometric measure).
>Incidentally i agree with Quine that for a journalistic report, direct 
>quotes in context are strongly indicated.
That's not Quine, that's Longley. It's excerpted from another message of 
the same era. This was an internal e-mail/newsgroup system which ran for 
about 8 years from 87-96. You should look into what's happening in the 
UK in that area. Look up evidence based practice and try to find out 
what was done before the PROBE preoject.
> Paraphrase, however is useful as an acknowledgment of a successful 
>communication and as a curative means to add additional interpretive 
>practice.  These become the very acts which create our language 
>distinguishing it from bird calls! Emotionally i love it when someone 
>*correctly* paraphrase me, for it is proof that i have communicated to 
>them ... i no longer need to merely guess ... i have finally cast my 
>bottle into the ocean of doubt and ambiguity and some other creature 
>has discovered it :)  When they can add a new idea not contained or 
>imagined in my original, then i must leap for joy as the very purpose 
>of my struggle with these multi-interpretive marks has finally become a 
>step (however minor) in the cultural dance.  So i must find Quine's 
>judgmental comment, at best they are creative acts, to be sourly 
>lacking in insight.
You miss the point.
>Unfortunately i don't have time to guess what connects all of the above 
>to what you say below.
>> -
>> This  is, I argue, why interviews, exams and similar devices  for
>> measuring  behaviour (e.g. minute taking in meetings) are  widely
>> subjected  to the  criticism that they are not reliable  measures
>> of  behaviour,  particularly when such measures are  supposed  to
>> comprise important elements in a human regression process (review
>> board) which  seeks to make inferences or projections about other
>> behaviour. It is also why continuous recording of what happens is
>> more  promising, since such measures provide more  representative
>> samples  under naturalistic conditions (ie the settings are  more
>> prone to generalise) - Such a record is a profile of behaviour.
>> -
>> These  principles are observable in the processes of operant  and
>> classical   conditioning.  In  the  latter  paradigms,   one   is
>> interested  in  the basic processes  of  association  ascertained
>> after  the acquisition process through observation  of  behaviour
>> following  the  presentation  of  a  CS  (the  whole  process  is
>> therefore one of recognition or recall). The touchstone of  rival
>> models  is the 'blocking effect' (Kamin 1968)  which  illustrates
>> that  elements  of a stimulus compound may come  not  to  control
>> behaviour  unless  those  elements  have  come  to  provide   new
>> information, ie change the conditional probability of the  event-
>> event  relationship.  In operant acquisition, one  tends  not  to
>> focus  on  the  response topography, but  records  the  class  of
>> behaviours comprising lever pressing as the operant. But when one
>> looks  at the extinction of this class of behaviours why does  it
>> take  so  long  to  reach  criterion? May  it  not  be  that  the
>> acquisition  of  the  operant is in fact no  different  from  the
>> configuring process in classical conditioning, ie the acquisition
>> process  is  the acquisition of multiple R-S  contingencies,  and
>> extinction involves a testing of all the elements:
>> -
>> In its most basic form each R-S* during acquisition differs, each
>> is  a  slightly different 'perspective'. This  might  comprise  a
>> slight  motor  variation  on  a CRF task,  such  as  a   slightly
>> different pressure on the lever above threshold, or angle of  the
>> body, in other cases it may be the ratio of presses on a FR or VR
>> schedule  (which  will increase resistance to  extinction).  Note
>> that  the element of behaviour here is lever press (p or p*)  and
>> the outcome is pellet delivery (q) ie
>> -
>>                    if 'p'ress then 'q'ualia.
>> -
>> Each CRF trial  RECORDS either 'p then q', 'p* then q', 'p** then
>> q'  and  so  on.  In fact  every  supra-criterion  R-S  variation
>> possible,  leads to a testing of almost all of the  contingencies
>> during extinction :
>> -
>>                   not 'q'ualia then not 'p'ress
>> -
>> but...perhaps  it will be  p*.... or perhaps p**, so each has  to
>> be tested/subjected to falsification (negation). After all,  each
>> variation or 'perspective' was reinforced during acquisition.  On
>> partial reinforcement schedules we interleave a little extinction
>> or impossible discrimination training.
>> -
>> Do each of these behaviours, dispositions, or properties, learned
>> only  in their particular context, occupy their own  location  or
>> node in the operant repertoire, only becoming extensional through
>> the  eyes of the observer? Or are they really configured  into  a
>> class   of  such  behaviours?.  Surely  all  trainers  have   had
>> experience  of  this  process whereby the  basic  elements  of  a
>> complex skill may be taught but the trainee just fails to pull it
>> all  together  coherently, and surely all assessors have had  the
>> problem of deciding whether the student 'really knows'.
>> -
>> The  same processes seem to be operating in 'group work'  of  all
>> sorts.  Here,  as  with  performance on  a  test,  the  behaviour
>> occurring in such settings, (which is predominantly verbal),  may
>> not  generalise  to  other situations  or  be  representative  of
>> general  performance  (other than to other similar  behaviour  in
>> groups or interviews!). Such context effects have been central to
>> Cognitive  Psychology for decades e.g. 'the encoding  specificity
>> principle'  of  Tulving and Thompson. Such processes seem  to  be
>> quite  basic to the very nature of habit formation (cf. notes  on
>> novelty,  opioids & habit formation). Similarly, if 'role  plays'
>> had  any substantial (generalisable) effects on  behaviour,  then
>> many  of our oscar winning actors would surely be more likely  to
>> be  roaming  cities killing and maiming, (or  living  in  idyllic
>> relationships). Acting is no less real or true behaviour than  is
>> any other.
>>  David Longley
>> July 7, 1991
>patty
-- 
David Longley
http://www.longley.demon.co.uk
===
Subject: Re: Turing Machines and Physical Computation
[. . .]
>>I get that computer behavior and human behavior are essentially 
>>different.  Why they are so different is certainly what we came here to 
>>discuss.
>And that's what Fragments is largely about as I have said many times.
David, does it every bother you that people don't pay attention to
what you say many times?
===
Subject: Re: Turing Machines and Physical Computation
 <ZDPod.21741$zx1.19672@newssvr13.news.prodigy.com>
 <41a3ef28$0$576$b45e6eb0@senator-bedfellow.mit.edu>
 <oeTod.49010$QJ3.3531@newssvr21.news.prodigy.com>
 <41a4a697$0$563$b45e6eb0@senator-bedfellow.mit.edu>
 <rzepd.22321$zx1.20354@newssvr13.news.prodigy.com>
 <A0DP8HDvfkpBFwVG@longley.demon.co.uk> <GtIpd.155286$R05.112941@attbi_s53>
 <C2rv8hFO03pBFwEi@longley.demon.co.uk> <41a83717.2872751@netnews.att.net[. 
. .]
>I get that computer behavior and human behavior are essentially
>different.  Why they are so different is certainly what we came here to
>discuss.
>>And that's what Fragments is largely about as I have said many times.
>David, does it every bother you that people don't pay attention to
>what you say many times?
Ignorant people irritate me. The world's full of them and you're 
definitely a member of that class. Ignorant people don't know that 
they're ignorant and take offence when others try to enlighten them. 
That's what makes them so ignorant and that's what keeps them ignorant 
(ask any teacher). Educated people behave differently. They try to learn 
from criticism. They actively look for criticism.
-- 
David Longley
===
Subject: Re: Turing Machines and Physical Computation
> Incidentally, computer-assisted proofs are gradually evolving from the
> type in which a large computation is delegated to the computer to the 
type
> where the *entire proof* is encoded in a data file governed by simple
> syntactical rules.  So to check the correctness of the proof, you just
> have to write a short program of your own to verify that the 
certificate
> has the advertised syntax.  Instead of having to worry about whether, 
say,
> a complex piece of software like Mathematica has some subtle bug in it
> that affects the correctness of your computation, you only have to worry
> that a very short piece of code that you can write yourself is executing
> as intended.
> -- 
THE SURVEYABILITY OF MATHEMATICAL PROOF:
A HISTORICAL PERSPECTIVE by O. B. Bassler
In the case of proof, global surveyability requires not just a 
recognition
of the sufficiency of the individual steps but in addition a recognition 
that
the individual steps compose together to form a proof as what I have 
referred
to above as a comprehensible whole. Speaking of proof, John Mayberry 
stresses  the role played by this requirement of comprehensibility when he 
remarks that,
'It may very well be, if the [formal] rules are formulated with sufficient
niceness and exactitude,that the question whether the inference from
A, B, C, . . . to D is in accordance
with those rules, can be settled by purely combinatorial or syntactic 
means.
But conviction can be compelled only if the logical force of the inference
A, B, C, . . . : therefore D
is felt, and that is not a matter of purely combinatorial insight.6'
6 John Mayberry, The Foundations of Mathematics in the Theory of Sets
(Cambridge: Cambridge University Press, 2000, pp. 108-109). 
===
Subject: Re: Turing Machines and Physical Computation
>>Your contributions have been noted. My second favor author is Joe Shipman
>>who I notice expresses his reservations mildly, we can not have the 
same
>>type of certainty whereas I used the cruder distrust. There is 
alway
>>some distrust of computer aided proofs.
>>http://www.cs.nyu.edu/pipermail/fom/1998-August/002076.html Joe Shipman:
>>In the absence of rigorous proofs of the correctness of computer 
systems
>>(including chip design, programming language, compiler, and operating
>>system),
>>we can not have the same type of certainty about the correctness of a
>>computer-aided proof as we can of a humanly surveyable proof.
> But Shipman isn't even saying here that we can't have the same kind of
> certainty, only that we need proofs of correctness of the computer 
systems
> in question.
I don't think so. And not because I'm much of a mathematician but because
my reading comprehension and interpretation are up to speed. I also find
that the use of the word trust is not too strong. I found these posts 
on
Mathforge, this one by N Megill:
An entirely different issue is not whether a computer-verified proof can 
be 
trusted but whether it can be grasped by a human. The book New Directions in 

the Philosophy of Mathematics by Thomas Tymoczko discusses this in relation 

to the four-color theorem. For mathematicians, three desirable 
characteristics of proofs is that they be convincing, formalizable, and 
surveyable (p. 247). The computer proof of the four-color theorem lacks 
surveyability, even though it may be correct in all other respects.
>>we can not have the same type of certainty about the correctness of a
>>computer-aided proof as we can of a humanly surveyable proof.
You would need to argue that surveyability does not contribute to
certainty in order to arrive at your conclusion But Shipman isn't even
saying here that we can't have the same kind of certainty...
I think both Shipman and Megill are saying that you don't have the
certainty that surveyability contributes. That grasping a proof
contributes to accepting the proof.
Megill continues, even though it may be correct in all other aspects.
These other apects I think are called by Shipman: correctness of computer 
systems (including chip design, programming language, compiler, and
operating system),
It seems fairly clear to me that both Shipman and Megill are not agreeing
to your assertion only that we need proofs of correctness of the computer 
systems in question. They think surveyability is a lcking necessary 
ingredient
especially with complex computer assisted proofs.
Apparently Megill thinks Tymoczko supports his statement by quoting
from Tymoczko's book: For mathematicians, three desirable characteristics
of proofs is that they be convincing, formalizable, and surveyable (p. 
247).
I think their point is that a proof of the four-color theorem that is
surveyable is more convincing than a proof which is not surveyable,
so that the two proofs do not have the same kind of certainty.
At least not to the minds of some mathematicians. Perhaps there is
an objective equivalence, if some mechanical verification process is
used but that is still going to require the contribution of some human 
comprehension that recurs to having first grasped the proof outline,
seems to me.
> By the way, I suspect that Shipman's objection is a minority objection,
> although admittedly I have only anecdotal evidence.  I have problems with
> his objection myself:
Well, at this point it does not seem to be that small of a minority. I did 
find
that there is debate surrounding this issue and you appear to be on one 
side.
THE SURVEYABILITY OF MATHEMATICAL PROOF:
A HISTORICAL PERSPECTIVE by O. B. Bassler
Abstract:This paper rejoins the debate surrounding Thomas Tymockzko's 
paper 
on the surveyability of proof, first published in the Journal of 
Philosophy,
and makes the claim that by attending to certain broad features of modern
conceptions of proof we may understand ways in which the debate surrounding 

the surveyability of proof has heretofore remained unduly circumscribed.
> 1. Implicitly, Shipman seems to regard processing each step of a proof in
> my brain in the conventional way as being the gold standard.  I don't
> see why this should be true.  If the goal is certainty that there is
> no error, then I might do better by relying on someone or something
> else that has a lower error rate than I have.  And then the distinction
> between computer-assisted and conventional proofs blurs.
This makes sense if one can rest assured that each step in your brain
in arriving at a proof can be mechanically faithfully represented without
translation error. It seems like this paragraph could get into creative
theorem provers and the validity of the Turing test, c.a.p. issues.
If your brain isn't the gold standard, in order for calculation to be
the gold standard, I think there is an assumption that a mathematician's
inventions can be reduced to an algorithmic approach. Actually, maybe
I shouldn't have responded to this part because I don't want to pursue it.
> 2. But maybe I've misinterpreted Shipman, and he accepts that processing 
a
> proof in my brain isn't the only gold standard.  After all, he 
doesn't
> say that the certainty of computer-aided proofs is *inferior*, only that
> it is *not the same type*.  And he also refers to the surrogate process
That is not how I understand Shipman in conjunction with reading Megill.
I am fairly sure they are taking the position that they are not the same 
type
and that surveyability does contribute some certainty to having confidence 
about/in a proof. Bassler says the controvery is alive and well about the
role of surveyability and a lot of opinions have been published including
Wittgenstein. I don't know what department you are in. But I think the
cognitive or computer science environment would tend to see this issue one 
way and prefer it to be a closed matter, though maybe not deliberately so.
> of checking proofs of correctness of computer systems.  In that case,
> he and I are mostly in agreement, except that I don't quite agree
> with the implicit suggestion that proofs of correctness of computer
> systems would give the same type of certainty as conventional proofs.
It doesn't seem to me that Shipman thinks that proofs of correctness
of computer systems would give the same type of certainty as
conventional proofs. I thought you didn't think this.
> But Shipman isn't even saying here that we can't have the same kind of
> certainty, only that we need proofs of correctness of the computer 
systems
> in question.
I think he is saying we can't have the same kind of certainty. And not
that we only need proofs or correctness. It seemed to me that this was
your position that you read into Shipman's statement. You write:
**> except that I don't quite agree
> with the implicit suggestion that proofs of correctness of computer
> systems would give the same type of certainty as conventional proofs.
which I don't see as consistent with the earlier part of your post.
Apparently, I am not understanding something, because I and
I think Megill and Shipman would agree with you on this last quote **.
> (Shipman doesn't state this explicitly; he only states the converse, but 
I
> think it's implicit.)  The problem is that having a proof of correctness
> doesn't guarantee that the physical execution of a computer won't be
> corrupted by a stray cosmic ray, or more generally that the physical
> computer actually behaves as designed on paper.  Note also that we never
> demand proofs of correctness of our brains.  So proofs of correctness
> aren't sufficient or necessary to achieve the same type of certainty.
This is getting convoluted :-) I would agree that proofs of correctness
are not sufficient. They may be necessary in the same way that future
generations of mathematicians reread proofs to see if there are any
undiscovered errors. The same type of certainty involves surveyability
and that is not going to be achieved in a complex computer aided proof.
That to me, seems rather independent of proving the computer is sound.
Necessary for same type of certainty are convincing, formalizable, and 
surveyable (p. 247).
> ---
> Incidentally, computer-assisted proofs are gradually evolving from the
> type in which a large computation is delegated to the computer to the 
type
> where the *entire proof* is encoded in a data file governed by simple
> syntactical rules.  So to check the correctness of the proof, you just
> have to write a short program of your own to verify that the 
certificate
> has the advertised syntax.  Instead of having to worry about whether, 
say,
> a complex piece of software like Mathematica has some subtle bug in it
> that affects the correctness of your computation, you only have to worry
> that a very short piece of code that you can write yourself is executing
> as intended.
> -- 
That is interesting. You see I thought this was connected to why
no complex piece of software can be verified as bugfree by an
algorithmic test. Eternal regress. I don't mean to be argumentative
but doesn't this involve writing a program to test the program that is
testing the program and so on. Maybe my doubt is out of context
because I'm not sure what a certificate entails (no spoofing?).
How do you know that the short piece of code is sufficient to
exploit any potential gap/error? It makes me kind of suspect that
surveyability is being derivitively transfered to the short program.
Maybe because I just don't understand how it works :-).
head to quit thinking, it is time to eat which makes me think of
Stephen 
===
Subject: Re: Turing Machines and Physical Computation
http://mygate.mailgate.org/mynews/comp/comp.theory/6421966862743ea97416a47fb
f
02de8d.48257%40mygate.mailgate.org
> An entirely different issue is not whether a
> computer-verified proof can be trusted but whether
> it can be grasped by a human.
And yet, we do not require, of the balance sheet of
a multibillion dollar corporation, that it be
graspable in total by a single human being, for it
to be considered auditable. The auditors instead
use computers to do their audit.
There are simply some things that computers are and
always will be better at doing than humans are,
handling large, repetitive data processing tasks
probably paramount above them.
To insist that for a proof to be surveyable it
must be surveyable by a single human, or even
surveyable by the collection of all technically
competent humans avaiable for the domain of
discourse, working as a group, it to limit human
understanding to comparatively small problems.
It is to indulge in some rather incredible hubris as
well, that if a task cannot be done by humans, it
cannot be done well enough _at all_ to be trusted
by humans.
In the case of something like the proof of the four
color mapping theorem, what is accessible for human
surveyability isn't the several billion part
proof, but the software which categorized and
exhausted that set of required subproofs and found
them all to be confirmed.
Even so, some future proof may require even software
too complex for any one human being or available
collection of human beings to graspr.
This is nothing new, we work with and trust such
software every day.
    [I was part of a 10^8 SLOC software project at
    Motorola (IRIDIUM), and was tangentially
    involved with one for the US Navy that contained
    6,000 linkable libraries of compiled modules,
    for two examples not likely to be anywhere close
    to the maximums in delivered, operational
    software today.]
To reject something merely because it is big is too
timid a behavior for any self-respecting human being
to indulge. Those who suggest that we should accept
such limitations in mathematical proofs should sleep
among the mice.
xanthian.
-- 
===
Subject: Re: Turing Machines and Physical Computation
> Turing was interested in the foundations of Mathematics. His paper
> is a response to Hilbert's program and in particular question #10. He 
> mentions
> Godel's result which also disproved question #10 but says his proof is 
> clearer.
Indeed, it is a lot clearer. Much clearer.
> Another way you can tell that Turing did not mean a 
> physical machine, but rather a mathematical object, an idea, was that he 
> gave the TM an infinite tape which was one-dimensional. This condition 
> cannot be physically realized in the ordinary usage of machine
Irrelevant.
This argument from dimensionality is nonsense. The one-dimension of
the Turing Machine is not a necessary property of the general theory
of computation. (However, some dimensionality will be required in
general!) This dimension can evidently be embedded in 3-space. This
means that it can be physically realized. Therefore, your argument is
not only partially but completely wrong. It has been refuted.
In other words, Turing would disagree with your revisionist reading.
--
Eray Ozkural
PS: A similar argument from dimensionality was conceived by Dr. Alex
Green on c.a.p. with an application to subjective experience of
vision, etc., but I am quite suspicious of the use of this approach. I
think it's nonsense to talk about the *unphysical* dimension of the
tapes in whatever computer. Surely, the only sensible dimensions for
machinery are the physical dimensions, and of course phase spaces that
might describe machine design etc. which are just as real (or almost
as real ;) All of these dimensions describe aspects of physical
reality, as do storage specifications of computation.
===
Subject: Re: Turing Machines and Physical Computation
>      Definition:   Common term for computer, usually when considered 
at 
>the hardware level. The Turing Machine, an early example of this usage, was 

>however neither hardware nor software, but only an idea. 
>http://www.hyperdictionary.com/dictionary/machine
It was a gedankenexperiment, and eminently realizable in physical
form, purposely so, to answer Hilbert's #10.
Someone remind me, are there any TM results that depend on an infinite
tape for positive results?
J.
===
Subject: Re: Turing Machines and Physical Computation
>>      Definition:   Common term for computer, usually when considered 
at
>>the hardware level. The Turing Machine, an early example of this usage, 
>>was
>>however neither hardware nor software, but only an idea.
>>http://www.hyperdictionary.com/dictionary/machine
> It was a gedankenexperiment, and eminently realizable in physical
> form, purposely so, to answer Hilbert's #10.
> Someone remind me, are there any TM results that depend on an infinite
> tape for positive results?
> J.
The only result I know of is that the infinite tape was required/expedient
to explain his proof of the undecidability of the halting problem.
In computability theory the halting problem is a decision problem which
can be informally stated as follows:
Given a description of an algorithm and its initial input, determine 
whether
the algorithm, when executed on this input, ever halts (completes). The
alternative is that it runs forever without halting.
Alan Turing proved in 1936 that a general algorithm to solve the halting
problem for all possible inputs cannot exist. We say that the halting 
problem
is undecidable.
SH: If Turing had given the TM a shorter, finite tape then it would not
be possible for the algorithm to run forever. It could stop because
it ran out of tape, just like a decidable algorithm stops when it
completes. How would you distinguish between the two reasons
for the tape halting? So that you could conclude that all tapes
that halt are decidable and all the rest are undecidable, that don't halt.
Maybe there is some way to query the tape and tell if it ran out of
tape and halted or it legitimately completed its computation and halted.
But explaining that would have made his paper more complex than needed
and Turing liked elegant solutions in general. So specifying the tape could
be as long as needed to complete any finite calculation was a neat solution
which eliminated wondering if the tape halted because it wasn't long 
enough.
Pi has an infinite expansion. The abstract TM with its infinite tape can
compute more finite digits of Pi than any physical PC. That is, one can
pick a some huge number of digits (which are still finite) which are
computable by a TM and not a PC. This comes under computations that
are called intractable for a PC (due to lack of resources). In the 
literature
I've come across connecting a calculation which cannot be completed due
to heat death of the universe with the computationally hard and the 
limits
of a PC. I will provide one supporting quote at the very end for this.*
One of the reasons a problem cannot be computed is insufficient
physical memory for a PC. The universe does not contain an infinite
supply of memory. Nor does the universe contain a sufficient supply of
finite memory so that a physical PC can compute as many finite digits
of Pi as a TM can compute. There is no upper bound on the number
of finite digits that a TM can compute and there is an upper bound
of the number of finite digits that a PC can compute. TMs don't have
the finite upper bound because they do have an ideal of an unending
tape (used as memory) and TMs do not have their computation
potential constrained by physical resources; PCs do. (memory subtance)
This is a theoretical point of interest. The issue arose in a theoretical
discussion. I have stated that making this distinction between what
PCs can do versus TMs has little to do with the practical use of PCs
to calculate problems useful to the human realm.
Eray denied the theoretical difference of TMs and PCs. That is the
issue under debate. There was never an issue about how practical
the distinction would be, how useful. Turing endowed his TM with
a tape that could solve problems that a PC could not. That is because
there is no physical world substitution for the tape (memory device) that
he used. If a PC could use the tape Turing describes, then it could
compute just as many finite digits of Pi as a TM. But the PC can't use
that type of potential memory. It doesn't physically exist. When they
built that physical TM, its memory could never have the power of
the tape Turing introduced in his 1936 paper:
ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM
The tape Turing imagined is like a perpetual motion machine, not real.
Overwordly,
Stephen
* http://www.hum.utah.edu/philosophy/Faculty/millgram/hha.htm
But before doing all these things, let me address a necessary preliminary: 
what hard means here. A problem is easy if you can do it in a reasonable 
amount of time and with a reasonable amount of effort, without overtaxing 
your memory, and so on. A problem is hard if it's not easy; it's hard in 
practice but not in principle if you could solve the problem on a bigger, 
faster computer that you could buy. (I mean hardness in practice to be 
sensitive to what resources are actually available to you when you're trying 

to solve the problem; one might say impractical for the problem solver 
instead.) A problem is hard in principle if buying a bigger, faster computer 

wouldn't help. (For instance, if buying a computer as big as the known 
the universe, and running it from the Big Bang until the heat death of the 
universe wouldn't solve the problem -- then it's hard in principle.)
SH: One can specify calculating some huge, huge, finite number of digits
of Pi on a TM that far exceeds the hard in principle description above
that applies to a PC. That is because the tape in the TM isn't part of
the universe and does not have physical universe constraints. The TM
can in principle calculate more finite digits of Pi than a TM because
the TM doesn't have physical universe restrictions and a real PC does,
so the PC could only calculate the finite digits of Pi within the age of 
the
universe; that is less than what a TM can do which operates outside of 
time.
That does not place the TM or its tape in the realm of mathematical 
Platonism either. It is just an abstract idea, not a philosophical 
commitment.
===
Subject: Re: Turing Machines and Physical Computation
>      Definition:   Common term for computer, usually when 
> considered at
>the hardware level. The Turing Machine, an early example of this 
>usage, was
>however neither hardware nor software, but only an idea.
>http://www.hyperdictionary.com/dictionary/machine
>> It was a gedankenexperiment, and eminently realizable in physical
>> form, purposely so, to answer Hilbert's #10.
>> Someone remind me, are there any TM results that depend on an 
>> infinite
>> tape for positive results?
>> J.
> The only result I know of is that the infinite tape was 
> required/expedient
> to explain his proof of the undecidability of the halting problem.
> In computability theory the halting problem is a decision problem 
> which
> can be informally stated as follows:
> Given a description of an algorithm and its initial input, determine 
> whether
> the algorithm, when executed on this input, ever halts (completes). 
> The
> alternative is that it runs forever without halting.
> Alan Turing proved in 1936 that a general algorithm to solve the 
> halting
> problem for all possible inputs cannot exist. We say that the halting 
> problem
> is undecidable.
> SH: If Turing had given the TM a shorter, finite tape then it would 
> not
> be possible for the algorithm to run forever. It could stop because
> it ran out of tape, just like a decidable algorithm stops when it
> completes. How would you distinguish between the two reasons
> for the tape halting? So that you could conclude that all tapes
> that halt are decidable and all the rest are undecidable, that don't 
> halt.
> Maybe there is some way to query the tape and tell if it ran out of
> tape and halted or it legitimately completed its computation and 
> halted.
Would it matter if querying the tape would alter what's on tape and 
hence the computation? The old computation is halted because it changed.
Halfly serious - since the discussions on Turing never halt and remain 
undecided.. all you need is a computer that can auto-update ie rewrite 
every moment rather randomly its own program on tape.. to pass the 
T-test. This would suggest that the more you program into it, the less 
chance there is that it will become intelligent. 
===
Subject: Re: Turing Machines and Physical Computation
:>      Definition:   Common term for computer, usually when considered 
at 
:>the hardware level. The Turing Machine, an early example of this usage, 
was 
:>however neither hardware nor software, but only an idea. 
:>http://www.hyperdictionary.com/dictionary/machine
: It was a gedankenexperiment, and eminently realizable in physical
: form, purposely so, to answer Hilbert's #10.
: Someone remind me, are there any TM results that depend on an infinite
: tape for positive results?
: J.
Sure.  Can the language a^n b^n for n>=0 be recognized by a Turing 
Machine, or recognized at all.  If you only have a finite tape
than there are members of this language you cannot recognize.
Stephen
===
Subject: Re: Turing Machines and Physical Computation
> <cyberguard1048-usenet@yahoo.com Perhaps what I am interested in is a 
class on the fuzzy boundary
> between FA and UTM, large FA's, equivalent to this here machine I am
> typing on.  I suspect many others would be interested in this same
> class, if indeed it is distinguishable in principle from either
> smaller FA's or UTMs.  I suppose it is distinguishable, but perhaps
> not interestingly so.
I think this is interesting, but I don't know much about it. I also
find the presence or topic of indistinguishability interesting.
>>http://plato.stanford.edu/entries/turing-machine/
>>A Turing machine is a kind of state machine.
>>Turing machines are not physical objects but mathematical ones.
> Fascinating, a non-physical state machine.
The definition of machine is not the usual one that involves something 
physical.
In college, an exercise is to use pencil paper and eraser to duplicate 
finding
a result of a Turing machine. This produces the same result as if your 
computer
simulated a Turing machine. This is an abstract type of machine and it does
not work like a physical machine. I can draw a car, but that doesn't help
me get to the store for groceries, because the car is a physical machine.
> I'm a big fan of SEP, but I believe the idealist interpretation they
I'm not sure what SEP stands for. I am not aware of anything
or transferring infinity in an ideal machine to a physical machine.
But that does not make Turing Machine theory inapplicable
to machines that can be realized.
SH: As far as I know, this has not been claimed, unless Eray
has misrepresented something said which I am not aware of
before I killfiled him.
There is a claim that there is at least one aspect of a Turing machine
which cannot be duplicated by a physical machine, however, and
Turing imagined it and placed it into his thought experiement.
The tape is potentailly infinitely long, because he says it can compute Pi
which is infinite to any degree of accuracy (number of digits).
It will not halt for the reason that it runs out of tape.
Turing states the ink supply is infinite.
The turing machine is given no power supply so it is most like a perpetual
motion idea in terms of continuing energy to move the tape.
I don't think there are any infinite supplies contained within the 
universe.
So those infinite ideas that he imagined and endowed to his TM are not
going to let you build a physical machine that never runs out of gas for 
instance.
PCs that we build need power supplies, and ram which is finite, not a
potentially infinite tape, and a cpu that operates within time. Time is not
mentioned as a factor in Turing's paper. Turing's TM can calculate a
huge number of digits of Pi, way past the capability of a PC that exists
within the physical universe and is constrained by the future of the 
universe.
Another way of looking at this is the abstract Super-Turing computers.
PCs must use truncations/approximations of real numbers some of which
are infinitely long. TMs don't compute with reals, the computation would
never get past computing the infinite value of the real needed for the rest
of the computation.
Super-Turing machines are also ideal machines. The theory is that they
are magically hand-waved so that they can perform calculations using
all the digits of a real number, so they infinitely precise and thus have
more power than an ordinary TM. That does not mean that this imagined
capability can be realized in a physically real device/machine. This 
discussion
of the capability of TMs vs. Super-TMs is entirely theoretical.
And it is quite implausible that these theoretical capabilities, conferred 
by
the imagination for use by abstract constructs (the meaning of machine
as used by Turing) will ever have a physical expression in reality, so this
is closer to magical thinking rather than science.
Other aspects of TMs have a good match with a physically constructed
PC. The ideas translate well. The ideas which involve using infinity in
the operation of your physical machine have no practical implementation.
It has not been observed, and science deals with observed things within
reality certainly more so than the speculation of physical machines endowed
with the power of infinity being built by humans.
That is what my argument with Eray boils down to. I said that because a
TM was an abstract construction which contained the imagined use of
infinity in Turing's thought experiment, that this abstraction* could 
perform
a process that could not be done physically. The burden of proof is on
those who want to claim that in the future we can build a physical computer
that uses an infinite anything for the source of its operation (or a part of 

it).
*The TM can compute more digits of an otherwise intractable number than
a PC, because it has no time or memory limitation which physical PCs have.
Star Trek time,
Stephen
===
Subject: Re: Turing Machines and Physical Computation
>> I'm a big fan of SEP, but I believe the idealist interpretation they
>I'm not sure what SEP stands for. 
The Stanford Encylopedia of Philosophy that you gave several URLs
from.
J.
===
Subject: Re: Turing Machines and Physical Computation
> I'm a big fan of SEP, but I believe the idealist interpretation they
>>I'm not sure what SEP stands for.
> The Stanford Encylopedia of Philosophy that you gave several URLs
> from.
> J.
http://www.alanturing.net/turing_archive/pages/Reference%20Articles/What%20i

s%20a%20Turing%20Machine.html#head 
by B.J. (Jack) Copeland
Commercially available computers are hard-wired to perform primitive
operations considerably more sophisticated than those of a Turing machine
--add, multiply, decrement, store-at-address, branch, and so forth. The
precise constitution of the list of primitives varies from manufacturer to
manufacturer. It is a remarkable fact that none of these computers can 
outdo
a Turing machine. Despite the Turing machine's austere simplicity, it is
capable of computing anything that any computer on the market can compute.
Indeed, since it is an abstract or notional machine, a Turing machine can
compute more than any physical computer. This is because
(1) the physical computer has access to only a bounded amount of memory,
and
(2) the physical computer's speed of operation is limited by various
real-world constraints.
It is sometimes said, incorrectly, that a Turing machine is necessarily
slow, since the head is continually shuffling backwards and forwards,
one square at a time, along a tape of unbounded length. But since a
Turing machine is an idealised device, it has no real-world constraints
on its speed of operation.
Copeland's qualification and publications:
http://www.phil.canterbury.ac.nz/people/copeland.shtml
http://www.phil.canterbury.ac.nz/people/copeland.shtml  Abstract:
Fuzzy logic extends deductive methods to situations in which the 
information available may be only partly or approximately true. Fuzzy logic 

has often been championed as a logic of vague terms, and it does indeed 
provide an intuitive analysis of what goes wrong in Sorites reasoning.
Distinguishably,
Stephen
===
Subject: Re: Turing Machines and Physical Computation
> I'm a big fan of SEP, but I believe the idealist interpretation they
>>I'm not sure what SEP stands for.
> The Stanford Encylopedia of Philosophy that you gave several URLs
> from.
> J.
> 
http://www.alanturing.net/turing_archive/pages/Reference%20Articles/What%20i
s
%20a%20Turing%20Machine.html#head 
> by B.J. (Jack) Copeland
> Commercially available computers are hard-wired to perform primitive
> operations considerably more sophisticated than those of a Turing machine
> --add, multiply, decrement, store-at-address, branch, and so forth. The
> precise constitution of the list of primitives varies from manufacturer 
to
> manufacturer. It is a remarkable fact that none of these computers can 
outdo
> a Turing machine. Despite the Turing machine's austere simplicity, it is
> capable of computing anything that any computer on the market can 
compute.
> Indeed, since it is an abstract or notional machine, a Turing machine can
> compute more than any physical computer. This is because
...
On the brakes! You're going off-track again.
--
Eray
===
Subject: Re: Turing Machines and Physical Computation
> I'm a big fan of SEP, but I believe the idealist interpretation they
>>I'm not sure what SEP stands for.
> The Stanford Encylopedia of Philosophy that you gave several URLs
> from.
> J.
which is why I've read Turing's 1936 paper that expounds on TMs
several times so that I could feel qualified to make a sensible comment.
I've grown out of thinking my imagination is a better source than fact.
Stephen 
===
Subject: Re: Turing Machines and Physical Computation
>> Hi Stephen,
>> Now, I recall that you made this strange argument about Turing
>> Machines being able to do things that real computers cannot, because
>> they have an infinite tape. What an observation.
> TMs are not meant to have physical constraints applied to them.
>> Another assertion.
If you had ever bothered to read and comprehend TMs abstract
constuction you would already know this is not an assertion.
Name a physical restraint applied to a TM, that limits its ability to
computability solve some problem.
You claim to have read Turing's 1936 paper. Well, visualizing this
hypothetical device, a Turing machine, one could imagine the tape
is like a typewriter tape, only infinitely long, and moving in both 
directions.
There are a finite number of keys on a typewriter satisfying the symbol
requirement. Physical machines have three dimensions.
Turing tells us we can think of the tape as being one-dimensional:
Computing is normally done by writing certain symbols on paper. We may
suppose this paper is divided into squares like a child's arithmetic book.
In elementary arithmetic the two-dimensional character of the paper is
sometimes used. But such a use is always avoidable, and I think that it 
will
be agreed that the two-dimensional character of paper is no essential of
computation. I assume then that the computation is carried out on
one-dimensional paper, i.e. on a tape divided into squares.
SH: What object that exists in the physical world is one dimensional?
The one dimensional object that I know about is a mathematical abstraction
called a line, which has no physical existence. A Turing Machine is a 
logical
mathematical abstract device/entity. That is why a TM fits in the same
category as set theory, potential and actual infinities,  which are also 
logical
mathematical abstract concepts/constructs. These are abstract ideas not
physical things like a PC. I've already provided several definitions of 
what
a TM is, hypothetical/abstract device with an (potential) infinitely long 
tape.
I did not make an assertion, rather I stated the definition. You could only
call it an assertion if you were unaware that I stated the well-known 
definition.
There is some philosophical debate about whether the tape is potentially
infinite or finitely unbounded. That philosophical debate does not impact
the basic definition that I used. A TM cannot have a physical constraint
because a TM is not physical! A PC has physical restraints because it
*is* physical! Because the idea of a TM provided a basis for developing
a physical computing device/machine, does not transfer in reverse the
physical properties and limitations of a physical machine to the abstract
machine just because it served as the conceptual basis for building the PC.
This all comes down to you not knowing the defintion of a TM as an
abstract device/idea, and not a physical device with physical constraints.
>> Ignoring physical plausibility directly opposes physicalism. When we
>> start talking about immaterial things, we are not making scientific
>> statements, we are making metaphysical ones, and that is not a good
>> sign if the metaphysical statements depict worlds that are not
>> physically similar to our own. It is a matter of degree, which we can
>> observe.
Ideas are immaterial things. Not every idea of two wizards riding
unicorns casting spells at each other is grounded in physical reality.
Physical theories, like quantum theory, are ideas about how reality works.
They are not reality itself. Formal mathematical systems are not required
to apply to reality. When there is an effort to make the formal system 
apply
to reality, the mathematics are not physical reality. This mathematics then 

is
an effort to represent reality as faithfully as possible. Mathematics is a
collection of abstract ideas often about physical reality, not physical 
reality
itself. This view is the opposite of mathematical platonism. It says that
mathematicians invent mathematics to describe observed regularities in
an applied sense, and that mathematicians can invent mathematics in a
pure sense without the intention of having the mathematics describe 
reality.
This is the opposite view mathematical platonism which has mathematicians
discovering pre-existing mathematical ideas in a non-material realm. I'm
not defending mathematical platonism at all. I am asserting that 
mathematical
platonism is independent of Cantor's set theory and actual infinity. Actual
infinity does not mean physically manifested, it is just another abstract 
concept.
Cantor's personal belief is independent of the value of set theory which 
was
used in part to develop fiber bundles.
Humans having abstract ideas is not considered dualistic or 
anti-physicalism.
I think along with many people that Cantor invented (not discovered) set
theory using his brain which produced ideas in his mind. That humans can
have ideas that are more concrete versus more abstract is not considered
an argument that consciousness is dualistic, thus anti-physical.
The word is abstract versus Platonic. Do you use abstract reasoning?
It is not necessary to drag in a physical PC to dismiss Platonism, just
use the word abstract, which is a type of counterfactual thinking, not
a claim for a realm outside the universe. Penrose is a mathematical
Platonist, but he doesn't claim that consciousness is beyond the scope
of physics or outside the universe. You have associated and generalized
Cantor's personal view/philosphy as if it were a requirement to develop
set theory. It is not. Perhaps there is a recursive relationship between
the abstract usage of an infinite tape and a TM, and the abstract usage
of infinity involved with set theory. But that does not lead to a dualistic
anti-physicalist description of consciousness; abstract reasoning has
never been considered the same as existing in a platonic realm. Abstract
ideas are a normal consequence of having physical brains producing a mind.
>> The question is: how far from reality? The further you talk, the less
>> real your statements are, not just as real!
That is a precise description of your predicament. You don't know
basic definitions and their are large gaps in your education about how
these ideas relate to each other. That is what you use to manufacture
your viewpoint and of course it is removed from reality. When you
keep reading criticisms from people who say you don't know the
definitions, it is really true. Definitions are a consenus reality and if
you don't know them it is your picture of reality which is randomized,
not the other people's picture which use accepted defintions and
share the same general knowledge such as the difference between
having abstract ideas and the conjecture of a Platonic realm of existence.
The real part is that you are either an arrogant, ignorant, though
intelligent young person, or you have a chemical brain imbalance which
handicaps your ability to reason abstractly and you have forgotten or
decided not to take your medication (though I don't know if meds help).
I'm not trying to insult you. This is an objective description which is
why you keep having it repeated to you in various posts in other 
newsgroups.
You try to pretend to be expert but keep making revealing remarks
which show their are big gaps in your education (not sure of the reason 
why),
some of your remarks are so fundamentally wrong that they are silly.
You keep writing as if Cantor's use of actual infinity makes a claim
about embedding infinity in the real physical world. It does not, his
actual infinity is just another abstract idea.
>> At this point, I should remind you the famous Levin quote. Levin was
>> Kolmogorov's student, so he probably has a better idea of this issue
>> than you have.
> A Turing
> Machine or potential sentence of a language (there is no pre-existing
> specification that the sentence has to be of finite length) is not of 
> this
> world.
>> Of which world is it then? (^_^) And you are saying this Platonist
>> talk is not theology! You have just murdered physicalism.
That quote means that a TM is an abstract concept, a non-material
mathematical object as opposed to a real world physical object
with 3 dimensions. It does not invoke Platonism! Platonism and
abstract reasoning are not the same thing. No human is required to
do abstract reasoning in an out of this universe platonic realm.
Well, that is it. I'm tired of reading your nonsense. You keep mixing in 
ideas
that are not relevant to the discussion and you don't know enought to know 
it.
>> The latter camp has the precise terminology.
> There is no
> physical time constraint applied to when the calculation has to be
> completed.
>> If you mean *any* computation. But a Turing Machine is a *particular*
>> computer. Not just *any* computer. Let's pay attention to our
>> language.
Of course I mean any computation. There are an infinite number of
Turing machines which match up with an infinite number of possible
computations. There is an infinite subset of computations that match
with a TM, that can be computed on a TM and not a PC because
a PC is physical, having a time constraint, which the TM is an infinite
number of cases does not. Because a Universal Turing Machine (UTM)
can be used to emulate this condition in infinitely many cases doesn't
make my statement imprecise. Because a PC can compute some of
same calculations that a TM computes, does not impute a physical
time constraint to the TM, just because PCs always have time constraints.
Bringing this up doesn't even make any sense.
You join Longley,
Stephen
===
Subject: Re: Applications of proper classes?
>Has the notion of a proper class been necessary to solve any problems 
in
>number theory or real analysis?
>> No.  ZFC has no proper classes, and (so far) all advances in
>> number theory and real analysis can be formalized in ZFC
>> (plus, perhaps, some large cardinal axioms).
>> The class of all sets would be a proper class.  It's just that sets are 
>> more interesting.
>Furthermore, it can be shown that anything you can do with proper classes
>(including super-classes of THESE, etc), can in a suitable sense already
>be done with sets plus large cardinals.
NBG is a conservative extension of ZF with proper classes.
Any theorem of NBG not explicitly involving proper classes
is also a theorem of ZF.
Large cardinals give MORE than proper classes.  Also, it
cannot be proved that large cardinals are consistent, with
or without choice.
The main advantages of proper classes are those of convenience.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Re: Applications of proper classes?
>Has the notion of a proper class been necessary to solve any problems 
in
>number theory or real analysis?
(See Wikipedia entry on Classes at
>http://en.wikipedia.org/wiki/Class_%28set_theory%29 )
Dan
>Download DC Proof 1.0 at http://www.dcproof.com
 No.  ZFC has no proper classes, and (so far) all advances in
>> number theory and real analysis can be formalized in ZFC
>> (plus, perhaps, some large cardinal axioms).
> The class of all sets would be a proper class.  It's just that sets are 
> more interesting.
> The class of all sets would be a proper class if ZFC had proper classes,
> but it doesn't.
How do we exclude the existence of a class whose properness (as I
understand it, representability as a set) is undecidable?
I'm speaking from ignorance, but can't a class C have the form if
(undecidable proposition) then x in C iff (definition yielding proper
class); otherwise x in C iff (definition yielding a set))?
Maybe I just misunderstand what you mean by ZFC doesn't have proper
classes (or maybe I don't understand what a class is!).
===
Subject: Re: Applications of proper classes?
>>  
> 
> 
>>Has the notion of a proper class been necessary to solve any problems 
in
>>number theory or real analysis?
>>(See Wikipedia entry on Classes at
>>http://en.wikipedia.org/wiki/Class_%28set_theory%29 )
>>Dan
>>Download DC Proof 1.0 at http://www.dcproof.com
> 
> No.  ZFC has no proper classes, and (so far) all advances in
> number theory and real analysis can be formalized in ZFC
> (plus, perhaps, some large cardinal axioms).
>>  
>> The class of all sets would be a proper class.  It's just that sets are 
>> more interesting.
>> The class of all sets would be a proper class if ZFC had proper classes,
>> but it doesn't.
> How do we exclude the existence of a class whose properness (as I
> understand it, representability as a set) is undecidable?
We don't have to.  There is no axiom of ZFC that talks about anything
being excluded.  The axioms give certain conditions for concluding that
sets exist with certain properties.
Note that the word set appears nowhere in the axioms of ZFC.  That's
because it's implicit that everything that exists in ZFC is what we
commonly call a set.  That is, for all X in an axiom is to be
understood as for every set X, and there exists an X means there
exists a set X.  There is no room for anything that doesn't exist.
> I'm speaking from ignorance, but can't a class C have the form if
> (undecidable proposition) then x in C iff (definition yielding proper
> class); otherwise x in C iff (definition yielding a set))?
You are no longer talking about ZFC, but about some different set theory
(perhaps NBG, which does have proper classes).
> Maybe I just misunderstand what you mean by ZFC doesn't have proper
> classes (or maybe I don't understand what a class is!).
ZFC does not have proper classes, because everything in ZFC is a set.
-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
===
Subject: November 25 is Infinite Clause day!!
Merry X-count-mas!
Do you believe in Infinite Clause?  Will he make it to your real place
in time and all of the infinite places and make something new for you?
Have you been good and read your maths text so you too can see
Infinite Clause? He's keeping a list and he's checking it twice!
CANTOR :
The uncountable number does not have the 1st digit of the 1st
countable number.
UTM(neN,d) :
ALL 10 digits are present in the 1st digit place an infinite number of
times, on the list of computable reals.
CANTOR :
The uncountable number does not have the 2nd digit of the 2nd
countable number.
UTM(neN,d) :
ALL 10 digits are present in the 2nd digit position (following every
possibility of the 1st digit) for an infinite number of reals.
CANTOR :
The uncountable number does not have the 3rd digit of the 3rd
countable number.
...
Start looking at reality sci.math not your text, not David Ullrich and
Barb Knox who *make money teaching texts*.  0.123... is on the list of
computable numbers infinite times with infinite possible tails after
the 123.  0.654... is there infinite times.  ALL PREFIXES of UNLIMITED
new sequence of digits that is not on the list of computable reals. 
ALL PERMUTATIONS ARE THERE.  The diag number must be SOME sequence of
digits, like 0.123..., like 0.654... ITS NOT A NEW NUMBER, the elves
went on strike!
Herc
--
If an infinite number of people toss a coin infinite times each, is it
possible for you to toss a coin infinite times in a new sequence?  Of
course not, that's the problem with uncountable infinity, unlike other
branches of mathematics the proofs don't support one another, they
just all make the same mistake.
===
Subject: Re: November 25 is Infinite Clause day!!
> Merry X-count-mas!
I prefer Happy Hall0ween from Count Dracula.
> Do you believe in Infinite Clause?  Will he make it to your real place
> in time and all of the infinite places and make something new for you?
> Have you been good and read your maths text so you too can see
> Infinite Clause? He's keeping a list and he's checking it twice!
Quite the 1-to-1 correspondence.  Will I get 2 turtle doves, 3 French
hens, ad nauseum?  (Only in department stores for 2 months, he
replies)
> 0.123... is on the list of
> computable numbers infinite times with infinite possible tails after
> the 123.  0.654... is there infinite times.  ALL PREFIXES of UNLIMITED
> LENGTH are ON THE LIST.
But this 1-to-1 correspondence is not constructed so easily.
The problem is that all of the occurrences of your finite prefix of
length L are after the 1st. L numbers!  Then if you say that the WHOLE
infinite decimal expansion is on the list as entry number L someone
can say No it isn't.
It is true that every subset is listed.  That is because the list of
finite sets is no bigger than the list of natural numbers.  But the
WHOLE Cantor number is not.  And if you are forced to say WHERE it is,
you are completing the 1-to-1 correspondence that cannot be (unlike
the above that is), and a contradiction arises.
C-B
5 cents, please.
The dr. is in.
> the elves went on strike!
Only when asked to do the impossible.  (Violates Mathematicians'
Union.)
> Herc
===
Subject: Re: November 25 is Infinite Clause day!!
Counting rationals is hard if you've never been shown.
Rationals
|   Numerator -->
|   1 2 3 4 5 6
| 1 1 2 4
| 2 3 5
| 3 6
v  Denominator
Rational 1 = Num(1)/Den(1) = 1/1 = 1.0
Rational 2 = Num(2)/Den(2) = 2/1 = 2.0
Rational 3 = Num(3)/Den(3) = 1/2 = 0.5
Rational 4 = Num(4)/Den(4) = 3/1 = 3.0
Rational 5 = Num(5)/Den(5) = 2/2 = 1.0
Rational 6 = Num(6)/Den(6) = 1/3 = 0.33333..
Reals
Real 1 = UTM(1, 0) = 0.123
Real 2 = UTM(2, 0) = 0.5
Real 3 = UTM(3, 0) = 0.33333..
Real 4 = UTM(4, 0) = 3.14159..
Real 5 = UTM(5, 0) = 2.7818..
Real 6 = UTM(6, 0) = 0.1111..
Every digit position is saturated with infinite possible digits, ergo
this list is complete.
An infinite number of people each toss a coin infinite times. Can you
come up with a new sequence of Heads and Tails?
Herc
===
Subject: Re: November 25 is Infinite Clause day!!
> Merry X-count-mas!
> Do you believe in Infinite Clause?  Will he make it to your real place
> in time and all of the infinite places and make something new for you?
> Have you been good and read your maths text so you too can see
> Infinite Clause? He's keeping a list and he's checking it twice!
> CANTOR :
> The uncountable number does not have the 1st digit of the 1st
> countable number.
> UTM(neN,d) :
> ALL 10 digits are present in the 1st digit place an infinite number of
> times, on the list of computable reals.
> CANTOR :
> The uncountable number does not have the 2nd digit of the 2nd
> countable number.
> UTM(neN,d) :
> ALL 10 digits are present in the 2nd digit position (following every
> possibility of the 1st digit) for an infinite number of reals.
> CANTOR :
> The uncountable number does not have the 3rd digit of the 3rd
> countable number.
> ...
> Start looking at reality sci.math not your text, not David Ullrich and
> Barb Knox who *make money teaching texts*.  0.123... is on the list of
> computable numbers infinite times with infinite possible tails after
> the 123.  0.654... is there infinite times.  ALL PREFIXES of UNLIMITED
> new sequence of digits that is not on the list of computable reals. 
> ALL PERMUTATIONS ARE THERE.  The diag number must be SOME sequence of
> digits, like 0.123..., like 0.654... ITS NOT A NEW NUMBER, the elves
> went on strike!
Where is THE LIST?
David Bernier
===
Subject: Re: November 25 is Infinite Clause day!!
> Merry X-count-mas!
> Do you believe in Infinite Clause?  Will he make it to your real place
> in time and all of the infinite places and make something new for you?
> Have you been good and read your maths text so you too can see
> Infinite Clause? He's keeping a list and he's checking it twice!
> CANTOR :
> The uncountable number does not have the 1st digit of the 1st
> countable number.
> UTM(neN,d) :
> ALL 10 digits are present in the 1st digit place an infinite number of
> times, on the list of computable reals.
> CANTOR :
> The uncountable number does not have the 2nd digit of the 2nd
> countable number.
> UTM(neN,d) :
> ALL 10 digits are present in the 2nd digit position (following every
> possibility of the 1st digit) for an infinite number of reals.
> CANTOR :
> The uncountable number does not have the 3rd digit of the 3rd
> countable number.
> ...
> Start looking at reality sci.math not your text, not David Ullrich and
> Barb Knox who *make money teaching texts*.  0.123... is on the list of
> computable numbers infinite times with infinite possible tails after
> the 123.  0.654... is there infinite times.  ALL PREFIXES of UNLIMITED
> new sequence of digits that is not on the list of computable reals. 
> ALL PERMUTATIONS ARE THERE.  The diag number must be SOME sequence of
> digits, like 0.123..., like 0.654... ITS NOT A NEW NUMBER, the elves
> went on strike!
> Where is THE LIST?
> David Bernier
Just don't ask him to send it as an attachment to an e-mail; it will
take forever to open.
===
Subject: Re: November 25 is Infinite Clause day!!
In sci.logic, George Dance
<georgedance@hotmail.com> Merry X-count-mas!
>> 
>> Do you believe in Infinite Clause?  Will he make it to your real place
>> in time and all of the infinite places and make something new for you?
>> 
>> Have you been good and read your maths text so you too can see
>> Infinite Clause? He's keeping a list and he's checking it twice!
>> 
>> CANTOR :
>> The uncountable number does not have the 1st digit of the 1st
>> countable number.
>> 
>> UTM(neN,d) :
>> ALL 10 digits are present in the 1st digit place an infinite number of
>> times, on the list of computable reals.
>> 
>> CANTOR :
>> The uncountable number does not have the 2nd digit of the 2nd
>> countable number.
>> 
>> UTM(neN,d) :
>> ALL 10 digits are present in the 2nd digit position (following every
>> possibility of the 1st digit) for an infinite number of reals.
>> 
>> CANTOR :
>> The uncountable number does not have the 3rd digit of the 3rd
>> countable number.
>> ...
>> Start looking at reality sci.math not your text, not David Ullrich and
>> Barb Knox who *make money teaching texts*.  0.123... is on the list of
>> computable numbers infinite times with infinite possible tails after
>> the 123.  0.654... is there infinite times.  ALL PREFIXES of UNLIMITED
>> new sequence of digits that is not on the list of computable reals. 
>> ALL PERMUTATIONS ARE THERE.  The diag number must be SOME sequence of
>> digits, like 0.123..., like 0.654... ITS NOT A NEW NUMBER, the elves
>> went on strike!
>> Where is THE LIST?
>> David Bernier
> Just don't ask him to send it as an attachment to an e-mail; it will
> take forever to open.
Is that a denumerable forever, or a non-denumerable forever? :-)
-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.
===
Subject: Re: November 25 is Infinite Clause day!!
> In sci.logic, George Dance
> <georgedance@hotmail.com>Merry X-count-mas!
>>Do you believe in Infinite Clause?  Will he make it to your real place
>>in time and all of the infinite places and make something new for you?
>>Have you been good and read your maths text so you too can see
>>Infinite Clause? He's keeping a list and he's checking it twice!
>>CANTOR :
>>The uncountable number does not have the 1st digit of the 1st
>>countable number.
>>UTM(neN,d) :
>>ALL 10 digits are present in the 1st digit place an infinite number of
>>times, on the list of computable reals.
>>CANTOR :
>>The uncountable number does not have the 2nd digit of the 2nd
>>countable number.
>>UTM(neN,d) :
>>ALL 10 digits are present in the 2nd digit position (following every
>>possibility of the 1st digit) for an infinite number of reals.
>>CANTOR :
>>The uncountable number does not have the 3rd digit of the 3rd
>>countable number.
>>...
>>Start looking at reality sci.math not your text, not David Ullrich and
>>Barb Knox who *make money teaching texts*.  0.123... is on the list of
>>computable numbers infinite times with infinite possible tails after
>>the 123.  0.654... is there infinite times.  ALL PREFIXES of UNLIMITED
>>new sequence of digits that is not on the list of computable reals. 
>>ALL PERMUTATIONS ARE THERE.  The diag number must be SOME sequence of
>>digits, like 0.123..., like 0.654... ITS NOT A NEW NUMBER, the elves
>>went on strike!
>Where is THE LIST?
>David Bernier
>>Just don't ask him to send it as an attachment to an e-mail; it will
>>take forever to open.
> Is that a denumerable forever, or a non-denumerable forever? :-)
I know you're just joking here but a non-denumerable set could
be finite.
--
Replace Roman numerals with digits to reply by email
===
Subject: Re: November 25 is Infinite Clause day!!
In sci.logic, David Bernier
<david250@videotron.ca>
<vLUnd.98560$De5.1797114@wagner.videotron.net>:
>> Merry X-count-mas!
>> Do you believe in Infinite Clause?  Will he make it to your real place
>> in time and all of the infinite places and make something new for you?
>> Have you been good and read your maths text so you too can see
>> Infinite Clause? He's keeping a list and he's checking it twice!
>> CANTOR :
>> The uncountable number does not have the 1st digit of the 1st
>> countable number.
>> UTM(neN,d) :
>> ALL 10 digits are present in the 1st digit place an infinite number of
>> times, on the list of computable reals.
>> CANTOR :
>> The uncountable number does not have the 2nd digit of the 2nd
>> countable number.
>> UTM(neN,d) :
>> ALL 10 digits are present in the 2nd digit position (following every
>> possibility of the 1st digit) for an infinite number of reals.
>> CANTOR :
>> The uncountable number does not have the 3rd digit of the 3rd
>> countable number.
>> ...
>> Start looking at reality sci.math not your text, not David Ullrich and
>> Barb Knox who *make money teaching texts*.  0.123... is on the list of
>> computable numbers infinite times with infinite possible tails after
>> the 123.  0.654... is there infinite times.  ALL PREFIXES of UNLIMITED
>> new sequence of digits that is not on the list of computable reals. 
>> ALL PERMUTATIONS ARE THERE.  The diag number must be SOME sequence of
>> digits, like 0.123..., like 0.654... ITS NOT A NEW NUMBER, the elves
>> went on strike!
> Where is THE LIST?
> David Bernier
Not to mention HERC777's understanding.  The question is not whether
the diagonal number's digits are within the list (most likely,
they are, unless The List consists of, say, numbers exclusively
in the Cantor set and the digits are taken from base 3 instead
of base 10), but whether the number itself is on the list.
For purposes of this post, I'll posit that the number is
represented Diag, and each digit can be fetched from a
function Diag(N), and digit N of entry M in The List can
be represented List(M,N); the M'th entry in The List can
be represented List(M).  (For technical reasons no entry
on the List can have an infinite trail of 9's.)
Now List can be computed by a variety of methods:
- a Turing machine, spitting out numbers.
- a wizard's ball.
- a function.
- enumeration of a countable set, such as the rather simple one T_10 =
  {.0} union {k/10^n; k > 0, k, n in J, (k mod 10) != 0, 10^(n-1) < k < 
10^n)}
  which contains all finite decimal expansions in [0,1)
It doesn't really matter.
We also need to construct Diag.  This is fairly simple,
and one can have many variants, so we need just pick one:
if List(N,N) = 4 then Diag(N) = 5 else Diag(N) = 4
The question is whether Diag is in the List or not.  This question
is too vague in some respects; a more formal version might be
Does an N exist such that Diag = List(N)?
and that needs improvement too; the final version is
Does an N exist such that for every positive M in J,
   Diag(M) = List(N,M)?
The answer clearly is no (if M=N Diag(N) != List(N,N) by construction).
It's also clear that T_10 contains every digit an infinite number of times,
in any digit slot one cares to specify.  T_10 is also dense;
Diag may not be on the list but T_10 will contain numbers arbitrarily
close to Diag.  However, Diag is simply not in T_10, in the same
manner that 1/3 is not in the set { (10^n - 1) / (3 * 10^n): n in J, n > 0}
= {.3, .33, .333, .3333, ...}, though again one can find elements
that are arbitrarily close to 1/3.
Cantor's first proof also addresses the issue, in a more abstract fashion.
If one wants to rescope the problem, one can state Diag(List) instead
of Diag and propagate forward as necessary; the English equivalent
might be
   for *any* List of reals, Diag is not on that particular List
which might work even better, though HERC777 has tried to insert Diag
into the postulated List (which won't work as it changes the List).
-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.
===
Subject: Re: November 25 is Infinite Clause day!!
In sci.logic, herc777@hotmail.com
<herc777@hotmail.com -Not to mention HERC777's understanding.  The question 
is not whether
> -the diagonal number's digits are within the list (most likely,
> -they are
> this is your 1st error, you overlook the fact all digit sequences to
> infinite length are computable the conclusion of Cantor's proof relies
> on the existence of a new sequence of digits which is clearly non
> existent.
Not sure all infinite digit sequences are computable.  One problem:
the set of IDS is a power set (10^N); card(10^N) > card(N).
Since all UTMs depend on integer codes (usually broken down into
instructions), this yields digit sequences that cannot be generated
by a TM.
> -We also need to construct Diag.  This is fairly simple,
> -and one can have many variants, so we need just pick one:
> -if List(N,N) = 4 then Diag(N) = 5 else Diag(N) = 4
> this is your second error, your construction can never be realised.
Correct, it's non-computable.  This doesn't mean it can't exist;
there are a large number of non-computable functions out there.
> its like saying construct the set of all sets that don't contain
> themselves, if its a member of itself its not a member, if its not a
> member of itself its a member.  you can theorise the construction but
> you cannot refute your basic premises of set theory because of one
> theorem, regardless if a contradiction is formed.
> this is all you are doing.
> let R = 1, 2, 3, 4...
> let p <> R1, p <> R2, p <> R3...
> therefore R is incomplete.
> your claim that p is a valid construction but its not rigorous.
Neither p nor Diag is a valid construction as such (in the sense that
it can be realized by a TM).  Diag merely needs to exist in some form.
Cf. Cantor's first proof.
> Its like your Godel proof, you are trying to prove ! (proof(X) <-> X)
> proof(X) <=> Exists a proof of X.
> so you allow this theorem, G <-> ! proof(G) but you don't allow this
> meta-consistency theorem X <-> proof(X).  Either of these theorems will
> work but they are mutually exclusive, no one here has shown a PROOF
> that ! (proof(X) <-> X).
That's because there is no such proof.  All Godel proved is that
there are true unprovable statements.
> In AI the meta-constistency theorem is called
> Truth Maintenance, only proven facts are allowed and never need to be
> revoked.  The alternate system is called belief revision.  Godel's
> proof only works in one automated theorem system.
> Same with the set of reals, assume all possible combinations of digits
> are present in a countable list, with UTM(n, 0) they actually are.  Now
> examine the construction of diag, it is a flawed construction!  QED.
That it is.  Diag cannot be constructed using a UTM.  However, that
doesn't mean it's not a real.
> -Does an N exist such that for every positive M in J,
> -  Diag(M) = List(N,M)?
> -The answer clearly is no (if M=N Diag(N) != List(N,N) by
> construction).
> This is your 3rd error.  Why is it when I write 'obviously' it gets
> picked up yet everyone uses 'clearly' around here.  What you all
> actually mean by 'clearly' is clearly it's in the text book!  Your
> construction breaks the premise of a complete list, it doesn't
> contradict it.
So you're stating that another member in the list is actually equal
to Diag?
Which one, then?  Or are you hypothesizing that List is not
a denumerable mapping?
> An infinite number of people toss a coin RANDOMLY an infinite number of
> times. After 10 people have tossed a coin 5 times each, the 1st 3
> places in the sequence are (tend to be) all covered.
> hhh..
> hht..
> hth..
> htt..
> thh..
> tht..
> tth..
> ttt..
> there are 2 duplicates due to random spread.
> After several million people have tossed a coin 30 times each, by the
> binomial distribution, all combinations have been covered up to 15
> tosses.
> 000000000000001..010101010
> 000000000000010..010101010
> 000000000000011..010101010
> 000000000000100..010101010
> 000000000000101..010101010
> 000000000000110..010101010
> 000000000000111..010101010
> 000000000001000..010101010
> ...
> 111111111111110..010101010
> 111111111111111..010101010
> <- - - - - - - ->|<- - - - - all combinations    still random
> covered
> The number of coins that get covered increases without bound,
> logarithmically. The length of 'covered combinations' is approx.
> log(people doing coin tosses). As the countable list approaches
> infinite length..
> As #people -> oo, number of digits covered -> log(oo).
> Therefore an infinite list contains all sequences of {H, T} of infinite
> length.
> 1 HTHTHTHTHT
> 2 HHHHHHHHH
> 3 TTTTTTTTTTT
> 4 THTHTHTHTHT
> ..
> Take the diagonal!  HHTH...
> Invert it!   TTHT..
> ALL SEQUENCES ARE PRESENT, TTHT.. is not a new sequence.
> INFINTIE people toss coins infinite times each, can you with 100%
> certainty form a NEW {H, T} sequence?  CLUE : NO!!  Inverting the
> diagonal is not a new sequnce of tosses, and modifying the real
> diagonal does not a new real make.
OK, dumb question.  Are you hypothesizing that card(2^N) = card(N)?
2^N can be mapped to the set of all fractional expansions, where
the n'th digit is 0 if n is not in the set S, and 1 if n is.
(Recall that the set S is in 2^N, therefore a *subset* of N.)
> Herc
-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.
===
Subject: Re: November 25 is Infinite Clause day!!
   <vLUnd.98560$De5.1797114@wagner.videotron.net>
   <sd0772-mug.ln1@sirius.athghost7038suus.net>
   <9kq772-t9i.ln1@sirius.athghost7038suus.net>
You are using the RESULTS of Cantors proof to substantiate Cantors
proof.
You haven't answered my question, an infinite number of people toss a
coin infinite times. Can you come up with a new sequence of Heads and
Tails?
Herc
===
Subject: Re: November 25 is Infinite Clause day!!
In sci.logic, herc777@hotmail.com
<herc777@hotmail.com You are using the RESULTS of Cantors proof to 
substantiate Cantors
> proof.
> You haven't answered my question, an infinite number of people toss a
> coin infinite times. Can you come up with a new sequence of Heads and
> Tails?
> Herc
I'm assuming you mean each tosses their own coin here and
that they toss their respective coins in unison.  (There are
some issues because of lightspeed propagation which are probably
not worth considering here.)
The answer is Yes.  Take the tosses a pair at a time.  (The reason
is because of the issue of .111... = 1.000...(2).)  Then diagonalize
in the more or less usual manner:
P{1} = [HT] TH TT TH TH TT HH TT HT
P{2} = TT [TH] TT TH HT HT HH HH HT
P{3} = TH TT [TH] TT TH TT TT TT HH
P{4} = TT HT TT [HH] TT TH TT HT TT
P{5} = TH HH TH HH [TT] HT TT HT HT
...
D = TH HT TH TH TH ...
where I've picked HT if the pair of tosses is TH, and TH otherwise.
D cannot be associated with any P{n}, although the n'th toss-pair of P{i}
for any i is likely to be equal to D[i] about 1/4 of the time, and any
finite prefix of D of length n will occur in the list once every 2^(2*n)
times.
It's an issue similar to the question:
   Does 1/3 occur in the set { (10^n - 1) / (3 * 10^n): n > 0, n in J}?
Let's call S = { (10^n - 1) / (3 * 10^n): n > 0, n in J}.  It's clear
that all elements of S are in the set of rational numbers, and
furthermore that the denominator must contain at least 1 of 3, 2, and 5.
Since the numerator == 9 (mod 10) it clearly isn't a multiple of 2 or 5.
However, it turns out 3 divides the numerator at least twice.  (It can
be more than twice; 999 = 3^3 * 37, for example.)  Therefore, one
for some integer N, and that gcd(N, 10^n) = 1, and furthermore that
3 divides N.
In order for 1/3 to be in S, it must also be writable in the form N / 10^n
for some N and n.  This is of course absurd, since it would mean that
3 would have to divide some power of 10.
Of course one can come arbitrarily close.  For any positive epsilon,
I can pick n = ceil(log_10(1/(3*epsilon)).
The term (10^n - 1) / (3 * 10^n) differs from 1/3 by the amount
1 / (3 * 10^n).  Since n >= log_10(1/(3*epsilon)),
10^n >= 1/(3*epsilon), 1/10^n <= 3*epsilon, 1/(3*10^n) <= epsilon,
and the limit is proved.
If you wish, you can define 1/3 by using the set S -- such is
vaguely similar to what Dedekind did some time back.  Of course
a better definition (and as it turns out closer to Dedekind's
defs anyway), were one to need to do such, is to define
two sets A and B:
A = {x: 3*x < 1}
B = {x: 3*x > 1}
and, after a suitable proof that an arbitrary element of a is
always less than an arbitrary element of b, one can squish 1/3
thereby.  Of course for a rational number this is overkill but
there are other numbers not so easily contained -- the positive
square root of 2 can be cut for example by:
C = {x: x < 0 or x^2 < 2}
D = {x: x > 0 and x^2 > 2}
Or one can pick the set U, where U = {2 * a(n) / p(n): n > 0, n in J}.
a(n) is the area of an inscribed regular n-sided polygon; p(n) is n
times the side of said polygon.  The elements of U will be dense
around a certain number, which turns out to be transcendental --
namely, pi.  But pi isn't in U.
-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.
===
Subject: Re: November 25 is Infinite Clause day!!
   <vLUnd.98560$De5.1797114@wagner.videotron.net>
   <sd0772-mug.ln1@sirius.athghost7038suus.net>
   <9kq772-t9i.ln1@sirius.athghost7038suus.net>
   <r88872-q0j.ln1@sirius.athghost7038suus.net>
-> You haven't answered my question, an infinite number of people toss
a
-> coin infinite times. Can you come up with a new sequence of Heads
and
-> Tails?
-
-
-The answer is Yes.  Take the tosses a pair at a time.  (The reason
-is because of the issue of .111... = 1.000...(2).)  Then diagonalize
-in the more or less usual manner:
-
-
-P{1} = [HT] TH TT TH TH TT HH TT HT
-P{2} = TT [TH] TT TH HT HT HH HH HT
-P{3} = TH TT [TH] TT TH TT TT TT HH
-P{4} = TT HT TT [HH] TT TH TT HT TT
-P{5} = TH HH TH HH [TT] HT TT HT HT
-...
-
-D = TH HT TH TH TH ...
-
-where I've picked HT if the pair of tosses is TH, and TH otherwise.
nice one, though I don't think infinite repeating heads is the same as
1 tail!
So does everyone who supports Cantor's proof here also believe that
Ghost has a valid technique to find a new sequence of Heads and Tails?
Herc
An infinite number of people toss a coin infinite times each. Can you
guarantee a new sequence of Heads and Tails?
===
Subject: Re: November 25 is Infinite Clause day!!
In sci.logic, herc777@hotmail.com
<herc777@hotmail.com -> You haven't answered my question, an infinite number 
of people toss
> -> coin infinite times. Can you come up with a new sequence of Heads
> and
> -> Tails?
> -The answer is Yes.  Take the tosses a pair at a time.  (The reason
> -is because of the issue of .111... = 1.000...(2).)  Then diagonalize
> -in the more or less usual manner:
> -P{1} = [HT] TH TT TH TH TT HH TT HT
> -P{2} = TT [TH] TT TH HT HT HH HH HT
> -P{3} = TH TT [TH] TT TH TT TT TT HH
> -P{4} = TT HT TT [HH] TT TH TT HT TT
> -P{5} = TH HH TH HH [TT] HT TT HT HT
> -...
> -D = TH HT TH TH TH ...
> -where I've picked HT if the pair of tosses is TH, and TH otherwise.
> nice one, though I don't think infinite repeating heads is the same as
> 1 tail!
> So does everyone who supports Cantor's proof here also believe that
> Ghost has a valid technique to find a new sequence of Heads and Tails?
What is this, a popularity contest? :-P
> Herc
> An infinite number of people toss a coin infinite times each. Can you
> guarantee a new sequence of Heads and Tails?
-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.
===
Subject: Re: November 25 is Infinite Clause day!!
   <vLUnd.98560$De5.1797114@wagner.videotron.net>
   <sd0772-mug.ln1@sirius.athghost7038suus.net>
   <9kq772-t9i.ln1@sirius.athghost7038suus.net>
   <r88872-q0j.ln1@sirius.athghost7038suus.net>
   <96c872-j7j.ln1@sirius.athghost7038suus.net>
Don't tell Xanthian... getting a group consensus is unmathematical, we
have to duke it out!
Barb Knox, George Green, Jesse James, John Savard, Dave B, you,
Ullrich, Will and dozens of others have all been vocally opposed to me
in their defence of Cantors diag proof.  Now they are all silent?  is
it apathy?
This is the bait :
> An infinite number of people toss a coin infinite times each. Can you
> guarantee a new sequence of Heads and Tails?
Herc : no, that's silly run some simulations and extrapolate to
infinity.
Ghost : take the diag as in Cantor's uncountabble real proof.
Jesse : hope they all toss heads.
If the answer is NO, which I think most if not many people will agree
it is, then Cantors' proof is as wrong as your Head Tail string
construction.
Do people here understand that believing in aleph0 and all this
cardinality theory entails that you believe you can answer this
question in the affirmative?
Herc
An infinite number of people toss a coin infinite times each. Can you
guarantee a new sequence of Heads and Tails?
===
Subject: Re: November 25 is Infinite Clause day!!
>In sci.logic, David Bernier
><david250@videotron.ca<vLUnd.98560$De5.1797114@wagner.videotron.net>:
> Merry X-count-mas!
Do you believe in Infinite Clause?  Will he make it to your real place
> in time and all of the infinite places and make something new for you?
Have you been good and read your maths text so you too can see
> Infinite Clause? He's keeping a list and he's checking it twice!
CANTOR :
> The uncountable number does not have the 1st digit of the 1st
> countable number.
UTM(neN,d) :
> ALL 10 digits are present in the 1st digit place an infinite number of
> times, on the list of computable reals.
CANTOR :
> The uncountable number does not have the 2nd digit of the 2nd
> countable number.
UTM(neN,d) :
> ALL 10 digits are present in the 2nd digit position (following every
> possibility of the 1st digit) for an infinite number of reals.
CANTOR :
> The uncountable number does not have the 3rd digit of the 3rd
> countable number.
> ...
> Start looking at reality sci.math not your text, not David Ullrich and
> Barb Knox who *make money teaching texts*.  0.123... is on the list of
> computable numbers infinite times with infinite possible tails after
> the 123.  0.654... is there infinite times.  ALL PREFIXES of UNLIMITED
> new sequence of digits that is not on the list of computable reals. 
> ALL PERMUTATIONS ARE THERE.  The diag number must be SOME sequence of
> digits, like 0.123..., like 0.654... ITS NOT A NEW NUMBER, the elves
> went on strike!
>> Where is THE LIST?
>> David Bernier
>Not to mention HERC777's understanding.
It certainly is true that his understanding of Cantor's diagonal proof
is flawed.
Cantor proves that a list, making all the real numbers correspond to the
counting integers, is not possible, by showing that one can construct a
new number not on any such list by picking a number which differs in the
N-th place from the N-th element on the list.
Yes, it may well be that this different number will be in the N-th place
for *other* numbers on the list, but that is not the point.
You can create a list that makes all the rational numbers correspond to
the counting integers, or all the computable reals correspond to the
counting integers. You can create a list, therefore, that contains
elements which begin with any finite sequence of digits, however large.
Cantor's proof only works because the number of digits in the expansion
of a real number is *an actual completed infinity*; some real numbers
have infinitely complex descriptions, and are therefore both
uncomputable and unnameable.
Anyways, this will soon come to an end. Having been 11 years old some
years ago, our indefatigable poster will eventually become interested in
girls and leave us alone.
John Savard
http://home.ecn.ab.ca/~jsavard/index.html
===
Subject: Re: Element of?
>As a layman I wonder if the basic meanings of all symbols used in axioms
>of set theory have already been completely and unambiguously clarified.
>>No.  The axioms frequently serve to *help* clarify the meanings of the 
>>symbols.  Also, any grammar rules *help* clarify the meanings. 
>>Ultimately, however, the meanings only exist unambiguously for a model 
>>of the axioms/symbols.
> A mathematics that is not just a self-satisfying game has to be
> carefully fitted into the whole knowledge of menkind. This requires at
> first properly chosen most elementary basics. The Atoms of mathematics
> are not the axioms but what one intends to express for instance with the
> symbol for element of.
That is, as I understand it, how math is actually developed.  The people 
who built ZF and ZFC versions of set theory *did* have an idea of what 
that symbol means.  However, that doesn't mean they started by defining 
it.  The purpose of the axioms is to describe what it means without 
referring to the model they had in mind.
>>I have no idea what this is supposed to mean.  Real and rational numbers 
>>are no more uncertain than integers.
> If physicists and engineers prefer integers, they often do so because
> these genuine numbers are absolutely precise. Calculation with reals or
> more precisely rationals always depends on the chosen accuracy.
To work with integers is to impose a level of accuracy as well.  Rarely 
do objects travel at integer velocities or cover integer distances.
>Given, zero does not exist as a rational number.
>>Zero *is* a rational number.  What made you think it is not?
> I gave several reasons in de.sci.mathematics
> Let me add a further one:
> Zero is supposed to be a neutral number without any sign. Can you
> imagine to divide a by b and yield a result without sign?
Perhaps you are not aware of either of the common definitions of a 
rational number.  Q = {m/n | m and n are integers, n =/= 0}  Choosing 
m=0, n=1 gives 0 as a rational number.
As for your issue with signs, if moving right is positive, and moving 
left is negative, does it make sense to refer to not moving as being 
right or left motion?
>Given, the concept of real numbers covers zero just in case of the
>potential infinity.
>>Huh?  What do you mean by covers zero and potential infinity?
> As did Weyl, I consider a continuum a sauce.
That makes no sense to me.  Granted, I haven't looked up Weyl's comment.
> The term potential infinity was introduced by Aristoteles and means
> infinity is a fiction outside the wealth of numbers. There is actually
> no infinite number.
Thus the fact that infinity is not a number.  Why obfuscate things with 
the term potential?
>Is there any justification for including or excepting a rational or real
>zero in physics?
>>I thought we were talking about math, which is a tool used in physics.
> I respect mathematics, but I am an engineer who demands flawless tools.
As an engineer you should know that we have no flawless tools.  More 
importantly, as an engineer it is very likely that you have little or no 
  understanding of the mentality behind most of mathematics.
>Couldn't reals be interpreted as integers divided by an denominator of
>infinite size? I would conclude from that: Reals are of quite different
>quality.
>>You can't have a denominator of infinite size, under any normal 
>>interpretation.  You think the reals of quite different quality from 
>>*what*?
> The entity of reals as a sauce is quite different from the notion of a
> number.
What do you mean by as a sauce?
> What are you referring to with IZ, IQ,
>>IR, and IC?
> Sorry for my awkward letters. I meant integer, rational, real, and
> complex numbers.
Those are usually just referred to as Z, Q, R, C.
-- 
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Re: Element of?
>> As a layman I wonder if the basic meanings of all symbols used in 
axioms
>> of set theory have already been completely and unambiguously 
clarified.
> No.  The axioms frequently serve to *help* clarify the meanings of the 
> symbols.  Also, any grammar rules *help* clarify the meanings. 
> Ultimately, however, the meanings only exist unambiguously for a model 
> of the axioms/symbols.
> A mathematics that is not just a self-satisfying game has to be
> carefully fitted into the whole knowledge of menkind. This requires at
> first properly chosen most elementary basics. The Atoms of mathematics
> are not the axioms but what one intends to express for instance with the
> symbol for element of.
This sounds reasonable, but I'm not sure it's true. Generations of
mathematicians have engaged quite deliberately in self-satisfying
games - read Hardy's A mathematician's apology. Then decades later
things (like abstract algebra, in particular finite fields) turn out
to have direct application to the real world, (quite possibly getting
classified by the US gumint as munitions [sic]). It's all part of
the unreasonable effectiveness of mathematics [or words very
Brian Chandler
http://imaginatorium.org
===
Subject: Re: Element of?
>>I thought we were talking about math, which is a tool used in physics.
> Of course, mathematics is much, much more than a tool used in 
physics.
Absolutely.
-- 
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Re: Cantor's diagonal proof wrong?
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iAIHHaI02526;
>> But now I'm starting to wonder if there may be value in looking at 
this
>> differently.  Might it be valid to think of division of 1 by 9 as an
>> infinite processes (algorithm) that produces closer and closer
>> approximations, yet is always unable to produce the actual point 
on
>> the line?
>> Nonesense. One can always divide a line segment into N equal parts by a
>> well known geometric construction. So getting the point on the segment
>> corresponding to k/N for k = 0,1,...N is trivial. Or equivalently choose
>> a unit length and lay out multiples of this length on an infinite ray.
>> Again one easily constructs points corresponding to k*N.  Why do you
>> complicate a very straightforward matter?
>Because most things in life that seem straightforward turn out later to be
>anything but that.  I enjoy finding the exceptions, so I search.  But like
>fishing, many times, there is just nothing to be caught.
>-- 
>Curt Welch                                            <a 
href=http://CurtWelch.Com/>http://CurtWelch.Com/</acurt@kcwc.com             
                           <a 
href=http://NewsReader.Com/>http://NewsReader.Com/</a>
Would you argue that 1/5 is also defined by an algorithm?
Probably not, because 1/5 = 0.2, a nice number. But why should the 
method of obtaining 1/5 differ from 1/9? It shouldn't. This occurs because 
of 
your choice of the decimal system. If you were to choose the system of 9 
digits, then 1/9 would be a nice 0.1 . Nut then 1/5, in this system, 
wouldn't be so nice anymore. Certainly, a theory about infinity, cannot 
depend on how many digits or symbols you use in your system.
facedancer
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> Would you argue that 1/5 is also defined by an algorithm?
Yes.  All mathematical operations are procedures or algorithms.
Some make reference to procedures that can never produce an answer, like
1/0, and some make reference to procedures that never terminate, like
.111....
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
> 1/0, and some make reference to procedures that never terminate, like
> .111....
1/9 terminates. Using a well known construction one can divide any 
length by an integer ( > 0 of course). It terminates very nicely think 
you. Once again you confuse the mathematics with the method of 
representing quantities. Don't quit your day job to take up a career in 
mathematics. You have not got the Right Stuff.
Bob Kolker
===
Subject: Re: Cantor's diagonal proof wrong?
>> Would you argue that 1/5 is also defined by an algorithm?
> Yes.  All mathematical operations are procedures or algorithms.
> Some make reference to procedures that can never produce an answer, like
> 1/0, and some make reference to procedures that never terminate, like
> .111....
Infinites can have the start and the end. Consider the reals in the closed 
area [0,1]. Zero is the start and 1 is the end. There are still infinite 
many reals. How are you so sure that 0.111... is NOT this type of infinite 
string?
Tapio
> -- 
> Curt Welch 
> http://CurtWelch.Com/
> curt@kcwc.com 
> http://NewsReader.Com/ 
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> Infinites can have the start and the end. Consider the reals in the
> closed area [0,1]. Zero is the start and 1 is the end. There are still
> infinite many reals. How are you so sure that 0.111... is NOT this type
> of infinite string?
> Tapio
This is a pefect example of how people let language confuse them.
If you take a finite sized line, and keep cuting it in half, you have the
description of an infinite procedure which (in the world of math and
geometry at least) has no end.  But the line does have two ends.
So then the guy above points to the end of the line and says, look,
there's an infinity with two ends!.
The ends he was pointing to has nothing to do with the end of the
infinite procedure under discussion.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
> So then the guy above points to the end of the line and says, look,
> there's an infinity with two ends!.
Yes, but somebody else have done that as they defined [0,1].
> The ends he was pointing to has nothing to do with the end of the
> infinite procedure under discussion.
How you can be so sure about that? :-)
Tapio
> -- 
> Curt Welch 
> http://CurtWelch.Com/
> curt@kcwc.com 
> http://NewsReader.Com/ 
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> So then the guy above points to the end of the line and says, look,
> there's an infinity with two ends!.
> Yes, but somebody else have done that as they defined [0,1].
> The ends he was pointing to has nothing to do with the end of the
> infinite procedure under discussion.
> How you can be so sure about that? :-)
I can't.  But I can pretend to be sure on Usenet. :)
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
> I can't.  But I can pretend to be sure on Usenet. :)
Has anyone told you that you are a horse's arse. No? O.K. You are a 
horse's arse.
Bob Kolker
===
Subject: Re: Cantor's diagonal proof wrong?
> I can't.  But I can pretend to be sure on Usenet. :)
So you admit you're a troll?
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> I can't.  But I can pretend to be sure on Usenet. :)
> So you admit you're a troll?
No, I'm not trolling.  But I'm sure some people see my posts like that.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
>> Would you argue that 1/5 is also defined by an algorithm?
>Yes.  All mathematical operations are procedures or algorithms.
>Some make reference to procedures that can never produce an answer, like
>1/0, and some make reference to procedures that never terminate, like
>.111....
Does 0.20000000.... terminate or not?  Does it make a difference if I
tell you that it was written in base 3?
Alan
-- 
Defendit numerus
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
>> Would you argue that 1/5 is also defined by an algorithm?
>Yes.  All mathematical operations are procedures or algorithms.
>Some make reference to procedures that can never produce an answer, like
>1/0, and some make reference to procedures that never terminate, like
>.111....
> Does 0.20000000.... terminate or not?
No.  But 0.2 does. :)
> Does it make a difference if I
> tell you that it was written in base 3?
No.
The ... langauge you used means does not terminate.
> Alan
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
:>
:>> Would you argue that 1/5 is also defined by an algorithm?
:>
:>Yes.  All mathematical operations are procedures or algorithms.
:>
:>Some make reference to procedures that can never produce an answer, 
like
:>1/0, and some make reference to procedures that never terminate, like
:>.111....
:> Does 0.20000000.... terminate or not?
: No.  But 0.2 does. :)
So does 1/5 terminate or not?  Does 1/5 = 0.2, or does it equal
0.20000000.....?   I would say that it equals both and the
idea of 1/5 'terminating' is not well defined. 
Stephen
===
Subject: Re: Cantor's diagonal proof wrong?
>::>> Would you argue that 1/5 is also defined by an algorithm?
>::>Yes.  All mathematical operations are procedures or algorithms.
>::>Some make reference to procedures that can never produce an answer, 
like
>:>1/0, and some make reference to procedures that never terminate, like
>:>.111....
>:> Does 0.20000000.... terminate or not?
>: No.  But 0.2 does. :)
>So does 1/5 terminate or not?  Does 1/5 = 0.2, or does it equal
>0.20000000.....?   I would say that it equals both and the
>idea of 1/5 'terminating' is not well defined. 
And after you are done answering that, does 2/3 terminate or not?
Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3).
Alan
-- 
Defendit numerus
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> And after you are done answering that, does 2/3 terminate or not?
> Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3).
the silly questions (and answers) back to the big picture of why on earth I
was talking about procedures that do or do not terminate.  You should read
that if you want to understand why I was talking about these things.
I believe math was created as a langauge for talking about what can, and
can not do, and know, about procedures.  But somewhere, it feels to me,
like it went off track and made up some rules of logic which violate the
laws of nature (the laws of procedures).  Did it go off track or not?  Is
it a bad thing even if it did?  I don't know for sure.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
:> And after you are done answering that, does 2/3 terminate or not?
:> Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3).
: the silly questions (and answers) back to the big picture of why on earth 

I
: was talking about procedures that do or do not terminate.  You should 
read
: that if you want to understand why I was talking about these things.
: I believe math was created as a langauge for talking about what can, and
: can not do, and know, about procedures.  But somewhere, it feels to me,
: like it went off track and made up some rules of logic which violate the
: laws of nature (the laws of procedures).  Did it go off track or not?  Is
: it a bad thing even if it did?  I don't know for sure.
Why do you believe this, and why do you think if went off track?
You have not actually given any evidence for your laws of nature
or that something has gone off track?  Despite your belief math
is clearly capable of discussing things that are not physically
possible.  Even something as simple as 
        358203583*2347922423 = 841034224524641609
is not a procedure that a human can complete.  Do you think
that you enough objects that you could arrange in a grid 358
million by 2.3 billion, then count up all the objects needed
to fill in the grid and verify that it is indeed 841 quintillion?
In any case, I thought you were interested in AI, and that you were
going to solve the AI problem?  I would expect a successful AI
to understand Cantor's proof.  I would also expect a successful
AI to understand why the diagonal argument does not apply to
the natural numbers or the rational numbers.  The fact that
you do not understand Cantor's proof does not mean that your
AI should not.
Stephen
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> :> And after you are done answering that, does 2/3 terminate or not?
> :> Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3).
> : get the silly questions (and answers) back to the big picture of why on
> : earth I was talking about procedures that do or do not terminate.  You
> : should read that if you want to understand why I was talking about
> : these things.
> : I believe math was created as a language for talking about what can,
> : and can not do, and know, about procedures.  But somewhere, it feels to
> : me, like it went off track and made up some rules of logic which
> : violate the laws of nature (the laws of procedures).  Did it go off
> : track or not?  Is it a bad thing even if it did?  I don't know for
> : sure.
> Why do you believe this, and why do you think if went off track?
> You have not actually given any evidence for your laws of nature
> or that something has gone off track?
Yeah, that's a good question to ask.  It's the question I've hard a hard
time finding a way to answer. I've put forth a lot of evidence, but I've
not put forth a convincing argument for anyone yet.
I present an argument that I hope some people will see shows how the logic
of the diagonal argument has problems.  This counter argument doesn't
depend on the definition of numbers, or reals, or anything said in set
theory.  It simply shows that any argument that takes the form of the
diagonal argument to prove that a value can not be in an infinite table is
invalid.  Maybe this will help some people open their minds to the
possibility that what they are looking at is just an illusion.  That there
might be more hidden behind the language than they every suspected.
The trick to this problem is that it's a matter of faith.  Like believing
in God, once you accept it as fact, it's hard to see anything else.  You
build a huge set of defensive arguments to support the initial belief, and
any single piece of evidence presented is easily ignored against the
fortress of defense built to support the first belief.  You can't even see
the other position until you are first willing to, if only for a second,
open a crack in your defenses and look with new eyes at the evidence - even
though it seems to contradict everything you believe in.
> Despite your belief math
> is clearly capable of discussing things that are not physically
> possible.  Even something as simple as
>         358203583*2347922423 = 841034224524641609
> is not a procedure that a human can complete.  Do you think
> that you enough objects that you could arrange in a grid 358
> million by 2.3 billion, then count up all the objects needed
> to fill in the grid and verify that it is indeed 841 quintillion?
The larger the problem, to more expensive in time and energy it becomes to
solve.  If it's a finite problem, then the only question is do you have
enough time and energy and matter to do it.  At some upper end of that
problem, there may not be enough matter in the universe to build a machine
that could solve the problem - but we don't really know the full nature of
the universe yet.
What we do know however, is that any infinite processes you specify will
require an infinite amount of time and energy, and will never complete.
> In any case, I thought you were interested in AI, and that you were
> going to solve the AI problem?
Yeah, I am.  And BTW, I don't need to shed light on this area of math in
order to solve AI.  This is not something I must resolve to finish my AI
work.  It's just something that because of my AI work, I was able to spot a
problem with.  So I became curious to further understand the nature of this
problem I spotted.
> I would expect a successful AI
> to understand Cantor's proof.  I would also expect a successful
> AI to understand why the diagonal argument does not apply to
> the natural numbers or the rational numbers.  The fact that
> you do not understand Cantor's proof does not mean that your
> AI should not.
I understand Cantor's proof.  It's trivial to understand.  What's much
harder to understand is why it's an invalid proof.  You might remember I
started this thread  by explaining I used to believe the proof.  I was
taught the proof some 25 years ago in school and thought it was a very cool
proof and understood the logic instantly even though the results were
surprising.  I spent the next 25 years believing it was an obviously valid
proof and when I've seen the same proof used in other fields like computer
science, I instantly believed the results of those proofs.  I no longer
believe that.  I know things know that I did not know 25 years ago.  I now
know what we are and why we do the things we do.  I know what language is
now.
My AI would have no problem believing that Cantor's proof is valid just
like I did and just like you do.  It would have no problem learning to talk
just like you do.  If it were educated the same way you were, it too would
be posting the type of messages you post in order to understand what this
fool named Curt thought he was talking about.  But with enough of the
proper education, it would also be able to understand what I was talking
about.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
>> :> And after you are done answering that, does 2/3 terminate or not?
>> :> Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3).
>> : get the silly questions (and answers) back to the big picture of why 
on
>> : earth I was talking about procedures that do or do not terminate.  You
>> : should read that if you want to understand why I was talking about
>> : these things.
>> : I believe math was created as a language for talking about what can,
>> : and can not do, and know, about procedures.  But somewhere, it feels 
to
>> : me, like it went off track and made up some rules of logic which
>> : violate the laws of nature (the laws of procedures).  Did it go off
>> : track or not?  Is it a bad thing even if it did?  I don't know for
>> : sure.
>> Why do you believe this, and why do you think if went off track?
>> You have not actually given any evidence for your laws of nature
>> or that something has gone off track?
>Yeah, that's a good question to ask.  It's the question I've hard a hard
>time finding a way to answer. I've put forth a lot of evidence, but I've
>not put forth a convincing argument for anyone yet.
>I present an argument that I hope some people will see shows how the logic
>of the diagonal argument has problems.  This counter argument doesn't
>depend on the definition of numbers, or reals, or anything said in set
>theory.  It simply shows that any argument that takes the form of the
>diagonal argument to prove that a value can not be in an infinite table is
>invalid.  
No, it does not show any such thing.
At _most_ it shows something of the form if you look at it that
way then the proof is invalid, which is simply irrelevant, because
the way you suggest people look at it is simply not consistent with
what the words in the statement of the theorem _mean_. _If_ one
interprets the reals are uncountable as meaning the moon is
made of green cheese then the reals are uncountable does indeed
become false, but this is simply silly because of that if -
the reals are uncountable does _not_ mean the moon is made
of green cheese.
>Maybe this will help some people open their minds to the
>possibility that what they are looking at is just an illusion.  That there
>might be more hidden behind the language than they every suspected.
>The trick to this problem is that it's a matter of faith.  Like believing
>in God, once you accept it as fact, it's hard to see anything else.  You
>build a huge set of defensive arguments to support the initial belief, and
>any single piece of evidence presented is easily ignored against the
>fortress of defense built to support the first belief.  You can't even see
>the other position until you are first willing to, if only for a second,
>open a crack in your defenses and look with new eyes at the evidence - 
even
>though it seems to contradict everything you believe in.
Find a mirror somewhere. This is a precise description of what the
rest of us see you doing: You started with an explanation why the
proof was wrong. You eventually agreed that your initial explanation
was totally bogus. But that doesn't seem to have had any effect on
your conviction that the proof is wrong - you just continue to invent
new explanations, gradually becoming more and more vague and less
relevant to what the theorem actually _says_.
>> [...]
>What we do know however, is that any infinite processes you specify will
>require an infinite amount of time and energy, and will never complete.
And this has no relevance whatever, because the statement of the
theorem has nothing to do with processes.
>> In any case, I thought you were interested in AI, and that you were
>> going to solve the AI problem?
>Yeah, I am.  And BTW, I don't need to shed light on this area of math in
>order to solve AI.  This is not something I must resolve to finish my AI
>work.  It's just something that because of my AI work, I was able to spot 
a
>problem with.  So I became curious to further understand the nature of 
this
>problem I spotted.
>> I would expect a successful AI
>> to understand Cantor's proof.  I would also expect a successful
>> AI to understand why the diagonal argument does not apply to
>> the natural numbers or the rational numbers.  The fact that
>> you do not understand Cantor's proof does not mean that your
>> AI should not.
>I understand Cantor's proof.  
If you think that the statement above about completing infinite
processes has some relevance then you don't even understand the
_statement_ of the theorem, much less the proof.
>It's trivial to understand.  What's much
>harder to understand is why it's an invalid proof.  You might remember I
>started this thread  by explaining I used to believe the proof.  I was
>taught the proof some 25 years ago in school and thought it was a very 
cool
>proof and understood the logic instantly even though the results were
>surprising.  I spent the next 25 years believing it was an obviously valid
>proof and when I've seen the same proof used in other fields like computer
>science, I instantly believed the results of those proofs.  I no longer
>believe that.  I know things know that I did not know 25 years ago.  I now
>know what we are and why we do the things we do.  I know what language is
>now.
>My AI would have no problem believing that Cantor's proof is valid just
>like I did and just like you do.  It would have no problem learning to 
talk
>just like you do.  If it were educated the same way you were, it too would
>be posting the type of messages you post in order to understand what this
>fool named Curt thought he was talking about.  But with enough of the
>proper education, it would also be able to understand what I was talking
>about.
************************
David C. Ullrich
===
Subject: Re: Cantor's diagonal proof wrong?
> I present an argument that I hope some people will see shows how the 
logic
> of the diagonal argument has problems. 
No it doesn't. YOU have problems. You do not understand the argument so 
you arrogantly assume it is the argument at fault and not your 
comprehension of the argument. You are a Legend in Your Own Mind.
Bob Kolker
===
Subject: Re: Cantor's diagonal proof wrong?
: I understand Cantor's proof.  It's trivial to understand.  What's much
: harder to understand is why it's an invalid proof.  
Sorry, but you do not understand it.  There is nothing invalid
about the proof.  All your talk about 'theology' is just
a rant.  Just declaring that infinite sets do not exist
does not affect any proofs that use infinite sets.  It is
just some weird bias on your part, or a lack of imagination,
or both.
A very short summary of Cantor's proof is that there
does not exist a surjection from the natural numbers to
the real numbers.  You whole argument is that the natural
numbers and the real numbers do not 'exist'.  Noone
agrees with that, and your arguments are all non-mathematical
in nature.
One of the earliest proofs in mathematics is that the number
of primes is infinite.  It is a 2000 year old proof.
According to you it must be invalid however because you do
not believe there are an infinite number of primes, because
you believe that mathmaticians should only concern themselves
with the finite, despite the fact they never have.
Stephen
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> because
> you believe that mathmaticians should only concern themselves
> with the finite, despite the fact they never have.
Well, at least you tried to understand.  I'll give you that.  You have
given me lots of good practice at expressing my ideas.
In 9th grade geometry class, we were asked to derive an equation for the
sum of the first N integers without looking up the answer first.  I figured
out how to do it by visualizing the area of a bar graph of the first N
integers.  The teacher asked me to explain to the class how I did it.  I
talked for 10 minutes, and nobody, including the teacher, understand how I
had used the area of a bar graph to derive the equation n(n+1)/2.  I just
had to sit down and shut up.  It does not surprise me that people are not
understanding what I'm trying to explain here either.  It has happened many
times in my life. :)
I have since learned much better ways to talk about the visualization which
was obvious to me back then, and I'm sure I could now explain where the
equation comes from to my old class so that they all understood.  One day,
maybe I'll figure out how to explain this stuff to people as well.
What I actually said is that infinity is two separate things, not one.
It's the algorithm which defines it, and it's the result of the algorithm
as it runs.
We can fully describe how to count with the natural numbers, but we can
never actually use that procedure to count to infinity.
When you manipulate the description for counting, you are working with one
type of infinity.  And there is no problem doing that.
But, if you make an argument based on the idea that what you manipulate is
the result of the counting procedure (i.e. all the natural numbers),
instead of the procedure itself, you are talking nonsense.
So, once again, there are two things that people confuse to be one when
they talk about infinity.  One thing is valid to work with (the algorithm),
the other thing (all the output) is not.
Every time I talk about the half that is invalid to work with, you are
unable to see I've been talking about only one of the two halves, and you
think I'm talking about the whole thing, and you think I have some problem
with working with all types of infinity.
You will never follow my logic until you understand I'm breaking the idea
of infinity into two halves and making different statements about each
half.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
> What I actually said is that infinity is two separate things, not one.
> It's the algorithm which defines it, and it's the result of the algorithm
> as it runs.
Hmm. Well, that isn't what mathematicians mean by 'infinity', because
(at least set theory based) maths is about sets and relations between
them, not about algorithms, or their outputs.
Anyway, it seems part of the problem is your insistence that this or
that doesn't exist, so - if you will - we can play the game I
mentioned earlier.
Can you tell us which of the following exist:
5
the empty set
e
1/9
-7.3
i
the Klein 4-group
the set of even integers
the set of reals
a circle of unit radius
an integer solution to the diophantine equation a^5+b^5=c^5
a heptagon with five sides all the same length, and four all different
the idea of proof by reductio ad absurdum
In at least some cases, if you said *where* these entities existed it
might be illuminating.
Brian Chandler
http://imaginatorium.org
===
Subject: Re: Cantor's diagonal proof wrong?
>[...]
>What I actually said is that infinity is two separate things, not one.
>It's the algorithm which defines it, and it's the result of the algorithm
>as it runs.
By infinity here we mean infinite sets, right? No, infinity is not
defined by an algorithm.
>We can fully describe how to count with the natural numbers, but we can
>never actually use that procedure to count to infinity.
That's true. And totally irrelevant.
>When you manipulate the description for counting, you are working with one
>type of infinity.  And there is no problem doing that.
>But, if you make an argument based on the idea that what you manipulate is
>the result of the counting procedure (i.e. all the natural numbers),
>instead of the procedure itself, you are talking nonsense.
>So, once again, there are two things that people confuse to be one when
>they talk about infinity.  One thing is valid to work with (the 
algorithm),
>the other thing (all the output) is not.
>Every time I talk about the half that is invalid to work with, you are
>unable to see I've been talking about only one of the two halves, and you
>think I'm talking about the whole thing, and you think I have some problem
>with working with all types of infinity.
>You will never follow my logic until you understand I'm breaking the idea
>of infinity into two halves and making different statements about each
>half.
_nobody_ is able to follow your logic. This includes a lot of very
smart people who've thought a lot about all these things. But
the only possibility is that you're right and _everyone_ else is
wrong.
Amusingly, that's the only possibility even though the source of the
problem is what various statements _mean_!
************************
David C. Ullrich
===
Subject: Re: Cantor's diagonal proof wrong?
> You will never follow my logic until you understand I'm breaking the idea
> of infinity into two halves and making different statements about each
> half.
If you learned how to express yourself clearly, maybe you would be 
better understood. Your use of language is sloppy. What you need are 
precise definitions and clear logic.
Bob Kolker
===
Subject: Re: Cantor's diagonal proof wrong?
> I believe math was created as a langauge for talking about what can, and
> can not do, and know, about procedures.  But somewhere, it feels to me,
> like it went off track and made up some rules of logic which violate the
> laws of nature (the laws of procedures).  Did it go off track or not?  Is
> it a bad thing even if it did?  I don't know for sure.
Before you criticize mathematics you should learn some math. You are an 
ignoramus and you have to basis for your criticisms.
All current mathematical theories appear to be logically correct. Not 
one internal inconsistency has been found since set theory has been 
purged of its logical problems. Mathematics has been clean for at least 
80 years.
If you can derive a logical contradcition from any current mathematical 
theory then do so, and get it published. You will become famous instantly.
Bob Kolker
===
Subject: Re: Cantor's diagonal proof wrong?
>> And after you are done answering that, does 2/3 terminate or not?
>> Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3).
> get
> the silly questions (and answers) back to the big picture of why on earth 

> was talking about procedures that do or do not terminate.  You should 
read
> that if you want to understand why I was talking about these things.
> I believe math was created as a langauge for talking about what can, and
> can not do, and know, about procedures.  But somewhere, it feels to me,
> like it went off track and made up some rules of logic which violate the
> laws of nature (the laws of procedures).  Did it go off track or not?  Is
> it a bad thing even if it did?  I don't know for sure.
When I see people say things like this about math (that it's a form of 
logic, or that it's only about procedures, or some such), I think of the 
writings of the great mathematician Salomon Bochner.  A quote that comes to 

mind, from his The Role of Mathematics in the Rise of Science, is: 
Mathematics is a form of poetry which transcends poetry in that it 
proclaims a truth; a form of reasoning which transcends reasoning in that it 

wants to bring about the truth it proclaims; a form of action, of ritual 
behaviour, which does not find fulfilment in the act but must proclaim and 
elaborate a poetic form of truth.
Michael 
===
Subject: Re: Cantor's diagonal proof wrong?
>> And after you are done answering that, does 2/3 terminate or not?
>> Clearly 2/3 is 0.66666..., but it is also 0.2 (in base 3).
> get
> the silly questions (and answers) back to the big picture of why on 
earth 
> I
> was talking about procedures that do or do not terminate.  You should 
read
> that if you want to understand why I was talking about these things.
> I believe math was created as a langauge for talking about what can, 
and
> can not do, and know, about procedures.  But somewhere, it feels to me,
> like it went off track and made up some rules of logic which violate 
the
> laws of nature (the laws of procedures).  Did it go off track or not?  
Is
> it a bad thing even if it did?  I don't know for sure.
> When I see people say things like this about math (that it's a form of 
> logic, or that it's only about procedures, or some such), I think of the 
> writings of the great mathematician Salomon Bochner.  A quote that comes 
to 
> mind, from his The Role of Mathematics in the Rise of Science, is: 
> Mathematics is a form of poetry which transcends poetry in that it 
> proclaims a truth; a form of reasoning which transcends reasoning in that 

it 
> wants to bring about the truth it proclaims; a form of action, of ritual 
> behaviour, which does not find fulfilment in the act but must proclaim and 

> elaborate a poetic form of truth.
This is most definitely an exaggeration. Not even science as a whole
should claim such a stronghold on truth.
To the activity described in this quote one could attribute delusional
or creative properties, or both. I would personally prefer creative.
When I think about a proof, I believe I am being creative. When I
hypothesize about what might possibly be true (for instance I have for
a lonf time imagined, like many other persons who believe that physics
*is* mathematical, that incompleteness must have something to do with
the uncertainty principle. And indeed Prof. Calude showed such a
relation!)
But *if* mathematics *creates* new truths, then it is delusional.
This was something that Godel argued against. Godel was a realist. He
thought we actually reached out to the truth out there using a
special faculty of perception, in whatever fashion it is physically
realized.
--
Eray Ozkural
PS: I couldn't find a good post in which I stated the relation I'd
hypothesized, but below, I suggest that uncertainty is the result of a
more specific law of complexity in action. Which seems to be precisely
what Calude showed. Unfortunately only a one way implication at the
moment, we need a bidirectional *bridge law* to bring the full effect.
And basically show just how relevant digital physics is! Now, I think,
it is time to study some mathematics!
===
Subject: Re: Cantor's diagonal proof wrong?
> When I see people say things like this about math (that it's a form of 
> logic, or that it's only about procedures, or some such), I think of the 
> writings of the great mathematician Salomon Bochner.  A quote that comes 
to 
> mind, from his The Role of Mathematics in the Rise of Science, is: 
> Mathematics is a form of poetry which transcends poetry in that it 
> proclaims a truth; a form of reasoning which transcends reasoning in that 

it 
> wants to bring about the truth it proclaims; a form of action, of ritual 
> behaviour, which does not find fulfilment in the act but must proclaim and 

> elaborate a poetic form of truth.
Bochner's gushings are nonsense. He was in a poetic mood and he made 
just about as much sense as most poems make, which is to say little or 
none.
Bob Kolker
===
Subject: Re: Cantor's diagonal proof wrong?
>> When I see people say things like this about math (that it's a form of 
>> logic, or that it's only about procedures, or some such), I think of the 

>> writings of the great mathematician Salomon Bochner.  A quote that comes 

>> to mind, from his The Role of Mathematics in the Rise of Science, is: 
>> Mathematics is a form of poetry which transcends poetry in that it 
>> proclaims a truth; a form of reasoning which transcends reasoning in that 

>> it wants to bring about the truth it proclaims; a form of action, of 
>> ritual behaviour, which does not find fulfilment in the act but must 
>> proclaim and elaborate a poetic form of truth.
> Bochner's gushings are nonsense.
That's overly harsh, and I have no idea why you'd react so negatively to 
such a comment.
> He was in a poetic mood and he made just about as much sense as most 
> poems make, which is to say little or none.
Ah, now I see.  You don't like poetry.  I think that's sad, although I doubt 

you'd agree.
Michael 
===
Subject: Re: Cantor's diagonal proof wrong?
> Bochner's gushings are nonsense. He was in a poetic mood and he made 
> just about as much sense as most poems make, which is to say little or 
> none.
But that's only because a poem is a song without the music :-(
Han de Bruijn
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> When I see people say things like this about math (that it's a form of
> logic, or that it's only about procedures, or some such), I think of the
> writings of the great mathematician Salomon Bochner.  A quote that comes
> to mind, from his The Role of Mathematics in the Rise of Science, is:
> Mathematics is a form of poetry which transcends poetry in that it
> proclaims a truth; a form of reasoning which transcends reasoning in that
> it wants to bring about the truth it proclaims; a form of action, of
> ritual behaviour, which does not find fulfilment in the act but must
> proclaim and elaborate a poetic form of truth.
Yeah, I like that.  It is poetry because it's the search for the most
elegant way to express the truth.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
>> When I see people say things like this about math (that it's a form of
>> logic, or that it's only about procedures, or some such), I think of the
>> writings of the great mathematician Salomon Bochner.  A quote that comes
>> to mind, from his The Role of Mathematics in the Rise of Science, 
is:
>> Mathematics is a form of poetry which transcends poetry in that it
>> proclaims a truth; a form of reasoning which transcends reasoning in 
that
>> it wants to bring about the truth it proclaims; a form of action, of
>> ritual behaviour, which does not find fulfilment in the act but must
>> proclaim and elaborate a poetic form of truth.
> Yeah, I like that.  It is poetry because it's the search for the most
> elegant way to express the truth.
It's a lot more than that, which was the point.
Michael 
===
Subject: Re: Cantor's diagonal proof wrong?
> Yeah, I like that.  It is poetry because it's the search for the most
> elegant way to express the truth.
Mathematics has little to do with empirical truth. There is nothing in 
mathematics that demands that it be in correspondence with facts. The 
only requirement is that it be internally consistent.
Bob Kolker
===
Subject: Re: Cantor's diagonal proof wrong?
:> Would you argue that 1/5 is also defined by an algorithm?
: Yes.  All mathematical operations are procedures or algorithms.
: Some make reference to procedures that can never produce an answer, like
: 1/0, and some make reference to procedures that never terminate, like
: .111....
You seem to be confusing numbers with their representations again.
1/5 is not a procedure.  It is a number.  .2 is also a number.
1/5 and .2 are different names for the same thing.  What procedure
is .2?
Stephen
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> :> Would you argue that 1/5 is also defined by an algorithm?
> : Yes.  All mathematical operations are procedures or algorithms.
> : Some make reference to procedures that can never produce an answer,
> : like 1/0, and some make reference to procedures that never terminate,
> : like .111....
> You seem to be confusing numbers with their representations again.
Try not to be confused by the way I talk about this stuff.  I understand
exactly what you mean when you call .2 a number.  I too talk that way.  I'm
trying to bring to light a lower level concept that is in math, but which
we are trained to ignore.
We are trained like parrots to say .2 is a number.  And, because I seem
not to be following my training, you keep coming back and saying - you 
are
not saying the right thing - you will not get your cracker if you keep
calling 1/5 a procedure - it's a number - awk.
> 1/5 is not a procedure.  It is a number.  .2 is also a number.
> 1/5 and .2 are different names for the same thing.  What procedure
> is .2?
Counting is a procedure.  When I ask a 6 year old what comes after 1 and he
says two, he has correctly performed the successor procedure.  We train
people to correctly apply the successor procedure so they can always
produce the next natural number given a starting number.
In this regard, it's reasonable to say that 4 represents  x x x x
iterations of the counting procedure.
I'm not saying it is wrong to call 1/5 a number.  I'm saying that behind
those words is a hidden procedure which is the true source of the meaning
of those words, but yet, we are taught to objectify the reference of the
sign and pretend it's an object, and not a procedure.  But behind all that
objectification done in the language, is a underlying foundation of
procedures.
.2 represents the results of dividing 1 by 5.  Division is the procedure
for finding the number when multiplied by 5, will produce 1.
Multiplication is the procedure for applying the addition procedure a fixed
number of times.  The addition procedure is multiple applications of the
counting procedure.
In the above problem, there is no counting number which we can find to
multiply 5 by in order to produce one.  So we invent a new notation called
decimal numbers to server as place holders (names) for the division
procedures which do not have answers.  .2 is the name for all division
procedures which have the same affect as trying to divide 2 by 10.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
> .2 represents the results of dividing 1 by 5.
???
1/5 represents the result of dividing 1 by 5. 
.2 represents the result of multiplying 2 by one tenth.
Phil
-- 
... one Marine noticed one of the prisoners was still breathing. A Marine 
can be heard saying on the pool footage provided to Reuters Television: 
He's ing faking he's dead. He faking he's ing dead.
The Marine then raises his rifle and fires into the man's head. 
The pictures are too graphic for us to broadcast, Sites said. 
===
Subject: Re: Cantor's diagonal proof wrong?
Just nit-picking here; I'm not concerned
that anyone will actually be misled.
>>.2 represents the results of dividing 1 by 5.
> ???
> 1/5 represents the result of dividing 1 by 5. 
> .2 represents the result of multiplying 2 by one tenth.
                                                *********  ten.
It's 2/10 that is 0.2, not 2/(1/10).
If you divide 2 by one tenth, the result is 20:
        2/(1/10) = 2 (1/10)^(-1) = 2(10) = 20.
> Phil
Dale
===
Subject: Re: Cantor's diagonal proof wrong?
> Just nit-picking here; I'm not concerned
> that anyone will actually be misled.
>>.2 represents the results of dividing 1 by 5.
> ???
> 1/5 represents the result of dividing 1 by 5. .2 represents the
> result of multiplying 2 by one tenth.
>                                                 *********  ten.
> It's 2/10 that is 0.2, not 2/(1/10).
> If you divide 2 by one tenth, the result is 20:
>       2/(1/10) = 2 (1/10)^(-1) = 2(10) = 20.
> Phil
> Dale
Phil said *multiplying* 2 by one tenth.  Not dividing.
Paul
-- 
Hanging on in quiet desperation is the English way.
The time is gone, the song is over.
Thought I'd something more to say.
===
Subject: Re: Cantor's diagonal proof wrong?
>>Just nit-picking here; I'm not concerned
>>that anyone will actually be misled.
>>.2 represents the results of dividing 1 by 5.
>???
>1/5 represents the result of dividing 1 by 5. .2 represents the
>result of multiplying 2 by one tenth.
>>                                                *********  ten.
>>It's 2/10 that is 0.2, not 2/(1/10).
>>If you divide 2 by one tenth, the result is 20:
>>      2/(1/10) = 2 (1/10)^(-1) = 2(10) = 20.
>Phil
>>Dale
> Phil said *multiplying* 2 by one tenth.  Not dividing.
> Paul
So he did. I was sure I read otherwise, and further inspection has me
wondering how on earth I read that.
Dale.
===
Subject: Re: Cantor's diagonal proof wrong?
> Just nit-picking here; I'm not concerned
> that anyone will actually be misled.
>>.2 represents the results of dividing 1 by 5.
> ???
> 1/5 represents the result of dividing 1 by 5. .2 represents the
> result of multiplying 2 by one tenth.
>                                                 *********  ten.
Why have you underlined empty space?
> It's 2/10 that is 0.2, not 2/(1/10).
Where do I say otherwise? 
 
> If you divide 2 by one tenth, the result is 20:
>       2/(1/10) = 2 (1/10)^(-1) = 2(10) = 20.
The Pope is Catholic, bears  in the woods, and Judith Chalmers 
has a passport. But what's any of that got to do with the price
of tea in China?
Phil
-- 
... one Marine noticed one of the prisoners was still breathing. A Marine 
can be heard saying on the pool footage provided to Reuters Television: 
He's ing faking he's dead. He faking he's ing dead.
The Marine then raises his rifle and fires into the man's head. 
The pictures are too graphic for us to broadcast, Sites said. 
===
Subject: Re: Cantor's diagonal proof wrong?
>>Just nit-picking here; I'm not concerned
>>that anyone will actually be misled.
>>.2 represents the results of dividing 1 by 5.
>???
>1/5 represents the result of dividing 1 by 5. .2 represents the
>result of multiplying 2 by one tenth.
>>                                                *********  ten.
> Why have you underlined empty space?
My mistake. On my screen, I had spaced over to what read one tenth.
I hadn't anticipated the ensuing formatting.
>>It's 2/10 that is 0.2, not 2/(1/10).
> Where do I say otherwise? 
Nope. It was my misreading of your text.
Endless apologies.
        ... the rest deleted ...
> Phil
Sorry. I misread.
Dale.
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> My mistake. On my screen, I had spaced over to what read one tenth.
You actually got the spacing correct (well within 1 character at least).
Anyone using a proportional font to read Usenet messages should not
complain about the spacing being wrong.  But it's true that there are so
many people doing that these days that you can seldom trust spacing to be
reproduced correctly when a message is displayed.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
 > My mistake. On my screen, I had spaced over to what read one 
tenth.
 > You actually got the spacing correct (well within 1 character at least).
 > Anyone using a proportional font to read Usenet messages should not
 > complain about the spacing being wrong.
This depends entirely on the proportional font you are using.
 >                                          But it's true that there are so
 > many people doing that these days that you can seldom trust spacing to 
be
 > reproduced correctly when a message is displayed.
As long as you are using non-proportional fonts there is no problem if 
other
people also set their font to non-proportional.  If you are using a
proportional font, there are people using a different proportional font
that will see something different.  In this world there is not a single
proportional font.
-- 
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, 
+31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
>    > My mistake. On my screen, I had spaced over to what read one
>  > tenth.
>    > You actually got the spacing correct (well within 1 character at
>  > least). Anyone using a proportional font to read Usenet messages
>  > should not complain about the spacing being wrong.
> This depends entirely on the proportional font you are using.
The correct way to display a usenet message requires you to use a
fixed-width font.  If you use a proportional font, you are not displaying
the message correctly according to the 20 year old standard which was never
changed and you deserve to have your tables, charts, and underlines, messed
up and you do not have a leg to stand on when you try to argue that someone
posted incorrectly when the message does display correctly in a fixed-width
font.
If you use a proportional font, many usenset messages will not display
correctly.  All my messages for example require a fixed-width font in order
for the signature to dispaly correctly.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
> Try not to be confused by the way I talk about this stuff.  I understand
> exactly what you mean when you call .2 a number.  I too talk that way.  
I'm
> trying to bring to light a lower level concept that is in math, but which
> we are trained to ignore.
> We are trained like parrots to say .2 is a number.  And, because I 
seem
That ration of 2 to 10 (or 1 to 5) is a rational number. A parrot would 
not understand that. Apparantly you don't either. Are you a parrot?
Bob Kolker
===
Subject: Re: Cantor's diagonal proof wrong?
:> 1/5 is not a procedure.  It is a number.  .2 is also a number.
:> 1/5 and .2 are different names for the same thing.  What procedure
:> is .2?
: Counting is a procedure.  When I ask a 6 year old what comes after 1 and 
he
: says two, he has correctly performed the successor procedure.  We 
train
: people to correctly apply the successor procedure so they can always
: produce the next natural number given a starting number.
: In this regard, it's reasonable to say that 4 represents  x x x x
: iterations of the counting procedure.
: I'm not saying it is wrong to call 1/5 a number.  I'm saying that behind
: those words is a hidden procedure which is the true source of the meaning
: of those words, but yet, we are taught to objectify the reference of the
: sign and pretend it's an object, and not a procedure.  But behind all 
that
: objectification done in the language, is a underlying foundation of
: procedures.
: .2 represents the results of dividing 1 by 5.  Division is the procedure
: for finding the number when multiplied by 5, will produce 1.
.2 also represents the result of adding .1 and .1.  It also represents
the result of taking the square root of .04.  It also represents
the result of summing (2^i)/10 for i=0 to infinity.  I can come up
with an infinite number of procedures that produce .2.  Claiming
that .2 is one and only one of those procedures seems silly.
If you claim that it is all of those procedures than you have
to claim that it is the result of both finite and infinite procedures,
which will be true of all numbers.
IMO, claiming that 1/5 is a finite procedure but 1/9 is not
makes no sense.  There is nothing infinite about dividing
something by 9.
Stephen
===
Subject: Re: Cantor's diagonal proof wrong?
> IMO, claiming that 1/5 is a finite procedure but 1/9 is not
> makes no sense.  There is nothing infinite about dividing
> something by 9.
He does not know the different between the numbers and the 
representation of the numbers in some base.
Bob Kolker
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> :> 1/5 is not a procedure.  It is a number.  .2 is also a number.
> :> 1/5 and .2 are different names for the same thing.  What procedure
> :> is .2?
> : Counting is a procedure.  When I ask a 6 year old what comes after 1
> : and he says two, he has correctly performed the successor 
procedure.
> : We train people to correctly apply the successor procedure so they can
> : always produce the next natural number given a starting number.
> : In this regard, it's reasonable to say that 4 represents  x x x x
> : iterations of the counting procedure.
> : I'm not saying it is wrong to call 1/5 a number.  I'm saying that
> : behind those words is a hidden procedure which is the true source of
> : the meaning of those words, but yet, we are taught to objectify the
> : reference of the sign and pretend it's an object, and not a procedure.
> : But behind all that objectification done in the language, is a
> : underlying foundation of procedures.
> : .2 represents the results of dividing 1 by 5.  Division is the
> : procedure for finding the number when multiplied by 5, will produce 1.
> .2 also represents the result of adding .1 and .1.  It also represents
> the result of taking the square root of .04.  It also represents
> the result of summing (2^i)/10 for i=0 to infinity.  I can come up
> with an infinite number of procedures that produce .2.  Claiming
> that .2 is one and only one of those procedures seems silly.
> If you claim that it is all of those procedures than you have
> to claim that it is the result of both finite and infinite procedures,
> which will be true of all numbers.
> IMO, claiming that 1/5 is a finite procedure but 1/9 is not
> makes no sense.  There is nothing infinite about dividing
> something by 9.
Yes, you what you say is true.
The point I've been making is not an issue about which procedures terminate
and which doesn't.  The point I've been making is the meaning behind all of
it is procedural in nature. That is the foundation which math grows out of.
And sometimes, the reference we make are to procedures which do not
terminate.
The fact that we make reference to procedures that do not terminate however
does not prevent us from talking in intelligent ways about the nature of
the procedure.  We know that an infinite series will approach a limit and
we don't have to perform the infinite procedure to know that.  If the limit
it approaches is 2.0, then we know that we can substitute the value 2.0 in
the equation where the infinite series was, and get the right answer - and
at the same time, change the procedure from an infinite one, to a finite
one (counting to two).
I have no problem, or confusion, about how we make good use of all the talk
about procedures that loop forever.
However, when you start talking about constructing a 1 to 1 mapping between
to procedures which loop forever, and tell me that it can't be done, I get
lost.
To talk about a 1 to 1 mapping between two finite sized sets of objects is
simple and clear.
To talk about 1 to 1 mapping between a finite set, and a procedure that
never terminates is also clear - it can't be done.
To talk about a 1 to 1 mapping between to infinite procedures, makes no
logical sense to me.  It's always possible to create the mapping, because
no matter how many objects are generated by each procedure, you can always
match them up as they are generated (this idea I'm beginning to see seems
to be related to the axiom of choice).
So, if the meaning behind all math concepts is ultimately derived from
procedures, and infinite procedures can always be 1 to 1 mapped, how is it
we are able to find the words to prove that some infinite procedures are
not 1 to 1 mapped with all other infinite procedures?
I just don't know how the words were found to prove that yet.  But there
are a lot of words used in the formal foundation of mathematics that
everyone says does prove it.  So I'm just curious how mathematicians did
what seems to be the impossible.
For example, if look at Cantor's diagonal argument with the belief that all
references to infinity are references to processes that don't terminate,
you see something very different than what is normally seen in the proof.
The proof starts with the assumption that there is a mapping.  This would
imply that there is a procedure able to sequentially generate all the
reals, one at a time.  Because a single real might in itself be represented
by an infinite procedure (to generate all the digits of PI for example),
you can think of the problem as a procedure for generating an infinite set
of procedures which in turn, generate each possible real.
Now the argument says you need another procedure running in parallel to
generate the diagonal.  This procedure takes each real generated by the
first procedure, and grabs the Nth digit from it.  Since for each real,
it's grapping a finite digit, it doesn't even need to let the procedure
continue generating all the digits.  So that's not a problem.
Now it takes the Nth digit, and changes it.  It continues in this fashion,
for as long as the real generating processes produces reals to work with.
We know for a fact that this diagonal number being generated will be
different from every past real generated.  But, at any time T, we have only
built N digits of the diagonal so far, and we have no way of knowing if the
partial real diagonal we have built so far, is located further down in the
table somewhere.
This procedure for generating reals, and generating the diagonal never
ends.  It can't end because the procedure for generating reals can never
end.
But yet, we now claim, that the existence of the above never ending
procedure, is proof that there must be some real that the first procedure
will never produce.
I hope all of you can see that when presented like this, (as never ending
procedures) the proof is no proof at all.  If not, I'll make the same style
argument for natural numbers...
Start with the counting procedure which produces all the natural numbers
starting with one (or zero).  Now, generate a number which is guaranteed
not to match any of the numbers produced so far.  We will do that by
picking N+1 where N is the highest number produced so far.  By obvious
inspection, N+1 is a number which is different from every number produced
so far.  Or, we can always generate a number which is not (yet) in the
table.
So does this prove that our procedure for producing natural numbers will
not be able to produce all the natural numbers?  Of course not.
So, this is the heart of my confusion about the diagonal argument.  It only
seems to make sense if you ignore the fact that all math is actually built
on the physical world notion of procedures, and procedures take time to
produce each new result, which means procedures which produce an infinite
number of results, can never terminate.
When, and why, was it ok for math to ignore its foundation in procedures?
What words can we point to that say, here is where mathematicians decided
to ignore the laws of nature that controls all procedures and make up their
own rules?
If that is what happened, I'd like to find those words and understand how
they violated the laws of nature.
OR, is it possible that a procedure based foundation of math can explain
why it's ok to say that sets with a cardinality greater than |N| exists?
If so, I'd like to see how that works, because the diagonal argument alone
doesn't seem to hold water when cast in a procedure based foundation.
Without a better understand of all the complex language used to justify the
foundation of math which leads to the hierarchy of cardinal sets, I can't
know for sure what happened.  However, my best bet at the moment is that
mathematics, over the centuries, lost track of its roots, and simply choose
to define a few laws of nature out of existence.  The most likely law they
choose to ignore is that nothing can be created without a physical agent
doing work and taking time.  That is, they assumed that they have machines
which could follow these logical procedures (definitions/axioms), and run
at infinite speed to produce an infinite amount of results in zero time.
And when that happened, it became valid to use logic such as NOT FOR ALL,
on infinite sets, as if it were a finite sized set.  Because in the real
world, NOT FOR ALL, is an invalid argument for an unspecified procedure in
a proof by contradiction argument.
But, like I have said, to uncover this mystery, I have to study the
language of math more.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
...
 > Start with the counting procedure which produces all the natural numbers
 > starting with one (or zero).  Now, generate a number which is guaranteed
 > not to match any of the numbers produced so far.  We will do that by
 > picking N+1 where N is the highest number produced so far.
But this is quite different from the procedure defined in the Cantor
argument in a subtle way.  The Cantor argument states what the n-th
decimal digit will be at step n.  This converges to a real number.
On the other hand, your sequence of numbers does not converge to a
natural number in any metric.
-- 
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, 
+31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
>  > Start with the counting procedure which produces all the natural
>  > numbers starting with one (or zero).  Now, generate a number which is
>  > guaranteed not to match any of the numbers produced so far.  We will
>  > do that by picking N+1 where N is the highest number produced so far.
> But this is quite different from the procedure defined in the Cantor
> argument in a subtle way.  The Cantor argument states what the n-th
> decimal digit will be at step n.  This converges to a real number.
> On the other hand, your sequence of numbers does not converge to a
> natural number in any metric.
Yes, that's very true.  And that's why people were quick to point out my
...111 from the first post in this thead was not an integer.  And in that
context, it was a very important and valid point which blew my argument out
of the water.
But in my message above, I was not trying to argue against the normal
Cantor Diagonal proof.  I was just showing that in the context of
procedures which never terminate, the argument structure that is accepted
as valid for Cantor's proof, becomes invalid.  It's not important that the
procedure converges to a single number in this context because since the
procedure never terminates, it will never converge.
And the real point of the above is not the example about the natrual
numbers, but instead the example about an infinite number of procedures,
each generating an infinite number of digits.  That does produce a single
answer as much as the real Cantor proof does.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
 >  > Start with the counting procedure which produces all the natural
 >  > numbers starting with one (or zero).  Now, generate a number which 
is
 >  > guaranteed not to match any of the numbers produced so far.  We 
will
 >  > do that by picking N+1 where N is the highest number produced so 
far.
 >
 > But this is quite different from the procedure defined in the Cantor
 > argument in a subtle way.  The Cantor argument states what the n-th
 > decimal digit will be at step n.  This converges to a real number.
 > On the other hand, your sequence of numbers does not converge to a
 > natural number in any metric.
...
 > And the real point of the above is not the example about the natrual
 > numbers, but instead the example about an infinite number of procedures,
 > each generating an infinite number of digits.  That does produce a 
single
 > answer as much as the real Cantor proof does.
But in Cantors proof there is not an infinite number of procedures,
each generating an infinite number of digits.
-- 
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, 
+31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> ...
>  > And the real point of the above is not the example about the natrual
>  > numbers, but instead the example about an infinite number of
>  > procedures, each generating an infinite number of digits.  That does
>  > produce a single answer as much as the real Cantor proof does.
> But in Cantors proof there is not an infinite number of procedures,
> each generating an infinite number of digits.
Yes, that's true.  But that's the source of my doubt.  I think the fact
that we were trained to believe that an infinite set can exist without a
processes to construct it is what makes us believe that the logic in
Cantor's proof is valid.
In this universe, so I believe, infinite sets can't actually exist.  We are
just pretending they do when we define them through the use of recursive
definitions.  Once you throw fact out the window, and both pretend they
exist, and make the decision that logic which applies to finite sets should
apply the same way to this new types of sets, then it seems that we do end
up with the ideas of cardinality of inifinite sets.
But, if this is the only reason cardinality becomes valid, then what was
the value of doing this?  What is the value to us to have this theory of
cardinality of sets which falls out of pretending something that doesn't
apply to our universe is real?  Is there a use outside of math to the
concept of cardinality?  I'm not asking this as proof we shouldn't do it,
I'm just currious if a use has been found.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
...
 > But in Cantors proof there is not an infinite number of procedures,
 > each generating an infinite number of digits.
 > Yes, that's true.  But that's the source of my doubt.  I think the fact
 > that we were trained to believe that an infinite set can exist without a
 > processes to construct it is what makes us believe that the logic in
 > Cantor's proof is valid.
But the whole point of mathematics initially was to idealise the physical
world.  That is what abstraction is about.  The (possiby discreet)
physical world is idealised to a continuum.  Mathematics is no longer
about processes, it is about axioms, definitions and consequences.  So
in essence it is purely theoretical without any immediate practical
results.  That there exist different kinds of infinities in the
mathematical world is the immediate result of the axioms and the logic.
And, in mathematics, infinite processes have no place.  Everything is
defined (and proven) with clear finite processes.
 > In this universe, so I believe, infinite sets can't actually exist.
But the set of natural numbers is infinite by its very definition.  So
in what way does that set not exist?
...
 > But, if this is the only reason cardinality becomes valid, then what was
 > the value of doing this?  What is the value to us to have this theory of
 > cardinality of sets which falls out of pretending something that doesn't
 > apply to our universe is real?  Is there a use outside of math to the
 > concept of cardinality?  I'm not asking this as proof we shouldn't do 
it,
 > I'm just currious if a use has been found.
I do not think there is actually use for it.  It is only one of the 
surprising
results we get when we use logical rigor.  And so I am surprised that you
attack the result, when (currently) there is no actual use for it.
The value?  Well, people wanted to know what the consequences were of
mathematical axioms, and so they found quite a few surprising consequences.
On the other hand, those very same axioms make it possible to numerically
solve partial differential equation, and whatever.  Pure mathematics
state when a scheme will converge to a solution, another branch of
mathematics states when a scheme will numerically result in a solution.
-- 
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, 
+31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
>  > But in Cantors proof there is not an infinite number of procedures,
>  > each generating an infinite number of digits.
>    > Yes, that's true.  But that's the source of my doubt.  I think the
>  > fact that we were trained to believe that an infinite set can exist
>  > without a processes to construct it is what makes us believe that the
>  > logic in Cantor's proof is valid.
> But the whole point of mathematics initially was to idealise the physical
> world.  That is what abstraction is about.  The (possiby discreet)
> physical world is idealised to a continuum.  Mathematics is no longer
> about processes, it is about axioms, definitions and consequences.  So
> in essence it is purely theoretical without any immediate practical
> results.  That there exist different kinds of infinities in the
> mathematical world is the immediate result of the axioms and the logic.
> And, in mathematics, infinite processes have no place.  Everything is
> defined (and proven) with clear finite processes.
>  > In this universe, so I believe, infinite sets can't actually exist.
> But the set of natural numbers is infinite by its very definition.  So
> in what way does that set not exist?
Well, I think I've answered that same question about 20 times in this
thread so far because it is so confusing.  It's kind of funny that I come
here and start a debate on Cantor's diagonal proof only to find out it's a
very common debate that new guys keep bringing up time and time again and
all the old timers are sick of it.  And in the processes, I too say
something odd (infinite sets don't exist), and now I'm forced to debate the
same point over and over again as if it were some type of payback for my
sins of starting this thread. :)
The idea is that only the description of the processes exists.  That
is what we manipulate when we talk about infinite sets.  The set itself
doesn't exist but we talk as if it did anyway - which is fine for almost
all cases until you start to use the type of logic as used in Cantor's
diagonal proof.
> ...
>  > But, if this is the only reason cardinality becomes valid, then what
>  > was the value of doing this?  What is the value to us to have this
>  > theory of cardinality of sets which falls out of pretending something
>  > that doesn't apply to our universe is real?  Is there a use outside of
>  > math to the concept of cardinality?  I'm not asking this as proof we
>  > shouldn't do it, I'm just currious if a use has been found.
> I do not think there is actually use for it.  It is only one of the
> surprising results we get when we use logical rigor.  And so I am
> surprised that you attack the result, when (currently) there is no actual
> use for it.
My attack is really a search for understanding more so than an attack.  
I
don't mind that all this exists.  And as I have learned since starting this
thread, as long as you are consistent in your use of these infinite sets,
all the logic is quite valid and consistent.  I just want to understand the
nature of the beast.
> The value?  Well, people wanted to know what the consequences were of
> mathematical axioms, and so they found quite a few surprising
> consequences. On the other hand, those very same axioms make it possible
> to numerically solve partial differential equation, and whatever.  Pure
> mathematics state when a scheme will converge to a solution, another
> branch of mathematics states when a scheme will numerically result in a
> solution.
I suspect that most areas of mathematics that deal with the concept of
infinity don't step over the line.  Everything I know about calculus and
differential equations doesn't seem to raise any red flags with me.  It's
only a few places I've seen that seem to have stepped over the line.  Most
notably, any time you use the argument NOT FOR ALL in the context of
infinity in a proof by contradiction.  And Cantor's proof is one pace that
happens.
The problem seems to be introduced with the axiom of infinity which is one
of the Zermelo-Fraenkel set theory axioms.  So anything build on this might
be in danger of stepping over the line.  But there is much I do not yet
know about the entire field of set theory and logic so I have not yet been
able to look at all that work with this notion of everything being
processed based.
I strongly suspect that if the axioms are not set up to allow a process
based foundation of all of mathematics, that it could be slightly modified
to still allow all the stuff in mathematics to be formally defined without
allowing anyone to step over the line by incorrectly working with the
concept of an infinite set.  But this is a year's worth of work to be done
in order to get up to speed with all the work done in that area so I could
actually debate the idea.
It might even be done by making no changes to the axioms, but by just
slightly changing the definition of their interpretation.
What would be the value of looking into this?  Well, if the current
practice allows for exploring abstractions that can't apply to this
universe, it might be interesting if we could figure out how to put up a
fence to make it clear what work was inside the fence, and what work was
exploring abstractions outside the fence.  I believe most of mathematics
would still be inside the fence.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
> ...
>  > And the real point of the above is not the example about the natrual
>  > numbers, but instead the example about an infinite number of
>  > procedures, each generating an infinite number of digits.  That does
>  > produce a single answer as much as the real Cantor proof does.
> But in Cantors proof there is not an infinite number of procedures,
> each generating an infinite number of digits.
> Yes, that's true.  But that's the source of my doubt.  I think the fact
> that we were trained to believe that an infinite set can exist without a
> processes to construct it is what makes us believe that the logic in
> Cantor's proof is valid.
> In this universe, so I believe, infinite sets can't actually exist. 
Can you give me an example in the physical universe of a FINITE set? 
Sure, you could show me 5 oranges and tell me that's a set of oranges, 
but I don't see any set there, only five oranges. Can you tell me what 
you mean by a set -- finite or infinite -- existing in the universe?
The objects of mathematics don't necessarily have to exist in the 
physical universe. Again with the oranges -- there are five of them, but 
can you show me something in the universe that's five? Not 5 oranges, 
but plain old 5. Where does it exist?
For that matter, where are truth, beauty, love, hate? These are 
concepts, abstractions. They have no physical existence either.
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> In this universe, so I believe, infinite sets can't actually exist.
> Can you give me an example in the physical universe of a FINITE set?
Yeah, I can.
Everyting in you mind is finite.  Sets don't exist in the external
universe.  They exist in the mind of the observer, and nothing in the mind
is infinite except time.
> The objects of mathematics don't necessarily have to exist in the
> physical universe.
They do.  They exist in our brains in a physical form.  If there were no
people, what do you think the purpose of a math book would be?  What do you
think the purpose of a proof would be?
Sure, we like to think of these things existing independently of us, 
but
they do not.  The books are independent, the marks on the pages are
independent of us, but without us, or some form of intelligence to
correctly interprete them, they have no meaning.
> Again with the oranges -- there are five of them, but
> can you show me something in the universe that's five? Not 5 oranges,
> but plain old 5. Where does it exist?
In the brains of a few billion people.  And if you damage the correct parts
of the brain, the idea of plain old 5 goes away.  If it were not physical,
how come we can make it go away by changing things in the physical
universe?
> For that matter, where are truth, beauty, love, hate? These are
> concepts, abstractions. They have no physical existence either.
Yes, once again, that is how we are taught to talk about those things.  But
as I have rambled on about, they are actually physical.  If you removed the
physical bodies of all the living things from the earth do you think the
rocks would start to hate each other?  There would be no hate because hate
is the behavior of physical stuff.
Now, I know there are a lot of people that will not be able to see things
this way.  This is no newsgroup to get in a long debate about this, but
this is what I believe to be true, and my work on this part of our
existence is what opened my eyes to how languge had confused us to how the
universe actually works, and that confusion I think is what got built into
some of these areas of math such as Cantor's diagonal proof.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
>  > Start with the counting procedure which produces all the natural
>  > numbers starting with one (or zero).  Now, generate a number which 
is
>  > guaranteed not to match any of the numbers produced so far.  We will
>  > do that by picking N+1 where N is the highest number produced so 
far.
> But this is quite different from the procedure defined in the Cantor
> argument in a subtle way.  The Cantor argument states what the n-th
> decimal digit will be at step n.  This converges to a real number.
> On the other hand, your sequence of numbers does not converge to a
> natural number in any metric.
> Yes, that's very true.  And that's why people were quick to point out my
> ...111 from the first post in this thead was not an integer.  And in that
> context, it was a very important and valid point which blew my argument 
out
> of the water.
> But in my message above, I was not trying to argue against the normal
> Cantor Diagonal proof.  I was just showing that in the context of
> procedures which never terminate, the argument structure that is accepted
> as valid for Cantor's proof, becomes invalid.  It's not important that 
the
> procedure converges to a single number in this context because since the
> procedure never terminates, it will never converge.
To clarify my understanding of your notion that an infinite procedure 
never terminates, do you agree or disagree that 
    3/10 + 3/100 + 3/1000 + 3/10000 + ... = 1/3?
In other words, do you believe in the convergence of infinite series? Or 
do you believe that the series can never be completed and therefore 
does not equal 1/3?
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> To clarify my understanding of your notion that an infinite procedure
> never terminates, do you agree or disagree that
>     3/10 + 3/100 + 3/1000 + 3/10000 + ... = 1/3?
> In other words, do you believe in the convergence of infinite series? Or
> do you believe that the series can never be completed and therefore
> does not equal 1/3?
Sure, I know it equals 1/3.  But let me point out what I think we are
really saying when we talk like that.
We are not in fact saying that you would get the value 1/3 by trying to
add all those numbers together.  What we are saying is that in a math
equation, where you find the infinite series above, you can substitute the
value of 1/3 without changing the meaning of the equation.
I can write Sum(n=1 oo | 3/(10^n)) and I can replace those words with 1/3
in any equation, and not change the value, the meaning, or the truth, of
that equation.
Because it is useful to treat these ideas as being equal in math equations,
we learn to think of them as being the same think.  That is, it's a lot
easier to understand the math if you think of the value of 1/3 being a
single entity with lots of different names.
But just because we treat the series as equal to all these other things
when doing math does not mean it is possible to actually add all the values
together.
So, for any concept which translates to a processes which never ends, we
know that the process can't actually exist, but it can still be useful to
talk, and think, as if it did.
However, I suspect the usefulness of these verbal and mental short-cuts
can, and have, come back to bite us.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
> To clarify my understanding of your notion that an infinite procedure
> never terminates, do you agree or disagree that
>     3/10 + 3/100 + 3/1000 + 3/10000 + ... = 1/3?
> In other words, do you believe in the convergence of infinite series? 
Or
> do you believe that the series can never be completed and therefore
> does not equal 1/3?
> Sure, I know it equals 1/3.  But let me point out what I think we are
> really saying when we talk like that.
> We are not in fact saying that you would get the value 1/3 by trying 
to
> add all those numbers together.  What we are saying is that in a math
> equation, where you find the infinite series above, you can substitute 
the
> value of 1/3 without changing the meaning of the equation.
> I can write Sum(n=1 oo | 3/(10^n)) and I can replace those words with 1/3
> in any equation, and not change the value, the meaning, or the truth, of
> that equation.
> Because it is useful to treat these ideas as being equal in math 
equations,
> we learn to think of them as being the same think.  That is, it's a lot
> easier to understand the math if you think of the value of 1/3 being a
> single entity with lots of different names.
> But just because we treat the series as equal to all these other things
> when doing math does not mean it is possible to actually add all the 
values
> together.
Well ok. Then real numbers exist. Every real number can be represented 
as a convergent infinite series. I would certainly agree with you that 
we can't get out a calculator and do the infinite sum, but that's not 
the point. If you agree with the 1/3 example above, then you must allow 
the existence of convergent series representing reals.
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
> Well ok. Then real numbers exist. Every real number can be represented
> as a convergent infinite series. I would certainly agree with you that
> we can't get out a calculator and do the infinite sum, but that's not
> the point. If you agree with the 1/3 example above, then you must allow
> the existence of convergent series representing reals.
Yes, my use of the nonexistence idea wasn't an attempt to say they idea
doesn't exist in math or that it isn't useful and valid.  I was talking
like that only to try and open people's eyes to a different way of looking
at the meaning behind all of the normal langauge we use in mathematics.
I think it's clear that ideas this poorly developed and this far out of the
maninstream can't be well communicated in a long rambling Usenet thread.
It's better left for a well thought out paper on the subject.
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
...
 > I can write Sum(n=1 oo | 3/(10^n)) and I can replace those words with 
1/3
 > in any equation, and not change the value, the meaning, or the truth, of
 > that equation.
...
 > So, for any concept which translates to a processes which never ends, we
 > know that the process can't actually exist, but it can still be useful 
to
 > talk, and think, as if it did.
But in mathematics it is *not* thought that the actual process of adding
infinitely many numbers together does exist.  In fact the axioms do not
specify what happens when you add infinitely many numbers together, and
the notation Sum(n=1 -> oo) 3/(10^n) is abuse of notation (that suggests
adding infinitely many numbers), because it is a shorthand for:
   lim(k=1 -> oo) sum(n=1 -> k) 3/(10^n)
where no infinite process is involved.  And indeed, limits do not involve
infinite processes either (it may be useful to think they are, but in the
definitions they are not).  It is precisely specified that
   lim(k=1 -> oo) sum(n=1 -> k) 3/(10^n) = 1/3
because for any eps>0 you take it is possible to find a k0 such that
   |sum(n=1 -> k0) 3/(10^n) - 1/3| < eps.
-- 
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, 
+31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/
===
Subject: Re: Cantor's diagonal proof wrong?
        
%IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w
>  > I can write Sum(n=1 oo | 3/(10^n)) and I can replace those words with
>  > 1/3 in any equation, and not change the value, the meaning, or the
>  > truth, of that equation.
> ...
>  > So, for any concept which translates to a processes which never ends,
>  > we know that the process can't actually exist, but it can still be
>  > useful to talk, and think, as if it did.
> But in mathematics it is *not* thought that the actual process of adding
> infinitely many numbers together does exist.
Yes, and I have no issue with infinite series.  I never said I did.  I just
said the infinite processes doesn't actually exist because it doesn't and
can't complete (and that confused some people as to what my point was
probably because they haven't read all my posts in this rather out of
control thread).  We simply use the logic of limits to determine, and
define, what numbers each series is equivalent to.  And that's great,
valid, and useful.
But in some places, like Cantor's diagonal proof, the idea of an infinite
sized set of natural numbers gets treated in a logical argument as if it
were a finite sized set, as if it did exist and as if were possible to
actually construct the entire diagonal anti-value.  If you look at that
argument about the infinite set of natural numbers like people look at
limits, you would not be so quick to assume that the application of proof
by contraction in the form used there is valid - as I tried to demonstrate
by showing how invalid the logic looks if you think of both the
construction of the mapping table and the construction of the diagonal
anti-value was done by a process which can never complete.
If you were given a description of the process which mapped the natural
numbers to the reals, say by a function such as f(n) = r, then you can
analyze what it will do without having to actually run the processes to
completion - as is commonly done with infinite series and which is the
foundation of calculus.
But in Cantor's argument, the actual mapping function is not given.  It's
only assumed to exist.  Then they argue that the diagonal anti-value can be
constructed for any mapping function provided - which is still a valid
idea.  But then they make a conclusion that is invalid.  They assume that
since the anti-diagonal value being constructed doesn't match any single
row, that it's valid to say that it doesn't match all the rows.  And as
valid and as logical as that sounds as that sounds, it's not at all valid
when you are constructing infinite sized real values in a infinite sized
table.  This is because it's impossible to construct the entire
anti-diagonal value, and any part you have constructed, can always be part
of a number that shows up further down in the table.
It's a subtle, but huge difference, because one way of looking at it,
Cantor's argument becomes totally invalid - and I believe, when the same
logic is removed from other similar arguments, all the support for
cardinality other than 0 would vanish from mathematics.  But if you
instead, choose to treat the construction of the anti-diagonal as something
that can complete, then the logical argument that it exists in no row of
the table is valid, and you get the results which have been well accepted
for 100 years.
But has it been accepted because people forgot what the real basis of all
this math was about and just got fooled by the language which sounded so
valid?  Or was it done intentionally for some reason?  And if so, what was
that reason?
-- 
Curt Welch                                            http://CurtWelch.Com/
curt@kcwc.com                                        http://NewsReader.Com/
===
Subject: Re: Cantor's diagonal proof wrong?
>>  > I can write Sum(n=1 oo | 3/(10^n)) and I can replace those words with
>>  > 1/3 in any equation, and not change the value, the meaning, or the
>>  > truth, of that equation.
>> ...
>>  > So, for any concept which translates to a processes which never ends,
>>  > we know that the process can't actually exist, but it can still be
>>  > useful to talk, and think, as if it did.
>> But in mathematics it is *not* thought that the actual process of adding
>> infinitely many numbers together does exist.
>Yes, and I have no issue with infinite series.  I never said I did. 
No, you've said you understand what they mean. And you've also said 
things about what they mean that show you _don't_ understand what
they mean.
> I just
>said the infinite processes doesn't actually exist because it doesn't and
>can't complete (and that confused some people as to what my point was
>probably because they haven't read all my posts in this rather out of
>control thread).  We simply use the logic of limits to determine, and
>define, what numbers each series is equivalent to.  And that's great,
>valid, and useful.
>But in some places, like Cantor's diagonal proof, the idea of an infinite
>sized set of natural numbers gets treated in a logical argument as if it
>were a finite sized set,
Huh?
> as if it did exist and as if were possible to
>actually construct the entire diagonal anti-value.  If you look at that
>argument about the infinite set of natural numbers like people look at
>limits,
If you look at that set like people look at limits you're
either misunderstanding limits or infinite sets or both. An
infinite sum is a certain limit, by definition. An infinite
set is not a limit.
> you would not be so quick to assume that the application of proof
>by contraction in the form used there is valid - as I tried to demonstrate
>by showing how invalid the logic looks if you think of both the
>construction of the mapping table and the construction of the diagonal
>anti-value was done by a process which can never complete.
>If you were given a description of the process which mapped the natural
>numbers to the reals, say by a function such as f(n) = r, then you can
>analyze what it will do without having to actually run the processes to
>completion - as is commonly done with infinite series and which is the
>foundation of calculus.
>But in Cantor's argument, the actual mapping function is not given.  It's
>only assumed to exist.  Then they argue that the diagonal anti-value can 
be
>constructed for any mapping function provided - which is still a valid
>idea.  But then they make a conclusion that is invalid.  They assume that
>since the anti-diagonal value being constructed doesn't match any single
>row, that it's valid to say that it doesn't match all the rows.  And as
>valid and as logical as that sounds as that sounds, it's not at all valid
>when you are constructing infinite sized real values in a infinite sized
>table.  This is because it's impossible to construct the entire
>anti-diagonal value, and any part you have constructed, can always be part
>of a number that shows up further down in the table.
For heaven's sake, you still haven't got the proof figured out? No,
it _can't_ be part of a number that shows up farther down in the
table.
>It's a subtle, but huge difference, because one way of looking at it,
>Cantor's argument becomes totally invalid - and I believe, when the same
>logic is removed from other similar arguments, all the support for
>cardinality other than 0 would vanish from mathematics.  But if you
>instead, choose to treat the construction of the anti-diagonal as 
something
>that can complete, then the logical argument that it exists in no row of
>the table is valid, and you get the results which have been well accepted
>for 100 years.
>But has it been accepted because people forgot what the real basis of all
>this math was about and just got fooled by the language which sounded so
>valid?  
No, it's accepted because it's valid. Obviously, clearly, very simply
seen to be valid. The fact that there exist people here who don't buy
it doesn't prove anything to the contrary.
>Or was it done intentionally for some reason?  And if so, what was
>that reason?
************************
David C. Ullrich
===
Subject: Re: Cantor's diagonal proof wrong?
> But in my message above, I was not trying to argue against the normal
> Cantor Diagonal proof.  I was just showing that in the context of
> procedures which never terminate, the argument structure that is accepted
> as valid for Cantor's proof, becomes invalid. 
The argument structure you put forth is a straw-man. It does not occur 
in any mathematical discourse but yours and I would not dignify your 
discourse with the description mathematical.
Your claim that your utter nonsense and balderdash correspondents to any 
mathematical reasoning is a bold face lie.
Bob Kolker
===
Subject: Yet Another Attempt at Refuting Cantor
http://mygate.mailgate.org/mynews/sci/sci.math/982edd331addbb950f57aad6336e7
4
5a.61944%40mygate.mailgate.org
http://arxiv.org/PS_cache/math/pdf/0403/0403169.pdf
-- 
===
Subject: Re: Yet Another Attempt at Refuting Cantor
>http://arxiv.org/PS_cache/math/pdf/0403/0403169.pdf
Do you want to know why it's wrong or is this just for amusement?
-- 
Wim Benthem
===
Subject: Re: Yet Another Attempt at Refuting Cantor
> http://arxiv.org/PS_cache/math/pdf/0403/0403169.pdf
Well, it's certainly a nice-looking paper. While I have no comment
on Theorems 1 - 3, it doesn't look to me as if your counterexample
in Theorem 4 is legitimate.  At the bottom of page 9 you use
an analogue of Cantor's work to show that the rationals are
nondenumerable, since the Cantor construction would provide a
rational, namely 1, that wasn't in the enumeration of the rationals.
Far from yielding a contradiction, all this shows is that your
proposed enumeration wasn't a list of *all* the rationals. Take
a look at what it would have to be; it would have to look like
this
... , 2/3, ... , 3/2, ... , 3/4, ... , 4/3, ...
(the order of appearance of, say, 2/3 and 3/2 could be reversed,
but that's immaterial in generating your intervals).  Now note
that no element in the first two ellipses could be in (0, 2),
no element in the second pair of ellipses could be in the
interval (2/3, 3/2), and so on.  In other words, 1 could not
appear anywhere in your list.  Whaddya know?  The Cantor
construction found a number not in your enumeration, just
as it promised to do.
Rick
===
Subject: Re: Yet Another Attempt at Refuting Cantor
 Discussion, linux)
>> http://arxiv.org/PS_cache/math/pdf/0403/0403169.pdf
> Well, it's certainly a nice-looking paper. While I have no comment
> on Theorems 1 - 3, it doesn't look to me as if your counterexample
> in Theorem 4 is legitimate.  
His?
Eamon Warnock's name doesn't appear on the paper, does it?
-- 
Jesse F. Hughes
Really, I'm not out to destroy Microsoft. That will just be a
completely unintentional side effect. -- Linus Torvalds
===
Subject: Re: Yet Another Attempt at Refuting Cantor
> http://arxiv.org/PS_cache/math/pdf/0403/0403169.pdf
A week later, they've published this:
http://es.arxiv.org/PS_cache/math/pdf/0403/0403288.pdf
Jose Carlos Santos
===
Subject: Re: Yet Another Attempt at Refuting Cantor
Among other interesting things in this paper, we learn that 0 is
transcendental and that an interval of rationals is compact in R.
-- 
I'm not interested in mathematics that might have anything
to do with reality. -- Russell Easterly, in sci.math
===
Subject: Re: what can we say about eigenvalues of (AB) and (BA)?
>> I hope I can find some numerical relationship between the eigenvalues of
>> AB and BA... for square matrices A and B...
>> I hope eig(AB)=eig(BA) or eig(AB)<eig(BA), etc.?
>If A or B is invertible, then AB and BA are conjugates,
>and have the same eigenvalues. By contunuity (preferably
>in the Zariski topology :-) ) the same is true even if
>both are singular.
We do not have to work so hard.  Consider the determinant
of 
                I       tA
                B       I
This is easily seen to be both |I - tAB| and |I - tBA|,
so even if A and B are not square, but the product is,
the non-zero characteristic roots are equal with equal
multiplicities.
                
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Re: what can we say about eigenvalues of (AB) and (BA)?
> I hope I can find some numerical relationship between the eigenvalues 
of
> AB and BA... for square matrices A and B...
> I hope eig(AB)=eig(BA) or eig(AB)<eig(BA), etc.?
>>If A or B is invertible, then AB and BA are conjugates,
>>and have the same eigenvalues. By contunuity (preferably
>>in the Zariski topology :-) ) the same is true even if
>>both are singular.
> We do not have to work so hard.  Consider the determinant
> of
> I tA
> B I
> This is easily seen to be both |I - tAB| and |I - tBA|,
> so even if A and B are not square, but the product is,
> the non-zero characteristic roots are equal with equal
> multiplicities. 
An alternative, probably inferior, argument is
that det(X) is expressible as a polynomial in
tr(X), tr(X^2), ...,
after which the result follows from tr(YX) = tr(XY).
 
-- 
Timothy Murphy  
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
===
Subject: Re: what can we say about eigenvalues of (AB) and (BA)?
> I hope I can find some numerical relationship between the eigenvalues 
of
> AB and BA... for square matrices A and B...
> I hope eig(AB)=eig(BA) or eig(AB)<eig(BA), etc.?
> If A or B is invertible, then AB and BA are conjugates,
> and have the same eigenvalues. By contunuity (preferably
> in the Zariski topology :-) ) the same is true even if
> both are singular.
Even if they are not square, but rectangular of opposite shape, the
_non-zero_ eigenvalues are the same.  In fact, the characteristic
polynomials are the same, save for a power of x (the variable). 
Amusing to prove.  One way uses the fact that A is any matrix, there
are invertible matrices P and Q so that PAQ has the form (I 0), I
being an identity and 0 a block of zeroes (or the transpose).
===
Subject: Re: what can we say about eigenvalues of (AB) and (BA)?
> If A and B are commutable, then exp_for_matrix(A) and exp_for_matrix(B) 
are 
> computable, right? 
If you meant commutable then yes, they are.
Jose Carlos Santos
===
Subject: REQ: formula logic help: for every 7th time do this TIA sal
Greets All
I'm wondering if anyone knows a simple formula for this problem:
Example:
If a button gets pressed every 7th,14th,21th,.. (every 7th iterration)
it will do this.  can someone help me with the logic.
I can program the formula I just need help with the formula Stuff example: 
(9x-2) 
TIA
===
Subject: Re: REQ: formula logic help: for every 7th time do this TIA sal
> Greets All
> I'm wondering if anyone knows a simple formula for this problem:
> Example:
> If a button gets pressed every 7th,14th,21th,.. (every 7th
iterration)
> it will do this.  can someone help me with the logic.
> I can program the formula I just need help with the formula Stuff
example: (9x-2)
> TIA
What you want is when x (mod 7) is congruent to 0.
You didn't say what you're programming in, so you'll have to look up
the syntax.
In Python, it would be   if (x % 7)==0: do_this
In Excel it would be     =IF(MOD(x,7)=0,do_this)
In VBA it would be       if (x mod 7) = 0: do_this
===
Subject: (Possibly) New model of compution
I apologise to those who read yesterdays post of the same topic. Turns out
rough sketches posted to their repository (so they said).
Anyway, again I was interested in feedback and wondered if it was a new
idea or not, or even if other people thought it was interesting and worth
polishing into a completed paper.
Anyway, I've now put it up on the following webpage:
http://home.cogeco.ca/~walton60/tree.pdf
So check it out and comment in this thread.
===
Subject: Re: (Possibly) New model of compution
> I apologise to those who read yesterdays post of the same topic. Turns 
out
> rough sketches posted to their repository (so they said).
> Anyway, again I was interested in feedback and wondered if it was a new
> idea or not, or even if other people thought it was interesting and worth
> polishing into a completed paper.
> Anyway, I've now put it up on the following webpage:
> http://home.cogeco.ca/~walton60/tree.pdf
> So check it out and comment in this thread.
I don't understand the first definition. You seem to be defining BT
via BT^2. What is BT^2 if not pair of elements from BT? You could
before the formula Section 2 line 2, indicate that b is representing
the colour (or color I suppose in American English) of the node x.
I guess you are trying to use a recursive definition. It is best to
set it out so that you define BT_0 as the empty set, and BT_{i+1} in
terms of BT_i. Then BT can be the union of all the BT_i.
However, you may find it easier to do something like the following:
One normally views trees as partial orders or graphs. 
In the first phrasing (and viewing our tree growing down), (T,>) is a
tree iff
1. < is a partial order (e.g. transitive, and here irreflexive)
2. for every r,s in T non equal, there exists some t in T greater than
or equal
to r and s.
(This should ensure it is connected)
3. For any s in T. the set {x in T:x>s} is totally ordered (by the
restriction of >.)
(this should ensure you don't have loops, so it is tree-like)
Then a binary tree is one where for each t in T, there are t_1,t_2 in
T incomparable so that for all x in T with x<t, either x < t_1 or x <
t_2.
A coloured tree is a tree T, a set of colours C and a function f : T
to C. I guess you are considering C={0,1}.
===
Subject: Re: (Possibly) New model of compution
> I don't understand the first definition. You seem to be defining BT
> via BT^2. What is BT^2 if not pair of elements from BT? You could
> before the formula Section 2 line 2, indicate that b is representing
> the colour (or color I suppose in American English) of the node x.
> I guess you are trying to use a recursive definition. 
Yes I guess technically i have a circular definition. I should have said
let TT be the unique set S such that S={x | etc. }. I'm pretty sure that
TT can be proven to exist in ZF. Anyway I more or less think you are
right. Even if TT does exist and behaves as i think it does its an awkward
formulation. I am going to take your advice and define things in the
manner you suggest.
===
Subject: Re: (Possibly) New model of compution
> I apologise to those who read yesterdays post of the same topic. Turns 
out
> rough sketches posted to their repository (so they said).
> Anyway, again I was interested in feedback and wondered if it was a new
> idea or not, or even if other people thought it was interesting and worth
> polishing into a completed paper.
> Anyway, I've now put it up on the following webpage:
> http://home.cogeco.ca/~walton60/tree.pdf
> So check it out and comment in this thread.
first, tell us what is the 'old model of compution'. what is
'compution' anyway? sound like a shorthand for 'computer pollution' to
me.
===
Subject: Re: (Possibly) New model of compution
> first, tell us what is the 'old model of compution'. what is
> 'compution' anyway? sound like a shorthand for 'computer pollution' to
> me.11
Old, in this context, means heretofore thought-of. I would like to know if
others have thought of using colored binary trees to model computation in
the manner that I have done.
I did say *possibly* new because i'm not sure.
===
Subject: how long does it take for a message to appear on the list?
is this a moderated newsgroup? how long does it for a message to appear? 
===
Subject: Re: how long does it take for a message to appear on the list?
> is this a moderated newsgroup?
No.
> how long does it for a message to appear? 
That depends upon your newsserver. With the one I use, it appears as
soon as I post it.
Jose Carlos Santos
===
Subject: Re: how long does it take for a message to appear on the list?
> is this a moderated newsgroup?
> No.
> how long does it for a message to appear? 
> That depends upon your newsserver. With the one I use, it appears as
> soon as I post it.
Well, it appears on *your* newsserver as soon as you post it. 
How long it takes for your message to appear on *my* newsserver 
depends on all the stuff between here and there. Could be seconds, 
could be minutes, hours, days....
-- 
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: how long does it take for a message to appear on the list?
>how long does it for a message to appear? 
>>That depends upon your newsserver. With the one I use, it appears as
>>soon as I post it.
> Well, it appears on *your* newsserver as soon as you post it. 
> How long it takes for your message to appear on *my* newsserver 
> depends on all the stuff between here and there. Could be seconds, 
> could be minutes, hours, days....
It looks as if you're saying that I see it as soon as I've posted it
because I'me seeing it with the same newsreader I've posted it from.
That's not true. In order to test what you seemed to be saying, I've
posted a test message (called Speed test) at the alt.test newsgroup
using another newsserver and right after I tried to see it using my
usual newsserver (news.uni-berlin.de). It was already there.
Jose Carlos Santos
===
Subject: Re: how long does it take for a message to appear on the list?
> how long does it for a message to appear?
>>That depends upon your newsserver. With the one I use, it appears as
>>soon as I post it.
> Well, it appears on *your* newsserver as soon as you post it. How
> long it takes for your message to appear on *my* newsserver depends
> on all the stuff between here and there. Could be seconds, could be
> minutes, hours, days....
> It looks as if you're saying that I see it as soon as I've posted it
> because I'me seeing it with the same newsreader I've posted it from.
> That's not true. In order to test what you seemed to be saying, I've
> posted a test message (called Speed test) at the alt.test newsgroup
> using another newsserver and right after I tried to see it using my
> usual newsserver (news.uni-berlin.de). It was already there.
That neither tests nor disproves what Gerry said.
Phil
-- 
... one Marine noticed one of the prisoners was still breathing. A Marine 
can be heard saying on the pool footage provided to Reuters Television: 
He's ing faking he's dead. He faking he's ing dead.
The Marine then raises his rifle and fires into the man's head. 
The pictures are too graphic for us to broadcast, Sites said. 
===
Subject: Re: how long does it take for a message to appear on the list?
...
 > how long does it for a message to appear? 
 > 
 > That depends upon your newsserver. With the one I use, it appears as
 > soon as I post it.
 > Well, it appears on *your* newsserver as soon as you post it. 
 > How long it takes for your message to appear on *my* newsserver 
 > depends on all the stuff between here and there. Could be seconds, 
 > could be minutes, hours, days....
Or never.
-- 
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, 
+31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/
===
Subject: ergodic matrix question
could you please recommend a (alebra/ advanced matrix algebra) book where i
can find a list of the properties of ergodic matrices?
does anybody know about these properties?
i know ergodic matrices are associated to markov chains but i am not 
looking
for markov vhains.
===
Subject: connected graphs
Assume there is a graph with n nodes
what is the least number of edges this graph should have in order to be 
connected?
thank you
dimitris 
===
Subject: Re: connected graphs
            days. My association with the Department is that of an alumnus.
>Assume there is a graph with n nodes
>what is the least number of edges this graph should have in order to be 
>connected?
I am unclear. Do you mean, what is the least number for which there
exist a graph with n nodes and that many edges which is connected, or
do you mean, what is the least number such that all graphs with n
nodes and that many edges (and no multiedges, presumably) are
connected?
In the former case, n-1 edges. Because any connected graph with n
nodes contains a spanning tree, and a tree with n nodes must have n-1
edges.
The latter case, I don't know off the top of my head.
-- 
It's not denial. I'm just very selective about
 what I accept as reality.
    --- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: connected graphs
            days. My association with the Department is that of an alumnus.
>>Assume there is a graph with n nodes
>>what is the least number of edges this graph should have in order to be 
>>connected?
>I am unclear. Do you mean, what is the least number for which there
>exist a graph with n nodes and that many edges which is connected, or
>do you mean, what is the least number such that all graphs with n
>nodes and that many edges (and no multiedges, presumably) are
>connected?
>In the former case, n-1 edges. Because any connected graph with n
>nodes contains a spanning tree, and a tree with n nodes must have n-1
>edges.
>The latter case, I don't know off the top of my head.
Oh, duh is me.
You can have a complete graph on n-1 nodes and an isolated node; add
one edge and this graph must be connected. So that gives 
(n-1 choose 2) + 1 as the least number of edges such that every graph
of n nodes and that many edges must be connected.
-- 
It's not denial. I'm just very selective about
 what I accept as reality.
    --- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: simple integral equations problem
I need to solve the following integral equation for f
int(f(x),x=x..x-mx+c)-w == 0
where w, c and m are positive constants and f(x) is defined when c>mx (I 
am not worried what it is outside of that range).  m is between 0 and 1.
How would I go about this?  I understand that the solution will not be 
unique.
Another simpler example is
int(f(x),x=x..2x)-w=0
where w is a positive constant.
How would I solve that?
Raphael
===
Subject: question about ergodic matrices
could you please recommend a (alebra/ advanced matrix algebra) book where i 

can find a list of the properties of ergodic matrices?
does anybody know about these properties?
i know ergodic matrices are associated to markov chains but i am not looking 

for markov vhains.
===
Subject: DeMorgan's Theorem Question
Can (A' B C) + (A B' C) + (A B C') be converted to (A' B C)(A B' C)(A B C')
using DeMorgans Theorem. I think not but I have been told that this can be
done. If it can be done can anyone explain how to do this. IMO even the
truth tables would be different.
TIA
===
Subject: Re: DeMorgan's Theorem Question
X+Y+Z = NOT(NOT(X).NOT(Y).NOT(Z))
> Can (A' B C) + (A B' C) + (A B C') be converted to (A' B C)(A B' C)(A B
C')
> using DeMorgans Theorem. I think not but I have been told that this can 
be
> done. If it can be done can anyone explain how to do this. IMO even the
> truth tables would be different.
===
Subject: Re: DeMorgan's Theorem Question
> X+Y+Z = NOT(NOT(X).NOT(Y).NOT(Z))
> Can (A' B C) + (A B' C) + (A B C') be converted to (A' B C)(A B' C)(A B
> C')
> using DeMorgans Theorem. I think not but I have been told that this can
be
> done. If it can be done can anyone explain how to do this. IMO even the
> truth tables would be different.
===
Subject: Re: DeMorgan's Theorem Question
> Can (A' B C) + (A B' C) + (A B C') be converted to (A' B C)(A B' C)(A B 
C')
> using DeMorgans Theorem.
No, inside that expression is A' A = 0.
Do you mean
        (A' + B + C)(A + B' + C)(A + B + C')
Then from that use distributive law of * over + to see what you get.
> I think not but I have been told that this can be done.
> If it can be done can anyone explain how to do this. IMO even the
> truth tables would be different.
===
Subject: Re: DeMorgan's Theorem Question
> Can (A' B C) + (A B' C) + (A B C') be converted to (A' B C)(A B' C)(A B 
> C')
> using DeMorgans Theorem. I think not but I have been told that this can 
be
> done. If it can be done can anyone explain how to do this. IMO even the
> truth tables would be different.
> TIA
I wanna talk about my dick.
===
Subject: Re: DeMorgan's Theorem Question
address@is.invalid says...
> Can (A' B C) + (A B' C) + (A B C') be converted to (A' B C)(A B' C)(A B 
C')
> using DeMorgans Theorem. I think not but I have been told that this can 
be
> done. If it can be done can anyone explain how to do this. IMO even the
> truth tables would be different.
No. You are basically saying that AND == OR. (A' B C)+(A B' C)+(A B C') 
is the equivalent of ((A'BC)'(AB'C)'(ABC')')'
DeMorgan's Theorem says that (AB)' = A' + B', or for three variables
(XYZ)' = X'+Y'+Z'   
If you substitute: 
         X=(A' B C)
         Y=(A B' C)
         Z=(A B C')
Using truth tables:
F1 = (A' B C) + (A B' C) + (A B C') 
F2' = (A'BC)'(AB'C)'(ABC')'
  A  B  C  |  X  Y  Z  | F1  
-----------+-----------+----
  0  0  0  |  0  0  0  |  0 
  0  0  1  |  0  0  0  |  0
  0  1  0  |  0  0  0  |  0
  0  1  1  |  1  0  0  |  1
  1  0  0  |  0  0  0  |  0
  1  0  1  |  0  1  0  |  1
  1  1  0  |  0  0  1  |  1
  1  1  1  |  0  0  0  |  0
  A  B  C  |  X' Y' Z' | F2'| F2    
-----------+-----------+----+----
  0  0  0  |  1  1  1  |  1 |  0
  0  0  1  |  1  1  1  |  1 |  0
  0  1  0  |  1  1  1  |  1 |  0
  0  1  1  |  0  1  1  |  0 |  1
  1  0  0  |  1  1  1  |  1 |  0 
  1  0  1  |  1  0  1  |  0 |  1
  1  1  0  |  1  1  0  |  0 |  1
  1  1  1  |  1  1  1  |  1 |  0
F1 = F2 Q.E.D.
-- 
  Keith
 
===
Subject: Re: how to solve integral equations?
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iAIIxLN12013;
>I need to solve the following integral equation for f
>int(f(x),x=x..x-mx+c)-w == 0
>where w, c and m are positive constants and f(x) is defined when c>mx (I 
>am not worried what it is outside of that range).  m is between 0 and 1.
>How would I go about this in maple for example?  I understand that the 
>solution will not be unique.
>Another simpler example is
>int(f(x),x=x..2x)-w=0
>where w is a positive constant.
>How would I solve that?
>Raphael
int(f(x),x=x..h(x))=w 
Turns into a functional equation.
Let us put int(f(x)dx)=F(x) 
We've got  F(h(x))-F(x)=w  ,rewritten:
           F(h(x))/w = F(x)/w +1  this is an Abel function.
           If we know which is the Abel function of h(x) say phi(x)|
           phi(h(x))=phi(x)+1  to a constant c .               (1)
           phi(x)+c = F(x)/w .
 For your second equation : F(2x)/w = F(x)/w + 1 ,   h(x)= 2x  (2)
  Show that ln(x)/ln(2)+ c verify (1) ,F(x)=w*(ln(x)/ln(2)+ c) (3),
 just have to derivate ...f(x)=...
Notice the Abel function of line ax+b is ln(x+b/(a-1))/ln(a) and you
can compute your first integral.
In fact we must take into account an invariant term for x->2x into (3)
Alain.
===
Subject: Calculus Question Phrasing Problem
I'm teaching myself calculus using a textbook. There is a problem in the 
book, stated as such:
Use the relation 1/(a^2-x^2) = (1/2a)( 1/[a+x] + 1/[a-x]) to find the nth 
derivative of 1/(a^2-x^2).
Now, I have solved the problem correctly, but I still don't understand the 
question (the method I used did not use the relationship given). I 
simply don't understand what use the above relationship is, and would 
like to understand what the question is driving at. Can anybody 
===
Subject: Re: Calculus Question Phrasing Problem
>I'm teaching myself calculus using a textbook. There is a problem in the 
>book, stated as such:
>Use the relation 1/(a^2-x^2) = (1/2a)( 1/[a+x] + 1/[a-x]) to find the nth 
>derivative of 1/(a^2-x^2).
>Now, I have solved the problem correctly, but I still don't understand the 

>question (the method I used did not use the relationship given). I 
>simply don't understand what use the above relationship is, and would 
>like to understand what the question is driving at. Can anybody 
The point is that it's quite easy to come up with a formula for the n'th 
derivative of 1/(a+x) or 1/(a-x), and not so easy to do the same for
1/(a^2-x^2) directly.
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
===
Subject: Re: Calculus Question Phrasing Problem
 <cnittd$otg$1@nntp.itservices.ubc.ca The point is that it's quite easy to 
come up with a formula for the n'th
> derivative of 1/(a+x) or 1/(a-x), and not so easy to do the same for
> 1/(a^2-x^2) directly.
===
Subject: Re: generalized birthday problem
> I'm going to try to get the exact formula for the expected number of
> people you need to put in the room to get m partnerships, using
> generating functions... not sure if there is a more direct approach.
> Let U(m,k) be the expected number of additional people you need to put
> in to get m more partnerships, starting with k people having all 
different
> birthdays, with n possible birthdays (equally likely).  
> We have U(0,k) = 0, and you want U(m,0).
> n = 1:
> U(m,0) = 2m
> n = 3:
> U(m,0) = 2m+1 - (1/9)^m
> n = 5:
> U(m,0) = 2m+2 - (18/25) (1/25)^m - (32/25) (9/25)^m
Using the Robert's previous post (heavily edited above), I came up with
the following Mathematica formula for U(m,0) in the case of odd n:
  u0[n_, m_] := 2m + ((n-1)/2) (1 - Sum[Binomial[n-2, j + (n-3)/2] *
      (2j-1)^(2m)/n^(2m+n-3) (j + (n-1)/2)^(j + (n-5)/2) *
      ((n+1)/2 - j)^((n-3)/2 - j), {j, (n-1)/2}]) /; OddQ[n];
I haven't proved this:  it's just the pattern I observed from computing
some of the U(m,0) for various n.  The values U(m,0) for n=365 are rational
numbers with large denominators.  E.g., the denominator of U(1,0)(in least
terms) is 5^276 73^360.  Here are some numerical values:
  Table[{m, N[u0[365, m]]}, {m, 0, 20}] // TableForm
 m    U(m,0)
------------
 0    0
 1   24.6166
 2   37.2807
 3   46.9626
 4   55.1566
 5   62.4225
 6   69.0401
 7   75.1727
 8   80.9249
 9   86.3689
10   91.5565
11   96.5267
12  101.31
13  105.929
14  110.404
15  114.751
16  118.982
17  123.11
18  127.142
19  131.087
20  134.953
When m becomes large, U(m,0) approaches 2m+182.  This table shows how
fast the convergence is
 m    2m+182-U(m,0)
-------------------
   0  182
  73   36.9641
 146   14.6738
 219    6.28539
 292    2.76473
 365    1.22931
 438    0.549139
 511    0.245802
 584    0.110124
 657    0.0493571
 730    0.0221257
 803    0.00991922
 876    0.00444707
 949    0.00199378
1022    0.00089389
1095    0.000400767
-Jim Ferry
===
Subject: test
        boundary=----=_NextPart_000_001D_01C4CD63.64A974E0
---------------------------------------------------------------------
-- 
Edward Hyman
===
Subject: Learning Higher Math
Hello everyone, I have a website that helps anyone learn higher math
and write correct proofs:
http://fsc729.ifreepages.com
Some of you might remember me from my previous posts, I have
completely redesigned my website. I've added a forum so others may
post their study and learning methods, I've made the site more user
friendly, and I've added how to learn other subjects as well.
Feel free to share your ideas on how to learn higher math.
John G
===
Subject: Re: Calculus Question Phrasing Problem
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iAIK13a18162;
>I'm teaching myself calculus using a textbook. There is a problem in the 
>book, stated as such:
>Use the relation 1/(a^2-x^2) = (1/2a)( 1/[a+x] + 1/[a-x]) to find the nth 
>derivative of 1/(a^2-x^2).
>Now, I have solved the problem correctly, but I still don't understand the 

>question (the method I used did not use the relationship given). I 
>simply don't understand what use the above relationship is, and would 
>like to understand what the question is driving at. Can anybody 
If you solved it as required, you would have noticed that
(d/dx)(1/[a+x}) = -1/[a+x]^2
and 
(d/dx)(-1/[a+x]^2) = (-1)(-2)/[a+x]^3
etc. etc. You could then posit a general formula, and prove it by 
induction.
phil
===
Subject: Re: How to determine f(x), given f(x)*f(-x) = -exp(kx^2) ?
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iAIK4iY18501;
>Hi all,
>How to determine f(x), given f(x)*f(-x) = -exp(kx^2) ? where x is a
>complex variable.
>Is there a method that I could use?
>Nischal
Let f(x) = i exp[kx^2/2]
Then f(-x) = f(x) and f(x)f(-x) = - exp[kx^2/2] q.e.d.
phil
===
Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
http://mygate.mailgate.org/mynews/comp/comp.theory/a8d13dcbdca7ae739b921bb2b
a
3db48e.48257%40mygate.mailgate.org
> Hmmm. Given of course the truth of the usual axiomatic system, by
> which you probably mean ZFC. Zenkin's argument is more like, well,
> this is a finite nonhalting program, that never says anything.
But it is no such thing!
It is a program with a single step!
(Really, it is not a program at all,
since there is no execution happening.)
Create a single number, which at every
position, differs from the corresponding
diagonal digit position, all at the same
single creation instance time, via a
simple substitution rule. Boom. Done.
We can do this, because our tool is _not_
a computer, but an abstract conceptual
device, entirely capable of fully parallel
operation.
Zenkin's confusion is in attempting to _map_
this to some sequential process, done digit
by digit, when it is no such thing.
Now, we've wasted, by MailGate's count, over
108 postings on this piece of utterly
meritless garbage with which you trolled the
newsgroup, one in a long series of similar
time wasting moron activities by you, become
a successful attempt by you to prove that a
PhD can be garnered with no danger of it
being accompanied by an education, or at
least it can if one chooses Turkey as host
educator.
You are certainly an embarrassment to your
university's degree program each time you
repeat the demonstration that you paid no
attention in class, yet ended up with a
PhD. That rather diminishes the value of
all the other students' PhDs, doesn't it,
if onlookers reasonably expect them to have
learned no more than you did, if they
expect that your result is the usual one.
Could you possibly find a less destructive
hobby, and preferably one not involving
technical newsgroups?
Chat engines have a splendid property exactly
matching the nil value of your contributions.
They are not communications of record, and
don't sit around in archives cluttering up the
planet.
Maybe you could implement EZzz's Everlasting
Elixer of Eternal Emptiness on some IRC
channel, where you could be the sole star and
center of attention?
This is not that place. and your role here in
comp.theory is profoundly not a starring one.
xanthian.
-- 
===
Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
> You are certainly an embarrassment to your
> university's degree program....
With these kinds of remarks, you only underwrite your mental illness.
You have understood absolutely nothing from my posts, and this is not
surprising. But *you* are an amazing specimen.
Your attitude became apparent when you failed to understand the real
content of the extremely simple k-means problem. Your goal is not to
understand. Your goal is to abuse people. AFAICT, you are merely a
Dolan, and most of them grow out of that behavior when they finish
high school.
Disgusted,
--
Eray
===
Subject: [OT]: Zenkin's paper on Cantor (reply of Dr. Zenkin)
http://mygate.mailgate.org/mynews/comp/comp.theory/5d5d8c0301dc08f7d9b40dc00
9
b08b71.48257%40mygate.mailgate.org
Hmmm, look whose killfile I escaped yet again.
>> You are certainly an embarrassment to your
>> university's degree program....
> With these kinds of remarks, you only underwrite
> your mental illness.
Well, no, being mentally ill doesn't make me either
incorrect, or stupid. I don't happen to suffer from
a mental illness with those effects; mine just
prevents functional day to day living.
You are, indeed, an embarrassment to your
university's PhD degree program, an indication that
it is possible to be granted a degree without in the
process acquiring anything approximating the
education normally accompanying that degree. You
reinforce this point with each tedious, trolling
conversation you begin and prolong in technical
newsgroups far past its normal merciful death of
intense inanity.
> You have understood absolutely nothing from my
> posts, and this is not surprising.
Indeed. A more clueful person than you might
begin to suspect that the cause is the lack of any
meaningful intellectual content in your posts.
What is surprising is that you continue posting
them.
> But *you* are an amazing specimen.
Each of us likes to believe that of himself or
herself, but in the case of an atypical monopolar
depressive, the ability to convince oneself that is
true of oneself has mostly gone missing.
 
> Your attitude became apparent when you failed to
> understand the real content of the extremely
> simple k-means problem.
Your reference escapes me, but if this is about more
of your meaningless drivel wherein you argue ad
infinitum et ad tedium with world class experts,
from your base territory of didn't pay attention in
class, I have to admit my eyes rather glaze over
the few times I make any effort to wade through the
reams of incomprehension you put forward as your
best approximation of intelligent thought.
> Your goal is not to understand.
There is little about you to understand. You are
suffused with your own importance to the point you
cannot recognize you have become merely an
impediment to the newsgroups where you participate.
This doesn't take great intellectual effort on my
part to realize, I just have to take notice of where
you are supporting positions even my own long gone
stale understanding of theory of computation knows
to be arrant nonsense. I can dependably deduce from
there that the material beyond my own skills which
is similarly classified by others whose opinions I
respect on long acquaintanceship with their writing
is _also_ you taking forceful positions in which you
are completely in error.
> Your goal is to abuse people.
Well, close. My goal is to teach people who are
unable to be quiet when they have nothing to say,
the virtues of acquiring that skill.
> AFAICT, you are merely a simpleton who knows how
Which merely shows that you are willing to extend
your habit of argument from ignorance far outside
the technical venues. First, I'm not a simpleton,
my work product is in use many, many places all
across the civilized world, and has been for several
decades.
accomplishment. I've had many professionals in the
fields where I give advice slap themselves virtually
on the forehead with a why didn't I think to pose
the query that way.
> There are so many like you, Mr.  Dolan, and most
> of them grow out of that behavior when they finish
> high school.
I doubt it. The lack of willingness to deal
cheerfully with the world's great oversupply of time
wasting morons, you prominent among them, is a very
common human attribute. People don't tend, over
time, to become more accepting of intense,
voluntarily indulged idiocy, but rather less so, as
the life remaining to be wasted by the eructions of
such imbeciles grows shorter.
> Disgusted
As well you should be. Now, what are the odds that
you will correct your behavior so that such public
pillorying of your misplaced and misguided efforts
is not a result you commonly encounter?  You'll
notice that I am far from the only person expressing
this reaction to you, even in this single news
thread, so raving at me isn't going to change the
general result a bit.
The only way you can reduce the negative reactions
you receive is to reduce the amount of self-indulgent,
grotesquely technically incompetent, trolled postings
you produce.
Are you capable of taking that step?
xanthian.
-- 
===
Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
 <4181a33d$1$fuzhry+tra$mr2ice@news.patriot.net>
 <a8d13dcbdca7ae739b921bb2ba3db48e.48257@mygate.mailgate.org> You are 
certainly an embarrassment to your
>> university's degree program....
>With these kinds of remarks, you only underwrite your mental illness.
>You have understood absolutely nothing from my posts, and this is not
>surprising. But *you* are an amazing specimen.
>Your attitude became apparent when you failed to understand the real
>content of the extremely simple k-means problem. Your goal is not to
>understand. Your goal is to abuse people. AFAICT, you are merely a
>Dolan, and most of them grow out of that behavior when they finish
>high school.
>Disgusted,
>Eray
This is false.
-- 
David Longley
===
Subject: Re: Platonism
   <cp20me$2hm2$1@msunews.cl.msu.edu>
   <41b562b5.46155894@netnews.att.net>
   <cp2lv8$ajr$1@mailhub227.itcs.purdue.edu>
   <41b682ed.59873690@netnews.att.net>
   <cp56qi$2e0f$1@msunews.cl.msu.edu>
   <41b7d968.71009450@netnews.att.net>
   <cpb6ge$1hq6$2@msunews.cl.msu.edu>
   <41ba63ad.92196009@netnews.att.net>
   <cpcler$2oce$1@msunews.cl.msu.edu>
   <q3lud.217683$HA.96588@attbi_s01>
   <cpcnae$2rqd$1@msunews.cl.msu.edu>
   <U9Aud.640080$mD.240943@attbi_s02>
   <cpfmfp$2df4$1@msunews.cl.msu.edu>
>I have been reading all this in sci.math so computer
>representations were never part of the question. In any case,
>I guess it depends on how you look at it. A typical computer's
>memory is just an enourmous ordered list of bytes, so you could
>always claim that anything stored in memory is ordered if you
>look at it at a low enough level.
>On the other hand a hash table is not usually considered as
>an ordered data structure. The elements are of course going
>to be stored in some order in memory, but this order has
>little to do with any order property of the actual elements.
>If you start to consider parallel/distributed systems than
>ordering possibilities become much more interesting.
The important thing in a computer is functional ordering.
At the hardware level it depends on the address logic.
At the software level it depends on the software.
I not sure about ordering in an abstract mathematical set
but wouldn't it be one of convention? In other words there
is not inherent order in a set of items except that imposed
by some protocol?
John Casey
===
Subject: Re: Platonism
>:> 
>:> :> Show up to class and take a picture.  That specifies a set.
>:> :> What's the order?
>:> 
>:> : ... the order of the time space coordinates of the the students in the 

>:> : picture.  You cannot represent a set in physical medium without 
picking 
>:> : up a time space bias.  However, there may be a possible exception; 
>:> : representing a set in a neural network.  Perhaps someone could shed 
some 
>:> : light on that.
>:> 
>:> But what is the order of the time space coordinates?  By order
>:> I mean a total ordering.  There is a first element, and each
>:> element except the last has a unique successor.  I do not see
>:> that being the case with a photo.   There are partial orderings
>:> in a photo, but those are also rather arbitrary.
>:> 
>:> http://www.discoverhuroncounty.com/phgal/turkeys/D-34-6.jpg
>:> http://www.feathersite.com/Poultry/Turkeys/BuffFlock.JPEG
>:> 
>:> Which is the first turkey?  Which is the last?  I imagine 
>:> somewhere someone has done studies on this.  Perhaps
>:> people come up with consistent answers but it would surprise
>:> me.
>:> 
>: Ok, you are right.  Representing a set in a physical medium does not 
>: imbue it with a total ordering.  I was actually coming from a computer 
>: science perspective.  Representing an unordered set in a computer is 
>: very difficult.  Representing an ordered set is a piece of cake.  Can we 

>: then say that an ordered set is more fundamental to a computer than a 
>: unordered set?
>I have been reading all this in sci.math so computer 
>representations were never part of the question.  In any case,
>I guess it depends on how you look at it.  A typical computer's
>memory is just an enourmous ordered list of bytes, so you could
>always claim that anything stored in memory is ordered if you
>look at it at a low enough level.
>On the other hand a hash table is not usually considered as
>an ordered data structure.  The elements are of course going
>to be stored in some order in memory, but this order has
>little to do with any order property of the actual elements.  
>If you start to consider parallel/distributed systems than
>ordering possibilities become much more interesting. 
>:> : Incidentally i have no use for the lester-set or its lesternality. 
>:> : forget in that world there are also lesterdifferences.
>:> 
>:> That is true, I forgot about the lesterdifferences.  The
>:> fact he insists on using his own private vocabulary makes
>:> it clear he is only interested in pontificating and not interested 
>:> in communicating.
>:> 
>:> Stephen
>: What irritates me is that at the point where i get familiar enough with 
>: Zick's mentations to grok them, that is the very point at which he 
>: fluffs me off and becomes abusive.  It seems to me that Zick's 
>: mentations and Longley's share that property.  When that happens enough 
>: times, you tend not to want to go there again.
>You do seem to have some strange ones in comp.ai.philosophy.  
>But then again that is true of all of usenet.  I think attempting 
>to understand Zick is pointless.  He has demonstrated aptly in
>this thread that there is nothing there to understand.  When
>somebody claims that the set {1,2,3} contains 0 you know there
>is something hoplessly wrong with them.
Glad you brought it up. Zero just doesn't photograph well.
===
Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
> You are certainly an embarrassment to your
> university's degree program....
>>With these kinds of remarks, you only underwrite your mental illness.
>>You have understood absolutely nothing from my posts, and this is not
>>surprising. But *you* are an amazing specimen.
>>Your attitude became apparent when you failed to understand the real
>>content of the extremely simple k-means problem. Your goal is not to
>>understand. Your goal is to abuse people. AFAICT, you are merely a
>>Dolan, and most of them grow out of that behavior when they finish
>>high school.
>>Disgusted,
>>--
>>Eray
> This is false.
Haha..
Well DL, there is no one understanding you better than Dolan.
> -- 
> David Longley 
===
Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
 <4181a33d$1$fuzhry+tra$mr2ice@news.patriot.net>
 <a8d13dcbdca7ae739b921bb2ba3db48e.48257@mygate.mailgate.org>
 <ogGFp6BAYAnBFwB6@longley.demon.co.uk>
 <419c4735$0$44080$5fc3050@dreader2.news.tiscali.nl> You are certainly an 
embarrassment to your
>> university's degree program....
>With these kinds of remarks, you only underwrite your mental illness.
>You have understood absolutely nothing from my posts, and this is not
>surprising. But *you* are an amazing specimen.
>Your attitude became apparent when you failed to understand the real
>content of the extremely simple k-means problem. Your goal is not to
>understand. Your goal is to abuse people. AFAICT, you are merely a
>Dolan, and most of them grow out of that behavior when they finish
>high school.
>Disgusted,
>--
>Eray
>> This is false.
>Haha..
>Well DL, there is no one understanding you better than Dolan.
>> --
>> David Longley
And there's an illustrative example of why it's false. Most people, at 
any age, don't take criticism well (if at all), and very very few 
actively seek it out even when it's in their own best interests to do 
so. It's an essential skill of the professional to court criticism. It's 
the sine qua non for learning.
Your grasp of the affective disorders and behaviour is as flawed as 
Ozkurals'.
-- 
David Longley
===
Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
http://mygate.mailgate.org/mynews/comp/comp.theory/2c885f7a77f555bf6891f72ec
d
ddefec.48257%40mygate.mailgate.org
> And there's an illustrative example of why it's
> false. Most people, at any age, don't take
> criticism well (if at all), and very very few
> actively seek it out even when it's in their own
> best interests to do so. It's an essential skill
> of the professional to court criticism. It's the
> sine qua non for learning.
Now compare and contrast this statement which you
profess to believe, with your utter incapability to
understand what dozens if not hundreds of people
are telling you when they say that your ravings are
misplaced in comp.ai.philosophy, a newsgroup which
has precisely no charter admitting discussions of
Skinnerian behaviorism, and draw what conclusions
your broken thinking processes can accomplish for
you using only that information.
A modification of your behavior would be most
welcome, but least expected, as the outcome of such
introspection.
HTH
xanthian.
-- 
===
Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
 <4181a33d$1$fuzhry+tra$mr2ice@news.patriot.net>
 <a8d13dcbdca7ae739b921bb2ba3db48e.48257@mygate.mailgate.org>
 <ogGFp6BAYAnBFwB6@longley.demon.co.uk>
 <419c4735$0$44080$5fc3050@dreader2.news.tiscali.nl>
 <Jpfd7EJk2LnBFw1T@longley.demon.co.uk>
 <2c885f7a77f555bf6891f72ecdddefec.48257@mygate.mailgate.org> And there's an 
illustrative example of why it's
>> false. Most people, at any age, don't take
>> criticism well (if at all), and very very few
>> actively seek it out even when it's in their own
>> best interests to do so. It's an essential skill
>> of the professional to court criticism. It's the
>> sine qua non for learning.
>Now compare and contrast this statement which you
>profess to believe, with your utter incapability to
>understand what dozens if not hundreds of people
>are telling you when they say that your ravings are
>misplaced in comp.ai.philosophy, a newsgroup which
>has precisely no charter admitting discussions of
>Skinnerian behaviorism, and draw what conclusions
>your broken thinking processes can accomplish for
>you using only that information.
As I've said in another thread, you neglect history to your own 
detriment. You should ask yourself just how much of Skinnerian 
behaviorism you know, and given that, how would you know whether any of 
what you say above is true or false?
>A modification of your behavior would be most
>welcome, but least expected, as the outcome of such
>introspection.
>HTH
>xanthian.
Nonsense.
Your environment has shaped you up to believe that rhetoric is more 
powerful than science. You should read the quotes I recently provided 
from one of the most influential mathematical logicians from the last 
century. You are just being abusive and there is not point to it. That's 
what makes it pathological.
As long as you don't look at the larger context, I predict you'll just 
go on being miserably deluded (and that's your description of yourself 
as well as my own observation on the basis of how you post). I have not 
interest in seeing you be miserable, hard though you may find that to 
believe.
Have a peek out at what I have advised you to read. It might clear the 
air a little for you. Look into some of the links that I have provided 
and see if you can follow the logic and practical developments. You'll 
need to look further and wider than you have so far.
-- 
David Longley
http://www.longley.demon.co.uk
===
Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
> You are certainly an embarrassment to your
> university's degree program....
>>With these kinds of remarks, you only underwrite your mental 
>>illness.
>>You have understood absolutely nothing from my posts, and this is 
>>not
>>surprising. But *you* are an amazing specimen.
>>Your attitude became apparent when you failed to understand the real
>>content of the extremely simple k-means problem. Your goal is not to
>>understand. Your goal is to abuse people. AFAICT, you are merely a
>>Dolan, and most of them grow out of that behavior when they finish
>>high school.
>>Disgusted,
>>--
>>Eray
> This is false.
>>Haha..
>>Well DL, there is no one understanding you better than Dolan.
> --
> David Longley
> And there's an illustrative example of why it's false. Most people, at 
> any age, don't take criticism well (if at all), and very very few 
> actively seek it out even when it's in their own best interests to do 
> so. It's an essential skill of the professional to court criticism. 
> It's the sine qua non for learning.
I just loath you, DL. You are on the same list with people who would say 
I'm an heretic and sinner, and am instructed that in order to understand 
that I should read the Koran.
Longley Uh Ackbar!! 
===
Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
Excuse me, but I want to make it really clear that compared to
Longley, Mr. Dolan is a reasonable person. As a matter of fact, I find
Mr. Dolan quite knowledgeable in a number of subjects, but he does not
seem to appreciate the importance of civil discussion on public
forums. Even when agreeing with somebody, he will assert irrelevant
personal statements. He seems to be on  an excess amount of caffeine
constantly.
When criticized properly, I will honor it. I have taken much of Mr.
Dolan's criticism seriously. However, he seems to know very little
about the debate about actual/potential infinity which is a
significant *philosophical* matter  going back to the ancient Greece.
Such matters are not to be left to those who accept the validity of
whatever philosophy is implicitly assumed in their particular school
of mathematics, *without* ever questioning. There is no need to state
that it will be pointless to discuss foundational subjects with
persons who cannot question them.
Incidentally, many people who call themselves constructivists come
around saying that constructivism has no trouble saying that a Turing
Machine has *actually* an infinite tape, which makes the discussions
all the more fuzzy! There are many versions of constructivism and
intuitionism, and it seems that we fail to acknowledge the *exact*
metaphysical assumptions in these versions when we use the suitcase
words.
I have had such a discussion with an extremely intelligent and
experienced mathematician. He told me that PCs are not Turing
Machines, because they have an infinite tape. I think he did not
know anything about descriptive complexity. This infinite portion of
the tape consists entirely of blank symbols, and therefore has
descriptive complexity O(1), which is easily realized by a physical
system. When I told him about Ullman's indefinite growing argument, he
objected But when the universe is filled up, it cannot grow any more!
Then, it is not infinite, to which I responded Yes, but there is
*nothing* that is larger than the universe. To assume the contrary
would be theology, which I despise.
>
> This is false.
>>Haha..
>>Well DL, there is no one understanding you better than Dolan.
> --
> David Longley
> And there's an illustrative example of why it's false. Most people, at 
> any age, don't take criticism well (if at all), and very very few 
> actively seek it out even when it's in their own best interests to do 
> so. It's an essential skill of the professional to court criticism. 
> It's the sine qua non for learning.
> I just loath you, DL. You are on the same list with people who would say 
> I'm an heretic and sinner, and am instructed that in order to understand 
> that I should read the Koran.
> Longley Uh Ackbar!!
===
Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
> I have had such a discussion with an extremely intelligent and
> experienced mathematician. He told me that PCs are not Turing
> Machines, because they have an infinite tape. I think he did not
> know anything about descriptive complexity. This infinite portion of
> the tape consists entirely of blank symbols, and therefore has
> descriptive complexity O(1), which is easily realized by a physical
> system. When I told him about Ullman's indefinite growing argument, he
> objected But when the universe is filled up, it cannot grow any more!
> Then, it is not infinite, to which I responded Yes, but there is
> *nothing* that is larger than the universe. To assume the contrary
> would be theology, which I despise.
Turing Machines are idealized which means they are not physically realized.
TMs are not meant to have physical constraints applied to them. That mixes
categories. Sometimes the question is asked, how many sentences can be
generated in some natural (say English) language? There are finitely many
words in the language, but the standard answer is that there are countably
infinite number of sentences, due to appending etc.
These kind of questions ask what is the potential in theory, not what is
practically possible. Observations like: a person can only articulated 
finitely
many sentences in a lifetime, or any sentence has to be uttered before
somebody dies or a machine wears out, or that how many sentences
potentially exist is related to how many people generate sentences over
the lifetime of humanity in the universe are not relevant, because that is
not the question being asked. Every process within the universe is finite
due to heat death of the universe, so that makes all such questions 
trivial,
if one interprets them to mean or apply to a physical reality. A Turing
Machine or potential sentence of a language (there is no pre-existing
specification that the sentence has to be of finite length) is not of this 
world.
The set of natural numbers is countably infinite and is has some use
theoretically. Would you claim infinite sets have no use because they
And the original description of a Turing Machine. It is common to call
this tape 'infinite' though some prefer finitely unbounded. There is no
physical time constraint applied to when the calculation has to be
completed. So there are calculations that a physical PC the size of
a galaxy could not complete before the universe ran out of power to
energize the computer. A Turing Machine can of course complete
such a calculation (because the calculation does not need to be infinite,
just finitely larger/longer in time that can be accomplished by any
physical device during the existence of the physical universe) because
the constraint of physical time is not applied to idealized situations.
Keeping those categories seperate, the idealized and the physical,
is definitional. The answer to theoretical questions is trivial and obvious
if you mix these categories. Mathematicians invented infinity without
the requirement that it be physically realized because it was useful.
Pure mathematics invents formal mathematical systems with no
requirement that this formal system represent any physical event or
> Yes, but there is
> *nothing* that is larger than the universe. To assume the contrary
> would be theology, which I despise.
When you say *nothing* you mean no physical something. Mathematical
objects need not be physical. Ideas may be generated physically, but the
idea of a unicorn can exist without the idea being physically manifested.
The mathematical idea of a circle exists. We do find physical objects
which remind of this mathematical idea. Pi is the ratio of a circumference
of a circle to the diameter. Even if you think of Pi as finitely unbounded,
there is still no last digit of Pi, there is still no last digit of Pi. So 
in theory
you can talk about the digit expansion of Pi after the decimal to a value
say, 10^10^10^10^ and so on to say millions of exponents of exponents
computer could calculate within the lifetime of the universe.
That does not make Pi theology. You will find Pi used in Physics.
You will find infinity used in quantum theory which makes theoretical
predictions which match experiments to 10^11 of real world accuracy.
Mathematics has nothing to do with theology. It certainly does not require
one to adopt mathematical platonism, a metaphysical realm outside the 
universe.
They do say that mathematics is 'unreasonably effective'. Mathematics 
starts
with observations of physical reality and then regularities are then 
*represented*.
Mathematics is a logical relationship to reality, it is idealistic/symbolic, 

especially when formalized, and is a useful tool to predict the behavior of 

physical reality.
It is _not_ the same as physical reality. And that is why concepts of
mathematics can have theoretical objects; mathematics as abstract thinking 
is not required to map one-to-one to existing physical events or objects.
I can imagine the successor function which adds one to the previous number
and which can generate the naturals 1,2,3,... and so on and so on into 
infiinity,
even though I cannot mentally grasp infinity. But you are trying to make 
this
analagous to grasping God or theology. I cannot visualize God as having a
1,2,3... foundation successor function so therefore having an abstract 
existence.
When you made this comparison, infinity and theology/God, to the size
of the physical universe, you crossed over from debating potential vs.
actual infinities to declaring abstract thinking is just theology in another 

guise.
> Yes, but there is
> *nothing* that is larger than the universe. To assume the contrary
> would be theology, which I despise.
Statements like this are going to appear to others as displaying gaps
in your background education. Abstract thinking in mathematics does
not assume that there is a physical object under discussion so that
_size_ (larger than the universe) is a pertinent factor. And it is
muddled to conflate the inability to grasp how a potential infinity
transforms into an actual infinity as a theological issue. Your statement
attacks mathematics using even the abstract concept of a *potential*
infinity as religious mumbo jumbo. A potential infinity is larger than
any aspect contained within the universe also. And actualized infinity
certainly has nothing to do with that infinity being manifested within
the physical universe.
Actualized infinity is another abstract mathematical construct 
conceptualizing completing a potential infinity, neither of which 
abstractions have physical size.
Your idea reminds me of, There can't be an acutalized infinity(number) 
because
it  would be too long to fit in the universe and nobody would live long 
enough
to write it down anyway. Two factors having nothing to do with the 
discussion.
After writing this, I think you may not have realized this.
-- 
Mathematics - this may surprise or shock
 some - is never deductive in its creation.
The mathematician at work makes vague
guesses, visualizes broad generalizations,
and jumps to unwarranted conclusions.
He arranges and rearranges his ideas,
and he becomes convinced of their truth
long before he can write down a logical
proof....The deductive stage, writing the
results down, and writing its rigorous proof
are relatively trivial once the real insight
arrives: it is more the draftsman's work not
the architect's. - Paul Halmos
SH: Achieving or having a mathematical insight is not the
same thing as having a religious/theological experience.
The idea that it is the same thing, is itself, a mystical claim.
Language is abstract and symbolic,
Stephen
===
Subject: Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
>> I have had such a discussion with an extremely intelligent and
>> experienced mathematician. He told me that PCs are not Turing
>> Machines, because they have an infinite tape. I think he did not
>> know anything about descriptive complexity. This infinite portion of
>> the tape consists entirely of blank symbols, and therefore has
>> descriptive complexity O(1), which is easily realized by a physical
>> system. When I told him about Ullman's indefinite growing argument, he
>> objected But when the universe is filled up, it cannot grow any more!
>> Then, it is not infinite, to which I responded Yes, but there is
>> *nothing* that is larger than the universe. To assume the contrary
>> would be theology, which I despise.
> Turing Machines are idealized which means they are not physically 
> realized.
> TMs are not meant to have physical constraints applied to them. That 
mixes
> categories. Sometimes the question is asked, how many sentences can be
> generated in some natural (say English) language? There are finitely many
> words in the language, but the standard answer is that there are 
countably
> infinite number of sentences, due to appending etc.
> These kind of questions ask what is the potential in theory, not what is
> practically possible. Observations like: a person can only articulated
> finitely
> many sentences in a lifetime, or any sentence has to be uttered before
> somebody dies or a machine wears out, or that how many sentences
> potentially exist is related to how many people generate sentences over
> the lifetime of humanity in the universe are not relevant, because that 
is
> not the question being asked. Every process within the universe is finite
> due to heat death of the universe, so that makes all such questions 
> trivial,
> if one interprets them to mean or apply to a physical reality. A Turing
> Machine or potential sentence of a language (there is no pre-existing
> specification that the sentence has to be of finite length) is not of 
this
> world.
> The set of natural numbers is countably infinite and is has some use
> theoretically. Would you claim infinite sets have no use because they
> universe?
> And the original description of a Turing Machine. It is common to call
> this tape 'infinite' though some prefer finitely unbounded. There is no
> physical time constraint applied to when the calculation has to be
> completed. So there are calculations that a physical PC the size of
> a galaxy could not complete before the universe ran out of power to
> energize the computer. A Turing Machine can of course complete
> such a calculation (because the calculation does not need to be infinite,
> just finitely larger/longer in time that can be accomplished by any
> physical device during the existence of the physical universe) because
> the constraint of physical time is not applied to idealized situations.
> Keeping those categories seperate, the idealized and the physical,
> is definitional. The answer to theoretical questions is trivial and 
> obvious
> if you mix these categories. Mathematicians invented infinity without
> the requirement that it be physically realized because it was useful.
> Pure mathematics invents formal mathematical systems with no
> requirement that this formal system represent any physical event or
>> Yes, but there is
>> *nothing* that is larger than the universe. To assume the contrary
>> would be theology, which I despise.
> When you say *nothing* you mean no physical something. Mathematical
> objects need not be physical. Ideas may be generated physically, but the
> idea of a unicorn can exist without the idea being physically manifested.
> The mathematical idea of a circle exists. We do find physical objects
> which remind of this mathematical idea. Pi is the ratio of a 
circumference
> of a circle to the diameter. Even if you think of Pi as finitely 
> unbounded,
> there is still no last digit of Pi, there is still no last digit of Pi. 
So
> in theory
> you can talk about the digit expansion of Pi after the decimal to a value
> say, 10^10^10^10^ and so on to say millions of exponents of exponents
> computer could calculate within the lifetime of the universe.
> That does not make Pi theology. You will find Pi used in Physics.
> You will find infinity used in quantum theory which makes theoretical
> predictions which match experiments to 10^11 of real world accuracy.
> Mathematics has nothing to do with theology. It certainly does not 
require
> one to adopt mathematical platonism, a metaphysical realm outside the
> universe.
> They do say that mathematics is 'unreasonably effective'. Mathematics 
> starts
> with observations of physical reality and then regularities are then
> *represented*.
> Mathematics is a logical relationship to reality, it is 
> idealistic/symbolic,
> especially when formalized, and is a useful tool to predict the behavior 
> of
> physical reality.
> It is _not_ the same as physical reality. And that is why concepts of
> mathematics can have theoretical objects; mathematics as abstract 
thinking
> is not required to map one-to-one to existing physical events or objects.
> I can imagine the successor function which adds one to the previous 
number
> and which can generate the naturals 1,2,3,... and so on and so on into
> infiinity,
> even though I cannot mentally grasp infinity. But you are trying to make
> this
> analagous to grasping God or theology. I cannot visualize God as having a
> 1,2,3... foundation successor function so therefore having an abstract
> existence.
> When you made this comparison, infinity and theology/God, to the size
> of the physical universe, you crossed over from debating potential vs.
> actual infinities to declaring abstract thinking is just theology in 
> another
> guise.
>> Yes, but there is
>> *nothing* that is larger than the universe. To assume the contrary
>> would be theology, which I despise.
> Statements like this are going to appear to others as displaying gaps
> in your background education. Abstract thinking in mathematics does
> not assume that there is a physical object under discussion so that
> _size_ (larger than the universe) is a pertinent factor. And it is
> muddled to conflate the inability to grasp how a potential infinity
> transforms into an actual infinity as a theological issue. Your statement
> attacks mathematics using even the abstract concept of a *potential*
> infinity as religious mumbo jumbo. A potential infinity is larger than
> any aspect contained within the universe also. And actualized infinity
> certainly has nothing to do with that infinity being manifested within
> the physical universe.
> Actualized infinity is another abstract mathematical construct
> conceptualizing completing a potential infinity, neither of which
> abstractions have physical size.
> Your idea reminds me of, There can't be an acutalized infinity(number)
> because
> it  would be too long to fit in the universe and nobody would live long
> enough
> to write it down anyway. Two factors having nothing to do with the
> discussion.
> After writing this, I think you may not have realized this.
> -- 
> Mathematics - this may surprise or shock
> some - is never deductive in its creation.
> The mathematician at work makes vague
> guesses, visualizes broad generalizations,
> and jumps to unwarranted conclusions.
> He arranges and rearranges his ideas,
> and he becomes convinced of their truth
> long before he can write down a logical
> proof....The deductive stage, writing the
> results down, and writing its rigorous proof
> are relatively trivial once the real insight
> arrives: it is more the draftsman's work not
> the architect's. - Paul Halmos
> SH: Achieving or having a mathematical insight is not the
> same thing as having a religious/theological experience.
> The idea that it is the same thing, is itself, a mystical claim.
> Language is abstract and symbolic,
> Stephen
http://www.disf.org/en/Voci/13.asp
Against both of these bastions of rationalist 18th century philosophy and 
its anti-metaphysical program, Cantor poses the distinction between relative 

actual infinity, or transfinite, as a mathematical notion ( CANTOR, 
III), 
and the absolute actual infinity, as a metaphysical and theological 
notion, typically attributed to the divine nature, and absolutely 
unreachable by pure mathematical knowledge. Unfortunately, Cantor thought 
his view on infinity as opposed to the Thomist conception, because of his 
insufficient knowledge of Aquinas' thought, together with the insufficient 
scholarship of some of his interlocutors. Therefore he was led to believe 
that he had to systematically oppose the Scholastic philosophical doctrine 
with his conception of actual infinity in mathematics. The necessity of 
actual infinity here re-appears in a sense that joins the Parmenidean 
instance with the Platonic instance. The necessity for the existence of 
actual infinity is so linked by Cantor with the necessity for its 
conceivability (Parmenidean instance), properly in relation to the rigorous 

definition of the notion of limits within the analytical calculus and 
regarding the definition of Dedekind of real number as the limit of a 
sequence of rational numbers not belonging to the sequence itself. These 
two notions in fact imply that, in order to let mathematics be founded on 
them in a really auto-consistent way (Parmenidean instance), the indefinite 

variation of the finite (potential infinity) requested by the notion of 
limit has to suppose the a priori complete determination of the domain 
of 
variation (Platonic instance). .82There is no doubt that we 
cannot do without 
the variable quantities within the sense of potential infinity; and that 
from this can be demonstrated the necessity for actual infinity. In order 
that there is a variable quantity in a mathematical theory, the domain 
of 
its variability must be, strictly speaking, known ahead of time through a 
definition. Thus, the said domain must not be itself something variable, 
otherwise every base founded for the study of mathematics would vanish.
Consequently, this domain is a definite set of values, and is thereby 
actually infiniteé (Cantor, 1886, p. 9).
SH: It is true that the term actual infinity has a historical basis 
which 
is theological. But that concluding sentence does not require that history 
nor Platonism. Under discussion is a mathematical domain, not a region of 
physical or religious space.
Cantor poses the distinction between relative actual infinity, or 
transfinite, as a mathematical notion (CANTOR, III), and the 
absolute 
actual infinity, as a metaphysical and theological notion, typically 
attributed to the divine nature, and absolutely unreachable by pure 
mathematical knowledge.
begin 666 arrow.gif
===
Subject: Kruskal's surreal integral
I'm looking for information on Kruskal's integral, which Conway mentions 
at the end of ONAG.  The most explicit mention I've found is at:
http://www.ics.uci.edu/~eppstein/cgt/surreal.html
> What follows is a definition of integration.
>         int_a^b f(t) dt = 
>     {
>         int_a^{b_L} f(t) dt + intd_{b_L}^b {f_L}(t) dt ,      
>         int_a^{b_R} f(t) dt + intd_{b_R}^b {f_R}(t) dt ,      
>         int_{a_R}^b f(t) dt + intd_a^{a_R} {f_L}(t) dt ,      
>         int_{a_L}^b f(t) dt + intd_a^{a_L} {f_R}(t) dt 
>     |       
>         int_a^{b_L} f(t) dt + intd_{b_L}^b {f_R}(t) dt ,      
>         int_a^{b_R} f(t) dt + intd_{b_R}^b {f_L}(t) dt ,      
>         int_{a_R}^b f(t) dt + intd_a^{a_R} {f_R}(t) dt ,      
>         int_{a_L}^b f(t) dt + intd_a^{a_L} {f_L}(t) dt 
>     }
> I used TEX notation for integrals, subscripts and superscripts.
> ``intd'' should be written as an integral sign with a capital `D'
> over it, in the middle.  It means direct integration, which means
> do not chop the domain into pieces.  Notice that some integrations
> in the above definition will go from right to left, which means you
> have to do the usual change of signs.
This integral is considered flawed because it yields e^w (w=omega) for 
the integral of e^x from 0 to w, rather than the expected e^w-1.
I don't understand what is meant by direct integration above, and it 
know any other sources where I can read about this integral?
===
Subject: Re: Kruskal's surreal integral
> I'm looking for information on Kruskal's integral, which Conway mentions 
> at the end of ONAG.  The most explicit mention I've found is at:
> http://www.ics.uci.edu/~eppstein/cgt/surreal.html
So as I read the Epilogue to the 2nd edition of ONAG, this is not
Kruskal's integral, but Norton's integral.  Kruskal did the
computations that showed it is not a good definition.  When Conway
speaks on surreals in person he also says something like this.
> What follows is a definition of integration.
> $
>         int_a^b f(t) dt = 
>     {
>         int_a^{b_L} f(t) dt + intd_{b_L}^b {f_L}(t) dt ,      
>         int_a^{b_R} f(t) dt + intd_{b_R}^b {f_R}(t) dt ,      
>         int_{a_R}^b f(t) dt + intd_a^{a_R} {f_L}(t) dt ,      
>         int_{a_L}^b f(t) dt + intd_a^{a_L} {f_R}(t) dt 
>     |       
>         int_a^{b_L} f(t) dt + intd_{b_L}^b {f_R}(t) dt ,      
>         int_a^{b_R} f(t) dt + intd_{b_R}^b {f_L}(t) dt ,      
>         int_{a_R}^b f(t) dt + intd_a^{a_R} {f_R}(t) dt ,      
>         int_{a_L}^b f(t) dt + intd_a^{a_L} {f_L}(t) dt 
>     }
> $
> I used TEX notation for integrals, subscripts and superscripts.
> ``intd'' should be written as an integral sign with a capital `D'
> over it, in the middle.  It means direct integration, which means
> do not chop the domain into pieces.  Notice that some integrations
> in the above definition will go from right to left, which means you
> have to do the usual change of signs.
> This integral is considered flawed because it yields e^w (w=omega) for 
> the integral of e^x from 0 to w, rather than the expected e^w-1.
> I don't understand what is meant by direct integration above,
The version in the Epilogue to ONAG has no direct integral (whatever
that is) but a dissected integral in which you dissect the interval
into finitely many subintervals, and choose a particular option for
each of the subintervals.
> and it 
> know any other sources where I can read about this integral?
I saved a few pages in hard copy (this was 1988, you know), but there is
nothing on the integral beyond the cgt web page already cited here.
-- 
G. A. Edgar                               
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: Kruskal's surreal integral
> So as I read the Epilogue to the 2nd edition of ONAG, this is not
> Kruskal's integral, but Norton's integral.  Kruskal did the
> computations that showed it is not a good definition.  When Conway
> speaks on surreals in person he also says something like this.
Oops, you're right.  I don't know why I latched on to the wrong name. 
That might be part of the reason I was having trouble finding anything.
> The version in the Epilogue to ONAG has no direct integral (whatever
> that is) but a dissected integral in which you dissect the interval
> into finitely many subintervals, and choose a particular option for
> each of the subintervals.
Yes, I see it now.  This is the same definition as on the cgt page, 
with some signs and limits reversed.  For some reason I thought the 
definition was not given in the epilogue.
> I saved a few pages in hard copy (this was 1988, you know), but there is
> nothing on the integral beyond the cgt web page already cited here.
===
Subject: Re: Evolving the Weaire-Phelan Structure using Surface Evolver
> For example one of the vertices in phelanc.fe should have the 
coordinates:
> 1 0 1.5
> But what you find is:
> 0.999205 0.000988 1.500509
Are these calculated values? If so they might be the result of
accumulated truncations.   Alternately this might be the reuslt
of base-10 to base-2 and back to base-10 rounding errors.
If possible try the calculation with higher precision ( REAL*16 
instead of REAL*8 in FORTRAN-speak ).  That should reduce but
probably not eliminate the errors.  A good contemporary textbook 
in Numerical Analysis will have a discussion of these issues.
tom
-- 
We have discovered a therapy ( NOT a cure ) 
for the common cold.  Play tuba for an hour.
===
Subject: Re: Analysis problem, need some help here...
> hi guys, I'm kinda stuck at this problem,
 > Let f: [a,b] -> R be continuous on [a,b]. If f(x) > 0 for all x in
 [a,b], show that inf{f(x): x in [a,b]} > 0.
> Thoughts:
 > Ok, since f is continuous on [a,b], that means f is a bounded function.
 since f(x) > 0 for all x in [a,b], that means 0 is a lower bound of f.
 therefore,
 > inf{f(x): x in [a,b]}(or in short, inf f) >= 0 since inf f is the
 greatest lower bound. How then do I show that inf f is not equal to 0?
The problem with proof inquiries is that one has to guess what has been
covered before the problem at hand.
My guess: You already know that a function continuous on [a,b] must be
bounded (on [a,b]).
Since f(x) is never zero, the function g: g(x)=1/f(x) is continuous on
[a,b], hence bounded -- say g(x)<=m (and m>0, of course). This means
f(x) >= what?
===
Subject: ITS, Inc. for Sale - Reply to Skidder von Cleese - POTENTIAL 
WINDFALL
The Estate of Eric Langjahr just recently became aware of the post by
Skidder von Cleese dated 9/12-13/04, in reply to the Estate's post
dated 8/31/04.  Potential purchasers of Eric's stock should know that
the Estate STANDS BEHIND EVERY WORD of its original post.
Skidder von Cleese was not one of the names Eric ever mentioned to
anyone in his biological family.  Perhaps Mr. von Cleese is simply
HIDING behind a phony name for his non-credible post.
Certainly, Mr. von Cleese paints a radically different picture of ITS
from the picture that Eric himself painted, in great detail, to his
family - and which was confirmed by other ITS staffers.
In his post, Mr. von Cleese RENDERED LEGAL OPINIONS to prospective
purchasers of my brother's stock in ITS.  Prospective purchasers
should know that Mr. von Cleese's words may have been ILLEGAL.  Mr.
von Cleese's post suggests that he lives in Nevada - but there do not
appear to be any Nevada-licensed attorneys with names similar to his. 
Therefore, Mr. von Cleese's conduct may constitute UNLICENSED practice
of law, which is a CRIME in many states - a FELONY in at least one
that I know of.  Illegally rendered legal opinions obviously have no
value or validity to anyone.
Mr. von Cleese, holding himself out in his post as agent on behalf of
ITS and Michael Pizzolla, appears to have tried to sabotage my
brother's estate's ability to sell Eric's stock for fair value.  Mr.
von Cleese did that by actively spreading false and misleading legal,
technical and other information, at my late brother's expense.
As a result, my brother's estate is duty-bound a) to try to mitigate
the damages caused by Mr. von Cleese and b) to set the record straight
in DEFENSE against Mr. Von Cleese's ATTACKS on my brother's legal
interests.   For those reasons, the Estate will try to correct Mr. von
Cleese's misleading descriptions of the applicable law and facts.
As an actual (non-Nevada) attorney, I am not at liberty to and will
not render specific legal advice to prospective purchasers of my
brother's stock in this forum.  Nonetheless, my brother's estate will
respond with a discussion of general legal theory, which should shed
some light on why the purchaser of my brother's stock in ITS may enjoy
a POTENTIAL WINDFALL.
Naturally, anyone who wants to be able to rely on specific legal
advice tailored to his specific factual situation should ALWAYS
consult with a competent, licensed attorney in the appropriate
jurisdiction.
Mr. von Cleese's pseudo-legal analysis indicates that it relies on
partnership law and cases.  But International Thoroughbred
Superhighway, INC. (a/k/a ITS, INC.) is NOT a partnership of human
beings.  Rather, ITS, INC. is a CORPORATION, a perpetual legal entity,
completely distinct from the various human beings who may, from time
to time, own some portion of it.  As a result, partnership law does
not apply to ITS at all.  The very different law of corporations
applies.
Mr. von Cleese's post also mistakenly suggests that the POTENTIAL
WINDFALL referred to in the Estate's original post hinges solely upon
missing stock share certificates.  Actually, the Estate never used
the word missing.  Mr. von Cleese's contention that alleged stock
certificates are missing ASSUMES that alleged stock certificates
were first issued and delivered and then, later, allegedly went
missing.
But, in fact, contrary to Mr. von Cleese's post on behalf of ITS and
Michael Pizzolla, to date, the corporate secretary of ITS has produced
no proof that ITS' corporate secretary ever issued OR delivered any
alleged stock certificates to the other alleged shareholder or that
any such alleged stock certificates subsequently went missing. 
According to public records, ITS's corporate secretary was Michael
Pizzolla.
In any event, the fact that the other alleged shareholder of ITS is
not in possession of any alleged stock certificates is hardly the only
basis for the POTENTIAL WINDFALL referred to in the Estate's original
post.  (There simply was no need to go into the other bases - until
Mr. von Cleese's post on behalf of ITS and Michael Pizzolla made it
necessary.)
There is a second form of legally valid proof of ownership of stock in
a corporation.  The corporate secretary is ALSO required to make a
notation in the corporate stock ledger documenting any shareholder's
ownership.
But, in fact, to date, ITS's corporate secretary has produced no proof
that the corporate secretary of ITS made any notations on the
corporate stock ledger to reflect the other alleged shareholder's
alleged ownership in ITS.  According to public records, ITS's
corporate secretary was Michael Pizzolla.
But there are still more bases for the POTENTIAL WINDFALL referred to
in the Estate's original post.
Mr. von Cleese's post suggests that a court of equity will always
honor ANY claim of alleged ownership, despite a lack of ANY proof of
ownership whatsoever, not just lack of what lawyers call legally
cognizable proof.  Obviously, that would invite just about everyone -
even Mr. von Cleese - to claim ownership.
In reality, what a court of equity MAY elect to do differently from a
court of pure law is to consider ALTERNATIVE but nonetheless
CREDITABLE forms of PROOF - IF, of course, alternative, creditable
forms of proof are produced by the party seeking to prove its alleged
ownership.  And, yes, it is that party who bears the burden of proof,
even in equity.
Mr. von Cleese's post also suggests that equity will always honor ANY
claim of alleged ownership, without any regard to the total
circumstances in which the allegation of ownership is made.  Not so.
Equitable relief is extraordinary relief, reserved to the sound
discretion of a court.  A party who seeks to cast aside legal
requirements and invoke equity in his behalf bears the BURDEN OF
PROVING to the court that it is JUST AND APPROPRIATE for the court to
do so UNDER ALL OF THE CIRCUMSTANCES of the case.
Although Mr. von Cleese is apparently unfamiliar with it, one of the
most well-known equitable principles is the DOCTRINE OF UNCLEAN HANDS.
 Under that doctrine, a court of equity will not come to the aid of
someone who has engaged in INTENTIONAL MISCONDUCT in the context in
which he is asking for equitable relief.  Since he is such a proponent
of equity, Mr. von Cleese might want to bone up on that important
equitable doctrine.
Under all of the applicable facts and circumstances of the case, the
Estate, in good faith, believes that equitable principles in no way
detract from the POTENTIAL WINDFALL that the purchaser of my brother's
stock may enjoy.
Additionally, a court sitting in equity just may be mindful that the
other alleged shareholder is not just an Ordinary Joe.  Rather, he has
held himself out as a lawyer who knowingly (by definition) assumed
important responsibilities as ITS's corporate secretary.
If the other alleged shareholder's allegations of ownership are
believed to be true, then that same alleged shareholder may have been
grossly negligent in the performance of his paid corporate secretarial
duties.  Certainly, there is ample legal precedent for a court to hold
a lawyer to higher standards than a non-lawyer.
 
I trust that this discussion of general legal theory has shed a little
light on several of the legal bases for my brother's estate's
acknowledgment that the purchaser of my brother's stock in ITS may
enjoy a POTENTIAL WINDFALL.
This reply to Mr. von Cleese is posted on behalf of Eric's father,
Charles Langjahr, in his capacity as co-administrator of Eric's
estate.
Janet Langjahr, as agent on behalf of
Charles Langjahr, Co-Administrator,
Estate of Eric Langjahr
its_stock@yahoo.com
===
Subject: Re: Homeomorphic non-isomorphic normed vector spaces
>>My question is in the topic: does there exist any two normed vector 
spaces
>>that are homeomorphic but not isomorphic ?
>> Yes.  Amazingly enough, all separable infinite-dimensional Banach spaces 

>> are homeomorphic.  See M.I. Kadec, Function Anal. App. 1 (1967) 53-62.
>Seperable and complete, so obviously completely metrizable.  I suppose
>it follows that they're Polish spaces, hence homeomorphic to {0,1}^N,
>N^N, and so on.  But how does the infinite dimensional part fit in? 
>(I.e. -- what's stopping a seperable finite dimensional Banach space
>from being Polish?  Do they even exist?)
What in the world are you talking about?  Polish spaces are not all
homeomorphic, and  R^n isn't homeomorphic to R^m if n <> m.
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
===
Subject: Re: Homeomorphic non-isomorphic normed vector spaces
>>My question is in the topic: does there exist any two normed vector 
spaces
>>that are homeomorphic but not isomorphic ?
> Yes.  Amazingly enough, all separable infinite-dimensional Banach 
spaces 
>> are homeomorphic.  See M.I. Kadec, Function Anal. App. 1 (1967) 53-62.
>Seperable and complete, so obviously completely metrizable.  I suppose
>it follows that they're Polish spaces, hence homeomorphic to {0,1}^N,
>N^N, and so on.  But how does the infinite dimensional part fit in? 
>(I.e. -- what's stopping a seperable finite dimensional Banach space
>from being Polish?  Do they even exist?)
> What in the world are you talking about?  Polish spaces are not all
> homeomorphic, and  R^n isn't homeomorphic to R^m if n <> m.
Uh oh.  This is very bad for my thesis.  :-(
'cid 'ooh
===
Subject: Re: Homeomorphic non-isomorphic normed vector spaces
> What in the world are you talking about?  Polish spaces are not all
> homeomorphic, and  R^n isn't homeomorphic to R^m if n <> m.
> Uh oh.  This is very bad for my thesis.  :-(
 
Better to find out sooner rather than later.
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2
===
Subject: Contractive maps
Hi everybody!
Is it true that a map that has only one fixed point on a convex set is
strictly contractive?
Fabio.
===
Subject: Re: Contractive maps
>Is it true that a map that has only one fixed point on a convex set
>is strictly contractive?
No. 
-- 
Shmuel (Seymour J.) Metz, SysProg and JOAT  <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action.  I reserve the
right to publicly post or ridicule any abusive E-mail.  Reply to
domain Patriot dot net user shmuel+news to contact me.  Do not
===
Subject: Re: Contractive maps
> Is it true that a map that has only one fixed point on a convex set is
> strictly contractive?
Consider a rotation of the unit disk.
-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
===
Subject: VonNeumann Gametheory applied to StockMarket; BMY, BCE, SBC
Portfolio of PAF as of 9NOV04
BCE  19,000  22.84  $433,960.00
BLS  10  27.39  $273.90
BMY  900  23.96  $21,564.00
SBC  2,500   26.23   $65,575.00
VZ  100  40.97 $4,097.00
total share-wealth-units **22,510**
realestate land 3APR03 of 3 lots $19,000.
science-art of pictures,porcelain etc starting JAN03 for $16,561.
realestate land 30JUL03 another lot $11,500.
I usually am more nervous with alot of cash on hand than I am fully
invested. The main reason being is that stocks put money to work whereas
cash withers and erodes away over time. I got to where I am now with
stocks, not with cash.
So today I bought 12,000 shares of BCE at 22.80 and bought 800 shares
BMY at 23.90.
But in the past year, one of my favorite sectors, well, I have only
two-- telecom and drugs, I have been under pressure to look for a new
sector to replace drugs. I think that Health care is at a juncture in
history. A juncture where the Pill taking is no longer the mainstream of
Healthcare and that in the near future-- Cloning and StemCell
transplanting will overtake pilltaking. With the Merck debacle showing
us that a blockbuster pill can become a reverse and let me coin the term
Companysmasher. Other companysmashers maybe Celebrex and someday soon
see Pfizer meltdown from its $250 billion capitalization cut in half
perhaps even in 1/4 with litigation so that the current price of approx
$30. per share becomes approx $8. per share.
And we see it also in the patheticness of pills. Who in the 19th century
would have ever thought that male erection was a medical problem worthy
of drug company research. Who would have thought that cholesterol is
worthy of taking pills when the better alternative is proper diet.
So the drug companies in the 20th century have chased commercial money
and not problems of a focus on health. Drug companies should focus
research on eliminating viruses or cancer or malaria or true diseases
and ills.
I believe the future has Cloning which Stem Cells is part and parcel of.
I believe in 100 years from now, the largest Health Care company will be
a company that Clones humans and probably not a USA company looking at
the pathetic aversion to cloning from people in power in the USA. So in
100 years from now probably some Oriental company will be the largest
healthcare company in the world and people from the USA going to the
Orient to get cloned.
I need to talk about Share-Wealth-Units in detail since this portfolio
was initiated in October of 2002 because of the contrast of VonNeumann
Gametheory in playing the Stockmarket and all other investors in the
Stockmarket.
All other investors would say great portfolio that it has grown from
$461,000 in October 2002 to the above approx $576,000. To the all other
investors the above portfolio was a success over those 2 years.
To me, who plays the stockmarket via VonNeumann Gametheory the above
portfolio has done poorly for the past 2 years. Poorly because the
share-wealth -units is the best gauge of performance and that has done
poorly
for the portfolio.
This was the portfolio on inception:
portfolio 3OCT2002
with inception capital of $461,000.
10/03/02 Buy 3,000  BCE  at  17.19 for  $51,562.
10/03/02 Buy 4,000 BMY    at 22.47  for $89,866.
10/03/02 Buy 100 BLS     at 20.85 for  $2115.
10/03/02 Buy 500 DT     9.09 for $4560.
10/03/02 Buy 40,000  Q    various days average 2.57 for $102,900.
10/03/02 Buy 1,000 SBC   at 21.60 for $21630.
10/03/02 Buy 10,000 SGP  at 17.82 for  $178,220.
10/03/02 Buy 200 VZ    at  32.70 for  $6546.
10/03/02 Buy 100 WYE     at 34.42 for  $3472.
If in October of 2002 I had bought nothing but SBC with the $461,000 it
would have been 461000/21.60 it would have been 21,300 shares. And
21,300 shares of SBC today would be worth $554,000. So, really, have I
improved in those 2 years? I would say only a tiny improvement. It is
the marker of SHARE-WEALTH-UNITS that is the most important.
Example: all of those people who bought Microsoft in the 1980s and 1990s
and bought it and sold it and then later bought it back, throughout that
time period they would have made money but each time they bought MSFT
and sold it to later buy it back would have seen a gigantic shrinkage in
the number of shares of Microsoft they could buy back, even though they
made a money dollar profits.
The wealth in the StockMarket is not in the paper money valuation. The
wealth is in the shares of stock, or the ownership of a company.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies
===
Subject: Re: 142857
In sci.math, Thomas Mautsch
<mautsch@math.ethz.ch>
<slrncp12jm.ieg.mautsch@tarski.math.ethz.ch>:
> schrieb kalikinkar <kalikinkar@gmail.com>:
>> 142857 X 1 = 142857
>> 142857 X 2 = 285714
>> 142857 X 3 = 428571
>> 142857 X 4 = 571428
>> 142857 X 5 = 714285
>> 142857 X 6 = 857142
> 052631578947368421 X 1 = 052631578947368421 
> 052631578947368421 X 2 = 105263157894736842 
>  ...
Here's a question.  In 1/7 we have 6 digits.  However, in 1/3
we only have 1 digit repeating.  1/2 and 1/5 are special cases.
1/11 has 2 digits.  1/13 has 6 digits.
1/17 has 16 digits; 1/19 (above) has 18.  1/23 has 22.
1/29 has 28 digits.
Is there a correlation between the p-group (or groups) and the
number of repeating digits in 1/p?  The only thing I can think of
might be something relating to 4k+1:
1/(4*1 - 1): 1 digit
1/(4*2 - 1): 6 digits
1/(4*3 - 1): 2 digits
1/(4*3 + 1): 6 digits
1/(4*4 + 1): 16 digits
1/(4*5 - 1): 18 digits
but that's not panning out.
There are also issues when one uses other bases.  In base 12, for
instance, we have
1/5 = .2497... (4 digits)
1/7 = .186A35... (6 digits)
1/11 = .1... (1 digit)
1/13 = .0B... (2 digits)
1/17 = .08579214B36429A7... (16 digits)
1/19 = .076B45... (6 digits)
This can't be new territory but I'm curious as to where to look.
(Side issue: in Linux one can use 'bc' with 'scale=20' and 
'obase=12' to get decimal -- bidecimal? -- expansions in base 12.)
-- 
#191, ewill3@earthlink.net -- insert random curiosity here
It's still legal to go .sigless.
===
Subject: Re: 142857
ewill@sirius.athghost7038suus.net says...
> [...]
> Here's a question.  In 1/7 we have 6 digits.  However, in 1/3
> we only have 1 digit repeating.  1/2 and 1/5 are special cases.
> 1/11 has 2 digits.  1/13 has 6 digits.
> 1/17 has 16 digits; 1/19 (above) has 18.  1/23 has 22.
> 1/29 has 28 digits.
7, 17, etc. have 10 as a primitive root, whereas 3, 13, etc.
have not.
earlier discussion of this.
Also, this may be of interest:
http://www.lrz-muenchen.de/~hr/numb/period.html
Finally, also note that 142+857 = 999.
Christer Ericson
Sony Computer Entertainment, Santa Monica
===
Subject: Re: 142857
In sci.math, Christer Ericson
<christer_ericson@NOTplayTHISstationBIT.sony.com>
<MPG.1bfb69f167b407a8989795@news.verizon.net>:
> ewill@sirius.athghost7038suus.net says...
>> [...]
>> Here's a question.  In 1/7 we have 6 digits.  However, in 1/3
>> we only have 1 digit repeating.  1/2 and 1/5 are special cases.
>> 1/11 has 2 digits.  1/13 has 6 digits.
>> 1/17 has 16 digits; 1/19 (above) has 18.  1/23 has 22.
>> 1/29 has 28 digits.
> 7, 17, etc. have 10 as a primitive root, whereas 3, 13, etc.
> have not.
> earlier discussion of this.
> Also, this may be of interest:
> http://www.lrz-muenchen.de/~hr/numb/period.html
> Finally, also note that 142+857 = 999.
I have a suspicion that's true for all repeaters that
have 10 as a primitive root; the first half of
the digits run until -1, and the second half mirror
the first.  For example:
1 = 1 (mod 7)
10 = 3 (mod 7)
10^2 = 30 = 2 (mod 7)
10^3 = 20 = -1 (mod 7)
10^4 = -3 (mod 7)
10^5 = -2 (mod 7)
(For repeaters that do not have 10 as a primitive root,
this may or may not be the case.)
> Christer Ericson
> Sony Computer Entertainment, Santa Monica
-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.
===
Subject: Re: 142857
> I have a suspicion that's true for all repeaters that
> have 10 as a primitive root; the first half of
> the digits run until -1, and the second half mirror
> the first.  For example:
> 1 = 1 (mod 7)
> 10 = 3 (mod 7)
> 10^2 = 30 = 2 (mod 7)
> 10^3 = 20 = -1 (mod 7)
> 10^4 = -3 (mod 7)
> 10^5 = -2 (mod 7)
Your suspicion is correct: if p=2k+1 is an odd prime, then
(10^k-1)(10^k+1)=10^(2k)-1=10^(p-1)-1 is 0 mod p by Fermat, so by the Gauss
lemma, one of 10^k-1, 10^k+1 must be 0 mod p.
LD
===
Subject: Re: 142857
> ...
> (Side issue: in Linux one can use 'bc' with 'scale=20' and
> 'obase=12' to get decimal -- bidecimal? -- expansions in base 12.)
Not bidecimal but duodecimal I think.
===
Subject: Re: 142857
In sci.math, George Cox
>> ...
>> (Side issue: in Linux one can use 'bc' with 'scale=20' and
>> 'obase=12' to get decimal -- bidecimal? -- expansions in base 12.)
> Not bidecimal but duodecimal I think.
Correct, duh.  I, duh, think I, uh, need more coffee... :-)
-- 
#191, ewill3@earthlink.net -- that's my excuse and I'm sticking to it ...
It's still legal to go .sigless.
===
Subject: Re: 142857
> Here's a question.  In 1/7 we have 6 digits.  However, in 1/3
> we only have 1 digit repeating.  1/2 and 1/5 are special cases.
> 1/11 has 2 digits.  1/13 has 6 digits.
> 1/17 has 16 digits; 1/19 (above) has 18.  1/23 has 22.
> 1/29 has 28 digits.
> Is there a correlation between the p-group (or groups) and the
> number of repeating digits in 1/p? 
Indeed, there is. 
1/7=0.142857142857142857...=142857/999999. So for 1/p (p not 2 and not 5),
you're looking for the least k such that 10^k-1 is divisible by p, i.e.,
the order of 10 in the multiplicative group of the field F_p of p elements.
Now Lagrange tells you this order divides p-1.
LD
===
Subject: Re: FLT
> Fermat's Last Theorem 
> Ben Ito 
> 11-8-04 
> I will prove Fermat's Last Theorem. 
> l. Introduction 
> I will show that Fermat's n=4 proof is invalid and prove Fermat's
> n>2 theorem. 
> 2. Fermat's Proof (n=4) 
> Fermat is using the integer solutions of n=2 to prove that n=4 does
> not form integer solutions. Fermat uses, 
> A^4 + B^4 = C^2 (equ l). 
> and  
> A^2 = 2uv, B^2 = u^2 - v^2, and C = u^2 + v^2 (equ 2), 
> (Shanks, p.141). Fermat's n=4 proof forms integer solutions.  Example,
> using u=2 and v=l, 
> A^2 = 4, B^2=3 and C= 5 (equ 3). 
> Inserting the results of equation 3 in equation l 
> 4^2 + 3^2 = 5^2. (equ 3) 
> Fermat's n=4 proof is invalid. 
> 3. Elliptic Curve 
> The derivation of the elliptic curve is described. The elliptic curve
> equation is derived using the integer solution equations of n=2
> (Osserman, p.21), 
> a = k(m^2 - n^2),  b = k2mn,  c = k(m^2 + n^2). (equ 4)
> The elliptic curve is only valid for n=2; therefore, an elliptic curve
> can not be used to prove FLT for n>2.
> 4. Fermat's Proof. 
> A circle transformation (z = c) is used in Fermat's equation n=2,  
> x^2 + y^2 = c^2. (equ 5) 
> The integer solution equations of n=2, 
> x = 2uv, y = u^2 - v^2, and z = u^2 + v^2 (equ 6a,b,c) 
> are used in equation 5, 
> (2uv)^2 + (u^2 -v^2)^2 = u^2 + v^2 = c.(equ 7) 
> Consequently, equations 6c and 7 forms the transformation equation z=c
> which can only be derived when n=2; therefore, only n=2 forms integer
> solutions. 
> 5. Conclusion 
> Fermat's n=4 proof forms integer solutions; therefore, Fermat's n=4
> proof is invalid.  
> Fermat's elliptic curves are derived using the integer solution
> equations of n=2; therefore, an elliptic curve can not be used to
> prove FLT when n>2. 
> The n=2 equation is transformed into a circle equation then the
> integer solutions of n=2 are used to derive the transformation
> equation z=c which  can only be formed when n=2. 
> 6. References 
> Robert Osserman. Fermat's Last Theorem (a supplement to the video).
> MSRI. 1994 
> Daniel Shanks. Solved and Unsolved Problems in Number Theory. Chelsea
> Pub. 1985.
I believe it is wise to seek the opinion of Professor Ken Ribet. He is
an expert in FLT. If he says yes you can then add as many feathers as
you like on your hat
> *-----------------------*
>   www.GroupSrv.com
> *-----------------------*
===
Subject: Re: Probability: what is jointly uniform? jointly Gaussian? What 
does that mean?
>Hi all
>I have seen jointly Gaussian many times. What does that mean? What does 
>that mean intuitively?
Jointly Gaussian can be characterized in many ways.
If the random variables have a density, the logarithm 
of the density is quadratic.  In any case, every linear
combination of the random variables is Gaussian, or the
random variables are linear combinations of independent
Gaussian random variables.  These are all equivalent.
There are many other properties, such as conditional 
distributions are Gaussian, and any linear combinations
which are uncorrelated are independent.
>What's the opposite side towards jointly Gaussian? non-jointly 
Gaussian, 
>what does that mean?
One can have Gaussian random variables which do not satisfy
the above properties.
>Any jointly uniform, jointly exponential, jointly Geometric 
R.Vs?
There are no useful forms of the above to my knowledge.
Many of them do not work at all.
>I just have a very hard time understanding these things?
>Suppose I am using computer to generate R.V.s, suppose I generate two 
>uniform R.V.s in [0, 1] on my PC, and then transform them to Gaussian 
R.V.s, 
>(I can do that by put the uniform R.V. go through F_inv(x) of Gaussian, how 

>to find the expression of inverse of Gaussian Distribution?), are they 
>jointly Gaussian? 
Your PC generates pseudo-random numbers, which are not
independent.  You would need independent random variables
to do this.  However, this is what is often done in practice.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Re: Probability: what is jointly uniform? jointly Gaussian? What 
does that mean?
>Suppose I am using computer to generate R.V.s, suppose I generate two 
>uniform R.V.s in [0, 1] on my PC, and then transform them to Gaussian 
>R.V.s, 
>(I can do that by put the uniform R.V. go through F_inv(x) of Gaussian, 
>how 
>to find the expression of inverse of Gaussian Distribution?), are they 
>jointly Gaussian? 
 
> Your PC generates pseudo-random numbers, which are not
> independent.  You would need independent random variables
> to do this.  However, this is what is often done in practice.
Yes, one often uses pseudo-random numbers. But surely calculating
th inverse of the Gaussian Distribution function is not what one
does. One uses, say, the Box-Muller transformation:
  <http://mathworld.wolfram.com/Box-MullerTransformation.html>
-- 
Mostly economics:  <http://www.dreamscape.com/rvien/#PublicationsForFun>
r           c
 v         s a           Whether strength of body or of mind, or wisdom, or
  i       m   p          virtue, are found in proportion to the power or 
wealth
   e     a     e         of a man is a question fit perhaps to be discussed 

by
    n   e       .        slaves in the hearing of their masters, but highly
     @ r         c m     unbecoming to reasonable and free men in search of
      d           o      the truth.    -- Rousseau
===
Subject: Re: about Jacobian: any none-square Jacobians?
>>If I want to find Jacobian of
>>x=f(u, v)
>>y=g(u, v)
>>I can find 2x2 Jacobian matrix J_xy_vs._uv
>>But what if I have the above equations, but I want to find J_uv_vs._xy?
>>Maybe Jacobian is just the determinant, so I can inverse the J_xy_vs._uv 
to 
>>find J_uv_vs._xy?
>>When I study Jacobian, I am always wondering about non-square Jacobian?
>>What if I have
>>x=f(u, v)
>>y=g(u, v)
>>z=h(u, v)
>>or
>>x=f(u, v, w)
>>y=g(u, v, w)
>>? 
>The primary object, I'd say, is the derivative, which is a linear
>transformation whose matrix is the Jacobian matrix (the m x n
>matrix of partial derivatives, for any differentiable function from 
>R^n to R^m).  Only in the case m = n can you talk about the Jacobian
>determinant.  In this case your function is locally invertible if the
>Jacobian determinant is nonzero, and (by the chain rule) the Jacobian
>matrix of the inverse function is the inverse of the Jacobian matrix
>of the original function.  Functions from R^n to R^m with n <> m
>can't be locally invertible.
You can form Jacobian matrices, and these multiply
appropriately for successions of transformations.  
Also, if one has a smooth parametrization of a 
low-dimensional surface in a higher-dimensional
space, such as a curve or 2-surface in three-space,
the low-dimensional measure can be found by integrating
the square root of the determinant of the matrix times
its transpose.  For example, if z = f(x, y) is a smooth
surface in 3-space, the Jacobian of x, y, z with respect
to x, y is
                1       0       f_x
                0       1       f_y
and one can verify that the determinant of this
matrix times its transpose is 1 + f_x^2 + f_y^2,
which gives the usual area formula.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Re: recursion proof help
> Let x1 = , and for n > 1, let x_n = sqrt(3*x_n-1 + 1).  Prove x_n < 4 for
> all n in natural numbers.
An inductive proof looks more appropriate.
-- 
Daniel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
===
Subject: Re: recursion proof help
> Let x1 = , and for n > 1, let x_n = sqrt(3*x_n-1 + 1).  Prove x_n < 4 
for
> all n in natural numbers.
> An inductive proof looks more appropriate.
> --
> Daniel W. Johnson
> panoptes@iquest.net
> http://members.iquest.net/~panoptes/
> 039 53 36 N / 086 11 55 W
Actually, this is one of those: proving something more ambitious turns
out to be easier. (I hope that x1 is a small number, such as 0 or 1, or
maybe even 2 or 3.)
In problems like this one, look at the fixed points, that is, numbers r
such that r = sqrt(3*r+1). In your case, there is only one such r
(hint: r < 3.303). Proving that all x_n are less than r can be proved
nicely by induction (and some easy algebra).
===
Subject: Re: recursion proof help
>> Let x1 = , and for n > 1, let x_n = sqrt(3*x_n-1 + 1).  Prove x_n < 4 
for
>> all n in natural numbers.
>> An inductive proof looks more appropriate.
>> --
>> Daniel W. Johnson
>> panoptes@iquest.net
>> http://members.iquest.net/~panoptes/
>> 039 53 36 N / 086 11 55 W
>Actually, this is one of those: proving something more ambitious turns
>out to be easier. (I hope that x1 is a small number, such as 0 or 1, or
>maybe even 2 or 3.)
Not really. If x_n <= 4 then x_(n+1) = f(x_n) <= f(4) = sqrt(13) <
sqrt(16) = 4, by the monotonicity of f(x)= sqrt(3*x + 1).
So, just a 1-liner proof.
Thomas
P.S I over-complicated that in a previous post.
>In problems like this one, look at the fixed points, that is, numbers 
r
>such that r = sqrt(3*r+1). In your case, there is only one such r
>(hint: r < 3.303). Proving that all x_n are less than r can be proved
>nicely by induction (and some easy algebra).
===
Subject: A math problem I made. Try to solve it! :-)
Hey, guys. I came up with this one 2 years ago, and I think it's kind
of interesting, because at first you are not sure where the information
to solve it actually comes from! And even when you find it out, there
are increasing levels of reference and self-reference. Well, you'll
see...
visit 
www.intermagix.com/MathProblem1.htm
Tell me what you think!
-Greg Magarshak
===
Subject: Re: A math problem I made. Try to solve it! :-)
>Hey, guys. I came up with this one 2 years ago
How many times to you plan to post this?
(Hint: Once is enough.)
-- 
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
                                  http://OakRoadSystems.com/
And if you're afraid of butter, which many people are nowa-
days, (long pause) you just put in cream.     --Julia Child
===
Subject: Re: A math problem I made. Try to solve it! :-)
>Hey, guys. I came up with this one 2 years ago, and I think it's kind
>of interesting, because at first you are not sure where the information
>to solve it actually comes from! And even when you find it out, there
>are increasing levels of reference and self-reference. Well, you'll
>see...
>visit 
>www.intermagix.com/MathProblem1.htm
>Tell me what you think!
These are problems of a well-known type. What you probably heard in
class that got you thinking was a problem just like this. The books of
Raymond Smullyan contain numerous examples of similar kinds of
problems (usually around truth-tellers and liars and variations
thereof); the census-taker problem is also of a similar nature, so you
will find a lot of examples in the works of Martin Gardner.
-- 
It's not denial. I'm just very selective about
 what I accept as reality.
    --- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Terminology problem
>>Hello
>>I am scratching my head on this, don't understand what it means.
>>it is a sequence theorem
>>Given the sequence {a_n} if have a function f(x) such that f(n)=a_n
> OK, so we start with a sequence of values {a_n}. We define
> a function f(x) which takes on those values whenever x is
> a natural number. In other words, we plot a sequence of
> points (n, a_n). We plot our sequence {a_n} as point on
> an x-y graph.
> Now we're going to define a function which goes through
> those points, connecting the dots. We haven't said anything
> yet about what happens to that function between the dots,
> and we don't know anything about this sequence yet.
>>and lim(x->Inf) f(x)=L
> Now we know something more about our connect-the-dots
> function, namely that it converges to a limit as x->Inf.
> Since f(x) is defined at all x>0, this means we also
> get converging sequences of values when we take x at
> in-between values.
>>then lim(x->Inf) a_n =L
> I assume that this limit was for n->inf, not x->inf.
>>it seams like there may be an implied connection between f(x) and
> f(n)
>>that I just don't know.
> It's making a connection between two different types
> of convergence. The connection between f(x) and f(n)
> is made by assumption at the start: f(x) takes on the
> values a_n at x=n.
> Review the definitions of those two types of convergence
> (function and sequence).
>            - Randy
ñlinksî to function and sequence.
===
Subject: differenciation
Hello
thought about this for long enough, can't figur out what facts were used
if
            x^3
f(x) = --------------
        x^4 + 10000
then
         -x^2 (x^4-30000)
f'(x) = ------------------
          (x^4 + 10000)^2
===
Subject: Re: differenciation
>Hello
>thought about this for long enough, can't figur out what facts were used
>           x^3
>f(x) = --------------
>       x^4 + 10000
>then
>        -x^2 (x^4-30000)
>f'(x) = ------------------
>         (x^4 + 10000)^2
Quotient Rule: (u/v)' = (u'v - uv')/v^2
Here u = x^3,         u' = 3x^2
     v = (x^4+10000), v' = 4x^3
(u/v)' = [ 3x^2*(x^4+10000) - x^3*4x^3 ] / (x^4+10000)^2
       = [ 3x^6 + 30000x^2 - 4x^6 ] / (x^4+10000)^2
       = (-x^6+30000x^2) / (x^4+10000)^2 
       = -x^2 (x^4+30000) / (x^4+10000)^2
 
-- 
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
                                  http://OakRoadSystems.com/
And if you're afraid of butter, which many people are nowa-
days, (long pause) you just put in cream.     --Julia Child
===
Subject: Re: differenciation
> Hello
> thought about this for long enough, can't figur out what facts were used
> if
>         x^3
> f(x) = --------------
>     x^4 + 10000
> then
>      -x^2 (x^4-30000)
> f'(x) = ------------------
>       (x^4 + 10000)^2
A method I like is first to get rid of all fractions like this:
f*(x^4+10000) = x^3.
Then, I differentiate using the product rule:
f*(4*x^3) + f'*(x^4 + 10000) = 3*x^2.
I can then solve for f'.
Martin Cohen
===
Subject: Re: differenciation
If you don't like the quotient rule or don't remember it try the
product rule on the following:
f(x) = (x^3) * (x^4 + 10000)^-1
--
Casey
===
Subject: Re: differenciation
Somebody pointed out to me the following mnemonic for the quotient
rule:
D(HI/HO) = (HO*D(HI) - HI*D(HO)) / HO^2
>If you don't like the quotient rule or don't remember it try the
>product rule on the following:
>f(x) = (x^3) * (x^4 + 10000)^-1
--
Casey
===
Subject: Re: differenciation
>Hello
>thought about this for long enough, can't figur out what facts were used
>           x^3
>f(x) = --------------
>       x^4 + 10000
>then
>        -x^2 (x^4-30000)
>f'(x) = ------------------
>         (x^4 + 10000)^2
Ever hear of the quotient rule??
===
Subject: Re: differenciation
>Hello
>thought about this for long enough, can't figur out what facts were used
>           x^3
>f(x) = --------------
>       x^4 + 10000
>then
>        -x^2 (x^4-30000)
>f'(x) = ------------------
>         (x^4 + 10000)^2
It's the quotient rule and some simple algebra.
Thomas
===
Subject: Derivative of Hadamard Product of Matrices
I am working on an image processing application. I am dealing with an 
equation of the form:
I = A**O,
where I, A, and O are N x N complex matrices and ** represents a 
Hadamard product (point-by-point multiplication).
I want to evaluate the sensitivity of I to variations in O for a fixed 
A.  I thus want to find the derivative of I w.r.t. O. (dI/dO)
On the surface, this appears trivial - dI/dO = A.  I am slightly leery 
about this though - first of all, I don't know if I can apply the same 
techniques to matrices as to scalars, and second, the measurements are 
complex.
I would thus really appreciate any advice or pointers to texts, papers, 
etc.
TIA,
Hal
===
Subject: Re: Derivative of Hadamard Product of Matrices
>I am working on an image processing application. I am dealing with an 
>equation of the form:
>I = A**O,
>where I, A, and O are N x N complex matrices and ** represents a 
>Hadamard product (point-by-point multiplication).
>I want to evaluate the sensitivity of I to variations in O for a fixed 
>A.  I thus want to find the derivative of I w.r.t. O. (dI/dO)
>On the surface, this appears trivial - dI/dO = A.  I am slightly leery 
>about this though - first of all, I don't know if I can apply the same 
>techniques to matrices as to scalars, and second, the measurements are 
>complex.
>I would thus really appreciate any advice or pointers to texts, papers, 
etc.
You would be wise to use differentials.  In that case,
        dI = A**dO.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Re: Derivative of Hadamard Product of Matrices
> I am working on an image processing application. I am dealing with an
> equation of the form:
> I = A**O,
> where I, A, and O are N x N complex matrices and ** represents a
> Hadamard product (point-by-point multiplication).
> I want to evaluate the sensitivity of I to variations in O for a
fixed
> A.  I thus want to find the derivative of I w.r.t. O. (dI/dO)
> On the surface, this appears trivial - dI/dO = A.  I am slightly
leery
> about this though - first of all, I don't know if I can apply the
same
> techniques to matrices as to scalars, and second, the measurements
are
> complex.
When in doubt, do it component-wise.
First, what do you mean by dI/dO? Because I_ij depends
only on O_ij, it would be meaningful to define dI/dO
as a matrix whos (ij)-th element is dI_ij/dO_ij. But
in a general matrix expression you'd want to represent
dI_ij/dO_pq  for all i, j, p, q.
OK, but let's take the definition I assume. Then it's
trivially obvious that dI_ij/dO_ij = A_ij, and thus
dI/dO (under that definition) = A.
          - Randy
===
Subject: The Zone
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA9Kg4G20673;
I've spent a considerable amount of time building expertise in my industry.  

It took me a long time (ten years), but today I can actually say that I 
understand the mathematical premise on which my industry is based.  (For all 

intents and purposes its equations reflect my industry's truth.)  Oddly 
enough, now that I do get it, I can't seem to take my mind off of the 
underlying beauty of equations' structures and their implications.  I dare 
call them somewhat stunning.  Which brings me to my point...
I am unemployed and today an interview fell through (it was quite bad).  The 

reason:  I've been spending some time thinking about the related equations I 

referenced above - in particular, how theory meshes with practice.  (Did I 
say I find this all quite fascinating?)  Anyway, when I ponder these Nobel 
Prize winning works, I enter into a fairly deep level of concentration, I 
call it the Zone. It leaves me seeming air-headed and unfocused on the 
gritty details of day-to-day experiences.
I can only assume that you folks at math.sci have had similar experiences.  

After all, many of you probably do this for a living.  No?  My question is:  

What do you do to shake yourself out of the zone quickly?
jcm001
I know what you're thinking...This guy is just looking for someone to 
comiserate with.  Well, that's probably true too.  Ha!  (But I am interested 

in your replies.  They might come in handy next time.)
===
Subject: Re: The Zone
>What do you do to shake yourself out of the zone quickly?
I don't remember it being much of a problem, but people usually
find just doing something else of a different nature does the trick.
Some like a bit of unintellectual entertainment such as a movie.
Talking to people about other things seems to bring one out. I
usually find that after a night's sleep, I at least have the option
of not turning back to the question I was considering the night
before, but going on with something else relatively unimpeded.
Keith Ramsay
===
Subject: Re: The Zone
> I've spent a considerable amount of time building expertise in my
industry.  It took me a long time (ten years), but today I can actually
say that I understand the mathematical premise on which my industry is
based.  (For all intents and purposes its equations reflect my
industry's truth.)  Oddly enough, now that I do get it, I can't seem
to take my mind off of the underlying beauty of equations' structures
and their implications.  I dare call them somewhat stunning.  Which
brings me to my point...
> I am unemployed and today an interview fell through (it was quite
bad).  The reason:  I've been spending some time thinking about the
related equations I referenced above - in particular, how theory meshes
with practice.  (Did I say I find this all quite fascinating?)  Anyway,
when I ponder these Nobel Prize winning works, I enter into a fairly
deep level of concentration, I call it the Zone. It leaves me seeming
air-headed and unfocused on the gritty details of day-to-day
experiences.
> I can only assume that you folks at math.sci have had similar
experiences.  After all, many of you probably do this for a living.
No?  My question is:  What do you do to shake yourself out of the zone
quickly?
> jcm001
> I know what you're thinking...This guy is just looking for someone to
comiserate with.  Well, that's probably true too.  Ha!  (But I am
interested in your replies.  They might come in handy next time.)
You mean absent-minded professor syndrome? It sounds foolish, but this
phenomenon really does cause me a lot of trouble sometimes. The trouble
usually takes the form of forgetting appointments, meetings, classes,
etc. I've occassionally had problems with girlfriends and family
members saying things like, If it were really important to you, you'd
remember, or, If you really loved me. . .
I've had a few years to learn how to cope with it. Essentially, you
plan your life around it. If there's a project you are working on that
you know will put you in the zone as you say, intentionally look at
your schedule and make sure you won't miss anything. If there is
something important, some detail of life that you don't want to ignore,
devise some mechanism that will remind you with sufficient force to pay
attention to that detail. (For example, write a computer program that
reminds you of events and commit to obeying it.)
Ultimately, you have to realize that absent-minded professor syndrome
is a choice. You need to decide what kind of life you wish to live: you
can bask in the depths of concentration and intellectual revery and
miss out on the fantastic aesthetics of the rest of the world, or you
can find a balance. It's not a moral issue. Either lifestyle is ok. But
one should make a conscious, intelligent decision about how one wishes
to live.
For me, personally, I do not want to forget to eat and bathe like many
of my friends. When the joy of intellectual activity causes my personal
hygeine and relationships with other people to suffer, it's time to
make some changes in my life. It's worth giving up the great things I
would otherwise discover--the books I'd write, the theorems I'd prove,
the programs I'd build are just not as important as my health and loved
ones.
===
Subject: Re: The Zone
> I've been spending some time thinking about the related equations I
> referenced above - in particular, how theory meshes with practice.
> (Did I say I find this all quite fascinating?)  Anyway, when I
> ponder these Nobel Prize winning works, I enter into a fairly deep
> level of concentration, I call it the Zone. It leaves me seeming
> air-headed and unfocused on the gritty details of day-to-day
> experiences.
> I can only assume that you folks at math.sci have had similar
> experiences.  After all, many of you probably do this for a living.
> No?  My question is: What do you do to shake yourself out of the zone
> quickly?
Get OUT of the zone? My usual goal is to get INTO the zone. 
I've found that the gritty details of day-to-day experiences 
don't get any worse after you've ignored them for a decade or two.
dave
PS -- Your machine may not have a carriage but it probably still has
a return; use it after <= 80 characters.
===
Subject: Re: The Zone
random@control.com says...
> I've spent a considerable amount of time building expertise in my 
industry.  It took me a long time (ten years), but today I can actually say 

that I understand the mathematical premise on which my industry is based.  
(For all intents and purposes its equations reflect my industry's truth.)  
Oddly enough, now that I do get it, I can't seem to take my mind off of 
the underlying beauty of equations' structures and their implications.  I 
dare call them somewhat stunning.  
Which brings me to my point...
> I am unemployed and today an interview fell through (it was quite bad).  
The reason:  I've been spending some time thinking about the related 
equations I referenced above - in particular, how theory meshes with 
practice.  (Did I say I find this all quite fascinating?)  Anyway, when I 
ponder these Nobel Prize winning works, I enter into a fairly deep level of 

concentration, I call it the Zone. It leaves me seeming air-headed and 
unfocused on the gritty details of day-
to-day experiences.
> I can only assume that you folks at math.sci have had similar experiences. 

 After all, many of you probably do this for a living.  No?  My question is: 
 
What do you do to shake yourself out of the zone quickly?
> jcm001
> I know what you're thinking...This guy is just looking for someone to 
comiserate with.  Well, that's probably true too.  Ha!  (But I am interested 

in your replies.  They might come in handy next time.)
This sounds like autism.
===
Subject: Square root problem with primes
p and q are odd primes. n is positive integer.
When does square root n^2-pq equal to integer?
i.e.  sqrt(n^2-pq)=integer ?
excluding trivial case n^2=pq+1 as sqrt(pq+1-pq)=1.
What is the common parametric equation for n as a function of p and q?
Tapio 
===
Subject: Re: Square root problem with primes
> p and q are odd primes. n is positive integer.
> When does square root n^2-pq equal to integer?
> i.e.  sqrt(n^2-pq)=integer ?
> excluding trivial case n^2=pq+1 as sqrt(pq+1-pq)=1.
> What is the common parametric equation for n as a function of p and q?
> Tapio
If
    sqrt(n^2-pq)=integer
then letting 'integer' = x so that
    n^2 - p*q = x^2
re-arranging and factoring
   (n - x)*(n + x ) = p*q
let
   p = n - x, q = n + x
then you get n in terms of p and q
  p + q = 2n
  q - p = 2x
e.g.
(p,q) = (1,1) then n = 1, x = 0
(p,q) = (1,3) then n = 2, x = 1
(p,q) = (1,5) then n = 3, x = 2
choose p and q prime
(3,5) then n = 4, x = 1
etc
Richard Miller
===
Subject: Re: where is error?
Robert Israel a .8ecrit :
>>I want to calculate I(u)=int(exp(-u*x^2/(1-x^2)),x=-1..1)
>>I have found with Maple I(u)=sqrt(pi)*KummerU(1/2,0,u)
> (for u > 0)
>>I would like find again this result with differential equation of 
>>KummerU(a,b,z) function.
>>KummerU(a,b,z) function is solution of equation: 
>>z*w''(z)+(b-z)*w'(z)-a*w(z)=0
>>So, I write I(u), I'(u), I''(u) and I introduce these functions in 
>>equation u*I''(u)-u*I'(u)-I(u)/2=0 (1)
>>I(u)=int(exp(-u*x^2/(1-x^2)),x=-1..1)
>>I'(u)=int(exp(...)*(-x^2)/(1-x^2),x=-1..1)
>>I''(u)=int(exp(...)*(x^4)/(1-x^2)^2,x=-1..1)
>>I obtain:
>>int(exp(...)*[u*x^4/(1-x^2)^2+u*x^2/(1-x^2)-1/2],x=-1..1)
>>=>int(exp(...)*[u*x^4/(1-x^2)^2+u*x^2/(1-x^2)-1/2],x=-1..1)
>>=>int(exp(...)*[u*x^4+u*x^2*(1-x^2)-(1-x^2)^2/2]/(1-x^2)^2,x=-1..1)
>>=>int(exp(...)*[u*x^4+u*x^2-u*x^4-1/2+x^2-x^4/2]/(1-x^2)^2,x=-1..1)
>>=>int(exp(...)*[u*x^2-1/2+x^2-x^4/2]/(1-x^2)^2,x=-1..1)
> So far so good.
>>But, the expression [u*x^2-1/2+x^2-x^4/2]/(1-x^2)^2 would be equal to 0.
> No, it isn't.  
> Hint: integrate by parts.
I don't know which integration can be done
> Robert Israel                                israel@math.ubc.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel 
> University of British Columbia            Vancouver, BC, Canada
===
Subject: Re: where is error?
|> Robert Israel a .8ecrit :
|>>I want to calculate I(u)=int(exp(-u*x^2/(1-x^2)),x=-1..1)
|>>I have found with Maple I(u)=sqrt(pi)*KummerU(1/2,0,u)
|> (for u > 0)
|>>I would like find again this result with differential equation of 
|>>KummerU(a,b,z) function.
|>>KummerU(a,b,z) function is solution of equation: 
|>>z*w''(z)+(b-z)*w'(z)-a*w(z)=0
|>>So, I write I(u), I'(u), I''(u) and I introduce these functions in 
|>>equation u*I''(u)-u*I'(u)-I(u)/2=0 (1)
|>>I(u)=int(exp(-u*x^2/(1-x^2)),x=-1..1)
|>>I'(u)=int(exp(...)*(-x^2)/(1-x^2),x=-1..1)
|>>I''(u)=int(exp(...)*(x^4)/(1-x^2)^2,x=-1..1)
 
|>>I obtain:
|>>int(exp(...)*[u*x^4/(1-x^2)^2+u*x^2/(1-x^2)-1/2],x=-1..1)
 
|>>=>int(exp(...)*[u*x^4/(1-x^2)^2+u*x^2/(1-x^2)-1/2],x=-1..1)
|>>=>int(exp(...)*[u*x^4+u*x^2*(1-x^2)-(1-x^2)^2/2]/(1-x^2)^2,x=-1..1)
|>>=>int(exp(...)*[u*x^4+u*x^2-u*x^4-1/2+x^2-x^4/2]/(1-x^2)^2,x=-1..1)
|>>=>int(exp(...)*[u*x^2-1/2+x^2-x^4/2]/(1-x^2)^2,x=-1..1)
  
|> So far so good.
  
|>>But, the expression [u*x^2-1/2+x^2-x^4/2]/(1-x^2)^2 would be equal to 
0.
|> No, it isn't.  
|> Hint: integrate by parts.
 
|> I don't know which integration can be done
 
Do the original integral by parts so that you differentiate the 
exp(...) and bring down a factor including u.  This should then be
the required combination of u I'(u) and u I''(u). 
> J:= Int(exp(-u*x^2/(1-x^2)),x=-1..1);
  J1:= student[intparts](J,exp(-u*x^2/(1-x^2))) assuming u>0;
  normal(combine(u*diff(J,u$2) - u*diff(J,u) - 1/2*J1));
                             /1     
                            |       
                            |   0 dx
                            |       
                           /-1      
 
 Robert Israel                                israel@math.ubc.ca
 Department of Mathematics        http://www.math.ubc.ca/~israel 
 University of British Columbia            Vancouver, BC, Canada
===
Subject: Re: where is error?
Robert B. Israel a .8ecrit :
> |> Robert Israel a .8ecrit :
> |>>I want to calculate I(u)=int(exp(-u*x^2/(1-x^2)),x=-1..1)
> |>>I have found with Maple I(u)=sqrt(pi)*KummerU(1/2,0,u)
> |> (for u > 0)
> |>>I would like find again this result with differential equation of 
> |>>KummerU(a,b,z) function.
> |>>KummerU(a,b,z) function is solution of equation: 
> |>>z*w''(z)+(b-z)*w'(z)-a*w(z)=0
> |>>So, I write I(u), I'(u), I''(u) and I introduce these functions in 
> |>>equation u*I''(u)-u*I'(u)-I(u)/2=0 (1)
> |>>I(u)=int(exp(-u*x^2/(1-x^2)),x=-1..1)
> |>>I'(u)=int(exp(...)*(-x^2)/(1-x^2),x=-1..1)
> |>>I''(u)=int(exp(...)*(x^4)/(1-x^2)^2,x=-1..1)
> |>>I obtain:
> |>>int(exp(...)*[u*x^4/(1-x^2)^2+u*x^2/(1-x^2)-1/2],x=-1..1)
> |>>=>int(exp(...)*[u*x^4/(1-x^2)^2+u*x^2/(1-x^2)-1/2],x=-1..1)
> |>>=>int(exp(...)*[u*x^4+u*x^2*(1-x^2)-(1-x^2)^2/2]/(1-x^2)^2,x=-1..1)
> |>>=>int(exp(...)*[u*x^4+u*x^2-u*x^4-1/2+x^2-x^4/2]/(1-x^2)^2,x=-1..1)
> |>>=>int(exp(...)*[u*x^2-1/2+x^2-x^4/2]/(1-x^2)^2,x=-1..1)
>   
> |> So far so good.
>   
> |>>But, the expression [u*x^2-1/2+x^2-x^4/2]/(1-x^2)^2 would be equal to 
0.
> |> No, it isn't.  
> |> Hint: integrate by parts.
> |> I don't know which integration can be done
> Do the original integral by parts so that you differentiate the 
> exp(...) and bring down a factor including u.  This should then be
> the required combination of u I'(u) and u I''(u). 
>>J:= Int(exp(-u*x^2/(1-x^2)),x=-1..1);
>   J1:= student[intparts](J,exp(-u*x^2/(1-x^2))) assuming u>0;
>   normal(combine(u*diff(J,u$2) - u*diff(J,u) - 1/2*J1));
>                              /1     
>                             |       
>                             |   0 dx
>                             |       
>                            /-1      
>  Robert Israel                                israel@math.ubc.ca
>  Department of Mathematics        http://www.math.ubc.ca/~israel 
>  University of British Columbia            Vancouver, BC, Canada
===
Subject: Re: Concept of Dual!?
Steve O.
>> I'm just learning digital electronics, taking an introductory class.
>> Everything is pretty clear so far, except the concept of a dual of a
>> logical function has me slightly puzzled.
>> I think it's like this:  If I take any true logical equation, and
>> reverse the operators, and interchange 1s and 0s, (and, do NOT switch
>> A to A' or vice-versa), the equation I get as a result is still true,
>> and is the dual of the original -- but the new equation is NOT the
>> equivalent of the orginal.  Is that right?
>> For example:
>> A + A' = 1
>> AA' = 0
>> Or another example:
>> A + 1 = 1
>> A0 = 0
>> Or, once more:
>> A + 0 = A
>> A1 = A
>> advance for all replies.
>> Steve O.
>> Spying On The College Of Your Choice -- How to pick the college that 
is the Best Match for a high school student's needs.
>> www.SpyingOnTheCollegeOfYourChoice.com
>These are NOT (with one exception) the dual equations, although they
>are all valid.  The dual expression is gotten by also complementing
>each variable.  So the dual of A + 0 = A is A'1 = A'.  The dual of A +
>B = C is A'B' = C'.  Note that the latter equation is not a tautology,
>which all of your examples were.  Now it is a characteristic of a
>tautology that if you replace each free variable by an arbitrary
>expression, you still get a tautology.  So if you begin with A + 0 = A
>and replace A by A', you get the tautology A' + 0 = A'.  If you now
>dualize, you get A1 = A, again a tautology.  Now A + B = C is a
>contingent expression; its truth depends on those of A, B and C, but A
>+ B = C for exactly the same values as A'B' = C' and NOT those that
>make AB = C true.
>So to repeat, to dualize an expression (or an equation), exchange 0
>and 1, exchange meet and join and complement each variable.
Spying On The College Of Your Choice -- How to pick the college that is 
the Best Match for a high school student's needs.
www.SpyingOnTheCollegeOfYourChoice.com
===
Subject: Re: Zorn's Lemma
> I have a question about a problem given to me. I need to give an example 
of a partially ordered set which contains a maximal element but also has a 
nonempty chain which has no upper bound. If anyone has any info on this I 
would greatly appreciate this.
The natural numbers plus an isolated element?
===
Subject: Re: Square root problem with primes
Oooops! excluding - of course- the trivial case n^2=m^2+pq as 
sqrt((m^+2pq)-pq) =m for any m in Z+.
>p and q are odd primes. n is positive integer.
> When does square root n^2-pq equal to integer?
> i.e.  sqrt(n^2-pq)=integer ?
> excluding trivial case n^2=pq+1 as sqrt(pq+1-pq)=1.
> What is the common parametric equation for n as a function of p and q?
> Tapio
===
Subject: Re: Square root problem with primes
 >p and q are odd primes. n is positive integer.
 >
 > When does square root n^2-pq equal to integer?
 >
 > i.e.  sqrt(n^2-pq)=integer ?
 >
 > excluding trivial case n^2=pq+1 as sqrt(pq+1-pq)=1.
 > Oooops! excluding - of course- the trivial case n^2=m^2+pq as 
 > sqrt((m^+2pq)-pq) =m for any m in Z+.
What other kind of cases do you expect?  If sqrt(n^2 - pq) is integer,
then n^2 - pq = m^2, so n^2 = m^2 + pq.
-- 
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, 
+31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/
===
Subject: Re: Square root problem with primes
>p and q are odd primes. n is positive integer.
 When does square root n^2-pq equal to integer?
(cut)
> What other kind of cases do you expect?  If sqrt(n^2 - pq) is integer,
> then n^2 - pq = m^2, so n^2 = m^2 + pq.
Well, I noticed I was quite tired  - midnight :-(
Actually I was playing fun with two well known problems and I found 
information in web that those problems are separate problems. See the thread 

Another Goldbach like conjecture (May 2002).
I had idea that those unsolved problems are actually one and the same 
problem, namely:
Goldbach conjecture and conjecture Every even positive integer can be 
written as a difference of two primes.
The idea was based on the simple algebra:
2n=p+q   =>
4n^2=p^2+q^2+2pq  =>
4n^2-4pq = p^2+q^2 -2pq =>
4(n^2-pq) = (p-q)^2 =>
2 sqrt(n^2-pq) = p-q
Thus if sqrt(n^2-pq) is integer m^2, as it is seen, then those two problems 

present the one and same problem as 2 m equals - of course to -
 p-q.
QED
Tapio
> dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, 
> +31205924131
> home: bovenover 215, 1025 jn  amsterdam, nederland; 
> http://www.cwi.nl/~dik/ 
===
Subject: Re: Square root problem with primes
Correct:
.... sqrt((m^2+pq)-pq))=sqrt(m^2)=m
Sorry :-(
Tapio
> Oooops! excluding - of course- the trivial case n^2=m^2+pq as 
> sqrt((m^+2pq)-pq) =m for any m in Z+.
>>p and q are odd primes. n is positive integer.
>> When does square root n^2-pq equal to integer?
>> i.e.  sqrt(n^2-pq)=integer ?
>> excluding trivial case n^2=pq+1 as sqrt(pq+1-pq)=1.
>> What is the common parametric equation for n as a function of p and q?
>> Tapio
===
Subject: Re: Square root problem with primes
Thus, I need other non-trivial solutions.
Tapio
> Correct:
> .... sqrt((m^2+pq)-pq))=sqrt(m^2)=m
> Sorry :-(
> Tapio
>> Oooops! excluding - of course- the trivial case n^2=m^2+pq as 
>> sqrt((m^+2pq)-pq) =m for any m in Z+.
>p and q are odd primes. n is positive integer.
> When does square root n^2-pq equal to integer?
> i.e.  sqrt(n^2-pq)=integer ?
> excluding trivial case n^2=pq+1 as sqrt(pq+1-pq)=1.
> What is the common parametric equation for n as a function of p and q?
> Tapio
===
Subject: Re: Functional analysis: space of bounded varation separable?
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA9MxfI00787;
>>Is it true that the space of real valued functions on R of bounded 
>>variation is separable?
> No, it isn't.  For example, the indicator functions of intervals
> [a,b] form an uncountable family, and the total variation of the
> difference of any two of them is at least 2.
>>I need to think about your argument, because, meanwhile, I found out: 
>>BV[0,1] is the dual of C[0,1] (via the pairing induced by the 
>>Riemann-Stieltjes-integral). Since C[0,1] is a separable Banach space 
>>(Stone-Weierstrass), so is its dual BV[0,1].
>>Now, how do these two statements fit together?
>The unit sphere of BV[0,1] is separable in the weak-* 
>topology as the dual of C[0,1].
Does it hold always?
I mean,is it true that the unit sphere of the dual of a separable
Banach space is separable in the weak-*topology?
Ian
 
===
Subject: Re: Functional analysis: space of bounded varation separable?
>>The unit sphere of BV[0,1] is separable in the weak-* 
>>topology as the dual of C[0,1].
>Does it hold always?
>I mean,is it true that the unit sphere of the dual of a separable
>Banach space is separable in the weak-*topology?
Yes. Let X be a separable Banach space.  The unit ball of X* is 
compact and metrizable in the weak-* topology (with one metric being 
d(f,g) = sum_n 2^(-n) |f(x_n) - g(x_n)| where x_n is a dense
sequence in the unit ball of X), and any compact metric space
is separable.  Given any weak-* dense sequence {f_n} in the unit 
ball of X*, let h_n = f_n/||f_n|| (removing any cases where f_n = 0). 
Then it's not hard to show {h_n} is weak-* dense in the unit sphere of 
X*.
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
===
Subject: Re: Equivalent sequences
   at 02:48 PM, gerard.evrard@free.fr (GE) said:
>Let's take the sequence u as : u(n+2)=(6-u(n+1))/u(n), and u(0), u(1)
>having whatever given values.
>If you plot on a plane the points whose coordinates are (u(n),
>u(n+1)) you get a curve (closed or not), not a surface.
Huh? You get neither a curve nor a surface.
>This suggests that there exist some kind of invariant like :
>   F(u(n), u(n+1)) = constant
What gives you that idea? The obvious invariant, F(u(n),((n+1)=0,
exists regardless of how you define u. Even if you impose
nontriviality requirements, there are still multiple functions that
are invariant.
-- 
Shmuel (Seymour J.) Metz, SysProg and JOAT  <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action.  I reserve the
right to publicly post or ridicule any abusive E-mail.  Reply to
domain Patriot dot net user shmuel+news to contact me.  Do not
===
Subject: Operator norm
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA9NPgp03148;
Hello all,
If T:E-->F is a linear operator between normed spaces,
then the operator norm of T is defined to be
||T||= sup{||Tx||: ||x||=1 }
Now,let T:H-->H,where H is a Hilbert space and let (e_n) be
complete orthonormal sequence in H.
Does it follow that ||T||=sup{||Te_n||} ?
If ||x||=1 in H,is there some n such that x=e_n?
===
Subject: Re: Operator norm
> Hello all,
> If T:E-->F is a linear operator between normed spaces,
> then the operator norm of T is defined to be
> ||T||= sup{||Tx||: ||x||=1 }
> Now,let T:H-->H,where H is a Hilbert space and let (e_n) be
> complete orthonormal sequence in H.
> Does it follow that ||T||=sup{||Te_n||} ?
> If ||x||=1 in H,is there some n such that x=e_n?
I think you can come up with counter-examples on R^2.  I'm not completely 
sure what you mean by complete orthogonal sequence--I'm assuming you 
mean 
a sequence whose range is a maximal orthonormal set.
-Bill97 
===
Subject: Re: Cauchy's Theorem
>I may have this wrong, but if I associate the components of the 
gradient
>  of
>a smooth scalar field with the real and imaginary parts of a complex
>function C(x,y) then isn't C necessarily analytic?
> No, it isn't.  You need the Cauchy-Riemann equations: if C = u + i v
> then du/dx = dv/dy and du/dy = - dv/dx.  Now if u = dS/dx and v = dS/dy
> you actually have du/dy = dv/dx, not - dv/dx, so you should take the
> complex conjugate of the gradient rather than the gradient itself:
> u = dS/dx and v = - dS/dy.  But even then, you'll only have one of
> the Cauchy-Riemann equations and not the other one:  du/dx = dv/dy
> says d^2 S/dx^2 + d^2 S/dy^2 = 0, i.e. S must be a harmonic function
> in order for this to work.
> For just an arbitrary smooth function S(x,y), you could easily have
> the gradient of S be 0 on your path but S not constant inside, e.g.
> think of a bump on an otherwise flat plane.
 
> Suppose I now consider the vector field which consists of the conjugate 
of
> grad(S(x,y)) - i.e. the contour field. As you point out, the complex
> function derived from this field is not necessarily analytic - I can see
> immediately that f(z) and hence df/dz would be ill-defined at local 
minima
> and maxima of S(x,y) for example. But is the condition that S be harmonic
> over-restrictive - i.e. all harmonic's correspond to analytic but not
> vice-versa?  In particular, isn't it a sufficient condition that df/dz be
> uniquely defined within the domain of interest - this is after all the
> source of the C-R equations. Isn't this the case for the contours of a
> smooth real-analytic* function S(x,y) in a domain with the minima and 
maxima
> of S(x,y) excluded?
No.  For example, S(x,y) = x^2 is as nice a function as you can get,
but the conjugate of grad(S(x,y)) is 2x = 2 Re(z) which is not an analytic
function of z in any domain.
> Alternatively, if S(x,y) is restricted to those functions which can be
> expressed as a linear combination of harmonic functions then how 
restrictive
> is that? What sort of (interesting) real analytical functions would be
> excluded by such a restriction?
A linear combination of harmonic functions is harmonic.  It is very 
restrictive.  For example, there are only two linearly independent 
harmonic functions in x and y that are homogeneous polynomials of 
degree n (where n is a positive integer), namely Re((x+iy)^n) and 
Im((x+iy)^n).
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
===
Subject: Re: Cauchy's Theorem
>In particular, isn't it a sufficient condition that df/dz be
> uniquely defined within the domain of interest - this is after all the
> source of the C-R equations. Isn't this the case for the contours of a
> smooth real-analytic* function S(x,y) in a domain with the minima and
maxima
> of S(x,y) excluded?
> No.  For example, S(x,y) = x^2 is as nice a function as you can get,
> but the conjugate of grad(S(x,y)) is 2x = 2 Re(z) which is not an 
analytic
> function of z in any domain.
It seems that my mental model of complex numbers is in serious need of
revision. I had always imagined complex numbers as a form of 2D co-ordinate
with well defined rules for division and multiplication.
It seemed natural to extend this to equating complex functions of z to
vector functions of x and y. If the components of such vector fields had
well defined and continuous derivatives then it seemed intuitive that the
df/dz for the corresponding complex function would be well defined and
continuous. I can see now that this is not a valid extension of the analogy
between vector fields and complex functions. In particular there is no
obvious analogy of df/dz for a vector field.
The thing that interested me was the notion of analytic continuity - the
idea that a function's behaviour over a small sub-domain defines its
behaviour over the entire domain. As far as I can see, complex continuity 
is
a completion of the idea of real-continuity rather than an extension of
real-continuity to 2 dimensions. Complex numbers are more (and less) than 
2D
co-ordinates or vectors.
As you initially said then, there is no clear application of the C-R 
Theorem
to extrapolating a 2D vector field from a sub-region to an arbitrary point.
Finding this out has however been interesting and enlightening.
But does the notion of analytic continuity for a function of 2 independent
variables have any meaning and if so what branch of mathematics would
concern itself with that?
Fred
===
Subject: Re: ? time-avg ODE soln
> The after uavg is a solution of the before differential 
equation.
> It doesn't have the same initial condition, though.  I didn't mean
> to imply that it did.
>  Ok, but how do I know the formula for the initial value of the
> time-averaged functon?
> So far I need to explicitly derrive solutions from both approach and then
> recognize
> the appropriate initial time-averaged function. This is quite time
> consuming, is there
> faster way to do that?
Well, one thing you can do is this.  If u_a(0) = int_0^T u(s) ds/T
where u is a solution of u'(t) = a u(t) + q(t) (with a <> 0), then 
by integrating that equation from 0 to T we get 
u_a(0) = (u(T) - u(0) - int_0^T q(s) ds)/(a T)
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
===
Subject: Integrate 1/sqrt(a + 1/x)dx
How do I integrate 1/sqrt(a + 1/x)dx
Thanx.
===
Subject: Re: Integrate 1/sqrt(a + 1/x)dx
> How do I integrate 1/sqrt(a + 1/x)dx
> Thanx.
I would multiply num and den by x to get
x/sqrt(ax^2+x) dx
  and then, being lazy, look it up.
I find that, if R = (a+bx+cx^2) (in Jeffrey's Handbook of Mathematical 
Formulas and Integrals, 4.3.4,
int(x dx/R^(1/2)) = R^(1/2)/c - (b/2c)int(dx/R^(1/2))
and
int(dx/R^(1/2)) = (3 different cases depending on the sign of c and 
4ac-b^2).
===
Subject: Re: Integrate 1/sqrt(a + 1/x)dx
> How do I integrate 1/sqrt(a + 1/x)dx
Very carefully ;-)
===
Subject: Re: Integrate 1/sqrt(a + 1/x)dx
> How do I integrate 1/sqrt(a + 1/x)dx
> Thanx.
It's messier than you may think. The answers are different when a>0 and
when a<0 (and the case a=0 is obvious).
Start with substituting sqrt(a+1/x)=t, and correctly replace dx.
Checking hint: you will have to integrate -1/(a*(t^2-a)) dx.
What is the result when a>0 and when a<0?
===
Subject: Re: Integrate 1/sqrt(a + 1/x)dx
 <Pine.WNT.4.58.0411092027190.-384297@satori.mcmaster.ca>
(A typo corrected...)
> How do I integrate 1/sqrt(a + 1/x)dx
> Thanx.
> It's messier than you may think. The answers are different when a>0 and
> when a<0 (and the case a=0 is obvious).
> Start with substituting sqrt(a+1/x)=t, and correctly replace dx.
> Checking hint: you will have to integrate -1/(a*(t^2-a)) dx.
Correction: -1/(a*(t^2-a)) dt.
> What is the result when a>0 and when a<0?
===
Subject: Re: Curves invariant under a rational map (Was: Re: Equivalent 
sequences)
>Very interesting, although no prior exposure to this topic... could
>not follow much even after reading several times,
Sorry. I have often noted that we have been 'separated by a common tongue'.
>When T(x,y) -(y,z(x,y)) is given,
>by what procedures are the solution and invariants obtained ? 
I don't know, in general. This is really starting to irritate me. 
Given a _finite_ group  G  acting on the field  R(x,y)  I have a 
pretty good sense of how to construct the subfield fixed by  G . But 
I can't see how to do it when  G  is simply an infinite cyclic group.
>How is invariant  ( xy(x+y) + (x^2+y^2)+ 7 (x+y)- 6)/(x y) pulled out of
>T(x,y)=(y,(6-x)/y) ? Is it just judicious trial and error?
Yes. Start with any  p = (u,v), compute a few dozen points  T^(n) (p),
substitute each one into the polynomial function  P(x,y) = sum a_ij x^i 
y^j
( 0 <= i,j <= 3, say), set each equal to zero, and then solve for
the  a_ij. Substitute in to  P  to get an actual polynomial which vanishes
on this orbit. Repeat with a different starting point  p.  Notice the
form of the polynomial  P  in each case. Guess the generic such polynomial
P  and then compute that   P o T   is a multiple of  P, so that the 
curves  P = 0  are all  T-invariant. Rearrange terms in  P  so that the
equation  P = 0  is equivalent to  F = constant,  where  F  does not
vary with  (u,v). 
other way: when  F  is given, the recursion may be computed.
> and how is  x^2 + x y -y^2 pulled out of T(x,y)=(y,y+x)?  
That one's different because  T  is linear. Diagonalize so that in new
coordinates  T  is  T(u,v) = ( a u, b v ). Then  |u|^(log(|b|)/log(|a|)) 
/|v|
is invariant. Now change back to original coordinates  (x,y).
>If this is really so, do we have to solve some PDEs ? If so, how then
>are the PDEs set up in terms of partial derivatives p,q,x,y,t etc.?
Now I am the one confused.
dave
===
Subject: Re: Curves invariant under a rational map (Was: Re: Equivalent 
sequences)
>If this is really so, do we have to solve some PDEs ? If so, how then
>are the PDEs set up in terms of partial derivatives p,q,x,y,t etc.?
> Now I am the one confused.
OP was suggesting a 2D parametric curve it seemed to me, and I am
looking at x(t),y(t) parameterizations of the invariant.
Wondering about the constant +1,-1 in discrete Fibonacci case as
special case of continuous variable flat singular solution.
I meant to say OP was asking in terms of a 2D parametric curve in  x
and y and it is  x^2 + x y -y^2  for Fibonacci T(x,y)=(y,y+x) case,
where solution is Fibonacci series, nay, Fibonacci continuous
function, Fibonacci(t).
Two additional cases as 2D parameterizations are given to illustrate 
an envelope between y=+1 and y=- 1 lines, that can be confirmed also 
using C discriminant: (x,y)= (t, -Cos[pi t]), (Cos[t] + 2
t,Sin[t]),(3t -8 Cos[t],Sin[t]).
The  first one is same as x^2+xy-y^2 after plugging in continuous
functions Fibonacci(t),Fibonacci(t+1) for x and y.
After all the hunting is over, to check if a proposed solution &
invariant combination is valid, its singular solution should be a flat
constant, in terms of partial derivatives and C discriminant.
That is about the best of tongue.
===
Subject: Einstein Revolutionary Theater Lazy Dog Society
See below on Minkowski metric discussion with Alex & Paul
To be adapted for the stage:
You are like some Kafka bureaucrat who is fixated on a literal 
interpretation of The Bible [JS]
Actually I'm not fixated at all.
You are sure fooling me. ;-)
I am simply pointing out that there is a perfectly consistent
alternative physical interpretation of GR, in addition to the 
traditional Einsteinian model.
You are wrong on that. In fact, you are not even wrong on that. This is 
your Quixotic Delusion. That's how any editor of a mainstream journal in 
relativity will view it. I guarantee it. Try it. Prove me wrong. 
Surprise me.
This means we can get all the essential physical content of GR, and the 
same formalism,
without an Einstein equivalence principle and without Einsteinian 
general relativity.
My psychoceramic meter just went off scale past the redline and exploded 
psychokinetically! I must say however that this is an improvement on Hal 
Puthoff's PV without PV sold in the Eric Davis USAF teleportation 
report. You at least admit you violate the equivalence principle (WEP 
not SEP is understood). Hal thinks he does not violate it even though 
consistent with GCT, nor does it agree with the pulsar 1913 + 16 data. 
In contrast, Einstein's GR with WEP & GCT, agrees with that data to one 
part in 10^14 (100 trillion!).
Also, part of what I am doing is exposing what I consider to be the 
*metaphorical character* of
some of Einstein's most fundamental ideas in relativity physics. So the 
shoe is really on the other foot.
All of theoretical physics has a metaphorical character. So what do 
you mean exactly?
Jack seems to think that anyone who argues critically against the 
canonical GR texts such as Misner,
Thorne, and Wheeler, or against any of Einstein's ideas regarding 
special and general relativity,
is somehow *ipso facto* a crackpot or an Einstein-basher with dark 
ulterior motives
and a low IQ. Unquestioning obedience to The Master has somehow become a 
Procrustean benchmark
for all right thinking in physics.
Psychoceramics warning: ... There are no inconsistencies in special 
relativity. Lorentzian Wormholes, p.204
Matt Visser.
But all I have really been saying is that we had been led to believe 
that the formalism of GR only
supports the classic Einsteinian model, based on the equivalence 
hypothesis, but this turns out not
to be the case -- because the LC connection of 1916 GR is linearly 
decomposable into a tensor and
a non-tensor.
There you go again with nonsense.
Theorem: Using only the methods of Einstein GR 1916, i.e. not allowing 
the additional adhoc excess baggage of Alex Poltorak's second affine 
connection, essentially a new theory beyond 1916 GR, there is no way to 
write
{L-C} = GCT tensor + non-GCT tensor.
BTW, the Minkowski metric nuv is not a GCT tensor. It is only a Poincare 
group tensor.
guv = nuv + huv
in the sense of spin 2 quantum field theory, huv and guv are NOT GCT 
tensors in perturbation theory out to a finite number of terms.
Feynman only gets GCT Einstein GR in a Poincare group vacuum instability 
like ODLRO BCS theory with a non-perturbative sum of an infinity of 
restricted Feynman diagrams. This is like my pre-inflationary Dirac Sea 
--> Higgs Ocean with a collapse of vacuum phase space volume explaining 
the Arrow of Time in the sense posed by Roger Penrose in Fashion, 
Faith and Fantasy. Zielinski's program here is not at all fashionable. 
It is pure fantasy.
I have not even asserted at this point that the alternative 
interpretation I am proposing is the uniquely
correct model -- just that it is available, is internally coherent, and 
is fully compatible with the GR
formalism.
I deny that it is internally coherent.
Neither have I said that the mere existence of this alternative model in 
and of itself necessarily invalidates
the Einsteinian approach and its equivalence hypothesis. However, 
there are additional arguments that
have been advanced in the literature on this subject that do raise 
serious questions about the original
heuristic motivation for GR and how seriously it is to be taken at this 
point.
Extraordinary claims require extraordinary proof.
If it is true that there is a viable alternative model for physically 
interpreting the formalism of GR
in its current stage of development, then I am simply suggesting that it 
might be interesting to
explore the ramifications of this alternative; although as a matter of 
fact these ramifications have
already been explored by many well-qualified authors in thoroughly 
competent papers published
in leading peer-reviewed journals such as Il Nuovo Cimento, the AJP, and 
Physical Review.
So judging by your reactions I think it is you who is thoroughly fixated 
and borderline irrational.
You seem to be desperately circling the wagons.
Z.
actually write - obscure though it is most of the time.
Minkowski Metric & Curvilinear Coordinates
People are sloppy. You can get away with using inconsistent units 
virtually in intermediate steps as long as one checks the final real 
equations dealing with the actual physics for dimensional consistency.
Example http://arXiv/gr-qc/9712019
pp 60-61:
However, since the metric tensor and the {L-C} connection in 1916 GR 
have direct physical meaning, I insist that a consistent choice be made 
at each step of the way. This means that guv must be dimensionless and 
{L-C} has dimension (length)^-1 (or (time)^-1). Curvature then has 
dimension (Area)^-1 (or (time)^-2).
Example 1. Plane polar coordinates on a truly FLAT plane.
You see in Carroll's discussion his guv is dimensionally inconsistent as 
are his Christoffel symbols. You have more freedom in pure math than in 
using the math in a physical model. Carroll's eq (3.22) is in fact
dl^2 = (drer + rdthetaetheta)^2 = grr(der)^2 + gthetatheta(detheta)^2
Where er and etheta are the radial and tangential orthonormal basis 
vectors in plane polar coordinates.
der = drer is an infinitesimal radial vector
detheta = rdthetaetheta is an infinitesimal tangential vector
The gij are mutually consistent dimensionless numbers!
Indeed
gij = Kronecker delta, i,j = r,theta
gij,k = 0
All Christoffel symbols = 0
Now if you used an arbitrary basis of dimensionless e1, e2 unit vectors 
that are not orthogonal, but still linearly independent
dl^2 = (dx1e1 + dx2e2)^2 = dx1^2 + dx2^2 + 2e1.e2dx1dx2
g11 = 1
g22 = 1
g21 = g12 = e1.e2 = cos(phi)
These are still linear transformations.
Let's start with
dl^2 = dx^2 + dy^2  = nijdx^idx^j
A flat metric on the plane.
Make a GCT
x -> x'(x,y)
y -> y'(x,y)
x,y, x',y' all have same physical dimensions i.e. length.
Define, the Jacobian matrix of this GCT as the dimensionless 2x2 matrix
X^x'x X^x'y
X^y'x X^y'y
where
X^x'x = x',x
X^x'y = x',y
X^y'x = y',x
X^y'y = y',y
, means partial derivative
Also there is its inverse matrix since det(Jacobian) =/= 0 for a proper 
GCT.
nij = Kronecker delta
i.e.
nij = 1 if i = j
nij = 0 if i =/= j
nij is a GCT metric tensor, therefore
ni'j' = Xi'^iXj'^jnij = Xi'^iXj'^inii = X^i'1X^j'1 + X^i'2Xj'2
So that in general the FLAT metric tensor has off-diagonal terms in 
curvilinear coordinates.
This generalizes obviously to N-dim space.
However, in orthogonal transformations
X^i'1X^j'1 + X^i'2Xj'2 = Kronecker Delta i.e. = 1 if i'=j' = 0 if i' =/= j'
Example,
x' = xcos@ + ysin@
y' = -xsin@ + ycos@
X^x'x = cos@
X^x'y = sin@
X^y'x = -sin@
X^y'y = cos@
X^x'xXx'x + X^x'yX^x'y = cos^2@ + sin^2@ = 1
the off diagonals are
cos@sin@ - sin@cos@ = 0
Example 2 GCT in 4D GR
Let's start with a curved space-time metric GCT tensor guv
I make the split
guv = nuv + huv
where nuv is DEFINED as the CONSTANT Minkowski metric MATRIX, i.e. 
-1,1,1,1 along diagonal , 0's off-diagonal
NOTE that nuv and huv need NOT be GCT tensors separately, ONLY THEIR SUM 
NEED BE!
Note that all the nuv, huv and guv are pure numbers no physical dimensions
The GCT scalar invariant is
ds^2 = guvdx^udx^v
The dx^u all have the same physical dimensions UNLIKE some of the purely 
formal math discussions you cite.
Also this is not perturbation theory. In no sense is huv << nuv.
Next make a multi-linear GCT tensor transformation X (note that X is 
dimensionless):
guv -> gu'v' = Xu'^uXv'^vguv = Xu'^uXv'^v(nuv + huv) = Xu'^uXv'^vnuv + 
Xu'^uXv'huv
OK define THE OBVIOUS GCT TENSORS
n'u'v' = Xu'^uXv'^vnuv
h'u'v' = Xu'^uXv'^vhuv
Therefore,
guv -> gu'v' = Xu'^uXv'^vguv = n'u'v' + h'u'v'
But the game is not over. I now define
gu'v' = nu'v' + hu'v'
Where AGAIN like originally
nu'v' is DEFINED as the CONSTANT Minkowski metric MATRIX, i.e. -1,1,1,1 
along diagonal , 0's off-diagonal.
Therefore
hu'v' = gu'v' - nu'v'
This is a well-defined algorithm associated with each GCT. It's a kind 
of additional compensating gauge transformation associated with the GCT.
Again note that
n'u'v'(GCT tensor) =/= nu'v'(non-GCT tensor)
h'u'v'(GCT tensor) =/= hu'v'(non-GCT tensor)
Hey, if Alex can do his shenanigans with a SECOND AFFINE CONNECTION 
completely different from the FIRST PLAIN VANILLA {L-C} connection, I 
can certainly do this.
Now in my MACRO-QUANTUM theory of EMERGENT EINSTEIN 1915 PLAIN VANILLA GR
hu'v'(non-GCT tensor) = Strain Tensor of the World Distortion Field
i.e.
huv = Lp^2(Higgs Ocean Goldstone Phase)(,u,v)  note it is dimensionless
(  ) = symmetrizer
The dimensionless ANHOLONOMIC FIELD is
Suv = Lp^2(Higgs Ocean Goldstone Phase)[,u,v]
[  ] is anti-symmetrizer
We can use {L-C} from guv to make L-C GCT covariant derivatives and get 
a torsion tensor with same dimensions as the {L-C} connection field itself.
The GCT X^u'u come from local phase transformations on the Goldstone 
phase field of the post-inflationary Higgs Ocean.
===
Subject: Re: Einstein Revolutionary Theater Lazy Dog Society
[snip good stuff]
> NOTE that nuv and huv need NOT be GCT tensors separately, ONLY THEIR SUM 
> NEED BE!
 
> Note that all the nuv, huv and guv are pure numbers no physical 
dimensions
> The GCT scalar invariant is
> ds^2 = guvdx^udx^v
> The dx^u all have the same physical dimensions UNLIKE some of the purely 
> formal math discussions you cite.
Houston, we have a problem!
   ds^2 = guv dx^u dx^v
should have been written 
   ds^2 = g_uv dx^u dx^v
Also the balance of dimensions requires,
   ds^2   =   g_uv      dx^u dx^v
 (scalar)   (dx^-2)      (dx^2)     (dx == contravariant projection).
because g_uv =   vectors e>_u.e>_v    = real lengths.
                 (scalar product)   
In my rocker ship, the determinant of g_uv is not
invariant, but rather a relative tensor g =|g_uv|. 
  Einstein simplified GR presentation in 1916 by setting
g = -1, implying invariance, but he didn't mean that as
any law, just to get GR off the ground simply.
Since g is a relative tensor then it has units.
[snip interesting stuff]
Ken S. Tucker
===
Subject: Re: Einstein Revolutionary Theater Lazy Dog Society
You have made several serious mistakes in your math.
Several of your assumptions have been previously been proven wrong as well.
Please review and correct them.
===
Subject: matrix operation
For two matrix A and B; I can understand that
SUM[k=0.. INFINITE] (A*B)^k = (I-AB)^(-1)
where SUM[k=0.. INFINITE] represents the summation from zero to positive
infinite, k is integer.
Then, it seems that there is no closed-form result for the following
expression.
SUM[k=0.. INFINITE] (A^k) *(B^k)
Any comments or any techniques to develop a closed-form final results
for your attention.
-- 
Yan ZHANG
http://www.ntu.edu.sg/home5/pg01308021
===
Subject: Re: matrix operation
> For two matrix A and B; I can understand that
> SUM[k=0.. INFINITE] (A*B)^k = (I-AB)^(-1)
> where SUM[k=0.. INFINITE] represents the summation from zero to positive
> infinite, k is integer.
> Then, it seems that there is no closed-form result for the following
> expression.
> SUM[k=0.. INFINITE] (A^k) *(B^k)
> Any comments or any techniques to develop a closed-form final results
> for your attention.
Consider the transformation 
that maps an nxn matrix X to the product AXB. 
Call this linear transformation on the space of nxn matrices D. 
If the spectral radius of D is smaller than 1, then 
(according to your first formula, with D instead of AB) 
        SUM[k=0.. INFINITE] (A^k) *(B^k) = (Id-D)^(-1) (I), 
i.e. you have to take the inverse of (Id-D) 
and to apply it to the nxn identity matrix I. 
*If* A and B can be diagonalized, 
        A = T_A * D_A * T_A^(-1)
        B = T_B * D_B * T_B^(-1), 
then 
        D(X) = T_A * D_A * [ T_A^(-1) * X * T_B ] * D_B * T_B^(-1), 
which means that D is equivalent to the transformation 
        X |--> D_A * X * D_B. 
The eigenvectors of this latter transformation are just 
the nxn matrices with all entries zero but one, 
and the corresponding eigenvalues are products of entries from 
the diagonal matrices D_A and D_B. 
Start with   T_A^(-1) * I * T_B = T_A^(-1) * T_B, 
divide the entry in the i-th row and j-th column by ( 1-D_A(i)*D_B(j) ) 
for all 1<= i,j <=n respectively, 
then multiply the resulting matrix by T_A from the left, 
and by T_B^(-1) from the right. 
That should be all. 
Example: 
        A := <<1,1>|<2,1>> . <<1,0>|<0,1/2>>   . <<1,1>|<2,1>>^(-1); 
        B :=                 <<1/3,0>|<0,1/2>>                     ; 
        T_A^(-1) . T_B = <<-1,1>|<2,-1>>; 
        <<-1/(1-1*1/3),1/(1-1/2*1/3)>|<2/(1-1*1/2),-1/(1-1/2*1/2)>> 
        = <<-3/2,6/5>|<4,-4/3>>; 
        
        sum(MatrixPower(A,k).MatrixPower(B,k),k=0..infinity)
        = T_A . <<-3/2,6/5>|<4,-4/3>> . T_B^(-1) 
        = <<9/10,-3/10>|<4/3,8/3>>; 
===
Subject: Re: matrix operation
>> For two matrix A and B; I can understand that
>> SUM[k=0.. INFINITE] (A*B)^k = (I-AB)^(-1)
>> where SUM[k=0.. INFINITE] represents the summation from zero to positive
>> infinite, k is integer.
>> Then, it seems that there is no closed-form result for the following
>> expression.
>> SUM[k=0.. INFINITE] (A^k) *(B^k)
> Consider the transformation 
> that maps an nxn matrix X to the product AXB. 
> Call this linear transformation on the space of nxn matrices D. 
> If the spectral radius of D is smaller than 1, then 
> (according to your first formula, with D instead of AB) 
>       SUM[k=0.. INFINITE] (A^k) *(B^k) = (Id-D)^(-1) (I), 
> i.e. you have to take the inverse of (Id-D) 
> and to apply it to the nxn identity matrix I. 
> *If* A and B can be diagonalized, 
  ^^^^
>       A = T_A * D_A * T_A^(-1)
>       B = T_B * D_B * T_B^(-1), 
> then 
>       D(X) = T_A * D_A * [ T_A^(-1) * X * T_B ] * D_B * T_B^(-1), 
> which means that D is equivalent to the transformation 
>       X |--> D_A * X * D_B. 
> The eigenvectors of this latter transformation are just 
> the nxn matrices with all entries zero but one, 
> and the corresponding eigenvalues are products of entries from 
> the diagonal matrices D_A and D_B. 
> Start with   T_A^(-1) * I * T_B = T_A^(-1) * T_B, 
> divide the entry in the i-th row and j-th column by ( 1-D_A(i)*D_B(j) ) 
> for all 1<= i,j <=n respectively, 
> then multiply the resulting matrix by T_A from the left, 
> and by T_B^(-1) from the right. 
> That should be all. 
I have been thinking a little bit more about this: 
Is there a direct way to describe the matrix 
        (Id-D)^(-1) (I) 
with 
        D: X |--> A * X * B 
in terms of the matrices A and B? 
The comments above say that when A and B are diagonalizable, 
we can w.l.o.g. restrict ourselves to the case that A and B 
are themselves diagonal matrices, and then the eigenvalue decomposition 
of the transformation D (and thus (Id-D)^(-1)) can be easily computed 
in terms of A and B. 
However, it seems to me that in case 
that A or B are not diagonal, but only in Jordan normal form, 
the eigenvalue decomposition of the transformation D is not that easily 
to be achieved. 
(Somehow this seems to be related to the fact that 
the diagonal matrices form an subalgebra of all the algebra of all 
matrices, while the subalgebra generated by all Jordan normal forms 
consists of *all* upper triangular matrices.) 
Does someone see a way to simplify the transformation (Id-D)^(-1) 
if A and B are given Jordan normal forms? 
What if B is assumed to be diagonalizable? 
===
Subject: Re: matrix operation
[ ... ]
> Then, it seems that there is no closed-form result for the following
> expression.
> SUM[k=0.. INFINITE] (A^k) *(B^k)
>> Consider the transformation 
>> that maps an nxn matrix X to the product AXB. 
>> Call this linear transformation on the space of nxn matrices D. 
>> If the spectral radius of D is smaller than 1, then 
>>      SUM[k=0.. INFINITE] (A^k) *(B^k) = (Id-D)^(-1) (I), 
>> i.e. you have to take the inverse of (Id-D) 
>> and to apply it to the nxn identity matrix I. 
>> *If* A and B can be diagonalized, 
>   ^^^^
>>      A = T_A * D_A * T_A^(-1)
>>      B = T_B * D_B * T_B^(-1), 
>> then 
>>      D(X) = T_A * D_A * [ T_A^(-1) * X * T_B ] * D_B * T_B^(-1), 
>> which means that D is equivalent to the transformation 
>>      X |--> D_A * X * D_B. 
>> The eigenvectors of this latter transformation are just 
>> the nxn matrices with all entries zero but one, 
>> and the corresponding eigenvalues are products of entries from 
>> the diagonal matrices D_A and D_B. 
>> Start with   T_A^(-1) * I * T_B = T_A^(-1) * T_B, 
>> divide the entry in the i-th row and j-th column by ( 1-D_A(i)*D_B(j) ) 
>> for all 1<= i,j <=n respectively, 
>> then multiply the resulting matrix by T_A from the left, 
>> and by T_B^(-1) from the right. 
> I have been thinking a little bit more about this: 
> Is there a direct way to describe the matrix 
>       (Id-D)^(-1) (I) 
> with 
>       D: X |--> A * X * B 
> in terms of the matrices A and B? 
> The comments above say that when A and B are diagonalizable, 
> we can w.l.o.g. restrict ourselves to the case that A and B 
> are themselves diagonal matrices, and then the eigenvalue decomposition 
> of the transformation D (and thus (Id-D)^(-1)) can be easily computed 
> in terms of A and B. 
I found that the answer is not much more difficult 
in case that A and B are not diagonalizable: 
Let w.l.o.g. A and B be in Jordan normal form. 
Then every Jordan block of A only transforms 
the corresponding rows of the matrix X amongst themselves, 
and every Jordan block of B does the same the corresponding columns of X. 
Thus it is sufficient to consider the transformation 
        X |--> A * X * B 
for the case that X is an (mxn)-matrix, 
A an (mxm) Jordan block with eigenvalue a, and 
B an (nxn) Jordan block with eigenvalue b. 
The diagonal part of A acts on X by multiplication by a; 
die off-diagonal with entries 1 acts by 
pushing the matrix X up, i.e. inserting a zero row as last row 
and deleting the first row instead. One can write this as 
        A * X = a X + X^ 
Similarly 
        X * B = b X + X> (X pushed to the right; 
                          the operations ^ and > commute), 
and 
        (Id-D)(X) = X - A * X * B = ab X + a X> + b X^ + X^>. 
compute the inverse of (Id-D), and apply it to the identity matrix. 
One can e.g. use the von Neumann series for (Id-D)^(-1) 
and the fact that the operations ^ and > are nil-potent. 
Does anyone see a nice way to turn this into a formula? 
The series  SUM[k=0.. INFINITE] (A^k) *(B^k)  converges 
iff all products of pairs of eigenvalues of A and B 
have absolute values smaller than 1. 
BTW: Similarly one can compute the sum 
        SUM[k=0.. INFINITE] (A^k)*(B^k)/k! = exp(D) (I) 
I would be interested to learn, 
whether these sums really occur in everyday mathematics. 
===
Subject: Re: matrix operation
> For two matrix A and B; I can understand that
> SUM[k=0.. INFINITE] (A*B)^k = (I-AB)^(-1)
> where SUM[k=0.. INFINITE] represents the summation from zero to positive
> infinite, k is integer.
I don't have an answer for your further question. But the above is
not true for all matrices. For convergence you need something like
the maximum of the absolute values of the eigenvalue of (A*B) is
less than unity.
Is there some special condition on matrices A and B that would
ensure A*B = B*A?
-- 
Mostly economics:  <http://www.dreamscape.com/rvien/#PublicationsForFun>
r           c
 v         s a           Whether strength of body or of mind, or wisdom, or
  i       m   p          virtue, are found in proportion to the power or 
wealth
   e     a     e         of a man is a question fit perhaps to be discussed 

by
    n   e       .        slaves in the hearing of their masters, but highly
     @ r         c m     unbecoming to reasonable and free men in search of
      d           o      the truth.    -- Rousseau
===
Subject: Re: matrix operation
>> For two matrix A and B; I can understand that
 
>> SUM[k=0.. INFINITE] (A*B)^k = (I-AB)^(-1)
 
>> where SUM[k=0.. INFINITE] represents the summation from zero to positive
>> infinite, k is integer.
>I don't have an answer for your further question. But the above is
>not true for all matrices. For convergence you need something like
>the maximum of the absolute values of the eigenvalue of (A*B) is
>less than unity.
That is exactly the criterion for convergence.
>> Then, it seems that there is no closed-form result for the following
>> expression.
>> SUM[k=0.. INFINITE] (A^k) *(B^k)
The closest I can come to it is this.  Suppose you can diagonalize B.
Then you can write B = sum_j r_j u_j v_j^T where r_j are the eigenvalues
of B, u_j and v_j are column vectors with v_i^T u_j = 1 for i=j, 0 
otherwise: u_j is an eigenvector for B for eigenvalue r_j, and v_j
is an eigenvector for B^T for eigenvalue r_j.  Now
B^k = sum_j r_j^k u_j v_j^T, so 
sum_{k=0}^infty A^k B^k = sum_j sum_{k=0}^infty r_j^k A^k u_j v_j^T
    = sum_j (I - r_j A)^(-1) u_j v_j^T
where for convergence it's sufficient to have
max {|r|: r an eigenvalue of A} max {|r|: r an eigenvalue of B} < 1.
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
===
Subject: Frieze Patterns
I have the following problem:
Consider the infinite strip pattern: ... TTTTTT ...
Prove that every element of the symmetry group of the above pattern
has the form t^n or (t^n)r where t is a suitable translation and r is
some suitable reflection.
I can see clearly why every element of the symmetry group has either
of those forms, when you take t to be the translation 'one unit' (or
one 'T') to the right or left and then r as a reflection through the
vertical line passing through any one of the T's, but how on earth do
I go about proving this rigorously?
Can any one help with proving this properly?
Gary.
===
Subject: Re: Is this system causal?
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id i9VD09i32277;
>x[n] --> [system] -->y[n]
>input                outpu
>System: y[n] + y[n+1] = x[n]
>A system is causal if for every choice of n0, the output sequence
>(y[n])  value  at the index n=n0, depends only on the input sequence
>(x[n]) values for n<=n0
>I think it's caual, but just want to corroborate. Any help is
>Luis
Rewriting your system gives y[n+1]=x[n]- y[n]
 Compare n+1 left hand ,n right hand ;
 so you're right to think it is causal system (time oriented), 
===
Subject: NEED HELP -- find a basis of a subspace!
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id i9VD0BV32375;
Can anybody help me to solve the following question:
Show that the plane defined by x-2y=0 is a 2-dimensional subspace of
R^3(R cubic). Give a basis for this subspace,and extend it to a  basis
for R^3
===
Subject: Re: NEED HELP -- find a basis of a subspace!
> Can anybody help me to solve the following question:
> Show that the plane defined by x-2y=0 is a 2-dimensional subspace of
> R^3(R cubic). Give a basis for this subspace,and extend it to a  basis
> for R^3
Your plane consists of all points of the form (2*y, y, z) for real y
and z. 
(2*y, y, z) = y*(2, 1, 0) + z*(0, 0, 1) which should suggest something.
Once you have a basis for the plane, almost any vector you add in will
give a basis for R^3.
-- 
Paul Sperry
Columbia, SC (USA)
===
Subject: Re: Please HELP me with this riddle
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id i9UMBNF28788;
maybe he used an icerope to kill himself
===
Subject: graphing caculator
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id i9UMBbq28915;
i'm in algebra and we do a lot of stuff with graphing the only problem
is I don't have one so is there a good online graphing
caculator!!!!???????
===
Subject: Any general idea on proving a topology that is a discrete 
topology?
===
Subject: Re: Any general idea on proving a topology that is a discrete 
topology?
            days. My association with the Department is that of an alumnus.
Prove that every singleton is open?
-- 
It's not denial. I'm just very selective about
 what I accept as reality.
    --- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Question about Gomory-Cut
for an application I have to find an integer solution.
I use the simplex-algorithm and the Gomory-cut.
It works fine, but it is to slowly.
There are multiply opportunities to make a Gomory-cut.
Which Gomory-cuts should I take,
to get a solution in shortest time ?
Where can I get information about that (in the internet) ?
Ulrich
===
Subject: Re: Question about Gomory-Cut
I would suggest to ask this in a different news group:
sci.op-research. 
But, let me give it a stab: I suspect you are using a pure 
cutting plane method based on an algorithm from Ralph Gomory 
in the 1960's. This method is known to be slow for many 
problems. However these cuts have received renewed interest 
by combining them with a branch-and-bound framework. (Search 
for branch-and-cut).
----------------------------------------------------------------
Erwin Kalvelagen
erwin@gams.com, http://www.gams.com/~erwin
----------------------------------------------------------------
> for an application I have to find an integer solution.
> I use the simplex-algorithm and the Gomory-cut.
> It works fine, but it is to slowly.
> There are multiply opportunities to make a Gomory-cut.
> Which Gomory-cuts should I take,
> to get a solution in shortest time ?
> Where can I get information about that (in the internet) ?
> Ulrich
===
Subject: percentage reporting
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA1KSkH05680;
Apologies if this is considered too basic a question. 
I have two power percentages before and after: 84.6%  and 98.2%. 
Is the correct way to report the change as 'an increase of 13.6%'. 
Or is it the percentage increased by (1-98.2/84.6) 16%.
jonjo
===
Subject: Re: percentage reporting
Surely it depends on the way the question is worded and how you answer the
question - e.g.
a)    The change in percentage power was from 84.6%  to 98.2%, i.e. 13.6% 
of
Max Power
b)    The power increased by (98.2 - 84.6)/84.6 x 100 % = 16.075 % - say
16.1 % of its original value.
--
Richard.
I have yet to see any problem, however complicated, which when looked at 
in
the right way, did not become still more complicated
Poul Anderson
> Apologies if this is considered too basic a question.
> I have two power percentages before and after: 84.6%  and 98.2%.
> Is the correct way to report the change as 'an increase of 13.6%'.
> Or is it the percentage increased by (1-98.2/84.6) 16%.
> jonjo
===
Subject: Re: percentage reporting
>Apologies if this is considered too basic a question. 
>I have two power percentages before and after: 84.6%  and 98.2%. 
>Is the correct way to report the change as 'an increase of 13.6%'. 
In English we would say 13.6 percentage points, which is 
understood to mean that the new number as a percent is 13.6 bigger 
than the old number as a percent.
-- 
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
                                  http://OakRoadSystems.com/
And if you're afraid of butter, which many people are nowa-
days, (long pause) you just put in cream.     --Julia Child
===
Subject: Re: percentage reporting
>Apologies if this is considered too basic a question. 
>I have two power percentages before and after: 84.6%  and 98.2%. 
>Is the correct way to report the change as 'an increase of 13.6%'. 
> In English we would say 13.6 percentage points, which is 
And in finance you could say an increase of 1,360 basis points.
-- 
Rich Carreiro                            rlcarr@animato.arlington.ma.us
===
Subject: First Post
lzaandam@cafes.net
http://www.newsfeeds.com The #1 Newsgroup Service in the World! >100,000 
Newsgroups
---= East/West-Coast Server Farms - Total Privacy via Encryption =---
===
Subject: Re: First Post
If this is your real email address, don't post it either in the body of the 

message or as the email address of the sender (you have done both).
There are harvester programs that search newsgroups for email addresses 
to 
If you want to publish your real email address, do what I do (have a look at 

my email address); many people use similar tricks. This allows a person to 
send you email (because they can fix the email address by hand) but fools 
Ohh ... and welcome to Usenet.
> lzaandam@cafes.net
> News==----
> http://www.newsfeeds.com The #1 Newsgroup Service in the World! >100,000 
> Newsgroups
> ---= East/West-Coast Server Farms - Total Privacy via Encryption =--- 
===
Subject: Re: First Post
>  .................If you want to publish your real email address, do what
I do (have a look at
>  my email address); many people use similar tricks. This allows a person
to
>  send you email (because they can fix the email address by hand) but 
fools
Having looked hard, How do I do this - I must be thick, please help.
--
Richard.
I have yet to see any problem, however complicated, which when looked at 
in
the right way, did not become still more complicated
Poul Anderson
===
Subject: Re: First Post
>  .................If you want to publish your real email address, do 
what
> I do (have a look at
>  my email address); many people use similar tricks. This allows a 
person
> to
>  send you email (because they can fix the email address by hand) but 
fools
> Having looked hard, How do I do this - I must be thick, please help.
> --
> Richard.
> I have yet to see any problem, however complicated, which when looked at 
in
> the right way, did not become still more complicated
> Poul Anderson
Somewhere in your newsreader there is a place where you type in your email 
address. You can enter any old thing in there you like.
===
Subject: Re: First Post
alt.math.undergrad:
>If this is your real email address, don't post it either in the body of the 

>message or as the email address of the sender (you have done both).
What nonsense. 
terrorists win by going even further to impede communications.
If you feel you must mung your address, at least follow Usenet 
standards and put .invalid at the end of it.
-- 
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
                                  http://OakRoadSystems.com/
And if you're afraid of butter, which many people are nowa-
days, (long pause) you just put in cream.     --Julia Child
===
Subject: Re: First Post
> alt.math.undergrad:
>>If this is your real email address, don't post it either in the body of 
>>the
>>message or as the email address of the sender (you have done both).
> What nonsense.
I hate to break it to you, but many therefore it is certainly not nonsense 
(unless you're one to believe you the only one that's right and it must be 
everyone else that's wrong.)
> terrorists win by going even further to impede communications.
Munging an email address in your client takes 10 seconds (tops).  Dealing 
more on communication.
> If you feel you must mung your address, at least follow Usenet
> standards and put .invalid at the end of it.
Usenet 'standards', Stan, for what it's worth, is to use your true and valid 

address that you check regularly, which disqualified .invalid.  The .invalid 

trick is simply one method by which one gracefully breaks camp with the 
standard.
Kirk was wrong, Stan.  The needs of the many REALLY DO outweigh the needs of 

the few (or the one) so your continued argument of not munging addresses, so 

you can easily just hit 'reply' to email someone, doesn't hold much water. 
===
Subject: Re: First Post
> What nonsense.
> terrorists win by going even further to impede communications.
> If you feel you must mung your address, at least follow Usenet
> standards and put .invalid at the end of it.
I use the domain mousepotato.com for my fake email address. This is a real
domain, but it always points back to the machine originating the DNS
networks elsewhere.
If people want to email me they need to request my adress in the group,
that's fine with me.
===
Subject: Re: First Post
>If this is your real email address, don't post it either in the body of 
the 
>message or as the email address of the sender (you have done both).
> What nonsense.
It's the opposite of nonsense; it's good advice. 
> terrorists win by going even further to impede communications.
Look, it's just simple defense. If someone wants to contact me and they 
can't figure out my email address, I'm not sure I want to hear from them 
anyway. (And please, terrorists?)
> If you feel you must mung your address, at least follow Usenet 
> standards and put .invalid at the end of it.
So you think the machines haven't figured that one out?
===
Subject: Re: First Post
> terrorists win by going even further to impede communications.
Terrorists?
> If you feel you must mung your address, at least follow Usenet
> standards and put .invalid at the end of it.
What is the advantage of that?
===
Subject: Re: First Post
> terrorists win by going even further to impede communications.
> Terrorists?
> If you feel you must mung your address, at least follow Usenet
> standards and put .invalid at the end of it.
> What is the advantage of that?
Then nobody, except machines, will mistake it for a real one.
===
Subject: Re: First Post
> terrorists win by going even further to impede communications.
> Terrorists?
> If you feel you must mung your address, at least follow Usenet
> standards and put .invalid at the end of it.
> What is the advantage of that?
> Then nobody, except machines, will mistake it for a real one.
So you think the machines haven't by now been programmed to remove 
.invalid?
===
Subject: Re: First Post
> Then nobody, except machines, will mistake it for a real one.
Machines cannot recognize .invalid at the end? 
===
Subject: Re: How did we get this?
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA2KG7m04360;
hi Vladimir...you are a life saver.I totally forgot that
a^2-b^2=(a+b)(a-b)
A bit off topic question here:
we get a^2-b^2 with (a-b)(a+b).But how do we get a^2+b^2?
>= (a + b)*(a^(2n-1) - a^(2n-2)*b + a^(2n - 3)*b^2 + ... - b^(2n-1)) 
>= (a^n + b^n)*(a^n - b^n)
*****************************************************
Perhaps this is off topic,but if you'd see this kind of formula first
time,would you be able to get(derive) (a^n + b^n)*(a^n - b^n) from 
(a + b)*(a^(2n-1) - a^(2n-2)*b + a^(2n - 3)*b^2 + ... - b^(2n-1))?
thank you very much for your help
bye
===
Subject: Re: How did we get this?
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA1Mcmp17221;
>>hi
>>-a^7b+2a^6b-4a^5b+8a^4b=
>>-a^4b(a^3-2a^2+4a-8)=
>>But how did we get the expression bellow from the expression above?
>>=-a^4b(a-2)(a^2+4)
>>thank you
>A useful identitiy to remember is
>a^2n - b^2n =
>= (a + b)*(a^(2n-1) - a^(2n-2)*b + a^(2n - 3)*b^2 + ... - b^(2n-1)) 
>= (a^n + b^n)*(a^n - b^n)
>Since a^n - b^n always has the factor a - b, this implies that
>a^(2n-1) - a^(2n-2)*b + a^(2n - 3)*b^2 + ... - b^(2n-1)
>must have the factor (a - b) and for even n > 0, also the factor
>a^n + b^n. In your case, n = 2, b = 2:
>a^4 - b^4 =
>= (a + b)*(a^3 - 2a^2 + 4a - 8) 
>= (a^2 + 4)*(a^2 - 4)
>This implies that
>a^3 - 2a^2 + 4a - 8 
>must have factors a^2 + 4 and a - 2.
>Another way: The cubic polynomial a^3-2a^2+4a-8 has integer
>coefficients. By the rational root theorem for polynomials with
>integer coefficients, the possible rational roots are r = p/q, where
p
>is a divider of the absolute term (8) and q is a divider of the
>cefficient at the highest power of a (1). The possible rational roots
>are +-8, +-4, +-2, +-1. Try each of them. If one of these numbers is
a
>root, factor it out -  divide a^3-2a^2+4a-8 by (a - r).
hi
Before I try to figure out your reply I need to know this
a^n - b^n=(a-b)(a^(n-1)+ a^(n-2)b +...ab^(n-2) + b^(n-1))
I tried 3^4 - 4^8=(3-4)(3^3+3^2*4+3*4^2+3*4^3+3*4^4+3*4^5+3*4^6+4^7)
but I got the wrong result!What am I doing wrong?
thank you
===
Subject: Re: How did we get this?
<SNIP hi
> Before I try to figure out your reply I need to know this
> a^n - b^n=(a-b)(a^(n-1)+ a^(n-2)b +...ab^(n-2) + b^(n-1))
> I tried 3^4 - 4^8=(3-4)(3^3+3^2*4+3*4^2+3*4^3+3*4^4+3*4^5+3*4^6+4^7)
> but I got the wrong result!What am I doing wrong?
> thank you
3^4 - 4^8 is not of the form a^n - b^n
3^4 - 4^4 or 3^8 - 4^8 are of this form.
===
Subject: Re: How did we get this?
> <SNIP hi
> Before I try to figure out your reply I need to know this
> a^n - b^n=(a-b)(a^(n-1)+ a^(n-2)b +...ab^(n-2) + b^(n-1))
> I tried 3^4 - 4^8=(3-4)(3^3+3^2*4+3*4^2+3*4^3+3*4^4+3*4^5+3*4^6+4^7)
> but I got the wrong result!What am I doing wrong?
> thank you
> 3^4 - 4^8 is not of the form a^n - b^n
> 3^4 - 4^4 or 3^8 - 4^8 are of this form.
Or you could rewrite
3^4 - 4^8
as
3^4 - (4^2)^4 = 3^4 - 16^4
===
Subject: Ideal problem
Could someone please give me a hint for the following problem?
Let I, J be ideals of a ring R.  Prove that if R is commutative and if I + J 

= R, then IJ = I intersection J.
I see that IJ is contained in the intersection of I and J, so it remains to 

===
Subject: Re: Ideal problem
            days. My association with the Department is that of an alumnus.
>Could someone please give me a hint for the following problem?
>Let I, J be ideals of a ring R.  Prove that if R is commutative and if I + 

J 
>= R, then IJ = I intersection J.
>I see that IJ is contained in the intersection of I and J, so it remains to 

You need to use the fact that I+J = R. And also, I assume, that R has
a 1 (though you do not state it explicitly, maybe it is a given in
your book/course).
Let i in I intersect J, and write 1 = a + b, with a in I, b in J.
Then i = i*1 = i(a+b) = (ia) + (ib).
Can you show that ia + ib is in IJ?
-- 
It's not denial. I'm just very selective about
 what I accept as reality.
    --- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Modules and Matricies
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA3009d24171;
Hey guys, I am having trouble with this question could someone please
1.
Let F be a field, let V and W be vector spaces over F, and let 
a:V-->V and b:W-->W be F-Linear transformations.
Let M=(V,a) and N=(W,b) be the F[x]-modules given by a and b
respectively. 
Write M(h) for the F[x]-module given by a^h for h>=1 and define N(h)
similarily.
Show that if Q:M-->N is an F[x]-module homomorphism, 
then Q:M(h)-->N(h) is also a module homomorphism for all h>=1.
All help will be appreciated!
===
Subject: Re: Modules and Matricies
> Hey guys, I am having trouble with this question could someone please
> 1.
> Let F be a field, let V and W be vector spaces over F, and let 
> a:V-->V and b:W-->W be F-Linear transformations.
> Let M=(V,a) and N=(W,b) be the F[x]-modules given by a and b
> respectively. 
> Write M(h) for the F[x]-module given by a^h for h>=1 and define N(h)
> similarily.
> Show that if Q:M-->N is an F[x]-module homomorphism, 
> then Q:M(h)-->N(h) is also a module homomorphism for all h>=1.
> All help will be appreciated!
I hope I'm understanding you correctly. If so . . .
Let p(x)%v = (p(a))(v) be the scalar mult on M as an F[X] module and
p(x)#v = (p(a^h))(v) the scalar mult on M(h) - I'm assuming h is a
positive integer.
All you need is Q(p(x)#v) = p(x)#Q(v).
But if p(x) = a_0 + a_1*x + . . . + a_n*x^n, let q(x) = a_0 + a_1*x^h +
. . .  + a_n*x^nh.
Then q(x)%v = p(x)#v.
-- 
Paul Sperry
Columbia, SC (USA)
===
Subject: Solving Solvable Quintics
Hello all,
For those who like solvable quintics, here's a paper that generalizes what
Viete did for the general cubic, namely reduce it to binomial form so as
to solve it with the extraction of a single cube root:
Solving Solvable Quintics Using One Fifth Root Extraction
ABSTRACT:  We prove that all irreducible but solvable equations
of degree n can be transformed in radicals into the binomial form
y^n+c=0 using a Tschirnhausen transformation of degree n-1.  The
resulting equation is then solvable by a single nth root
extraction.  In particular, we illustrate the method using the
solvable quintic.
Mathematics Subject Classification.  Primary: 12E12; Secondary: 11D09
http://www.geocities.com/titus_piezas/morequintics.html
Just click at the link above.  It's the 2nd pdf file.
(P.S. Pardon if this appears twice.)
-Titus (tpiezasIII@uap.edu.ph -> remove III for email)
===
Subject: Re: Solving Solvable Quintics
> Hello all,
> For those who like solvable quintics, here's a paper that generalizes 
what
> Viete did for the general cubic, namely reduce it to binomial form so as
> to solve it with the extraction of a single cube root:
> Solving Solvable Quintics Using One Fifth Root Extraction
> ABSTRACT:  We prove that all irreducible but solvable equations
> of degree n can be transformed in radicals into the binomial form
> y^n+c=0 using a Tschirnhausen transformation of degree n-1.  The
> resulting equation is then solvable by a single nth root
> extraction.  In particular, we illustrate the method using the
> solvable quintic.
> Mathematics Subject Classification.  Primary: 12E12; Secondary: 11D09
> http://www.geocities.com/titus_piezas/morequintics.html
> Just click at the link above.  It's the 2nd pdf file.
> (P.S. Pardon if this appears twice.)
> -Titus (tpiezasIII@uap.edu.ph -> remove III for email)
didnt galois show that there is no formula method to solve quintics? 
===
Subject: Re: Solving Solvable Quintics
>> Hello all,
>> For those who like solvable quintics, here's a paper that generalizes 
>> what
>> Viete did for the general cubic, namely reduce it to binomial form so as
>> to solve it with the extraction of a single cube root:
>> Solving Solvable Quintics Using One Fifth Root Extraction
>> ABSTRACT:  We prove that all irreducible but solvable equations
>> of degree n can be transformed in radicals into the binomial form
>> y^n+c=0 using a Tschirnhausen transformation of degree n-1.  The
>> resulting equation is then solvable by a single nth root
>> extraction.  In particular, we illustrate the method using the
>> solvable quintic.
>> Mathematics Subject Classification.  Primary: 12E12; Secondary: 11D09
>> http://www.geocities.com/titus_piezas/morequintics.html
>> Just click at the link above.  It's the 2nd pdf file.
>> (P.S. Pardon if this appears twice.)
>> -Titus (tpiezasIII@uap.edu.ph -> remove III for email)
> didnt galois show that there is no formula method to solve quintics?
No general formula.
Some quintics can be solved, for example
x^5 - 1 = 0  can be factored (solved). There are lots of other non-trivial 
examples - (x-a)(x-b)(x-c)(x-d)(x-e)=0 can be expanded into a quintic, and 
its solutions are obviously a,b,c,d,e.
I assume this paper provides what its title suggests - an algorithm for 
solving those that can be solved.
===
Subject: Re: Solving Solvable Quintics
> didnt galois show that there is no formula method to solve quintics?
> No general formula.
> Some quintics can be solved, for example
> x^5 - 1 = 0  can be factored (solved). There are lots of other non-trivial 

> examples - (x-a)(x-b)(x-c)(x-d)(x-e)=0 can be expanded into a quintic, and 

> its solutions are obviously a,b,c,d,e.
> I assume this paper provides what its title suggests - an algorithm for 
> solving those that can be solved.
     Yes, you assume correctly. Some points. First, we did mention 
Solving
SOLVABLE Quintics.  The term is mathematical shorthand for solvable in 
a
finite number of arithmetic operations and root extractions. When the term 
is
used it is understood we are NOT talking about the general case.
     
     Second, in the abstract, it was pointed out we are dealing with 
solvable
but irreducible quintics so nothing as easy as simply factoring it into
lower degrees with rational coefficients.
     
     Third, we should be careful when we say, there is no general formula 
to
solve the general quintic.  There is no general formula IN RADICALS (root
extractions).  But if we use hypergeometric functions, there is.  It's an 
old
result by Felix Klein (1849-1925).
     
     So the statement should be, There is no general formula IN RADICALS 
to
solve the general quintic but there IS a general formula IN HYPERGEOMETRIC
FUNCTIONS that can solve the general quintic with symbolic coefficients.
See paper for details.
     
     A bit wordy perhaps, but we have to split hairs in mathematics.
--Titus
===
Subject: 'divide and conquer' method to find eigenvalues
hi,
could anyone tell me how a 4 x 4 matrix's eigenvalues could be found using 
the 'divide & conquer' method?  i know it has something to do with finding 
the eigenvalues of the sub-matrices, so would it involve finding the 
eigenvalues of the 4 sub 2 x 2 matrices contained in the 4 x 4?
===
Subject: College math
College math software or books please
===
Subject: Re: College math
> College math software or books please
give me lots of money please
===
Subject: Re: College math
>> College math software or books please
> give me lots of money please
You included a verb. You already make more sense than the OP.
===
Subject: Re: College math
> College math software or books please
You need to start with a question.  Are you looking for recommendations? 
  Ones to avoid?  Something else?
Hint: a verb and question mark would help.
-- 
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Fundamental Theorem of Calculus
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA3EdQb01507;
If F(x) = the integral evaluated from x^2 to x^3 of 1/(1+x^5)dx,
Compute F'(x).
===
Subject: Re: Fundamental Theorem of Calculus
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA3Ilhr27874;
If G(x) is the integral evaluated from a to x, then G'(x) is 
1/(1+x^5). Your integral is F(x) = G(x^3) - G(x^2), so F'(x) =
G'(x^3)3x^2 - G'(x^2)2x = 3x^2 /(1 + x^15) - 2x/(1 + x^10).
>If F(x) = the integral evaluated from x^2 to x^3 of 1/(1+x^5)dx,
>Compute F'(x).
===
Subject: Re: Fundamental Theorem of Calculus
> If G(x) is the integral evaluated from a to x, then G'(x) is 
> 1/(1+x^5). Your integral is F(x) = G(x^3) - G(x^2), so F'(x) =
> G'(x^3)3x^2 - G'(x^2)2x = 3x^2 /(1 + x^15) - 2x/(1 + x^10).
This is homework. Much better to give a hint rather than the whole 
solution.
===
Subject: Re: Fundamental Theorem of Calculus
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA3FuLY10219;
>If F(x) = the integral evaluated from x^2 to x^3 of 1/(1+x^5)dx,
>Compute F'(x).
The trick is  we do not have to compute the integral:
let us put indefinite integral  of 1/(1+u^5)du =G(u)
then F(x)=G(x^3)-G(x^2)  and F'(x)= 3x^2*G'(x^3)- ...
come on ,
Alain.
===
Subject: Re: Fundamental Theorem of Calculus
> If F(x) = the integral evaluated from x^2 to x^3 of 1/(1+x^5)dx,
> Compute F'(x).
1/(1 + (3x^2)^5) - 1/(1 + (2x)^5)
===
Subject: Re: Fundamental Theorem of Calculus
> If F(x) = the integral evaluated from x^2 to x^3 of 1/(1+x^5)dx,
> Compute F'(x).
> 1/(1 + (3x^2)^5) - 1/(1 + (2x)^5)
  I don't think so ... First of all, there are too many 'x's
  in the problem description. Define G(x) to be the integral 
  from 0 to x of 1/(1 + t^5) with respect to t. Then the 
  Fundamental Theorem of Calculus says (among other things)
  that G'(x) = 1/(1 + x^5). To answer the question posed above,
  note that F(x) = G(x^3) - G(x^2), so a couple of applications
  of the Chain Rule will give
     F'(x) = 3x^2 G'(x^3) - 2x G'(x^2)
           = 3x^2/(1 + x^15) - 2x/(1 + x^10)
===
Subject: Re: Fundamental Theorem of Calculus
> If F(x) = the integral evaluated from x^2 to x^3 of 1/(1+x^5)dx,
> Compute F'(x).
 1/(1 + (3x^2)^5) - 1/(1 + (2x)^5)
>   I don't think so ... First of all, there are too many 'x's
Agreed.
>   in the problem description. Define G(x) to be the integral
>   from 0 to x of 1/(1 + t^5) with respect to t. Then the
>   Fundamental Theorem of Calculus says (among other things)
>   that G'(x) = 1/(1 + x^5). To answer the question posed above,
>   note that F(x) = G(x^3) - G(x^2), so a couple of applications
>   of the Chain Rule will give
>      F'(x) = 3x^2 G'(x^3) - 2x G'(x^2)
>            = 3x^2/(1 + x^15) - 2x/(1 + x^10)
===
Subject: Re: Fundamental Theorem of Calculus
> If F(x) = the integral evaluated from x^2 to x^3 of 1/(1+x^5)dx,
> Compute F'(x).
> 1/(1 + (3x^2)^5) - 1/(1 + (2x)^5)
Way wrong.
===
Subject: Re: JSH update
> Looks like he's dropped sci.math.  His latest stuff is being posted to
> just
> alt.math.undergrad.  Basically, the same old stuff, but aimed at
students.
> -- 
> --Tim Smith
> He has a history of this over the last 11 years!
> He'll be back with a vengeance again, and again, and ...!
> I think he is dressing up for Halloween as a solved FLT Proof.  Of
course
> no-one, including JSH himself, will know what the hell that means! :-)
> HH!
Most of the time, it could just be reduced to 0 = 1, I think.
   Shedar
===
Subject: Re: Is there a pattern I keep missing?
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA43MR610913;
>2x^6-8x^5-18x^4+72x^3=2x^3(x-3)(x+3)(x-4)
>About that result.How am I supossed to know that in order to get the
>correct result I must also multiply with (x-4)?
>or -3x^5-15x^4+3x^3+15x^2=-3x^2(x-1)(x+1)(x+5)
>or x^3-3x-2=(x+1)^2(x-2)
>or 4x^2+17x+4=(4x+1)(x+4)
>Is there some pattern I keep missing or is this just trial and error
>type of thing?How am I suppose to know that (4x+1)(x+4) gives you
>4x^2+17x+4?I need some advice how to be able to figure this things
>out
>thank you
Assume that a cubic polynomial has 3 real roots +r, -r and s. The
polynomial is then
x^3 + b*x^2 + c*x + d = (x - r)*(x + r)*(x - s) =
                      = (x^2 - r^2)*(x - s) =
                      = x^3 - s*r^2*x^2 - r^2*x + r^2*s
 
pattern, you can say right away that one factor is (x + b) and the
remaining 2 factors are (x +- sqrt(|c|)).
In your 1st example:
2x^6 - 8x^5 - 18x^4 + 72x^3 = 2x^3(x^3 - 4x^2 - 9x + 36)
b = -4, c = -9, d = 36 = (-4)*(-9) = bc and the factors are
(x + b) = (x - 4)
(x +- sqrt(|c|)) = (x +- sqrt(9)) = (x +- 3)
In your 2nd example:
-3x^5 - 15x^4 + 3x^3 + 15x^2 = -3x^2(x^3 + 5x^2 - x - 5)
b = 5, c = -1, d = -5 = 5*(-1) = bc and the factors are
(x + b) = (x + 5)
x +- sqrt(|c|)) = (x +- sqrt(1)) = (x +- 1)
Assume that a cubic polynomial has 3 real -r, -r, +2r. The polynomial
is then
x^3 + b*x^2 + c*x + d = (x + r)*(x + r)*(x - 2r) =
                      = (x^2 + 2rx + r^2)*(x - 2r) =
                      = x^3 - 3r^2*x - 2r^3
The pattern is b = 0, c = -3*r^2 (-3 times full square), d = -2*r^3
two factors (x + cbrt(-d/2)) and the remaining factor is
(x - 2*cbrt(-d/2)).
In your 3rd example:
x^3 - 3x - 2
b = 0, c = -3*1^2 (-3 times full square), d = -2*1^3 (-2 times full
cube) and the factors are
(x + cbrt(-d/2))^2 = (x + cbrt(1))^2 = (x + 1)^2
(x - 2*cbrt(-d/2)) = (x - 2*cbrt(1)) = (x - 2)
Assume that a quadratic polynomial has 2 real roots r, s. The
polynomial is then
x^2 + b*x + c = (x - r)*(x - s) =
              = x^2 - (r + s)*x + rs
sometimes guess the two factors, if you see 2 numerical factors of the
term c that add up to minus the term b.
In your 4th example:
4x^2 + 17x + 4 = 4*(x^2 + 17/4*x + 1)
b = 17/4 = 4 + 1/4 = 16/4 + 1/4 = 17/4, c = 1 = (-4)*(-1/4) and the
factors are
(x - r) = (x + 4)
(x - s) = (x + 1/4)
4*(x^2 + 17/4*x + 1) = 4*(x + 4)*(x + 1/4) = (x + 4)*(4x + 1)
===
Subject: Re: Is there a pattern I keep missing?
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA43MSQ10933;
>2x^6-8x^5-18x^4+72x^3=2x^3(x-3)(x+3)(x-4)
>About that result.How am I supossed to know that in order to get the
>correct result I must also multiply with (x-4)?
>or -3x^5-15x^4+3x^3+15x^2=-3x^2(x-1)(x+1)(x+5)
>or x^3-3x-2=(x+1)^2(x-2)
>or 4x^2+17x+4=(4x+1)(x+4)
>Is there some pattern I keep missing or is this just trial and error
>type of thing?How am I suppose to know that (4x+1)(x+4) gives you
>4x^2+17x+4?I need some advice how to be able to figure this things
out
>thank you
See the previous message about patterns.
To find factors a polynomial of degree n, you have to find its roots
(numbers that make the polynomial zero). The polynomial equals to zero
if and only if at least one factor equals to zero. The fundamental
theorem of algebra says that a polynomial of degree n has exactly n
roots. However, these roots do not necessarily have to be real
numbers, in general, they are complex numbers. For example, the
quadratic polynomial x^2 + 1 does not have any real roots, because for
any real number, x^2 >= 0 and x^2 + 1 > 0, so it cannot equal to zero.
(The complex roots of this polynomial are +i and -i, where
i^2 = -1.)
You are trying to factor quadratic or cubic polynomials (or perhaps
higher order polynomials). Hence, you have to find roots of these
polynomials. For quadratic polynomials, the solution is simple. Let
a*x^2 + b*x + c be the quadratic polynomial you want to factor,
a != 0. Put
a*x^2 + b*x + c = 0
x^2 + b/a*x + c/a = 0
Now, using the formula (A + B)^2 = A^2 + 2AB + B^2, try to get this
pattern at the left side of the equation. This means putting
A = x and B = b/(2a), which gives us the terms A^2 and 2AB. Since the
term B^2 = b^2/(4a^2) is not there, add and subtract it:
x^2 + b/a*x + b^2/(4a^2) - b^2/(4a^2) + c/a = 0
x^2 + b/a*x + b^2/(4a^2) = b^2/(4a^2) - c/a
(x + b/(2a))^2 = (b^2 - 4ac)/(4a^2)
x + b/(2a) = +- sqrt(b^2 - 4ac)/2a
x = [-b +- sqrt(b^2 - 4ac)]/2a 
The expression D = b^2 - 4ac is called the discriminant of the
quadratic polynomial and the polynomial has real roots if and only if
D >= 0. Applying this to your last example
4x^2 + 17x + 4 = 0
x = [-17 +- sqrt(17^2 - 64)]/8 = [-17 +- sqrt(289 - 64)]/8 = 
  = [-17 +- sqrt(225)]/8 = [-17 +- 15]/8 =
  = -32/8 or -2/8 = -4 or -1/4
These are the roots of the quadratic polynomial, so we can write it as
4x^2 + 17x + 4 = 4*(x - (-4))*(x - (-1/4) = 
               = 4*(x + 4)*(x + 1/4) =
               = (x + 4)*(4x + 1)
There are similar (but much more complicated) formulas for the 3rd and
4th degree polynomials - Cardano's formulas for the 3rd degree
polynomials, even though Cardano did not find them, but only published
them, giving credit to the author. A cubic polynomial with real
(rational, integer) coefficients can have either 3 real roots or one
real root and 2 complex roots (just graph some cubic polynomials and
look at how many points they intersect the x-axis - 3 or 1 points, the
real roots). When a cubic polynomial has 3 real roots, Cardano's
formulas give two of them as sums of 3rd roots of complex numbers.
This makes the formulas of little practical use.   
A cubic (or higher order) polynomial with integer coefficient can (but
does not have to have) rational root or roots. If it does, the
rational root has to be r = p/q, where p is a divider of the absolute
term and q is a divider of the coefficient at the highest power. This
is known as the rational root theorem. Trying this on one of your
examples:
2x^6 - 8x^5 - 18x^4 + 72x^3 = 0
2x^3*(x^3 - 4x^2 - 9x + 36) = 0
This 6-degree polynomial has tripple root x = 0, which can be easily
factored out. The remainig factor is a cubic polynomial
x^3 - 4x^2 - 9x + 36
As the coefficient at the highest power is 1 (which has only factors
+1 or -1), the rational roots (if they exist) can be only:
x = +-1, +-2, +-3, +-4, +-6, +-9, +-12, +-18, +-36
All these numbers have to be tried (or until 3 roots are found) by
substituting them (both positive and negative) into the above cubic
polynomial to see, if it becomes zero. Or you can keep trying until
you find the first root, say +3, divide the cubic polynomial by
(x - 3) and then find the roots of the remaining quadratic factor as
above.
Finally, there are various numerical methods to find the roots. If the
numerical method gives you a root that looks like a rational
(periodic) number or an integer, you can try to factor this root out
of the polynomial. For example, there is the method of chords. You
pick 2 numbers x0 and x1, preferrably such that the polynomial values
are  P(x0) < 0 band P(x1) > 0, so that some root is in between, and
calculate the intersection of the line passing through the 2 points
(x0, P(x0)) and (x1, P(x1)). The equation of the line passing through
these 2 points is:  
y - P(x0) = [P(x0) - P(x1)]/(x0 - x1)*(x - x0)
or
y - P(x1) = [P(x1) - P(x0)]/(x1 - x0)*(x - x1)
I will use the 1st form. Putting y = 0 and calculating x (and denoting
it as x2):
x2 = x0 - P(x0)*(x0 - x1)/[P(x0) - P(x1)]
The intersection of the line with the x-axis is (x2, 0) and x2 is
hopefully closer to the root than x0 or x1 (it surely is if P(x0) < 0
and P(x1) > 0). Now you calculate P(x2) to see if P(x2) < 0 or
P(x2) > 0. If P(x2) < 0, you pick the numbers x2 and x1 and repeat the
process to calculate x3. If P(x2) > 0, you pick the numbers x0 and x2
and repeat the process to calculate x3. In a few repetitions, the
numbers x2, x3, x4, ... stop changing which means that you got the
root. If it is an integer, you can easily see it. It is best done on a
computer, for example in a spreadsheet such as MS Excel. There is also
a faster converging method of tangents, but to find the equation of a
tangent to a cubic polynomial (and then its intersection with the
===
Subject: Re: Is there a pattern I keep missing?
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA4Eh3s06124;
hi
Can this be explained without the use of polynomials?I'm in first year
of highschool and we will learn polynomials only in third year
(BTW I'm thinking of learning about polynomials nontheless :))
thank you for your input...it will come in handy either way
===
Subject: Re: operators
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA4D6iN28091;
I was wondering if you could show me a proof for relation no. 1 or
point me to a source? 
they look very interesting. is there a name for these relations?
>>hi,
>>given an operator O that has the property O^{n} f(x)=f(x), where n
is
>>some integer, how would one start to go about finding the
eigenvalues
>>of O? 
>>also, anyone happen to come across an operation that changes the
>>argument of a function eg. from g(x) to g(ax+b)?
>I consider x as a simple variable.
>Some useful relations:exp(c d/dx)*x =x+c, exp(a d/dx)*f(x)=f(x+a).
>1Á)exp(((a-1)x+b)d/dx)*g(x)=g(ax +b) ,
>2Á)(exp((1/x-x)d/dx)^2 = Id ,
>3Á)(exp(h(x)-x))d/dx))^[n] =Id 
;h(x)=(x+tan(t))/(1-x*tan(t)) for 
>  tan(n.t)=0 ,positive integer,t=+/- Pi/n .
>  2 and 3Á are built upon cyclic functions.
>Maybe a generalization to multidimensionnal objects is possible.
>friendly,Alain.
===
Subject: Re: operators
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA43MRZ10900;
using the matrix representation for this operator(i.e. O(x.y) = O
delta(x-y)), how would one find the hermitian conjugate?
>>hi,
>>given an operator O that has the property O^{n} f(x)=f(x), where n
is
>>some integer, how would one start to go about finding the
eigenvalues
>>of O? 
>>also, anyone happen to come across an operation that changes the
>>argument of a function eg. from g(x) to g(ax+b)?
>I consider x as a simple variable.
>Some useful relations:exp(c d/dx)*x =x+c, exp(a d/dx)*f(x)=f(x+a).
>1Á)exp(((a-1)x+b)d/dx)*g(x)=g(ax +b) ,
>2Á)(exp((1/x-x)d/dx)^2 = Id ,
>3Á)(exp(h(x)-x))d/dx))^[n] =Id 
;h(x)=(x+tan(t))/(1-x*tan(t)) for 
>  tan(n.t)=0 ,positive integer,t=+/- Pi/n .
>  2 and 3Á are built upon cyclic functions.
>Maybe a generalization to multidimensionnal objects is possible.
>friendly,Alain.
===
Subject: Solving g(x ,y)=f(x , y)+ x f(a , y) +b.y    ; a, b, g  known  
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA4D6jQ28146;
For you,
A kind of inverse..
g and f  R*R -> R , a # 1 ;
function g(x ,y)is known and g(a ,y)defined ,f(x ,y)? 
Alain.
===
Subject: Re: Solving g(x ,y)=f(x , y)+ x f(a , y) +b.y    ; a, b, g  known 
 
> For you,
> A kind of inverse..
> g and f  R*R -> R , a # 1 ;
> function g(x ,y)is known and g(a ,y)defined ,f(x ,y)? 
> Alain.
      By some contorted calculations which you don't want to see (and I 
may not remember), here's one solution.
f(x,y)  =  g(x,y) - by + (x/(a+1))(by - g(a,y)).
I've no idea whether that solution is unique, but anyway it's better 
than nothing.  :-)
            Ken Pledger.
===
Subject: Question about powers
[Apologies if this appears more than once. I am having problems
getting this thing to work.]
Hi
I have a couple of questions about powers which maybe someone can help
with?
If b and c are whole numbers with no common divisor then a ^ (b/c)
would seem to have c (complex) values. E.g. 2 ^ (2/3) has three
(complex) values, 12 ^ (15/7) has seven values, etc. Correct?
Now consider a ^ (nb/nc), where n is another whole number.
Clearly a ^ (nb/nc) = a ^ (b/c), which should have c different values.
On the other hand,
a ^ (nb/nc)  = a ^ (nb * 1/nc) =  (a ^ nb) ^ (1/nc)
which would appear to have nc different values. Something's not right
here? Is some step in the above equation failing? Which one, and why?
A related question:
If x is an irrational number, then how many (complex) values does a ^
x have? I thought at first it should have infinitely many, but now I'm
not so sure. Any thoughts?
Matt
===
Subject: Re: Question about powers
much notational as anything else. The way my brain works, a^(b/c) MUST
be identical to a^(bn/cn) because the ^ operator has no way of
peeking inside the brackets to see how its operand is arrived at.
For example, if I write a computer function f(p,q) to calculate p^q
then I inevitably get identical results for f(a,b/c) and f(a,bn/cn),
HOWEVER the function is defined.
But if I define a^(b/c) equal to f(a,b,c) (with f some suitable
function), THEN it is clear to me that f(a,b,c) need not necessarily
be the same as f(a,bn,cn).
Regarding the other strand of my original question, I also think I've
now figured out how a^(x*y) is not necessarily equal to (a^x)^y in the
e^{x*log(a)}, along with all that multi-valued logs, exp(it) = cos(t)
+ i sin(t), and other funky stuff... So that's cool!
matt
===
Subject: Re: Question about powers
> [Apologies if this appears more than once. I am having problems
> getting this thing to work.]
> Hi
> I have a couple of questions about powers which maybe someone can help
> with?
> If b and c are whole numbers with no common divisor then a ^ (b/c)
> would seem to have c (complex) values. E.g. 2 ^ (2/3) has three
> (complex) values, 12 ^ (15/7) has seven values, etc. Correct?
> Now consider a ^ (nb/nc), where n is another whole number.
> Clearly a ^ (nb/nc) = a ^ (b/c), which should have c different values.
> On the other hand,
> a ^ (nb/nc)  = a ^ (nb * 1/nc) =  (a ^ nb) ^ (1/nc)
> which would appear to have nc different values. Something's not right
> here? Is some step in the above equation failing? Which one, and why?
The standard rules of exponents that work with positive integer 
exponents on arbitrary bases do not all work with arbitrary rational or 
real exponents without a lot of extra restrictions on the bases one is 
permitted to have.
> A related question:
> If x is an irrational number, then how many (complex) values does a ^
> x have? I thought at first it should have infinitely many, but now I'm
> not so sure. Any thoughts?
For positive real a and real x, the equations y = a^x and
log(y) = x*log(a) are taken to be equivalent ( assuming any positive 
real base for the logarithms other than 1). Everything else is special 
cases.
> Matt
===
Subject: Re: Question about powers
            days. My association with the Department is that of an alumnus.
>[Apologies if this appears more than once. I am having problems
>getting this thing to work.]
>I have a couple of questions about powers which maybe someone can help
>with?
>If b and c are whole numbers with no common divisor then a ^ (b/c)
>would seem to have c (complex) values. E.g. 2 ^ (2/3) has three
>(complex) values, 12 ^ (15/7) has seven values, etc. Correct?
>Now consider a ^ (nb/nc), where n is another whole number.
>Clearly a ^ (nb/nc) = a ^ (b/c), which should have c different values.
Nothing clear about it! What is your definition of a^{b/c}?
Usually, the definition is that a^{b/c} is equal to the c-th root of
a^b. But this is not a map from C x Q to C; that is, it is not true
that if p and q are rationals and p=q, then a^p = a^q. It depends on
the representation of p and of q.
If, on the other hand, your definition of a^{b/c} is ->restricted<- to
the case when gcd(b,c)=1, then:
>On the other hand,
>a ^ (nb/nc)  = a ^ (nb * 1/nc) =  (a ^ nb) ^ (1/nc)
is wrong, because a^{nb/nc} is not the same thing as
(a^{nb})^{1/nc}. It would be equal to (a^{b})^{1/c}, because you need
to change the exponent nb/nc to b/c.
>If x is an irrational number, then how many (complex) values does a ^
>x have? I thought at first it should have infinitely many, but now I'm
>not so sure. Any thoughts?
Usually, you define a^x as e^{x*log(a)}. Since log is a multivalue
function with arbitrarily many values, yes.
-- 
It's not denial. I'm just very selective about
 what I accept as reality.
    --- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Question about powers
> ... it is not true
> that if p and q are rationals and p=q, then a^p = a^q. It depends on
> the representation of p and of q.
and later
> Usually, you define a^x as e^{x*log(a)}. 
hmmm... if p=q, and yet a^p is not identical to a^q, then I do not see
how the ^ operator can have any meaningful definition at all, whether
it be the one given or any other?
===
Subject: Re: Question about powers
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA71kuR26482;
>> ... it is not true
>> that if p and q are rationals and p=q, then a^p = a^q. It depends
on
>> the representation of p and of q.
>and later
>> Usually, you define a^x as e^{x*log(a)}. 
>hmmm... if p=q, and yet a^p is not identical to a^q, then I do not
see
>how the ^ operator can have any meaningful definition at all, whether
>it be the one given or any other?
Yes, that's the point -- that the function  f(z) = z^p  is not 
well-defined (as an complex analytic function of z) if p is not 
an integer.  In the second definition, log(z) is not a well-defined 
analytic function of non-zero z: it has infinitely many branches 
where the values differ by multiples of  2pi*i. 
Todd Trimble 
===
Subject: Re: Question about powers
            days. My association with the Department is that of an alumnus.
>> ... it is not true
>> that if p and q are rationals and p=q, then a^p = a^q. It depends on
>> the representation of p and of q.
>and later
>> Usually, you define a^x as e^{x*log(a)}. 
->IN THE COMPLEX NUMBERS<-, and the exponentiation function already
takes into account the normal form restriction given above.
>hmmm... if p=q, and yet a^p is not identical to a^q, then I do not see
>how the ^ operator can have any meaningful definition at all, whether
>it be the one given or any other?
Then you aren't paying attention to the definition.
It is ->PERFECTLY LEGITIMATE<- to define an operation based on a
normal form for the elements of the sets. The rational numbers have
multiple representations, but it is legitimate to define an operation
in terms of a single such representation. 
-- 
It's not denial. I'm just very selective about
 what I accept as reality.
    --- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Question about powers
>> ... it is not true
>> that if p and q are rationals and p=q, then a^p = a^q. It depends on
>> the representation of p and of q.
>and later
>> Usually, you define a^x as e^{x*log(a)}. 
>hmmm... if p=q, and yet a^p is not identical to a^q, then I do not see
>how the ^ operator can have any meaningful definition at all, whether
>it be the one given or any other?
Consider (-11)^1 and (-11)^(2/2). Some authors (not all) would 
consider the second one undefined on the reals because it implies 
the square root of a negative.
I can't think of any example where p=q, a^p and a^q are both 
defined on the reals, but a^p =/= a^q.
-- 
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
                                  http://OakRoadSystems.com/
And if you're afraid of butter, which many people are nowa-
days, (long pause) you just put in cream.     --Julia Child
===
Subject: Formula help to re-arrange
How do I re-arrange the following formula to find y
2 ln y = ln (2x - 5) + C
I can get an answer on my calculator, but have no idea on how to arrive at
the answer. Can anyone help?
TIA
===
Subject: Re: Formula help to re-arrange
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA63BFM12253;
>How do I re-arrange the following formula to find y
>2 ln y = ln (2x - 5) + C
>I can get an answer on my calculator, but have no idea on how to
arrive at
>the answer. Can anyone help?
>TIA
ln(y^2)=ln(2x-5)+c
apply exp to both sides, 
y^2=(2x-5)*exp(c)
y=sqrt(2x-5)*exp(c/2)
===
Subject: Re: Formula help to re-arrange
>How do I re-arrange the following formula to find y
>2 ln y = ln (2x - 5) + C
http://smjg.port5.com/faqs/usenet/xpost.html
-- 
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
                                  http://OakRoadSystems.com/
And if you're afraid of butter, which many people are nowa-
days, (long pause) you just put in cream.     --Julia Child
===
Subject: Re: Formula help to re-arrange
>How do I re-arrange the following formula to find y
>2 ln y = ln (2x - 5) + C
>I can get an answer on my calculator, but have no idea on how to arrive at
>the answer. Can anyone help?
Don't post the same question twice to two different newsgroups. If it
is appropriate for both, then crosspost.
-- 
It's not denial. I'm just very selective about
 what I accept as reality.
    --- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: HELP
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA4JKU903033;
What has two heads with one body,is used by young and old,has no legs
but can move,,can be used to measure and can be used inside out?
===
Subject: Re: HELP
> What has two heads with one body,is used by young and old,has no legs
> but can move,,can be used to measure and can be used inside out?
      Please post riddles in the  rec.puzzles  news group rather than a 
mathematical group.
            Ken Pledger.
===
Subject: ring of continous functions
Maybe someone can lend a hand with the following:
Let R be the ring of all continous functions on [0,1] into the real numbers 

and define M_x = {f in R | f(x)=0}. Prove that if M is a maximal ideal in R, 

then M = M_x for some x.
===
Subject: Re: ring of continous functions
> Let R be the ring of all continous functions on [0,1] into the real 
numbers 
> and define M_x = {f in R | f(x)=0}. Prove that if M is a maximal ideal in 

R, 
> then M = M_x for some x.
Hint: If M is contained in no M_x, then for every x in [0,1] there is an 
f_x in M such that f_x(x) is not zero. By continuity, (f_x)^2 > 0 in a 
neighborhood Nx of x. This implies (here's where you need to know something 

about [0,1]) there is an f in M that is positive on all of [0,1].
===
Subject: Re: ring of continous functions
alt.math.undergrad:
> Maybe someone can lend a hand with the following:
> Let R be the ring of all continous functions on [0,1] into the real 
numbers 
> and define M_x = {f in R | f(x)=0}. Prove that if M is a maximal ideal in 

R, 
> then M = M_x for some x.
Suppose that M is maximal.  For each f in M let Z(f) = 
{x : f(x) = 0}, the zero-set of f.  Show that for any finite
set {f_1, f_2, ..., f_n} in M, the intersection of the
Z(f_i) is non-empty, and conclude that the intersection of
all Z(f), f in M, is non-empty.  Let this intersection be Z,
and let M_Z = {f in R : Z(f) contains Z}.  Use the
maximality of M to show that M = M_Z and that Z must be a
singleton.
Brian
===
Subject: REQ, Need College math software or books please
===
Subject: Re: REQ, Need College math software or books please
Are you asking for:
- recommendations?
- donations?
--
Casey
===
Subject: question on continuity of function
When
f(z)=|z|^3 / z ,if z=/=0
=0, if z=0
How to prove f(z) is continuous on C-{0}?
My trying : If z=/=0, then f(z)=|z|*bar(z) 
so, by letting z=x+y*i (x,y in reals, x+y*i=/=0))
f(z)=x*sqrt(x^2 + y^2) - y*sqrt(x^2 + y^2)*i
in here, since 
real part x*sqrt(x^2 + y^2) and imaginary part -y*sqrt(x^2 + y^2)
are both partially differentiable w.r.t x&y ,
real part x*sqrt(x^2 + y^2) and imaginary part -y*sqrt(x^2 + y^2)
are continuous on R^2. So that f(z) is also continuous on C-{0}.
Is my trying corect ? 
If there is nice way to see f(z)is continuous on C-{0},
please post that.
 
===
Subject: Re: question on continuity of function
> When
> f(z)=|z|^3 / z ,if z=/=0
> =0, if z=0
> How to prove f(z) is continuous on C-{0}?
This is clearly the quotient of two functions continuous on C-{0} ...
===
Subject: question on continuity of function
When
f(z)=|z|^3 / z ,if z=/=0
=0, if z=0
How to prove f(z) is continuous on C-{0}?
My trying : If z=/=0, then f(z)=|z|*bar(z) 
so, by letting z=x+y*i (x,y in reals, x+y*i=/=0))
f(z)=x*sqrt(x^2 + y^2) - y*sqrt(x^2 + y^2)*i
in here, since 
real part x*sqrt(x^2 + y^2) and imaginary part -y*sqrt(x^2 + y^2)
are both partially differentiable w.r.t x&y ,
real part x*sqrt(x^2 + y^2) and imaginary part -y*sqrt(x^2 + y^2)
are continuous on R^2. So that f(z) is also continuous on C-{0}.
Is my trying corect ? 
If there is nice way to see f(z)is continuous on C-{0},
please post that.
 
===
Subject: Lin alg- Eigenvectors & lin. transformations
Hi!
Need some guidance:
Let T: P2 -> P3 be the transformation that maps a polynomial p(t) into the 
polynomial (t+5)p(t).
The image of p(t) = 2-t+t^2 is 10 -3t +4t^2 +t^3.
Can someone explain me the algorithm of finding the matrix T relative to the 

bases{1,t,t^2} and {1,t,t^2,t^3}?
Ronny Mandal
===
Subject: Re: Lin alg- Eigenvectors & lin. transformations
> Hi!
> Need some guidance:
> Let T: P2 -> P3 be the transformation that maps a polynomial p(t) into the 

> polynomial (t+5)p(t).
> The image of p(t) = 2-t+t^2 is 10 -3t +4t^2 +t^3.
> Can someone explain me the algorithm of finding the matrix T relative to 
the 
> bases{1,t,t^2} and {1,t,t^2,t^3}?
I am going to assume that the coefficients of a vector, with resepct to
a given basis, are arranged as a column.  (In ancient times, it would
be called a column vector.) Thus the example p(t) that you gave would
be represented thus:
2
-1
1
and the image Tp(t) would be represented thus:
10
-3
4
3 .
The matrix of T will have 4 rows and 3 columns.  I think it is mose
easily put together one column at a time.  Note that the columns
correspond to the members of thegiven basis of P2.  Call them e0, e1,
e2.  The first column of the matrix of T displays the coefficients of
the vector Te0, with respect to the basis of P3.  And so it goes.
-- 
Chris Henrich
God just doesn't fit inside a single religion.
===
Subject: Numbers with Non-Decreasing Digits
I've been unable to solve the following problem for a while know: How
many n-digit numbers a[1]a[2]...a[n] exits where 1 <= a[i] <= i for i
= 1, 2, ..., n and a[1] <= a[2] <= ... <= a[n]? For example, with n =
3 there are five numbers:
111
112
122
113
123
Somebody suggested in transforming the problem by adding a n + 1 to
the numbers in the sequence. For example, for the above sequence I
would have
1114
1124
1224
1134
1234
The sum of the differences between adjacent digits is always the same
(in the above case, 3). However, this new sequence is just as
perplexing as the original one. Can anybody shed some light on this?
Bernd
===
Subject: Re: Numbers with Non-Decreasing Digits
there is another way to write c(n), for n>= 3, im not sure if it can be any 

help ( maybe if your writing a program involving it), anyway,
case n = 2, c(2) =  2;
case n = 3, c(3) = Sum (i=2 to 3) [i] = 2+3=5;
case n = 4, c(4) = Sum (j=3 to 4) [ Sum (i=2 to j) [i] ]= 2 + 3 + 2 + 3 + 4 

= 14;
case n = 5, c(5) = Sum (k=4 to 5) [ Sum (j=3 to k) [ Sum (i=2 to j) [i] ] ] 

= 42;
and so on.
> I've been unable to solve the following problem for a while know: How
> many n-digit numbers a[1]a[2]...a[n] exits where 1 <= a[i] <= i for i
> = 1, 2, ..., n and a[1] <= a[2] <= ... <= a[n]? For example, with n =
> 3 there are five numbers:
> 111
> 112
> 122
> 113
> 123
> Somebody suggested in transforming the problem by adding a n + 1 to
> the numbers in the sequence. For example, for the above sequence I
> would have
> 1114
> 1124
> 1224
> 1134
> 1234
> The sum of the differences between adjacent digits is always the same
> (in the above case, 3). However, this new sequence is just as
> perplexing as the original one. Can anybody shed some light on this?
> Bernd
===
Subject: Re: Numbers with Non-Decreasing Digits
There are countably infinitely many such numbers.
Consider the number 19
Add a digit to get 119
Again, you get 1119
11119
111119
1111119,
and so on.
How many times can you do this ?
Aleph null.
Filling in the gaps with the rest of the possible numbers under
consideration still gives aleph null, because aleph null + aleph null =
aleph null.
===
Subject: Re: Numbers with Non-Decreasing Digits
alt.math.undergrad:
> There are countably infinitely many such numbers.
> Consider the number 19
> Add a digit to get 119
> Again, you get 1119
> 11119
> 111119
> 1111119,
None of these meets the requirements of Bernd's problem: one
of the requirements is that the number in position k be at
most k.
And even if they did satisfy the conditions, your answer
shows that you've completely missed the point of the
question, which is to count how many solutions of length n
there are _for_each_n_.
[...]
===
Subject: Re: Numbers with Non-Decreasing Digits
alt.math.undergrad:
> I've been unable to solve the following problem for a while know: How
> many n-digit numbers a[1]a[2]...a[n] exits where 1 <= a[i] <= i for i
> = 1, 2, ..., n and a[1] <= a[2] <= ... <= a[n]? For example, with n =
> 3 there are five numbers:
> 111
> 112
> 122
> 113
> 123
Let c(n) be the number of valid strings of length n.  By
direct enumeration we find that c(1) = 1, c(2) = 2, c(3) =
5, and c(4) = 14.  (For that matter, it's fairly clear that
c(0) = 1, since the empty string vacuously satisfies the
conditions.)  The sequence 1, 1, 2, 5, 14, ... is familiar:
it's the Catalan numbers C_n, which are defined by the
recurrence
   C_0 = 1
   C_{n+1} = Sum[i=0 to n; C_i * C_{n-i}].
   
If you can show that the numbers c(n) satisfy this
recurrence, you'll have c(n) = C_n for each n >= 0, and it's
well known that C_n = C(2n, n) / (n + 1), where C(2n, n) is
the binomial coefficient '2n choose n'.
To show this, suppose that a[1]...a[n] is a valid string of
length n.  Let m = max{i : a[i] = i}.  (E.g., if n = 5 and
the string is 11334, then m = 3; if the string is 11344, 
m = 4; and if the string is 12222, then m = 1.)  It's easy
to see that a[1]...a[m-1] is a valid string of length m-1.
(If m = 1, it's the empty string, which is the unique valid
string of length 0.)  Set b[i] = a[i] - m + 1 for 
i = m+1, ..., n.  By hypothesis m <= a[i] < i for 
i = m+1, ..., n, so 1 <= b[i] <= i - m, so b[m+1]...b[n] is
a valid string of length n-m.
For each value of m from 1 to n there are c(m-1) ways to
choose a[1]...a[m-1]; a[m] must be m; and there are c(n-m)
ways to choose b[m+1]...b[n].  Adding m-1 to the b[i] yields
c(n-m) possible choices of a[m+1]...a[n], so altogether
   c(n) = Sum[m=1 to n; c(m-1) * c(n-m)]
        = Sum[i=0 to n-1; c(i) * c(n-1-i)],
        
which is the Catalan recurrence.
[...]
Brian
===
Subject: Re: Numbers with Non-Decreasing Digits
> alt.math.undergrad:
> I've been unable to solve the following problem for a while know: How
> many n-digit numbers a[1]a[2]...a[n] exits where 1 <= a[i] <= i for i
> = 1, 2, ..., n and a[1] <= a[2] <= ... <= a[n]? For example, with n =
> 3 there are five numbers:
> 111
> 112
> 122
> 113
> 123
> Let c(n) be the number of valid strings of length n.  By
> direct enumeration we find that c(1) = 1, c(2) = 2, c(3) =
> 5, and c(4) = 14.  (For that matter, it's fairly clear that
> c(0) = 1, since the empty string vacuously satisfies the
> conditions.)  The sequence 1, 1, 2, 5, 14, ... is familiar:
> it's the Catalan numbers C_n, which are defined by the
> recurrence
>    C_0 = 1
>    C_{n+1} = Sum[i=0 to n; C_i * C_{n-i}].
>    
> If you can show that the numbers c(n) satisfy this
> recurrence, you'll have c(n) = C_n for each n >= 0, and it's
> well known that C_n = C(2n, n) / (n + 1), where C(2n, n) is
> the binomial coefficient '2n choose n'.
> To show this, suppose that a[1]...a[n] is a valid string of
> length n.  Let m = max{i : a[i] = i}.  (E.g., if n = 5 and
> the string is 11334, then m = 3; if the string is 11344, 
> m = 4; and if the string is 12222, then m = 1.)  It's easy
> to see that a[1]...a[m-1] is a valid string of length m-1.
> (If m = 1, it's the empty string, which is the unique valid
> string of length 0.)  Set b[i] = a[i] - m + 1 for 
> i = m+1, ..., n.  By hypothesis m <= a[i] < i for 
> i = m+1, ..., n, so 1 <= b[i] <= i - m, so b[m+1]...b[n] is
> a valid string of length n-m.
For the string 12222, shouldn't m = 2?
 
> For each value of m from 1 to n there are c(m-1) ways to
> choose a[1]...a[m-1]; a[m] must be m; and there are c(n-m)
> ways to choose b[m+1]...b[n].  Adding m-1 to the b[i] yields
> c(n-m) possible choices of a[m+1]...a[n], so altogether
>    c(n) = Sum[m=1 to n; c(m-1) * c(n-m)]
>         = Sum[i=0 to n-1; c(i) * c(n-1-i)],
>         
> which is the Catalan recurrence.
> [...]
> Brian
===
Subject: Re: Numbers with Non-Decreasing Digits
alt.math.undergrad:
>> alt.math.undergrad:
> I've been unable to solve the following problem for a while know: How
> many n-digit numbers a[1]a[2]...a[n] exits where 1 <= a[i] <= i for i
> = 1, 2, ..., n and a[1] <= a[2] <= ... <= a[n]? For example, with n =
> 3 there are five numbers:
> 111
> 112
> 122
> 113
> 123
>> Let c(n) be the number of valid strings of length n.  By
>> direct enumeration we find that c(1) = 1, c(2) = 2, c(3) =
>> 5, and c(4) = 14.  (For that matter, it's fairly clear that
>> c(0) = 1, since the empty string vacuously satisfies the
>> conditions.)  The sequence 1, 1, 2, 5, 14, ... is familiar:
>> it's the Catalan numbers C_n, which are defined by the
>> recurrence
>>    C_0 = 1
>>    C_{n+1} = Sum[i=0 to n; C_i * C_{n-i}].
>> If you can show that the numbers c(n) satisfy this
>> recurrence, you'll have c(n) = C_n for each n >= 0, and it's
>> well known that C_n = C(2n, n) / (n + 1), where C(2n, n) is
>> the binomial coefficient '2n choose n'.
>> To show this, suppose that a[1]...a[n] is a valid string of
>> length n.  Let m = max{i : a[i] = i}.  (E.g., if n = 5 and
>> the string is 11334, then m = 3; if the string is 11344, 
>> m = 4; and if the string is 12222, then m = 1.)  It's easy
>> to see that a[1]...a[m-1] is a valid string of length m-1.
>> (If m = 1, it's the empty string, which is the unique valid
>> string of length 0.)  Set b[i] = a[i] - m + 1 for 
>> i = m+1, ..., n.  By hypothesis m <= a[i] < i for 
>> i = m+1, ..., n, so 1 <= b[i] <= i - m, so b[m+1]...b[n] is
>> a valid string of length n-m.
> For the string 12222, shouldn't m = 2?
Yes; sorry about that.
>> For each value of m from 1 to n there are c(m-1) ways to
>> choose a[1]...a[m-1]; a[m] must be m; and there are c(n-m)
>> ways to choose b[m+1]...b[n].  Adding m-1 to the b[i] yields
>> c(n-m) possible choices of a[m+1]...a[n], so altogether
>>    c(n) = Sum[m=1 to n; c(m-1) * c(n-m)]
>>         = Sum[i=0 to n-1; c(i) * c(n-1-i)],
>> which is the Catalan recurrence.
My pleasure: it's a nice problem.
Brian
===
Subject: Re: I have a riddle I need help solving..!!
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA63BGX12305;
A man is walking down a road and comes to a fork in the road. There
are two paths he can take. He knows that at the end of one path is the
town of truths where everybody always tells the truth. At the end of
the other path is the town of lies where everyone always lies. the
problem is he doesn't know which path leads to which town. There is
another man standing where the road breaks into two different ones.
The first man can ask the second man ONE question. What is the
question?
===
Subject: Re: I have a riddle I need help solving..!!
> A man is walking down a road and comes to a fork in the road. There
> are two paths he can take. He knows that at the end of one path is the
> town of truths where everybody always tells the truth. At the end of
> the other path is the town of lies where everyone always lies. the
> problem is he doesn't know which path leads to which town. There is
> another man standing where the road breaks into two different ones.
> The first man can ask the second man ONE question. What is the
> question?
He should ask for the telephone number of a taxi firm.  The taxi driver
can then be _told_ where to go, no further questions are needed.
===
Subject: Re: I have a riddle I need help solving..!!
>A man is walking down a road and comes to a fork in the road. There
> are two paths he can take. He knows that at the end of one path is the
> town of truths where everybody always tells the truth. At the end of
> the other path is the town of lies where everyone always lies. the
> problem is he doesn't know which path leads to which town. There is
> another man standing where the road breaks into two different ones.
> The first man can ask the second man ONE question. What is the
> question?
The question is: If I were to ask you which path leads to the town of 
truths, which one would you indicate.
===
Subject: Re: I have a riddle I need help solving..!!
 
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi
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 VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
>>A man is walking down a road and comes to a fork in the road. There
>> are two paths he can take. He knows that at the end of one path is the
>> town of truths where everybody always tells the truth. At the end of
>> the other path is the town of lies where everyone always lies. the
>> problem is he doesn't know which path leads to which town. There is
>> another man standing where the road breaks into two different ones.
>> The first man can ask the second man ONE question. What is the
>> question?
> The question is: If I were to ask you which path leads to the town of 
> truths, which one would you indicate.
That's just where you came from.
-- 
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: systems of quads
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA63BG712320;
hey i need to solve this system using a computer - so any method needs
to be programming frendly if possiable.
R^2 = Mx(X) + Y^2 + Z^2
R^2 = X^2 + My(Y) + Z^2
R^2 = X^2 + Y^2 + Mz(Z)
so
Mx(X) + Y^2 + Z^2 = X^2 + My(Y) + Z^2 = X^2 + Y^2 + Mz(Z)
where R, Mx, My, Mz, are known
Just if you where wondering this is the intersection of 3 circles in
3D space I think
Thnx
===
Subject: Re: systems of quads
> hey i need to solve this system using a computer - so any method needs
> to be programming frendly if possiable.
> R^2 = Mx(X) + Y^2 + Z^2
> R^2 = X^2 + My(Y) + Z^2
> R^2 = X^2 + Y^2 + Mz(Z)
> so
> Mx(X) + Y^2 + Z^2 = X^2 + My(Y) + Z^2 = X^2 + Y^2 + Mz(Z)
> where R, Mx, My, Mz, are known
> Just if you where wondering this is the intersection of 3 circles in
> 3D space I think
> Thnx
if i'm reading your nomenclature, then it seems you have a system of 
non-linear simultaneous equations.   The usual methods to solve these on a 
computer are both recursive and are the Gauss-Seidel and Jacobi methods. 
Look them up on Google.
===
Subject: Re: systems of quads
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA81ms417156;
>> hey i need to solve this system using a computer - so any method
needs
>> to be programming frendly if possiable.
>> R^2 = Mx(X) + Y^2 + Z^2
>> R^2 = X^2 + My(Y) + Z^2
>> R^2 = X^2 + Y^2 + Mz(Z)
>> so
>> Mx(X) + Y^2 + Z^2 = X^2 + My(Y) + Z^2 = X^2 + Y^2 + Mz(Z)
>> where R, Mx, My, Mz, are known
>> Just if you where wondering this is the intersection of 3 circles
in
>> 3D space I think
>> Thnx
>if i'm reading your nomenclature, then it seems you have a system of 
>non-linear simultaneous equations.   The usual methods to solve these
on a 
>computer are both recursive and are the Gauss-Seidel and Jacobi
methods. 
>Look them up on Google.
the algebraic approach is by Groebner bases. You may have a look at
the excellent book by Cox-Little-OShea ideals varieties and
algorithms. For complicated problems, you may look at the software
Macaulay2 which is easily useable. It don't understand what are the
Mx(.), etc ... For the case you submit, if i understand weel you
should be able to do it by hand after you read Cox-Little-OShea. 
===
Subject: riddle
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA6CdQa24350;
I have hands but no thumbs. I have arms but can not reach. I run all
day but have no legs. what am I.
===
Subject: Re: riddle
> I have hands but no thumbs. I have arms but can not reach. I run all
> day but have no legs. what am I.
      Please post riddles in the  rec.puzzles  news group rather than a 
mathematical group.
            Ken Pledger.
===
Subject: Re: riddle
>I have hands but no thumbs. I have arms but can not reach. I run all
>day but have no legs. what am I.
This is NOT a riddle group . It IS a math group.
===
Subject: Re: riddle
>I have hands but no thumbs. I have arms but can not reach. I run all
>day but have no legs. what am I.
> This is NOT a riddle group . It IS a math group.
Perhaps the answer (which you have not guessed, and I do not know) has
something to do with math.
===
Subject: Re: riddle
> I have hands but no thumbs. I have arms but can not reach. I run all
> day but have no legs. what am I.
Um... a clock with arms?
(Anything to do with undergraduate mathematics?)
===
Subject: Re: riddle
svkcdw1tn68c@legacy...
> I have hands but no thumbs. I have arms but can not reach. I run all
> day but have no legs. what am I.
A watch ?
===
Subject: numerically determine area and circumference
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA71kvi26532;
Hey guys
I'm currently making a small game (simple space game) and have a
problem with numerically determine area and circumference of a ball
that is being thrown.
The problem is presented here:
http://trasigkondensator.tripod.com/ball.htm
===
Subject: Re: numerically determine area and circumference
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA81mqQ17017;
>Hey guys
>I'm currently making a small game (simple space game) and have a
>problem with numerically determine area and circumference of a ball
>that is being thrown.
>The problem is presented here:
href=http://trasigkondensator.tripod.com/ball.htm>http://trasigkondensat
or.tripod.com/ball.htm</a>
Put the coordinate system at the disk center and the x-axis through
the point A = (r, 0), where the ball touches the disk. Denote S = (-r,
0) the point, where the string is attached to the disk. If the angle
of the string with the x-axis is pi/2 <= psi <= 3pi/2, the disk does
not affect the ball. Let s = pi*r be the string length. The part of
the area enclosed by the moving ball left of the point S is a
semicircle with area A0 = pi*s^2/2 = pi^3*r^2/2 and arc length L0 =
2pi*s/2 = pi^2*r. Denote B =(xB, yB) the ball and its coordinates at
some angle fi = AOB. The coordinates of the tangency point T =(xT, yT)
of the string with the disk are  
xT = r*cos(fi)
yT = r*sin(fi)
The tangent length t = TB of the string from the tangency point T to
the ball B is
t = s - (pi - fi)*r = pi*r - (pi - fi)*r = fi*r
Since the tangent length of the string TB is perpendicular to OT, the
coordinates of the ball are
xB = xT + t*cos(fi - pi/2) = r*[cos(fi) + fi*sin(fi)]
yB = yT + t*sin(fi - pi/2) = r*[sin(fi) - fi*cos(fi)]
The area A1 enclosed by the ball right of the point B is then
        +r                  pi
A1 = 2 * S yB(xB) dxB = 2 * S yB(fi) * dxB/dfi dfi 
        -r                  0 
or
         s                        pi
A1 = 2 * S [xB(yB) + r] dyB = 2 * S xB(fi) * dyB/dfi dfi 
         0                        0 
The 1st equation is slightly suspicious, because yB(xB) is not a well
defined function of xB (recall the apple shape of the desired area A),
but both equations leads to the same result. Using the 1st equation:
dxB/dfi = -r*sin(fi) + r*sin(fi) + r*fi*cos(fi) = r*fi*cos(fi)
         pi
A1 = 2 * S  r*[sin(fi) - fi*cos(fi)] * r*fi*cos(fi)] dfi = 
         0
             pi
    = 2r^2 * S  [fi^2*cos^2(fi) - fi*sin(fi)*cos(fi)] dfi = 
             0
            pi
    = r^2 * S  [fi^2*(1 + cos(2*fi)) - fi*sin(2*fi)] dfi = 
            0
           2pi
    = r^2 * S  [(mu/2)^2*(1 + cos(mu)) - mu/2*sin(mu)] dmu/2 = 
            0
             2pi
    = r^2/8 * S  [mu^2*(1 + cos(mu)) - 2mu*sin(mu)] dmu = 
              0
(integration by parts)
                                                                      
= r^2/8 * [mu^3/3 + 2*mu*cos(mu) + (mu^2 - 2)*sin(mu) -
                                   2pi
            - 2sin(mu) + 2mu*cos(mu)] 
                                    0                 
    = r^2/8 * [(2pi)^3/3 + 4pi + 4pi] = pi*r^2*(pi^2/3 + 1)
and the total area A is
A = A0 + A1 = pi^3*r^2/2 + pi*r^2*(pi^2/3 + 1) =
            = pi*r^2*(5/6*pi^2 + 1)
Using the 2nd equation:
dyB/dfi = r*cos(fi) - r*cos(fi) + r*fi*sin(fi) = r*fi*sin(fi)
         pi
A1 = 2 * S r*[1 + cos(fi) + fi*sin(fi)] * r*fi*sin(fi) dfi = 
         0
             pi
    = 2r^2 * S  [fi^2*sin^2(fi) + fi*sin(fi)*cos(fi) + fi*sin(fi)] dfi
= 
             0
            pi
    = r^2 * S  [fi^2*(1 - cos(2*fi)) + fi*sin(2*fi) +
            0
             pi
    + 2r^2 * S fi*sin(fi)] dfi = 
             0
             2pi
    = r^2/8 * S  [mu^2*(1 - cos(mu)) + 2mu*sin(mu)] dmu + 2r^2*pi  
              0
    = r^2/8 * [mu^3/3 - 2*mu*cos(mu) - (mu^2 - 2)*sin(mu) +
                                   2pi
            + 2sin(mu) - 2mu*cos(mu)] + 2pi*r^2 
                                    0
  
    = r^2/8 * [(2pi)^3/3 - 4pi - 4pi] + 2pi*r^2 =
    = pi*r^2*(pi^2/3 - 1) + 2pi*r^2 = pi*r^2*(pi^2/3 + 1)
which is the same result as from the first equation. BTW, this result
is in agreement with the numerical integration of A1 for r = 1.
The perimeter L of the total area A is the perimeter L0 of the
semicircle left of the point S plus the length of the curve yB(xB)
right of the point S.
        +r 
L1 = 2 * S sqrt(1 + (yB/dxB)^2] dxB
        -r
dyB/dxB = dyB/dfi * dfi/dxB = r*fi*sin(fi)/(r*fi*cos(fi)) = tan(fi)
sqrt(1 + tan^2(fi)) = 1/cos(fi)
         pi                                 pi 
L1 = 2 * S 1/cos(fi)*r*fi*cos(fi) df = 2r * S fi dfi = 2r*pi
         0                                  0
The total perimeter L is
L = L0 + L1 = pi^2*r + 2pi*r = 2pi*r*(pi/2 + 1)
===
Subject: numerically determine area and circumference
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA71kwm26566;
Hey guys
I'm currently making a small game (simple space game) and have a
problem with numerically determine area and circumference of a ball
that is being thrown.
The problem is presented here:
http://trasigkondensator.tripod.com/ball.htm
===
Subject: close harmonian/bla
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA71l0326644;
how do i solve:
does u=exp(2xy)cos(x^2-y^2) have close harmonian?
===
Subject: algebraic geometry
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 

1.9  primary) id iA76FAu15071;
If F is a field of characteristic p, then how could one show that 
every line passing through 0 is a tangent line to the curve y=x^(p+1)?