mm-1739
 
===
Subject: Re: Epistemology 201: The Science of Science
Allan C Cybulskie said:
>Substance is an entity? I thought maybe entities were made of
>substances.....but substances seem like continuous fields, rather than 
discrete entities.
Some entities are made of other entities. Like water is made of hydrogen and 
oxygen.
> Water is often referred to as a substance, and yet the existence of water 
seems
> to depend of the existence of hyrdogen and oxygen, which are more 
like
> objects.
> Are you saying that hyrdogen and oxygen aren't substnaces, yet water 
is?
> That must be some really good stuff you're smoking. Canadian, is it?
> I'll pick up the replies to me later, but in following the thread 
(although
> perhaps it shifted a bit, since I haven't read all of these yet) it seems
> like Tony's comment is a reply to Bob's initial definition which is that
> something is a substance if it doesn't depend on other substances for its
> existence.  By that definition, water is not a substance, since it 
depends
> on hydrogen and oxygen for its existence.
See? Allan understood what I was saying.
as opposed to substances was a fortuitous ambiguity in my speaking and 
elicited 
a good response from Wolf that made me think a minute. That was good. :)
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
>> robert j. kolker said:
>Substance is an entity? I thought maybe entities were made of
>>substances.....but substances seem like continuous fields, rather 
than
>discrete
>>entities.
>>Some entities are made of other entities. Like water is made of 
hydrogen
>and oxygen.
>>Bob Kolker
> Water is often referred to as a substance, and yet the existence of
>water seems
>> to depend of the existence of hyrdogen and oxygen, which are more like
>objects.
>> Are you saying that hyrdogen and oxygen aren't substnaces, yet water is?
>> That must be some really good stuff you're smoking. Canadian, is it?
>I'll pick up the replies to me later, but in following the thread 
(although
>perhaps it shifted a bit, since I haven't read all of these yet) it seems
>like Tony's comment is a reply to Bob's initial definition which is that
>something is a substance if it doesn't depend on other substances for its
>existence.  By that definition, water is not a substance, since it depends
>on hydrogen and oxygen for its existence.
And by that definition substance must be circular.
===
Subject: Re: Epistemology 201: The Science of Science
Lester Zick said:
>> robert j. kolker said:
>Substance is an entity? I thought maybe entities were made of
>>substances.....but substances seem like continuous fields, rather 
than
>discrete
>>entities.
>>Some entities are made of other entities. Like water is made of 
hydrogen
>and oxygen.
>>Bob Kolker
> Water is often referred to as a substance, and yet the existence of
>water seems
>> to depend of the existence of hyrdogen and oxygen, which are more 
like
>objects.
 Are you saying that hyrdogen and oxygen aren't substnaces, yet water 
is?
 That must be some really good stuff you're smoking. Canadian, is it?
>I'll pick up the replies to me later, but in following the thread 
(although
>perhaps it shifted a bit, since I haven't read all of these yet) it 
seems
>like Tony's comment is a reply to Bob's initial definition which is that
>something is a substance if it doesn't depend on other substances for 
its
>existence.  By that definition, water is not a substance, since it 
depends
>on hydrogen and oxygen for its existence.
> And by that definition substance must be circular.
Or mathematical.
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
>robert j. kolker said:
>Substance is an entity? I thought maybe entities were made of
>substances.....but substances seem like continuous fields, rather than
> discrete
>entities.
>>Some entities are made of other entities. Like water is made of 
hydrogen
>>and oxygen.
>>Bob Kolker
>Water is often referred to as a substance, and yet the existence of
> water seems
>to depend of the existence of hyrdogen and oxygen, which are more like
> objects.
>>Are you saying that hyrdogen and oxygen aren't substnaces, yet water is?
>>That must be some really good stuff you're smoking. Canadian, is it?
> I'll pick up the replies to me later, but in following the thread 
(although
> perhaps it shifted a bit, since I haven't read all of these yet) it seems
> like Tony's comment is a reply to Bob's initial definition which is that
> something is a substance if it doesn't depend on other substances for its
> existence.  By that definition, water is not a substance, since it 
depends
> on hydrogen and oxygen for its existence.
Yes, those mathematiker boys have trouble staying on topic.  They 
seem to do fine when a reply can be quoted from a textbook, but 
are completely lost outside of their Hilbert space.
-- 
I consider it quite possible that physics cannot be based
on the field concept, i. e., on continuous structures. In that
case nothing remains of my entire castle in the air,
gravitation theory included, [and of] the rest of modern physics.
        -- Albert Einstein in a 1954 letter to Michele Besso.
===
Subject: Re: Epistemology 201: The Science of Science
>robert j. kolker said:
>Substance is an entity? I thought maybe entities were made of
>substances.....but substances seem like continuous fields, rather than
> discrete
>entities.
>>Some entities are made of other entities. Like water is made of 
hydrogen
>>and oxygen.
>>Bob Kolker
>Water is often referred to as a substance, and yet the existence of
> water seems
>to depend of the existence of hyrdogen and oxygen, which are more like
> objects.
>>Are you saying that hyrdogen and oxygen aren't substnaces, yet water is?
>>That must be some really good stuff you're smoking. Canadian, is it?
> I'll pick up the replies to me later, but in following the thread 
(although
> perhaps it shifted a bit, since I haven't read all of these yet) it seems
> like Tony's comment is a reply to Bob's initial definition which is that
> something is a substance if it doesn't depend on other substances for its
> existence.  By that definition, water is not a substance, since it 
depends
> on hydrogen and oxygen for its existence.
Water IS hydrogen and oxygen, so hydrogen and oxygen are closer to the 
substance of water. But of what is hydrogen and oxygen made? Well, there 
are the subatomic components so they are close to the basic stuff of 
water etc.
So a substance the basic stuff of which things consist. What we see are 
the appearences of the substance, i.e. its form, in the Aristotelean 
sense. The not well defined notion of substance is that which stands 
a search for -substance- in the old metaphysical meaning of the word. 
String or brane theory would be the ultimate theory accounting for the 
existence and composition of things that are perceived or inferred.
See:
http://www.bris.ac.uk/Depts/Philosophy/UG/Studyguide/meta.html
Identity, Substance and Ontology
What is it for the table in my office to be the same as the one I used 
to have at home, or the same as the one that was in my office yesterday? 
  Judgements that some entity A is identical to some entity B seem to be 
either trivial or false: if A really is the same as B then we are just 
saying that A is A which is analytically true and knowable a priori, 
since it is a law of logic that everything is identical to itself; on 
the other hand if we are saying that some entity A is identical to 
another entity B, then this is surely false for no two entities can be 
the same entity.  The classic discussion of identity statements is 
FregeÍs ïSense and ReferenceÍ.
Identity and necessity
Saul Kripke, in Naming and Necessity, put forward the revolutionary view 
that identity statements can be necessary but knowable only a 
posteriori.  He also argued that there exist truths which are contingent 
but knowable a priori.  See the collection, Naming, Necessity, and 
Natural Kinds, edited by S.P.Schwartz, and N.Salmon, Frege's Puzzle; see 
also Part II of J.Kim and E.Sosa (eds), Metaphysics: An Anthology.
Identity over Time
What is it for the table in my office to be the same table as the one 
that was there yesterday?  If I replace one of the legs, is it the same 
table?  What if I replace all of them?  In virtue of what can an entity 
be said to persist while its properties and parts change?  There are two 
main theories of persistence, the endurance theory which says that 
ordinary objects are wholly present at moments in time and that we can 
see numerically identical objects at different times, and the perdurance 
theory which says that ordinary objects, like tables, are stretched out 
in time just as they are in space having 'temporal parts' just as they 
have spatial parts.
M.Loux, Metaphysics: a contemporary introduction, chapter 6
S.Shoemaker, ïIdentity, Properties and 
CausalityÍ, in his Identity, 
Cause and Mind
D.M.Armstrong, ïIdentity through TimeÍ, in P.van 

Inwagen (ed.), Time and 
Cause
M.Johnston, ïIs there a Problem about 
Persistence?Í, Proceedings of the 
Aristotelian Society Supplementary Volume, 1987
D.Lewis, On the Plurality of Worlds, pp. 202-206
P.van Inwagen, '4D Objects', Nous 1990
J.Kim and E.Sosa (eds), Metaphysics: An Anthology, Part V
Substance
Are objects mere bundles of properties or do the properties inhere in 
some substratum?  Substance has been defined as that which does not 
depend for its existence on anything else.
Bob Kolker
===
Subject: Re: Epistemology 201: The Science of Science
>>robert j. kolker said:
>Substance is an entity? I thought maybe entities were made of
>>substances.....but substances seem like continuous fields, rather 
than
>> discrete
>>entities.
>>Some entities are made of other entities. Like water is made of 
hydrogen
>and oxygen.
>>Bob Kolker
>Water is often referred to as a substance, and yet the existence of
>> water seems
>>to depend of the existence of hyrdogen and oxygen, which are more like
>> objects.
>Are you saying that hyrdogen and oxygen aren't substnaces, yet water is?
>That must be some really good stuff you're smoking. Canadian, is it?
>> I'll pick up the replies to me later, but in following the thread 
(although
>> perhaps it shifted a bit, since I haven't read all of these yet) it 
seems
>> like Tony's comment is a reply to Bob's initial definition which is that
>> something is a substance if it doesn't depend on other substances for 
its
>> existence.  By that definition, water is not a substance, since it 
depends
>> on hydrogen and oxygen for its existence.
>Water IS hydrogen and oxygen, so hydrogen and oxygen are closer to the 
>substance of water. But of what is hydrogen and oxygen made? Well, there 
>are the subatomic components so they are close to the basic stuff of 
>water etc.
>So a substance the basic stuff of which things consist. What we see are 
>the appearences of the substance, i.e. its form, in the Aristotelean 
>sense. The not well defined notion of substance is that which stands 
>a search for -substance- in the old metaphysical meaning of the word. 
>String or brane theory would be the ultimate theory accounting for the 
>existence and composition of things that are perceived or inferred.
Yes, well, Bob, I imagine there is a definite fullness here to your
pseudo scientific ruminations wherein you acknowledge no idea
concerning the substance of your substances or the truth of your
facts wherein there is no finite tautological regression to self
contradictory alternatives to define an otherwise infinite regression
of substances. That was Aristotle's problem as well. The notion of
substance is not well defined because there is no substantial limit to
the regress. 
>See:
>http://www.bris.ac.uk/Depts/Philosophy/UG/Studyguide/meta.html
>Identity, Substance and Ontology
>What is it for the table in my office to be the same as the one I used 
>to have at home, or the same as the one that was in my office yesterday? 
>  Judgements that some entity A is identical to some entity B seem to be 
>either trivial or false: if A really is the same as B then we are just 
>saying that A is A which is analytically true and knowable a priori, 
>since it is a law of logic that everything is identical to itself; on 
>the other hand if we are saying that some entity A is identical to 
>another entity B, then this is surely false for no two entities can be 
>the same entity.  The classic discussion of identity statements is 
>Frege.89s ëSense and Reference.89.
>Identity and necessity
>Saul Kripke, in Naming and Necessity, put forward the revolutionary view 
>that identity statements can be necessary but knowable only a 
>posteriori.  He also argued that there exist truths which are contingent 
>but knowable a priori.  See the collection, Naming, Necessity, and 
>Natural Kinds, edited by S.P.Schwartz, and N.Salmon, Frege's Puzzle; see 
>also Part II of J.Kim and E.Sosa (eds), Metaphysics: An Anthology.
>Identity over Time
>What is it for the table in my office to be the same table as the one 
>that was there yesterday?  If I replace one of the legs, is it the same 
>table?  What if I replace all of them?  In virtue of what can an entity 
>be said to persist while its properties and parts change?  There are two 
>main theories of persistence, the endurance theory which says that 
>ordinary objects are wholly present at moments in time and that we can 
>see numerically identical objects at different times, and the perdurance 
>theory which says that ordinary objects, like tables, are stretched out 
>in time just as they are in space having 'temporal parts' just as they 
>have spatial parts.
>M.Loux, Metaphysics: a contemporary introduction, chapter 6
>S.Shoemaker, ëIdentity, Properties and Causality.89, in 
his Identity, 
>Cause and Mind
>D.M.Armstrong, ëIdentity through Time.89, in P.van Inwagen 
(ed.), Time and 
>Cause
>M.Johnston, ëIs there a Problem about Persistence?.89, 
Proceedings of the 
>Aristotelian Society Supplementary Volume, 1987
>D.Lewis, On the Plurality of Worlds, pp. 202-206
>P.van Inwagen, '4D Objects', Nous 1990
>J.Kim and E.Sosa (eds), Metaphysics: An Anthology, Part V
>Substance
>Are objects mere bundles of properties or do the properties inhere in 
>some substratum?  Substance has been defined as that which does not 
>depend for its existence on anything else.
>Bob Kolker
===
Subject: Re: Epistemology 201: The Science of Science
robert j. kolker said:
> 1. Consistency
> Consistency is a requirement for any theory. With incosistent theories 
> all well formed formulae can be derived as theorems.
> 2. Science
> Science as a method? It is no guarantee of truth at all. The only thing 
> we can be sure of is when a scientific theory is falsified empirically. 
> There is no way to determine whether all the predictions of a scientific 
> theory will be empirically corroberated.
The fact that science works is evidence of the consistency in the universe. 

So 
much for concise language.
> Bob Kolker
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
>> It is quite well defined as a class of undefined's
>Is the set of integers defined? Yes or No.
Well, Bob, I find you curiously demanding when it comes to other
peoples' definitions but singularly unforthcoming when it comes to
your own definitions, as in the universal implications of true facts.
Integers are certainly well defined. Whether that means your sets of
integers are well defined or even defined in all respects is another
matter requiring definition of sets. If your sets are in a one to one
mapping with the integers in them I would be inclined to say that they
are poorly defined.
>                                                                   As a one 

to one onto mapping 
>well defined? Yes or No.
One to one mapping is no better defined than things mapped or for that
matter things counted since you assume counting and mapping are more
basic than things counted or mapped. Basically my problem with modern
mathematikers is they profess their interest in counting countables
but then spend all their time counting uncountable countables.
===
Subject: Re: Epistemology 201: The Science of Science
robert j. kolker said:
> That's the definition? I thought it was the cardinality of the set of 
> rationals.........
> The rationals and the integers have the same cardinality. A one to one 
> function from the integers onto the rationals can be defined. Countable 
> is countable.
> Bob Kolker
Right Bob, that's why that definition is anything but exact or precise. If I 

say a chicken is a bird, does that tell you what a bird is? What proof do we 

have that Cantor's cardinality measure is valid for infinite sets? Is it 
empirically verified, or simply assumed as an axiom?
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
Daryl McCullough said:
> aeo6 says...
>robert j. kolker said:
>> You are a flat-earther. Aleph-0 is perfectly well defined. It is the 
>> cardinality of the set of integers.
>That's the definition? I thought it was the cardinality of the set of 
>rationals.........
>Okay, maybe not PERFECTLY well defined......
> Yes, it is perfectly well defined.
>    The cardinality of the rationals 
>  = The cardinality of the naturals
>  = The cardinality of the integers
>  = The cardinality of the finite character strings
>  = The cardinality of the finite ordinals
>  = The cardinality of the even integers
>  = The cardinality of the prime numbers
>  = aleph_0
> --
> Daryl McCullough
> Ithaca, NY
As well defined as animal
Animal=
The group that includes people
the group that include fish
the group that includes birds
the group that includes amphibians
the group that includes chickens
the group that includes termites
the group that includes worms
=animals
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
Lester Zick said:
>Lester Zick said:
>> 
>>Lester Zick said:
> 
>Wolf Kirchmeir said:
>> 
>> 
>AFAIK, photons don't have as structure, so if you have 
evidence to the 
>contrary, I'd like to know where I can find it.
>>Photons have wavelength and polarization.
>>True, but I don't think that's what Lester has in mind. Of 
course, I 
>could be wrong - it's hard to tell what Lester has in mind. 
His talk 
>about accounting for spin in mechanical terms suggests to 
me that he 
>thinks a photon is like a little spinning billiard ball.
>>Lester, care to explain? Curious minds want to know.
>>A couple of weeks back, Wolf, I posed to curious minds whether 
they
>>for the electron and proton apparently represents the radial 
integral
>>of radial velocity? To date the question has been studiously 
ignored
>>by all these curious minds.
>>Well, if you could explain what a radial integral is, and what 
a radial 
>velcoity is, and how you integrate an radial velocity, and why 
this is a 
>defintion or calculation of spin, I might understand what you 
have in mind.
>> 
>> 
>> Yeah, Wolf. Fortunately I don't need you to do my thinking for 
me but
>> you badly seem to need someone to do it for you. I guessed I 
deserve
>> it when I ask the incompetent. Maybe in your next profession 
you can
>> learn some physics before pontificating about curious minds 
and
>> 
>> 
>> You have your own definition of spin, which is not the same as 
the 
>> are length^2/time. But the units of the expression you quote (1/2 
>> h/2pi)includes mass. So either you have discovered something that 
other 
>> physicists haven't discovered, in which case a write up of your 
>> experimental setup and data is in order; or else you are 
producing 
>> pseudo-scientific claptrap.
>> 
>Actually, Lester, I think Wolf has a point. When I was going 
through your 
>previously you had been speaking purely of spacetime and geometry. 
My original 
>question was about the source of mass, and yet somehow your model 
seems to 
>introduce mass spontaneously without mention. Of course, I may have 
your out, 
>but only if you pay attention to the question I asked. :)
> 
> Oh, I paid attention all right, Tony. But I'm wondering if the mass
> you're referring to isn't the analytical masslet m0 ~ 10^-50kg-sec
> I used to define the origin of Planck's constant rather than mass 
in
> absolute terms.
>> 
>>I meant mass in absolute terms. I am not sure where the value for this 
masslet 
>>comes from except as a retrofitted number. The question as to what 
causes mass 
>>at all is what I was asking. What causes this masslet?
>> 
>> by rotational frequency dependent on the ratio of radial velocity in
>> relation to c. I have no explanation as to why there are only the two
>> stable ratios, one for the electron and one for the proton. Electron
>> mass is probably minimal and reflects some fundamental aspect of
>> interaction with the plenum. I imagine the proton is some convoluted
>> tiny flux of geometrically stable interaction with the plenum.
>If protons and electrons only differ in radius/energy, then how do you 
account 
>for opposite charge? Doesn't the proton have a more complex structure 
than an 
>electron?
> Existence of positrons and negative protons shows charge is a
> symmetric property. I don't explain it exactly at present but I'm
> pretty sure it's explicable. I assume proton structure is a reflection
> amount to.
Things yet to figger out, as always, indeed. And the band plays on......
>   
> itself is defined in terms of the magnitude of radially directed
> r/t2pi in relation to tangential r/t which is just the constant
> velocity of translation of light through space. The radial velocity
> r/t2pi can be of any finite magnitude and defines the radius of
>> 
>>This part makes sense to me. Higher energy/mass means higher frequency 
(as in 
>>light), which means smaller radius, assuming constant tangetntial 
velocity. I 
>>am not sure what eveidence there is for higher mass being associated 
with 
>>smaller radius, but it still makes abstract sense.
>> 
>> Yeah, it's basically a similar approach as used in macro angular
>> mechanics except that it accounts for constant rotational velocity
>> cases more accurately or more precisely or whatever. When the
>> tangential velocity is fixed, reducing the radius of rotation
>> increases energy with a constant spin whereas conventional macro
>> angular mechanics shows decreasing angular momentum L = r x p.
>> even though spin looks a lot like angular momentum.
>That makes some sense. If rotation is always at a constant radial 
velocity then 
>they would all fit in some way like clockwork gears.  I can see that...
> Good, Tony. It's a considerably more sensible explanation than trying
> to see things in terms of macro angular momentum and oscillating
> rotating transverse waveforms the way classical physics did. Macro
> angular mechanics mainly deals in rigid body rotation of fixed radius
> which yields incorrect interpretations of rr/t mechanics for angular
> momentum as r changes.
>> is Heisenberg's uncertainty relation. That's why this relation is what
>> it is and what it identifies. People were just so used to visualizing
>> the low mass electron as tiny and the high mass proton as large that
>> they never stopped to regress alternatives tautologically.
>That seems like a strange visualization, if it's widespread, since the 
electron 
>cloud occupies most of the space of an atom, and the protons are all 
squished 
>on the surface of the nucleus. Also, given the known realtionship between 
>frequency and energy for photons, one might think that idea was extended 
to 
> Of course, you're right, Tony. I can only say that my own previous
> interpretations rested on a direct equivalence between mass and size.
> Even today people look askance when I say it. So I'm not surprized.
I guess it's true that in the macro world, bigger and similar means heavier. 

I 
guess that's one of the backwards kind of things one has to get used to with 

QM.
>   
> I use m0 simply to postulate some characteristic property of the
> energy is any other terms. Wolf's point seemed to be that analysis 
of
> Planck's constant depended on mass, but that isn't correct. I only
> have to regress mass to the plenum in some way consistent with
>I still say that an object moving with a radial velocity of c, despite 
the 
>radius, would at any given moment have a translational velocity of c and 
>it could acquire relativistic mass that would look like absolute mass 
from 
>outside the rotation. The relativistic mass would probably be 
proportional to 
>the frequency, as with free photons.
> Okay. I can see the plausibility of your idea. But that assumes
> relativistic mass increase is the explanation and not what is to be
> explained mechanically by means of other residual constants and
> interactions. Nor would your explanation explain Planck's constant or
of c. I can't comment on Planck's constant at this time, but I am not sure 
you 
explained Planck's constant anyway. It seemed like you derived the unit 
masslet 
using Planck's constant, creating a new relation, but not deriving the 
constant 
from more elementary constants and relations. Did I miss something?
>>Are you sure that there is no relativistic component creating the mass 
from a 
>>masslesslet, due to the tangential velocity of c? Wouldn't the radial 
>>acceleration then be proportional to the mass as always?
>> 
>> dependent on the bidirectional average velocity of rotation in
>> relation to the velocity of light. At velocity this increases because
>> reverse side the rotation is shorter but the average for both
>This sounds like you are saying there is radial inertia that acts 
according to 
>anyone else have comment on this?
> relations as well.
relativistically due to tangential velocity of c would wreck any of what you 

have. I think you might find it helps. Come on, try it. First one's 
free.....
>> structure to mention of the kind in photons that causes cosmic
>> indefinitely. (I considered mentioning this earlier because I just
>> knew someone would think of it but decided it would complicate the
>> subject unnecessarily at this point.)
>what everyone says about photons. I don't think photons have any distinct 
>structure, but are 2D wavelets passing through 3D space, and they are not 
>supposed to slow down, since their masslessness requires that any energy 
they 
>have be translated into velocity of c. I think it's exactly this initial 
>masslessness you should be looking at for the source of mass IMHO.
> Well, there's a very good reason they all look similar, Tony. I think
> waves of light in space rotating only because they are half and not
> whole waves like photons. 
But electrons absorb photons and turn them into increased energy levels, and 

release photons when they drop back down to lower energy levels. How can a 
half 
electrons and protons have a level of 3D structure that is missing in 
photons, 
but of course I've never seen either one in detail. 
> And of course you're at liberty to visualize
> spin characteristics, Planck's constant and the Heisenberg uncertainty
> same wavelength.
So to speak ;)
Yes I share your desire for a more mechanical explanation tying all 
these 
constants together and explaining things in simplest possible form. Of 
course, I have been working, when I find time, on other things too....:)
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
>Lester Zick said:
>>Lester Zick said:
> 
>Lester Zick said:
>> 
>>Wolf Kirchmeir said:
> 
> 
>AFAIK, photons don't have as structure, so if you have 
evidence to the 
>>contrary, I'd like to know where I can find it.
>Photons have wavelength and polarization.
>True, but I don't think that's what Lester has in mind. Of 
course, I 
>>could be wrong - it's hard to tell what Lester has in mind. 
His talk 
>>about accounting for spin in mechanical terms suggests 
to me that he 
>>thinks a photon is like a little spinning billiard ball.
>>Lester, care to explain? Curious minds want to know.
>A couple of weeks back, Wolf, I posed to curious minds 
whether they
>for the electron and proton apparently represents the radial 
integral
>of radial velocity? To date the question has been studiously 
ignored
>by all these curious minds.
>Well, if you could explain what a radial integral is, and what 
a radial 
>>velcoity is, and how you integrate an radial velocity, and why 
this is a 
>>defintion or calculation of spin, I might understand what you 
have in mind.
> 
> 
> Yeah, Wolf. Fortunately I don't need you to do my thinking for 
me but
> you badly seem to need someone to do it for you. I guessed I 
deserve
> it when I ask the incompetent. Maybe in your next profession 
you can
> learn some physics before pontificating about curious minds 
and
> 
> 
> You have your own definition of spin, which is not the same as 
the 
> are length^2/time. But the units of the expression you quote 
(1/2 
> h/2pi)includes mass. So either you have discovered something 
that other 
> physicists haven't discovered, in which case a write up of your 
> experimental setup and data is in order; or else you are 
producing 
> pseudo-scientific claptrap.
> 
>>Actually, Lester, I think Wolf has a point. When I was going 
through your 
>>previously you had been speaking purely of spacetime and geometry. 
My original 
>>question was about the source of mass, and yet somehow your model 
seems to 
>>introduce mass spontaneously without mention. Of course, I may 
have your out, 
>>but only if you pay attention to the question I asked. :)
>> 
>> Oh, I paid attention all right, Tony. But I'm wondering if the 
mass
>> you're referring to isn't the analytical masslet m0 ~ 10^-50kg-sec
>> I used to define the origin of Planck's constant rather than mass 
in
>> absolute terms.
> 
>I meant mass in absolute terms. I am not sure where the value for 
this masslet 
>comes from except as a retrofitted number. The question as to what 
causes mass 
>at all is what I was asking. What causes this masslet?
> 
> by rotational frequency dependent on the ratio of radial velocity in
> relation to c. I have no explanation as to why there are only the two
> stable ratios, one for the electron and one for the proton. Electron
> mass is probably minimal and reflects some fundamental aspect of
> interaction with the plenum. I imagine the proton is some convoluted
> tiny flux of geometrically stable interaction with the plenum.
>>If protons and electrons only differ in radius/energy, then how do you 
account 
>>for opposite charge? Doesn't the proton have a more complex structure 
than an 
>>electron?
>> Existence of positrons and negative protons shows charge is a
>> symmetric property. I don't explain it exactly at present but I'm
>> pretty sure it's explicable. I assume proton structure is a reflection
>> amount to.
>Things yet to figger out, as always, indeed. And the band plays on......
>>   
>> itself is defined in terms of the magnitude of radially directed
>> r/t2pi in relation to tangential r/t which is just the constant
>> velocity of translation of light through space. The radial 
velocity
>> r/t2pi can be of any finite magnitude and defines the radius of
> 
>This part makes sense to me. Higher energy/mass means higher 
frequency (as in 
>light), which means smaller radius, assuming constant tangetntial 
velocity. I 
>am not sure what eveidence there is for higher mass being associated 
with 
>smaller radius, but it still makes abstract sense.
> 
> Yeah, it's basically a similar approach as used in macro angular
> mechanics except that it accounts for constant rotational velocity
> cases more accurately or more precisely or whatever. When the
> tangential velocity is fixed, reducing the radius of rotation
> increases energy with a constant spin whereas conventional macro
> angular mechanics shows decreasing angular momentum L = r x p.
> even though spin looks a lot like angular momentum.
>>That makes some sense. If rotation is always at a constant radial 
velocity then 
>>they would all fit in some way like clockwork gears.  I can see that...
>> Good, Tony. It's a considerably more sensible explanation than trying
>> to see things in terms of macro angular momentum and oscillating
>> rotating transverse waveforms the way classical physics did. Macro
>> angular mechanics mainly deals in rigid body rotation of fixed radius
>> which yields incorrect interpretations of rr/t mechanics for angular
>> momentum as r changes.
>>  
> is Heisenberg's uncertainty relation. That's why this relation is 
what
> it is and what it identifies. People were just so used to visualizing
> the low mass electron as tiny and the high mass proton as large that
> they never stopped to regress alternatives tautologically.
>>That seems like a strange visualization, if it's widespread, since the 
electron 
>>cloud occupies most of the space of an atom, and the protons are all 
squished 
>>on the surface of the nucleus. Also, given the known realtionship 
between 
>>frequency and energy for photons, one might think that idea was extended 
to 
>> Of course, you're right, Tony. I can only say that my own previous
>> interpretations rested on a direct equivalence between mass and size.
>> Even today people look askance when I say it. So I'm not surprized.
>I guess it's true that in the macro world, bigger and similar means 
heavier. I 
>guess that's one of the backwards kind of things one has to get used to 
with 
>QM.
Now at least we know what we're talking about for a change.
   
>> I use m0 simply to postulate some characteristic property of the
>> energy is any other terms. Wolf's point seemed to be that analysis 
of
>> Planck's constant depended on mass, but that isn't correct. I only
>> have to regress mass to the plenum in some way consistent with
>>I still say that an object moving with a radial velocity of c, despite 
the 
>>radius, would at any given moment have a translational velocity of c and 
>>it could acquire relativistic mass that would look like absolute mass 
from 
>>outside the rotation. The relativistic mass would probably be 
proportional to 
>>the frequency, as with free photons.
>> Okay. I can see the plausibility of your idea. But that assumes
>> relativistic mass increase is the explanation and not what is to be
>> explained mechanically by means of other residual constants and
>> interactions. Nor would your explanation explain Planck's constant or
>of c. I can't comment on Planck's constant at this time, but I am not sure 

you 
>explained Planck's constant anyway. It seemed like you derived the unit 
masslet 
>using Planck's constant, creating a new relation, but not deriving the 
constant 
>from more elementary constants and relations. Did I miss something?
Actually not, Tony. This is a reasonable assessment.  I have regressed
Planck's constant to more fundamental considerations without proving
those more fundamental considerations independently. So in this sense
I have only regressed Planck's constant without explaining it in more
fundamental terms.
However I would add that proof of the regression relies on the self
characteristics defined in terms of Planck's constant. In other words
there is no other interpretation consistent with the facts of constant
suggest that in the absence of any other constant such as c, m0 is
proven by regression through Planck's constant. Planck's constant is
properties are regressed in terms of the natural constant c to m0.
 
>Are you sure that there is no relativistic component creating the 
mass from a 
>masslesslet, due to the tangential velocity of c? Wouldn't the radial 
>acceleration then be proportional to the mass as always?
> 
> dependent on the bidirectional average velocity of rotation in
> relation to the velocity of light. At velocity this increases because
> reverse side the rotation is shorter but the average for both
>>This sounds like you are saying there is radial inertia that acts 
according to 
>>anyone else have comment on this?
>> relations as well.
>relativistically due to tangential velocity of c would wreck any of what 
you 
>have. I think you might find it helps. Come on, try it. First one's 
free.....
I have been considering the idea, Tony. The problem is that it does
kinda wreck things by mucking them up. Let's say I go along with the
maybe, moving at c. So we have zero mass transmogrified into finite
mass through an infinite increase in mass. So far so good.
spin and produce the uncertainty constant inversely proportional to
mass? Then how does relativistic rotational latency correlate with
relativistic mass increase dependent on velocity because that latency
certainly correlates with something? These are exactly the kinds of
considerations we have to explain in terms of one another and not by
resort to ad hoc possibilities.
 
> structure to mention of the kind in photons that causes cosmic
> indefinitely. (I considered mentioning this earlier because I just
> knew someone would think of it but decided it would complicate the
> subject unnecessarily at this point.)
>>  
>>what everyone says about photons. I don't think photons have any 
distinct 
>>structure, but are 2D wavelets passing through 3D space, and they are 
not 
>>supposed to slow down, since their masslessness requires that any energy 
they 
>>have be translated into velocity of c. I think it's exactly this initial 
>>masslessness you should be looking at for the source of mass IMHO.
>> Well, there's a very good reason they all look similar, Tony. I think
>> waves of light in space rotating only because they are half and not
>> whole waves like photons.
 
>But electrons absorb photons and turn them into increased energy levels, 
and 
>release photons when they drop back down to lower energy levels. How can a 

half 
>electrons and protons have a level of 3D structure that is missing in 
photons, 
>but of course I've never seen either one in detail. 
This is exactly why I don't like analogical reasoning, Tony, because
I'm dealing at a speculative level below what I can explain
mechanically. How all this happens depends on what various terms are
taken to indicate exactly. And so far all I've done is sketch in the
basics in mechanical terms. I think you're probably right that protons
at least have some structure, and that is something I think topology
could work on constructively. But I don't see the lack of information
on such issues as anything more than missing links. I don't think they
represent any kind of basic threat to the ideas I've developed so far.
>> And of course you're at liberty to visualize
>> spin characteristics, Planck's constant and the Heisenberg uncertainty
>> same wavelength.
>So to speak ;)
>Yes I share your desire for a more mechanical explanation tying all 
these 
>constants together and explaining things in simplest possible form. Of 
>course, I have been working, when I find time, on other things too....:)
Go for it, Tony. No one ever said science is content with what it
knows. I think for my part though I'm pretty much content with what I
do know of the mechanics involved because I have other tools in the
fire as well.
===
Subject: Re: Epistemology 201: The Science of Science
Lester Zick said:
>Lester Zick said:
>> 
>>Lester Zick said:
> 
>Lester Zick said:
>> 
>>Wolf Kirchmeir said:
> 
> 
>AFAIK, photons don't have as structure, so if you have 
evidence to the 
>>contrary, I'd like to know where I can find it.
>Photons have wavelength and polarization.
>True, but I don't think that's what Lester has in mind. Of 
course, I 
>>could be wrong - it's hard to tell what Lester has in 
mind. His talk 
>>about accounting for spin in mechanical terms suggests 
to me that he 
>>thinks a photon is like a little spinning billiard ball.
>>Lester, care to explain? Curious minds want to know.
>A couple of weeks back, Wolf, I posed to curious minds 
whether they
>for the electron and proton apparently represents the 
radial integral
>of radial velocity? To date the question has been 
studiously ignored
>by all these curious minds.
>Well, if you could explain what a radial integral is, and 
what a radial 
>>velcoity is, and how you integrate an radial velocity, and 
why this is a 
>>defintion or calculation of spin, I might understand what 
you have in mind.
> 
> 
> Yeah, Wolf. Fortunately I don't need you to do my thinking 
for me but
> you badly seem to need someone to do it for you. I guessed I 
deserve
> it when I ask the incompetent. Maybe in your next profession 
you can
> learn some physics before pontificating about curious minds 
and
> 
> 
> You have your own definition of spin, which is not the same as 
the 
> are length^2/time. But the units of the expression you quote 
(1/2 
> h/2pi)includes mass. So either you have discovered something 
that other 
> physicists haven't discovered, in which case a write up of 
your 
> experimental setup and data is in order; or else you are 
producing 
> pseudo-scientific claptrap.
> 
>>Actually, Lester, I think Wolf has a point. When I was going 
through your 
>>previously you had been speaking purely of spacetime and 
geometry. My original 
>>question was about the source of mass, and yet somehow your 
model seems to 
>>introduce mass spontaneously without mention. Of course, I may 
have your out, 
>>but only if you pay attention to the question I asked. :)
>> 
>> Oh, I paid attention all right, Tony. But I'm wondering if the 
mass
>> you're referring to isn't the analytical masslet m0 ~ 
10^-50kg-sec
>> I used to define the origin of Planck's constant rather than mass 
in
>> absolute terms.
> 
>I meant mass in absolute terms. I am not sure where the value for 
this masslet 
>comes from except as a retrofitted number. The question as to what 
causes mass 
>at all is what I was asking. What causes this masslet?
> 
> by rotational frequency dependent on the ratio of radial velocity 
in
> relation to c. I have no explanation as to why there are only the 
two
> stable ratios, one for the electron and one for the proton. 
Electron
> mass is probably minimal and reflects some fundamental aspect of
> interaction with the plenum. I imagine the proton is some 
convoluted
> tiny flux of geometrically stable interaction with the plenum.
>> 
>>If protons and electrons only differ in radius/energy, then how do you 
account 
>>for opposite charge? Doesn't the proton have a more complex structure 
than an 
>>electron?
>> 
>> Existence of positrons and negative protons shows charge is a
>> symmetric property. I don't explain it exactly at present but I'm
>> pretty sure it's explicable. I assume proton structure is a reflection
>> amount to.
>Things yet to figger out, as always, indeed. And the band plays on......
>>   
>> itself is defined in terms of the magnitude of radially directed
>> r/t2pi in relation to tangential r/t which is just the constant
>> velocity of translation of light through space. The radial 
velocity
>> r/t2pi can be of any finite magnitude and defines the radius of
> 
>This part makes sense to me. Higher energy/mass means higher 
frequency (as in 
>light), which means smaller radius, assuming constant tangetntial 
velocity. I 
>am not sure what eveidence there is for higher mass being 
associated with 
>smaller radius, but it still makes abstract sense.
> 
> Yeah, it's basically a similar approach as used in macro angular
> mechanics except that it accounts for constant rotational velocity
> cases more accurately or more precisely or whatever. When the
> tangential velocity is fixed, reducing the radius of rotation
> increases energy with a constant spin whereas conventional macro
> angular mechanics shows decreasing angular momentum L = r x p.
> even though spin looks a lot like angular momentum.
>> 
>>That makes some sense. If rotation is always at a constant radial 
velocity then 
>>they would all fit in some way like clockwork gears.  I can see 
that...
>> 
>> Good, Tony. It's a considerably more sensible explanation than trying
>> to see things in terms of macro angular momentum and oscillating
>> rotating transverse waveforms the way classical physics did. Macro
>> angular mechanics mainly deals in rigid body rotation of fixed radius
>> which yields incorrect interpretations of rr/t mechanics for angular
>> momentum as r changes.
>>  
> is Heisenberg's uncertainty relation. That's why this relation is 
what
> it is and what it identifies. People were just so used to 
visualizing
> the low mass electron as tiny and the high mass proton as large 
that
> they never stopped to regress alternatives tautologically.
>> 
>>That seems like a strange visualization, if it's widespread, since the 
electron 
>>cloud occupies most of the space of an atom, and the protons are all 
squished 
>>on the surface of the nucleus. Also, given the known realtionship 
between 
>>frequency and energy for photons, one might think that idea was 
extended to 
>> 
>> Of course, you're right, Tony. I can only say that my own previous
>> interpretations rested on a direct equivalence between mass and size.
>> Even today people look askance when I say it. So I'm not surprized.
>I guess it's true that in the macro world, bigger and similar means 
heavier. I 
>guess that's one of the backwards kind of things one has to get used to 
with 
>QM.
> Now at least we know what we're talking about for a change.
Change is good :)
>    
>> I use m0 simply to postulate some characteristic property of the
>> energy is any other terms. Wolf's point seemed to be that 
analysis of
>> Planck's constant depended on mass, but that isn't correct. I 
only
>> have to regress mass to the plenum in some way consistent with
>> 
>>I still say that an object moving with a radial velocity of c, despite 
the 
>>radius, would at any given moment have a translational velocity of c 
and 
>>it could acquire relativistic mass that would look like absolute mass 
from 
>>outside the rotation. The relativistic mass would probably be 
proportional to 
>>the frequency, as with free photons.
>> 
>> Okay. I can see the plausibility of your idea. But that assumes
>> relativistic mass increase is the explanation and not what is to be
>> explained mechanically by means of other residual constants and
>> interactions. Nor would your explanation explain Planck's constant or
>of c. I can't comment on Planck's constant at this time, but I am not 
sure you 
>explained Planck's constant anyway. It seemed like you derived the unit 
masslet 
>using Planck's constant, creating a new relation, but not deriving the 
constant 
>from more elementary constants and relations. Did I miss something?
> Actually not, Tony. This is a reasonable assessment.  I have regressed
> Planck's constant to more fundamental considerations without proving
> those more fundamental considerations independently. So in this sense
> I have only regressed Planck's constant without explaining it in more
> fundamental terms.
> However I would add that proof of the regression relies on the self
> characteristics defined in terms of Planck's constant. In other words
> there is no other interpretation consistent with the facts of constant
> suggest that in the absence of any other constant such as c, m0 is
> proven by regression through Planck's constant. Planck's constant is
> properties are regressed in terms of the natural constant c to m0.
I suspect that Planck's constant and c are intimately connected and 
determined 
by similar underlying variables. It is quite likely that the values are 
basically arbitrary, and vary from universelet to universelet, depending on 

their particular dimensional characteristics. So, if that is the case, there 

may be no direct derivation of certain root variables, but only 
identification 
and measurement of them, and integration of them into a cohesive picture. 
Sounds like that's what you're working on. Keep up the good work, you crank 

;)
Yeah I know, takes one to know one. Consider it a complement! :)
>Are you sure that there is no relativistic component creating the 
mass from a 
>masslesslet, due to the tangential velocity of c? Wouldn't the 
radial 
>acceleration then be proportional to the mass as always?
> 
> dependent on the bidirectional average velocity of rotation in
> relation to the velocity of light. At velocity this increases 
because
> reverse side the rotation is shorter but the average for both
>> 
>>This sounds like you are saying there is radial inertia that acts 
according to 
>>anyone else have comment on this?
>> 
>> relations as well.
>relativistically due to tangential velocity of c would wreck any of what 
you 
>have. I think you might find it helps. Come on, try it. First one's 
free.....
> I have been considering the idea, Tony. The problem is that it does
> kinda wreck things by mucking them up. Let's say I go along with the
> maybe, moving at c. So we have zero mass transmogrified into finite
> mass through an infinite increase in mass. So far so good.
> spin and produce the uncertainty constant inversely proportional to
> mass? Then how does relativistic rotational latency correlate with
> relativistic mass increase dependent on velocity because that latency
> certainly correlates with something? These are exactly the kinds of
> considerations we have to explain in terms of one another and not by
> resort to ad hoc possibilities.
Good questions and worth answering, although I am not sure what you mean by 

the 
rotational latency. As far as what makes them go, rotating, orbiting or 
simply 
reason to stop moving. What makes a piano string go? How long would it go 
with 
no friction?
> structure to mention of the kind in photons that causes cosmic
> indefinitely. (I considered mentioning this earlier because I just
> knew someone would think of it but decided it would complicate the
> subject unnecessarily at this point.)
>>  
>>what everyone says about photons. I don't think photons have any 
distinct 
>>structure, but are 2D wavelets passing through 3D space, and they are 
not 
>>supposed to slow down, since their masslessness requires that any 
energy they 
>>have be translated into velocity of c. I think it's exactly this 
initial 
>>masslessness you should be looking at for the source of mass IMHO.
>> 
>> Well, there's a very good reason they all look similar, Tony. I think
>> waves of light in space rotating only because they are half and not
>> whole waves like photons.
>But electrons absorb photons and turn them into increased energy levels, 
and 
>release photons when they drop back down to lower energy levels. How can 
a half 
>electrons and protons have a level of 3D structure that is missing in 
photons, 
>but of course I've never seen either one in detail. 
> This is exactly why I don't like analogical reasoning, Tony, because
> I'm dealing at a speculative level below what I can explain
> mechanically. How all this happens depends on what various terms are
> taken to indicate exactly. And so far all I've done is sketch in the
> basics in mechanical terms. I think you're probably right that protons
> at least have some structure, and that is something I think topology
> could work on constructively. But I don't see the lack of information
> on such issues as anything more than missing links. I don't think they
> represent any kind of basic threat to the ideas I've developed so far.
Probably not, though I think another type of mechanics is in order besides 
the 
macro kind. I don't know enough about QM to critique it, but what I do know 

makes some sense. Still, I agree that there is a basic mechanical picture of 

what's going on that is still quite hazy. I think you have some good ideas 
happening. Still a lot of missing links we have to identify, but maybe fewer 

than we think.
I read somewhere how protons are dodecahedrons. Did you ever notice there 
are 
two electrons missing from the shell progression? What makes a neutron stick 

together, really, and where can we find the next stable nucleus or electron 

shell? There are some interesting ideas about, indeed, if one looks.
>> And of course you're at liberty to visualize
>> spin characteristics, Planck's constant and the Heisenberg uncertainty
>> same wavelength.
>So to speak ;)
>Yes I share your desire for a more mechanical explanation tying all 
these 
>constants together and explaining things in simplest possible form. 
Of 
>course, I have been working, when I find time, on other things too....:)
> Go for it, Tony. No one ever said science is content with what it
> knows. I think for my part though I'm pretty much content with what I
> do know of the mechanics involved because I have other tools in the
> fire as well.
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
>Lester Zick said:
>>Lester Zick said:
> 
>Lester Zick said:
>> 
>>Lester Zick said:
> 
>Wolf Kirchmeir said:
>> 
>> 
>AFAIK, photons don't have as structure, so if you have 
evidence to the 
>contrary, I'd like to know where I can find it.
>>Photons have wavelength and polarization.
>>True, but I don't think that's what Lester has in mind. 
Of course, I 
>could be wrong - it's hard to tell what Lester has in 
mind. His talk 
>about accounting for spin in mechanical terms 
suggests to me that he 
>thinks a photon is like a little spinning billiard ball.
>>Lester, care to explain? Curious minds want to know.
>>A couple of weeks back, Wolf, I posed to curious minds 
whether they
>>for the electron and proton apparently represents the 
radial integral
>>of radial velocity? To date the question has been 
studiously ignored
>>by all these curious minds.
>>Well, if you could explain what a radial integral is, and 
what a radial 
>velcoity is, and how you integrate an radial velocity, and 
why this is a 
>defintion or calculation of spin, I might understand what 
you have in mind.
>> 
>> 
>> Yeah, Wolf. Fortunately I don't need you to do my thinking 
for me but
>> you badly seem to need someone to do it for you. I guessed 
I deserve
>> it when I ask the incompetent. Maybe in your next 
profession you can
>> learn some physics before pontificating about curious minds 
and
>> 
>> 
>> You have your own definition of spin, which is not the same 
as the 
>> are length^2/time. But the units of the expression you quote 
(1/2 
>> h/2pi)includes mass. So either you have discovered something 
that other 
>> physicists haven't discovered, in which case a write up of 
your 
>> experimental setup and data is in order; or else you are 
producing 
>> pseudo-scientific claptrap.
>> 
>Actually, Lester, I think Wolf has a point. When I was going 
through your 
>previously you had been speaking purely of spacetime and 
geometry. My original 
>question was about the source of mass, and yet somehow your 
model seems to 
>introduce mass spontaneously without mention. Of course, I may 
have your out, 
>but only if you pay attention to the question I asked. :)
> 
> Oh, I paid attention all right, Tony. But I'm wondering if the 
mass
> you're referring to isn't the analytical masslet m0 ~ 
10^-50kg-sec
> I used to define the origin of Planck's constant rather than 
mass in
> absolute terms.
>> 
>>I meant mass in absolute terms. I am not sure where the value for 
this masslet 
>>comes from except as a retrofitted number. The question as to what 
causes mass 
>>at all is what I was asking. What causes this masslet?
>> 
>> by rotational frequency dependent on the ratio of radial velocity 
in
>> relation to c. I have no explanation as to why there are only the 
two
>> stable ratios, one for the electron and one for the proton. 
Electron
>> mass is probably minimal and reflects some fundamental aspect of
>> interaction with the plenum. I imagine the proton is some 
convoluted
>> tiny flux of geometrically stable interaction with the plenum.
> 
>If protons and electrons only differ in radius/energy, then how do 
you account 
>for opposite charge? Doesn't the proton have a more complex structure 
than an 
>electron?
> 
> Existence of positrons and negative protons shows charge is a
> symmetric property. I don't explain it exactly at present but I'm
> pretty sure it's explicable. I assume proton structure is a 
reflection
> amount to.
>>Things yet to figger out, as always, indeed. And the band plays 
on......
>   
> itself is defined in terms of the magnitude of radially 
directed
> r/t2pi in relation to tangential r/t which is just the constant
> velocity of translation of light through space. The radial 
velocity
> r/t2pi can be of any finite magnitude and defines the radius of
>> 
>>This part makes sense to me. Higher energy/mass means higher 
frequency (as in 
>>light), which means smaller radius, assuming constant tangetntial 
velocity. I 
>>am not sure what eveidence there is for higher mass being 
associated with 
>>smaller radius, but it still makes abstract sense.
>> 
>> Yeah, it's basically a similar approach as used in macro angular
>> mechanics except that it accounts for constant rotational velocity
>> cases more accurately or more precisely or whatever. When the
>> tangential velocity is fixed, reducing the radius of rotation
>> increases energy with a constant spin whereas conventional macro
>> angular mechanics shows decreasing angular momentum L = r x p.
>> even though spin looks a lot like angular momentum.
> 
>That makes some sense. If rotation is always at a constant radial 
velocity then 
>they would all fit in some way like clockwork gears.  I can see 
that...
> 
> Good, Tony. It's a considerably more sensible explanation than trying
> to see things in terms of macro angular momentum and oscillating
> rotating transverse waveforms the way classical physics did. Macro
> angular mechanics mainly deals in rigid body rotation of fixed radius
> which yields incorrect interpretations of rr/t mechanics for angular
> momentum as r changes.
>  
>> is Heisenberg's uncertainty relation. That's why this relation is 
what
>> it is and what it identifies. People were just so used to 
visualizing
>> the low mass electron as tiny and the high mass proton as large 
that
>> they never stopped to regress alternatives tautologically.
> 
>That seems like a strange visualization, if it's widespread, since 
the electron 
>cloud occupies most of the space of an atom, and the protons are all 
squished 
>on the surface of the nucleus. Also, given the known realtionship 
between 
>frequency and energy for photons, one might think that idea was 
extended to 
> 
> Of course, you're right, Tony. I can only say that my own previous
> interpretations rested on a direct equivalence between mass and size.
> Even today people look askance when I say it. So I'm not surprized.
>>I guess it's true that in the macro world, bigger and similar means 
heavier. I 
>>guess that's one of the backwards kind of things one has to get used to 
with 
>>QM.
>> Now at least we know what we're talking about for a change.
>Change is good :)
>>    
> I use m0 simply to postulate some characteristic property of 
the
> energy is any other terms. Wolf's point seemed to be that 
analysis of
> Planck's constant depended on mass, but that isn't correct. I 
only
> have to regress mass to the plenum in some way consistent with
> 
>I still say that an object moving with a radial velocity of c, 
despite the 
>radius, would at any given moment have a translational velocity of c 
and 
>it could acquire relativistic mass that would look like absolute mass 
from 
>outside the rotation. The relativistic mass would probably be 
proportional to 
>the frequency, as with free photons.
> 
> Okay. I can see the plausibility of your idea. But that assumes
> relativistic mass increase is the explanation and not what is to be
> explained mechanically by means of other residual constants and
> interactions. Nor would your explanation explain Planck's constant or
>>of c. I can't comment on Planck's constant at this time, but I am not 
sure you 
>>explained Planck's constant anyway. It seemed like you derived the unit 
masslet 
>>using Planck's constant, creating a new relation, but not deriving the 
constant 
>>from more elementary constants and relations. Did I miss something?
>> Actually not, Tony. This is a reasonable assessment.  I have regressed
>> Planck's constant to more fundamental considerations without proving
>> those more fundamental considerations independently. So in this sense
>> I have only regressed Planck's constant without explaining it in more
>> fundamental terms.
>> However I would add that proof of the regression relies on the self
>> characteristics defined in terms of Planck's constant. In other words
>> there is no other interpretation consistent with the facts of constant
>> suggest that in the absence of any other constant such as c, m0 is
>> proven by regression through Planck's constant. Planck's constant is
>> properties are regressed in terms of the natural constant c to m0.
>I suspect that Planck's constant and c are intimately connected and 
determined 
>by similar underlying variables. It is quite likely that the values are 
>basically arbitrary, and vary from universelet to universelet, depending on 

>their particular dimensional characteristics. So, if that is the case, 
there 
>may be no direct derivation of certain root variables, but only 
identification 
>and measurement of them, and integration of them into a cohesive picture. 
>Sounds like that's what you're working on. Keep up the good work, you crank 

;)
>Yeah I know, takes one to know one. Consider it a complement! :)
is determined by m0 and c (and pi and circular rotation of course).
  
>>Are you sure that there is no relativistic component creating the 
mass from a 
>>masslesslet, due to the tangential velocity of c? Wouldn't the 
radial 
>>acceleration then be proportional to the mass as always?
>> 
>> dependent on the bidirectional average velocity of rotation in
>> relation to the velocity of light. At velocity this increases 
because
>> reverse side the rotation is shorter but the average for both
> 
>This sounds like you are saying there is radial inertia that acts 
according to 
>anyone else have comment on this?
> 
> relations as well.
>>relativistically due to tangential velocity of c would wreck any of what 
you 
>>have. I think you might find it helps. Come on, try it. First one's 
free.....
>> I have been considering the idea, Tony. The problem is that it does
>> kinda wreck things by mucking them up. Let's say I go along with the
>> maybe, moving at c. So we have zero mass transmogrified into finite
>> mass through an infinite increase in mass. So far so good.
>> spin and produce the uncertainty constant inversely proportional to
>> mass? Then how does relativistic rotational latency correlate with
>> relativistic mass increase dependent on velocity because that latency
>> certainly correlates with something? These are exactly the kinds of
>> considerations we have to explain in terms of one another and not by
>> resort to ad hoc possibilities.
>Good questions and worth answering, although I am not sure what you mean by 

the 
>rotational latency. As far as what makes them go, rotating, orbiting or 
simply 
>reason to stop moving. What makes a piano string go? How long would it go 
with 
>no friction?
Well friction is a very interesting subject in its own right, Tony.
But saying boundary velocity rotates is not the same as saying
equal-and-opposite reactions because there is a lack of constant
angular momentum requiring explanation. This isn't just something
which can be put off to some other speculative cause like ethereal
vibration.
Rotational latency has several potential implications but the one I'm
translation through space in ways directly related to relativistic
effects.
  
>> structure to mention of the kind in photons that causes cosmic
>> indefinitely. (I considered mentioning this earlier because I just
>> knew someone would think of it but decided it would complicate the
>> subject unnecessarily at this point.)
>  
>what everyone says about photons. I don't think photons have any 
distinct 
>structure, but are 2D wavelets passing through 3D space, and they are 
not 
>supposed to slow down, since their masslessness requires that any 
energy they 
>have be translated into velocity of c. I think it's exactly this 
initial 
>masslessness you should be looking at for the source of mass IMHO.
> 
> Well, there's a very good reason they all look similar, Tony. I think
> waves of light in space rotating only because they are half and not
> whole waves like photons.
>>  
>>But electrons absorb photons and turn them into increased energy levels, 
and 
>>release photons when they drop back down to lower energy levels. How can 
a half 
>>electrons and protons have a level of 3D structure that is missing in 
photons, 
>>but of course I've never seen either one in detail. 
>> This is exactly why I don't like analogical reasoning, Tony, because
>> I'm dealing at a speculative level below what I can explain
>> mechanically. How all this happens depends on what various terms are
>> taken to indicate exactly. And so far all I've done is sketch in the
>> basics in mechanical terms. I think you're probably right that protons
>> at least have some structure, and that is something I think topology
>> could work on constructively. But I don't see the lack of information
>> on such issues as anything more than missing links. I don't think they
>> represent any kind of basic threat to the ideas I've developed so far.
>Probably not, though I think another type of mechanics is in order besides 

the 
>macro kind. I don't know enough about QM to critique it, but what I do know 

>makes some sense. Still, I agree that there is a basic mechanical picture 
of 
>what's going on that is still quite hazy. I think you have some good ideas 

>happening. Still a lot of missing links we have to identify, but maybe 
fewer 
>than we think.
>I read somewhere how protons are dodecahedrons. Did you ever notice there 
are 
>two electrons missing from the shell progression? 
No, I can't say as I have. At least not that I remember.
>                                                                            

     What makes a neutron stick 
>together, really,
Well, this is a question that turns out to have a really fascinating
and very specific mechanical answer, Tony. But I'm reluctant to get
into it all at present because there is also a tenuous connection to
planetary perihelion advance involved as well. All will be revealed in
the fullness of time I hope.
>                          and where can we find the next stable nucleus or 

electron 
>shell? There are some interesting ideas about, indeed, if one looks.
> And of course you're at liberty to visualize
> spin characteristics, Planck's constant and the Heisenberg 
uncertainty
> same wavelength.
>>So to speak ;)
Yes I share your desire for a more mechanical explanation tying all 
these 
>>constants together and explaining things in simplest possible form. 
Of 
>>course, I have been working, when I find time, on other things 
too....:)
>> Go for it, Tony. No one ever said science is content with what it
>> knows. I think for my part though I'm pretty much content with what I
>> do know of the mechanics involved because I have other tools in the
>> fire as well.
===
Subject: Re: Epistemology 201: The Science of Science
Lester Zick said:
>robert j. kolker said:
>> 
>> 
>> And the mind and mental effects are non material factors unless you
>> propose to show an equal-and-opposite-ideas rule for the mind.So 
what?
>> 
>> What does equal and opposite mean when those words refer to ideas. Are 
>> you talking about one idea being the logical negation of the other? In 
>> that case the ideas are not equal. Are you talking about a pair of 
>> vectors of equal magnitude but pointing in opposite directions? What 
the 
>> hell are you talking about?
>> 
>> Bob Kolker
>> 
>> 
>Duh! Tautological regression to self-contradictory alternatives, doofus! 
>Everybody knows that!
> Don't forget the finite Tony. Lest we be accused of undefined
> infinite countation.
Oh yeah, finite, though I am not so worried about infinite regressions.....I 

rather look forward to them!
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
>Lester Zick said:
>>robert j. kolker said:
> 
> 
> And the mind and mental effects are non material factors unless you
> propose to show an equal-and-opposite-ideas rule for the mind.So 
what?
> 
> What does equal and opposite mean when those words refer to ideas. Are 
> you talking about one idea being the logical negation of the other? In 
> that case the ideas are not equal. Are you talking about a pair of 
> vectors of equal magnitude but pointing in opposite directions? What 
the 
> hell are you talking about?
> 
> Bob Kolker
> 
> 
>>Duh! Tautological regression to self-contradictory alternatives, doofus! 
>>Everybody knows that!
>> Don't forget the finite Tony. Lest we be accused of undefined
>> infinite countation.
>Oh yeah, finite, though I am not so worried about infinite 
regressions.....I 
>rather look forward to them!
Just don't hold your breath, Tony. Infinite regressions take a while.
It turns out they require an uncountable number of breaths to finish.
===
Subject: Re: Epistemology 201: The Science of Science
Lester Zick said:
>Lester Zick said:
>> 
>>robert j. kolker said:
> 
> 
> And the mind and mental effects are non material factors unless 
you
> propose to show an equal-and-opposite-ideas rule for the mind.So 
what?
> 
> What does equal and opposite mean when those words refer to ideas. 
Are 
> you talking about one idea being the logical negation of the other? 
In 
> that case the ideas are not equal. Are you talking about a pair of 
> vectors of equal magnitude but pointing in opposite directions? What 
the 
> hell are you talking about?
> 
> Bob Kolker
> 
> 
>>Duh! Tautological regression to self-contradictory alternatives, 
doofus! 
>>Everybody knows that!
>> 
>> Don't forget the finite Tony. Lest we be accused of undefined
>> infinite countation.
>> 
>> 
>Oh yeah, finite, though I am not so worried about infinite 
regressions.....I 
>rather look forward to them!
> Just don't hold your breath, Tony. Infinite regressions take a while.
> It turns out they require an uncountable number of breaths to finish.
Unless you hold your breath!!!
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
>Lester Zick said:
>>Lester Zick said:
> 
>robert j. kolker said:
>> 
>> 
>> And the mind and mental effects are non material factors unless 
you
>> propose to show an equal-and-opposite-ideas rule for the mind.So 
what?
>> 
>> What does equal and opposite mean when those words refer to ideas. 
Are 
>> you talking about one idea being the logical negation of the other? 
In 
>> that case the ideas are not equal. Are you talking about a pair of 
>> vectors of equal magnitude but pointing in opposite directions? 
What the 
>> hell are you talking about?
>> 
>> Bob Kolker
>> 
>> 
>Duh! Tautological regression to self-contradictory alternatives, 
doofus! 
>Everybody knows that!
> 
> Don't forget the finite Tony. Lest we be accused of undefined
> infinite countation.
> 
> 
>>Oh yeah, finite, though I am not so worried about infinite 
regressions.....I 
>>rather look forward to them!
>> Just don't hold your breath, Tony. Infinite regressions take a while.
>> It turns out they require an uncountable number of breaths to finish.
>Unless you hold your breath!!!
Known as the breathless hush of anticipation. Not an infinite regress
however.
===
Subject: Re: Epistemology 201: The Science of Science
Lester Zick said:
>Lester Zick said:
>> 
>>Lester Zick said:
> 
>robert j. kolker said:
>> 
>> 
>> And the mind and mental effects are non material factors unless 
you
>> propose to show an equal-and-opposite-ideas rule for the 
mind.So what?
>> 
>> What does equal and opposite mean when those words refer to 
ideas. Are 
>> you talking about one idea being the logical negation of the 
other? In 
>> that case the ideas are not equal. Are you talking about a pair 
of 
>> vectors of equal magnitude but pointing in opposite directions? 
What the 
>> hell are you talking about?
>> 
>> Bob Kolker
>> 
>> 
>Duh! Tautological regression to self-contradictory alternatives, 
doofus! 
>Everybody knows that!
> 
> Don't forget the finite Tony. Lest we be accused of undefined
> infinite countation.
> 
> 
>>Oh yeah, finite, though I am not so worried about infinite 
regressions.....I 
>>rather look forward to them!
>> 
>> Just don't hold your breath, Tony. Infinite regressions take a while.
>> It turns out they require an uncountable number of breaths to finish.
>> 
>> 
>Unless you hold your breath!!!
> Known as the breathless hush of anticipation. Not an infinite regress
> however.
I still like the giant pile of children's extra fingers that awaits us at 
the 
end......
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
Lester Zick said:
> On 15 Mar 2005 08:38:31 -0800, stevendaryl3016@yahoo.com (Daryl
>Lester Zick says...
>>And bijective mapping is about the comparison of non differentiable
>>countable undefined cardinalities.
>There is a definition of the cardinality of an infinite set,
>so calling it undefined is incorrect.
>Actually, there are two definitions:
>   Definition 1 (due to Russell, I think): The cardinality of a set A
>   is the proper class of all sets A' such that there exists a bijection
>   between A and A'.
>   Definition 2 (due to Von Neumann, I think): The cardinality of a set 
A
>   is the smallest Von Neumann ordinal alpha such that there is a
>   bijection between A and the set of ordinals less than alpha.
> But either way the actual cardinality of an infinite set is undefined
> whether you can compare that lack of definition to other lack of
> definitions or not. It's the same analytical technique employed by
> L'Hospital's Rule except that countable points are not differentiable.
Ah, but does this matter? If the discrete points are defined by a 
differentiable continuous function, can't one take the saturation of the 
infinity to be the inverse of the derivative of that function, or something 

similar? Ooh I'm going to catch it now!
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
>Lester Zick said:
>> On 15 Mar 2005 08:38:31 -0800, stevendaryl3016@yahoo.com (Daryl
>>Lester Zick says...
>And bijective mapping is about the comparison of non differentiable
>countable undefined cardinalities.
There is a definition of the cardinality of an infinite set,
>>so calling it undefined is incorrect.
Actually, there are two definitions:
   Definition 1 (due to Russell, I think): The cardinality of a set A
>>   is the proper class of all sets A' such that there exists a 
bijection
>>   between A and A'.
   Definition 2 (due to Von Neumann, I think): The cardinality of a set 
A
>>   is the smallest Von Neumann ordinal alpha such that there is a
>>   bijection between A and the set of ordinals less than alpha.
>> But either way the actual cardinality of an infinite set is undefined
>> whether you can compare that lack of definition to other lack of
>> definitions or not. It's the same analytical technique employed by
>> L'Hospital's Rule except that countable points are not differentiable.
>Ah, but does this matter? If the discrete points are defined by a 
>differentiable continuous function, can't one take the saturation of the 
>infinity to be the inverse of the derivative of that function, or something 

>similar? Ooh I'm going to catch it now!
a differentiable continuous function, it is the continuous function
that is differentiable and not the countable discrete points. I don't
quite understand what you mean by saturation of the infinity to be
the inverse of the derivative. Countable points just have no defined
functional derivative or, more to the point, they have an infinity of
functional derivatives.
===
Subject: Re: Epistemology 201: The Science of Science
> Ah, but does this matter? If the discrete points are defined by a 
> differentiable continuous function, 
How is a discrete point defined by a differentiable continuous function. 
Could you please define what you mean by that? What does that phrase 
mean? What is the domain of the function. What is its co-domain. What 
are the metrics (differentiability only pertains to functions from 
metric spaces to metric spaces where subtraction and additon are 
defined). Give us an example of a differentiable continuous function 
(specify domain and co-domain, please) which defines a discrete point.
Bob Kolker
===
Subject: Re: Epistemology 201: The Science of Science
>> Ah, but does this matter? If the discrete points are defined by a 
>> differentiable continuous function, 
>How is a discrete point defined by a differentiable continuous function. 
>Could you please define what you mean by that? What does that phrase 
>mean? What is the domain of the function. What is its co-domain. What 
>are the metrics (differentiability only pertains to functions from 
>metric spaces to metric spaces where subtraction and additon are 
>defined). Give us an example of a differentiable continuous function 
>(specify domain and co-domain, please) which defines a discrete point.
This is how it works, Bob: you give me the definition of a fact, I
give you the definition of a point.
===
Subject: Re: Epistemology 201: The Science of Science
robert j. kolker said:
> 
> Ah, but does this matter? If the discrete points are defined by a 
> differentiable continuous function, 
> How is a discrete point defined by a differentiable continuous function. 
> Could you please define what you mean by that? What does that phrase 
> mean? What is the domain of the function. What is its co-domain. What 
> are the metrics (differentiability only pertains to functions from 
> metric spaces to metric spaces where subtraction and additon are 
> defined). Give us an example of a differentiable continuous function 
> (specify domain and co-domain, please) which defines a discrete 
point.
> Bob Kolker
Here I am trying to help you Bob, and what do I get?
I mean that each discrete value is the value of a continuous function f(x) 
at 
discrete values of x. Take the size of the set of integers to be unit 
infinity, 
or aleph. The set of evens is bijectionable (like the new word?) using the 
function f(x)=2*x, since each even is twice the size of its corresponding 
integer. The derivative (slope of ascent) of f(x) is 2, the inverse of which 

is 
1/2, which is the ratio of the size of that set to aleph. 
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
> I mean that each discrete value is the value of a continuous function f(x) 

at 
> discrete values of x. Take the size of the set of integers to be unit 
infinity, 
> or aleph. The set of evens is bijectionable (like the new word?) using the 

> function f(x)=2*x, since each even is twice the size of its corresponding 

> integer. The derivative (slope of ascent) of f(x) is 2, the inverse of 
which is 
> 1/2, which is the ratio of the size of that set to aleph. 
The function f(x) = 2*x restricted to the integers is not 
differentiable. what does lim (h -> 0) [(f(x + h) - f(x))/h] mean when 
were resrict the function to the integers? It makes sense over the 
rationals or the reals but not the integers.
Bob Kolker
===
Subject: Re: Epistemology 201: The Science of Science
robert j. kolker said:
> 
> I mean that each discrete value is the value of a continuous function 
f(x) at 
> discrete values of x. Take the size of the set of integers to be unit 
infinity, 
> or aleph. The set of evens is bijectionable (like the new word?) using 
the 
> function f(x)=2*x, since each even is twice the size of its 
corresponding 
> integer. The derivative (slope of ascent) of f(x) is 2, the inverse of 
which is 
> 1/2, which is the ratio of the size of that set to aleph. 
> The function f(x) = 2*x restricted to the integers is not 
> differentiable. what does lim (h -> 0) [(f(x + h) - f(x))/h] mean when 
> were resrict the function to the integers? It makes sense over the 
> rationals or the reals but not the integers.
> Bob Kolker
No kidding Bob, but I just demonstrated using as clear a method as Cantor 
that 
the set of integers is twice the size of the evens, which is what I've been 

saying all along. When I get this all fleshed out you're going to kick 
yourself 
for not grabbing the purse and running while you could.
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
> No kidding Bob, but I just demonstrated using as clear a method as Cantor 

that 
> the set of integers is twice the size of the evens, which is what I've 
been 
> saying all along. When I get this all fleshed out you're going to kick 
yourself 
> for not grabbing the purse and running while you could.
What do you mean by size? Yes, your demo is true on the rationals and 
the reals. It does not hold when restricted to the integers. Why? 
Because there is no definition of derivative on the integers. For there 
to be you must have a dense ordering. The derivative of f at x is -by 
definition- lim(h->0) (f(x + h) - f(x))/h.  Hows is that defined on the 
integers? That is what a derivative is, so you have demonstrated nothing 
on the integers.
Sure enough 2*x is twice x (by definition duh!). So what? This hs 
nothing to do with sizes of sets.
Bob Kolker
===
Subject: Re: Epistemology 201: The Science of Science
robert j. kolker said:
> No kidding Bob, but I just demonstrated using as clear a method as 
Cantor that 
> the set of integers is twice the size of the evens, which is what I've 
been 
> saying all along. When I get this all fleshed out you're going to kick 
yourself 
> for not grabbing the purse and running while you could.
> What do you mean by size? Yes, your demo is true on the rationals and 
> the reals. It does not hold when restricted to the integers. Why? 
> Because there is no definition of derivative on the integers. For there 
> to be you must have a dense ordering. The derivative of f at x is -by 
> definition- lim(h->0) (f(x + h) - f(x))/h.  Hows is that defined on the 
> integers? That is what a derivative is, so you have demonstrated nothing 
> on the integers.
> Sure enough 2*x is twice x (by definition duh!). So what? This hs 
> nothing to do with sizes of sets.
> Bob Kolker
It works fine on the integers if the mapping function is continuous for 
reals. 
I already showed that.
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
> It works fine on the integers if the mapping function is continuous for 
reals. 
> I already showed that.
If the mapping function has domain = set of integers in what way is it 
continuous. Are you talking about a function defined on the integers or 
a function defined on the reals? If it is defined on the reals what does 
it have to do with a restriction to the integers?
All you have said is that the slope of the curve y = 2*x is 2. So what?
Bob Kolker
===
Subject: Re: Epistemology 201: The Science of Science
robert j. kolker said:
> 
> It works fine on the integers if the mapping function is continuous for 
reals. 
> I already showed that.
> If the mapping function has domain = set of integers in what way is it 
> continuous. Are you talking about a function defined on the integers or 
> a function defined on the reals? If it is defined on the reals what does 
> it have to do with a restriction to the integers?
> All you have said is that the slope of the curve y = 2*x is 2. So what?
> Bob Kolker
I already ing explained this to you rock head. If one applies a 
continuous 
function f(x) to discrete integral values, one will get discrete values from 

the function. Are you saying I can apply f(x)=x^2 to reals but not integers? 

Can I tell what f(x)=x^2-4x+7 is at x=913? Does it matter that 913 is an 
integer?
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
 <Tr_Td.59101$Am3.1863567@news20.bellglobal.com> 
<cvu3oq$12pv$1@msunews.cl.msu.edu>
 <cCNVd.47970$Vf6.1313829@news20.bellglobal.com> 
<d08aa4$sgn$2@msunews.cl.msu.edu>
 <rk6Xd.10104$fW4.285555@news20.bellglobal.com> 
<d0j3vd$fdl$3@msunews.cl.msu.edu>
 <hiXYd.1152$Ai3.16087@news20.bellglobal.com> 
<CNZYd.1996$Ai3.62670@news20.bellglobal.com>
 <MPG.1c9f839f9c0d45329898de@newsstand.cit.cornell.edu> 
<d14ed9010j5@drn.newsguy.com>
 <4235e31c.38492205@netnews.att.net> <d14qiu02gog@drn.newsguy.com>
 <42360b87.44041345@netnews.att.net> <39moaoF63lffvU3@individual.net>
 <4236fb50.49404197@netnews.att.net> <d173270aq9@drn.newsguy.com>
 <423731fa.54129391@netnews.att.net> 
<MPG.1ca219a67a8192c0989941@newsstand.cit.cornell.edu>
 <39r33gF4c6552U1@individual.net> 
<MPG.1ca243ffb3073c99989955@newsstand.cit.cornell.edu>
 <39re1dF62csvaU1@individual.net> 
<MPG.1ca253473b987380989964@newsstand.cit.cornell.edu robert j. kolker said:
> 
> 
> 
> 
> I mean that each discrete value is the value of a continuous function 
f(x) at 
> discrete values of x. Take the size of the set of integers to be unit 
infinity, 
> or aleph. The set of evens is bijectionable (like the new word?) using 
the 
> function f(x)=2*x, since each even is twice the size of its 
corresponding 
> integer. The derivative (slope of ascent) of f(x) is 2, the inverse of 
which is 
> 1/2, which is the ratio of the size of that set to aleph. 
> 
> The function f(x) = 2*x restricted to the integers is not 
> differentiable. what does lim (h -> 0) [(f(x + h) - f(x))/h] mean when 
> were resrict the function to the integers? It makes sense over the 
> rationals or the reals but not the integers.
> 
> Bob Kolker
> 
> No kidding Bob, but I just demonstrated using as clear a method as Cantor 

that 
> the set of integers is twice the size of the evens, which is what I've 
been 
> saying all along. When I get this all fleshed out you're going to kick 
yourself 
> for not grabbing the purse and running while you could.
Your proposal doesn't really work.
Consider your map x->2*x restricted to the rationals.  This map is a 
bijection from the rationals to itself, but according to your logic this 
would imply that the rationals had half as many elements as itself.
===
Subject: Re: Epistemology 201: The Science of Science
Creighton Hogg said:
> robert j. kolker said:
> 
> 
> 
> 
> I mean that each discrete value is the value of a continuous 
function f(x) at 
> discrete values of x. Take the size of the set of integers to be 
unit infinity, 
> or aleph. The set of evens is bijectionable (like the new word?) 
using the 
> function f(x)=2*x, since each even is twice the size of its 
corresponding 
> integer. The derivative (slope of ascent) of f(x) is 2, the inverse 
of which is 
> 1/2, which is the ratio of the size of that set to aleph. 
> 
> The function f(x) = 2*x restricted to the integers is not 
> differentiable. what does lim (h -> 0) [(f(x + h) - f(x))/h] mean when 
> were resrict the function to the integers? It makes sense over the 
> rationals or the reals but not the integers.
> 
> Bob Kolker
> 
> No kidding Bob, but I just demonstrated using as clear a method as 
Cantor that 
> the set of integers is twice the size of the evens, which is what I've 
been 
> saying all along. When I get this all fleshed out you're going to kick 
yourself 
> for not grabbing the purse and running while you could.
> Your proposal doesn't really work.
> Consider your map x->2*x restricted to the rationals.  This map is a 
> bijection from the rationals to itself, but according to your logic this 
> would imply that the rationals had half as many elements as itself.
Are you saying that the set of all rationals with even numerators includes 
all 
rationals? I didn't think so. I'd say........about half. QED
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
 <d08aa4$sgn$2@msunews.cl.msu.edu> 
<rk6Xd.10104$fW4.285555@news20.bellglobal.com>
 <d0j3vd$fdl$3@msunews.cl.msu.edu> 
<hiXYd.1152$Ai3.16087@news20.bellglobal.com>
 <CNZYd.1996$Ai3.62670@news20.bellglobal.com> 
<MPG.1c9f839f9c0d45329898de@newsstand.cit.cornell.edu>
 <d14ed9010j5@drn.newsguy.com> <4235e31c.38492205@netnews.att.net>
 <d14qiu02gog@drn.newsguy.com> <42360b87.44041345@netnews.att.net>
 <39moaoF63lffvU3@individual.net> <4236fb50.49404197@netnews.att.net>
 <d173270aq9@drn.newsguy.com> <423731fa.54129391@netnews.att.net>
 <MPG.1ca219a67a8192c0989941@newsstand.cit.cornell.edu> 
<39r33gF4c6552U1@individual.net>
 <MPG.1ca243ffb3073c99989955@newsstand.cit.cornell.edu> 
<39re1dF62csvaU1@individual.net>
 <MPG.1ca253473b987380989964@newsstand.cit.cornell.edu>
 <Pine.LNX.4.58.0503161338340.23675@clivia.hep.wisc.edu>
 <MPG.1ca259eba8c9b727989968@newsstand.cit.cornell.edu Creighton Hogg said:
> 
> robert j. kolker said:
> 
> 
> 
> 
> I mean that each discrete value is the value of a continuous 
function f(x) at 
> discrete values of x. Take the size of the set of integers to be 
unit infinity, 
> or aleph. The set of evens is bijectionable (like the new word?) 
using the 
> function f(x)=2*x, since each even is twice the size of its 
corresponding 
> integer. The derivative (slope of ascent) of f(x) is 2, the 
inverse of which is 
> 1/2, which is the ratio of the size of that set to aleph. 
> 
> The function f(x) = 2*x restricted to the integers is not 
> differentiable. what does lim (h -> 0) [(f(x + h) - f(x))/h] mean 
when 
> were resrict the function to the integers? It makes sense over the 
> rationals or the reals but not the integers.
> 
> Bob Kolker
> 
> No kidding Bob, but I just demonstrated using as clear a method as 
Cantor that 
> the set of integers is twice the size of the evens, which is what I've 
been 
> saying all along. When I get this all fleshed out you're going to kick 
yourself 
> for not grabbing the purse and running while you could.
> 
> Your proposal doesn't really work.
> Consider your map x->2*x restricted to the rationals.  This map is a 
> bijection from the rationals to itself, but according to your logic this 
> would imply that the rationals had half as many elements as itself.
> 
> Are you saying that the set of all rationals with even numerators includes 

all 
> rationals? I didn't think so. I'd say........about half. QED
The map x->2*x doesn't map the set of all rationals to the set of all 
rationals with even numerators.  Afterall, 1/6 * 2 = 1/3.  You get back 
*all* the rationals this way, since for any rational y/x, y/(2x) is also a 
rational.
===
Subject: Re: Epistemology 201: The Science of Science
Creighton Hogg said:
> Creighton Hogg said:
> 
> robert j. kolker said:
> 
> 
> 
> 
> I mean that each discrete value is the value of a continuous 
function f(x) at 
> discrete values of x. Take the size of the set of integers to be 
unit infinity, 
> or aleph. The set of evens is bijectionable (like the new word?) 
using the 
> function f(x)=2*x, since each even is twice the size of its 
corresponding 
> integer. The derivative (slope of ascent) of f(x) is 2, the 
inverse of which is 
> 1/2, which is the ratio of the size of that set to aleph. 
> 
> The function f(x) = 2*x restricted to the integers is not 
> differentiable. what does lim (h -> 0) [(f(x + h) - f(x))/h] mean 
when 
> were resrict the function to the integers? It makes sense over the 
> rationals or the reals but not the integers.
> 
> Bob Kolker
> 
> No kidding Bob, but I just demonstrated using as clear a method as 
Cantor that 
> the set of integers is twice the size of the evens, which is what 
I've been 
> saying all along. When I get this all fleshed out you're going to 
kick yourself 
> for not grabbing the purse and running while you could.
> 
> Your proposal doesn't really work.
> Consider your map x->2*x restricted to the rationals.  This map is a 
> bijection from the rationals to itself, but according to your logic 
this 
> would imply that the rationals had half as many elements as itself.
> 
> Are you saying that the set of all rationals with even numerators 
includes all 
> rationals? I didn't think so. I'd say........about half. QED
> The map x->2*x doesn't map the set of all rationals to the set of all 
> rationals with even numerators.  Afterall, 1/6 * 2 = 1/3.  You get back 
> *all* the rationals this way, since for any rational y/x, y/(2x) is also a 

> rational.
Okay, I can see that, in the context of Cantorian orderings. I don't think 
that 
changes my argument, though. For any given rational value in x and 2*x, the 

second set will have half as many elements before it as the first set. The 
first set can be said to saturate the rationals fully, while second only 
saturates it half as quickly. I understand the logic behind the mappings but 

I 
think it causes problems due to unnatural orderings imposed arbitrarily. I 
mean, it seems obvious to me that there are more rationals in [0,2] than in 

[0,1]. This is basically what is at issue in my mind.
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
 <rk6Xd.10104$fW4.285555@news20.bellglobal.com> 
<d0j3vd$fdl$3@msunews.cl.msu.edu>
 <hiXYd.1152$Ai3.16087@news20.bellglobal.com> 
<CNZYd.1996$Ai3.62670@news20.bellglobal.com>
 <MPG.1c9f839f9c0d45329898de@newsstand.cit.cornell.edu> 
<d14ed9010j5@drn.newsguy.com>
 <4235e31c.38492205@netnews.att.net> <d14qiu02gog@drn.newsguy.com>
 <42360b87.44041345@netnews.att.net> <39moaoF63lffvU3@individual.net>
 <4236fb50.49404197@netnews.att.net> <d173270aq9@drn.newsguy.com>
 <423731fa.54129391@netnews.att.net> 
<MPG.1ca219a67a8192c0989941@newsstand.cit.cornell.edu>
 <39r33gF4c6552U1@individual.net> 
<MPG.1ca243ffb3073c99989955@newsstand.cit.cornell.edu>
 <39re1dF62csvaU1@individual.net> 
<MPG.1ca253473b987380989964@newsstand.cit.cornell.edu>
 <Pine.LNX.4.58.0503161338340.23675@clivia.hep.wisc.edu>
 <MPG.1ca259eba8c9b727989968@newsstand.cit.cornell.edu>
 <Pine.LNX.4.58.0503161402510.23675@clivia.hep.wisc.edu Creighton Hogg said:
> 
> robert j. kolker said:
> 
> 
> 
> 
> I mean that each discrete value is the value of a continuous 
function f(x) at 
> discrete values of x. Take the size of the set of integers to be 
unit infinity, 
> or aleph. The set of evens is bijectionable (like the new word?) 
using the 
> function f(x)=2*x, since each even is twice the size of its 
corresponding 
> integer. The derivative (slope of ascent) of f(x) is 2, the 
inverse of which is 
> 1/2, which is the ratio of the size of that set to aleph. 
> 
> The function f(x) = 2*x restricted to the integers is not 
> differentiable. what does lim (h -> 0) [(f(x + h) - f(x))/h] mean 
when 
> were resrict the function to the integers? It makes sense over the 
> rationals or the reals but not the integers.
> 
> Bob Kolker
> 
> No kidding Bob, but I just demonstrated using as clear a method as 
Cantor that 
> the set of integers is twice the size of the evens, which is what 
I've been 
> saying all along. When I get this all fleshed out you're going to 
kick yourself 
> for not grabbing the purse and running while you could.
> 
> Your proposal doesn't really work.
> Consider your map x->2*x restricted to the rationals.  This map is a 
> bijection from the rationals to itself, but according to your logic 
this 
> would imply that the rationals had half as many elements as itself.
> 
> Are you saying that the set of all rationals with even numerators 
includes all 
> rationals? I didn't think so. I'd say........about half. QED
> The map x->2*x doesn't map the set of all rationals to the set of all 
> rationals with even numerators.  Afterall, 1/6 * 2 = 1/3.  You get back 
> *all* the rationals this way, since for any rational y/x, y/(2x) is also a 

> rational.
I should probably expand on this and say that the rationals are a field 
under the usual operations of addition and multiplication, which is the 
essence of what I'm trying to illustrate above.
===
Subject: Re: Epistemology 201: The Science of Science
Daryl McCullough said:
> aeo6 says...
>What I object to is the use of numbers in orderings that are not 
consistent, 
>using a technique that is not proven, to derive results that contradict 
more 
>basic notions of set theory
> Well, nobody is using numbers in orderings that are not consistent, and
> nobody is deriving results that contradict more basic notions of set 
theory.
If the ordering of quantities is inconsistent with the quantities, then the 

ordering is inconsistent. If a set contains all the elements of another set, 

plus some more, then it has more. These are pretty basic concepts that I 
shouldn't have to repeat.
>> I already told you, the result is not there are an infinite number
>> of naturals between any two naturals, the result is that for the
>> particular ordering that I defined, there are infinitely many
>> naturals between any two naturals.
>Right, but do you believe the conclusion based on that ordering?
> The conclusion is: There exists an ordering R(x,y) on naturals
> such that between any two naturals x and y, there exists infinitely
> many naturals z such that R(x,z) and R(z,y).
> Yes, I believe that. I constructed such an ordering.
>Is it true that there are infinitely many naturals between
>any two naturals, given the normal meaning of between for
>natural numbers?
> Of course not.
>If not, then  you should examine the train of thought that led to
>that conclusion
> Nothing led to that conclusion. You are misquoting the conclusion.
> I came to the conclusion that
> There exists an ordering R(x,y) on naturals
> such that between any two naturals x and y,
> there exists infinitely many naturals z such
> that R(x,z) and R(z,y).
You are not being honest. Here is your actual quote:
I didn't say that. I said that for that particular *ordering* on the
naturals, there exists infinitely many naturals between
any pair of naturals. It is a property of the ordering, not the
underlying set.
Now, do you believe that there exists infinitely many naturals between
any pair of naturals or not? If not, how do you explain the conclusion you 
drew?
>because you either have a faulty premise, or a flawed method of
>calculation.
> Or, what's actually the case, you misquoted the conclusion.
Or not.
>This is yet another example of how artificial orderings that
>violate the natural order of real numbers can lead to erroneous
>results.
> The only erroneous results are due to you.
Try again.
>>Doesn't a result like that give you any kind of twinge?
>> 
>> The fact that you left out the qualifying phrase for that particular
>> ordering certainly does.
>That is my point. The ordering is the problem. The conclusion is wrong,
> My conclusion wasn't wrong---yours was.
Read your quoted conclusion above, again.
>> Because you say things like
>> 
>>   anyone who argues there are as many rationals as integers is nuts, 
>>   in my mind
>> 
>> Why should anyone show you respect when you don't show any respect?
>I have already made my case regarding the integers and rationals.
> You have said nothing other than the obvious fact that the integers
> are a proper subset of the rationals. Nobody disagrees with that.
I never mentioned the subset relationship with respect to the integers and 
rationals. I mentioned that with respect to [0,1] and [0,2] for whatever 
numerical set you want over that domain. With respect to integers and 
rationals, I pointed to the consistent patter throughout the real number 
system 
that between any two naturals there are an infinite number of distinct 
rationals, but not vice versa. The logic concluding that there are 
infinitely 
many more rationals than integers is far more starightforward than your 
bijections and artificial orderings, and the preference by mathematicians 
for 
the latter is a form of confusion I'd like to see dispelled.
You misquote yourself and misrepresent my position. Not a good sign.
>If, for EVERY unit on the real number line there is exactly ONE
>integer and INFINITE  rationals, in EVERY unit, then it is a
>no-brainer to conclude that overall, there are an infinity of
>rationals for every integer, not one.
> Nobody disagrees with the fact that, with the usual ordering
> on the rationals, there are infinitely many rationals between
> any two integers.
Are there infinitely many integers between any two rationals? No? What does 

that tell you about the relative numbers integers and rationals?
>To draw a conclusion so drastically different from the blatantly
>obvious,
> The only conclusion is that there exists a bijection between
> the rationals and the integers. That's provably the case.
More bijection genuflection. What does this case prove? Certainly nothing 
about 
the relative numbers of integers and rationals.
>using a technique that is assumed to work because it
>looks neat,
> That's not true.
Why else is it accepted?
>but is not supported by any empirical evidence or
>application to reality,
> That's not true.
What empirical evidence can you offer?
>and to accept that conclusion over what
>makes sense, to me, makes no sense.
> What conclusion? The concusion is that there is a bijection
> between the integers and the rationals. You disagree with that
> conclusion?
No, just with the conclusion that there exists infinitely many naturals 
between any pair of naturals
>I believe I am entitled to that opinion.
> I don't see how the word entitled applies here. You are almost
> completely ignorant of the subject that you are criticizing, and
> your criticism doesn't amount to a hill of beans. The complaints
> you have are almost all complaints about *word* choice, which is
> about the silliest basis for criticizing mathematics.
see that you have not been reading too carefully, as I've been making a 
clear 
case that the conclusions drawn from bijection are overdrawn. Do you now 
agree 
that the conclusion you drew, which I quoted, is incorrect?
> --
> Daryl McCullough
> Ithaca, NY
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
robert j. kolker said:
> Now, do you believe that there exists infinitely many naturals 
between
> any pair of naturals or not? If not, how do you explain the conclusion 
you 
> drew?
> It depends on what is meant by between. Let R be any linear 
> (transitive, non-symmetric and non-reflexive and trichotomy holds wrt R) 
> ordering (dense or not). We say z is R-between x and y if and only if 
> R(x,z) and R(z,y). There are many linear dense ordering on a countable 
> set, hence many linear dense orderings on the integers.
> Bob Kolker
Have you been paying attention Bob? In the context of numbers, between 
means 
greater than one and less than the other, in quantity. By this deifnition, 
does 
the conclusion make any sense?
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
 <MPG.1c965cd95cdcbb1198989c@newsstand.cit.cornell.edu>
 <d0j6n8$fdl$5@msunews.cl.msu.edu>
 <ErXYd.1221$Ai3.17974@news20.bellglobal.com>
 <vcb3buz8x3u.fsf@beta19.sm.ltu.se>
 <lSXYd.1567$Ai3.23620@news20.bellglobal.com>
 <d11tnk$1lio$2@msunews.cl.msu.edu>
 <MPG.1c9f8785dcad079f9898e0@newsstand.cit.cornell.edu>
 <d14i8401dgn@drn.newsguy.com>
 <MPG.1c9faeb58f9b3c7d9898fd@newsstand.cit.cornell.edu>
 <d14qc002ftc@drn.newsguy.com>
 <MPG.1c9fc00f82a6e1f798990a@newsstand.cit.cornell.edu>
 <d14ubv02ti7@drn.newsguy.com>
 <MPG.1c9fd85594f28221989913@newsstand.cit.cornell.edu>
 <d156h50n8q@drn.newsguy.com>
 <MPG.1ca0d893c7fb49ba98991f@newsstand.cit.cornell.edu>
 <d175tf0l65@drn.newsguy.com>
 <MPG.1ca12a58f596fab6989940@newsstand.cit.cornell.edu>
 <39r2t1F6495bnU1@individual.net>
 <MPG.1ca2422c4a8ff2bc989954@newsstand.cit.cornell.edu>
 Discussion, linux)
[Newsgroups trimmed a bit.  Sorry if Tony's group was omitted.]
> robert j. kolker said:
>> Now, do you believe that there exists infinitely many naturals 
between
>> any pair of naturals or not? If not, how do you explain the 
conclusion you 
>> drew?
>> It depends on what is meant by between. Let R be any linear 
>> (transitive, non-symmetric and non-reflexive and trichotomy holds wrt R) 

>> ordering (dense or not). We say z is R-between x and y if and only if 
>> R(x,z) and R(z,y). There are many linear dense ordering on a countable 
>> set, hence many linear dense orderings on the integers.
>> Bob Kolker
> Have you been paying attention Bob? In the context of numbers, between 
means 
> greater than one and less than the other, in quantity. By this deifnition, 

does 
> the conclusion make any sense?
Bull.  In this context, between meant greater[1] than one and less
Daryl has been absolutely precise on this point.
No one here has said that *according to the usual ordering*, there
exists infinitely many naturals between any two naturals.  That is
obviously false.
Daryl constructed a particular (non-standard) ordering in which it is
true.  What is so hard to understand about that?
Look: A linear order on a set S is any relation R satisfying the
following (for all x, y, z in S): 
  (1) x R x
  (2) x R y and y R z implies x R z
  (3) x R y and y R x implies x = y
  (4) x R y or y R x
Daryl showed that there is a relation on N satisfying (1) - (4) and
also the additional property that for all x, y such that x R y, there
exist infinitely many z such that x R z and z R y.  There isn't a damn
thing controversial about that claim.
Unless you're too thick to realize that there are non-standard linear
orders on N.
Footnotes: 
[1]  I hesitate to use the words greater and less here, since they
might imply the standard order.  I mean nothing more than stand in
the relation this way or that, loosely speaking.
-- 
I'm starting to absorb information [...] as I find myself more and
more fascinated by my own prime counting function.  The rate of
information absorption is on an exponential scale, like it always has
been for me for things I'm interested in. -- James S. Harris
===
Subject: Re: Epistemology 201: The Science of Science
>[Newsgroups trimmed a bit.  Sorry if Tony's group was omitted.]
Yeah, well, we've gotten along perfectly well without trimming the
newsgroups that I put on the thread originally so if you want to
broadcast to your own selected groups I suggest you find another
thread to do it on.
===
Subject: Re: Epistemology 201: The Science of Science
Lester Zick said:
>[Newsgroups trimmed a bit.  Sorry if Tony's group was omitted.]
> Yeah, well, we've gotten along perfectly well without trimming the
> newsgroups that I put on the thread originally so if you want to
> broadcast to your own selected groups I suggest you find another
> thread to do it on.
That's alright. Nobody's interested in infinity ayway ;)
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
>>[Newsgroups trimmed a bit.  Sorry if Tony's group was omitted.]
> Yeah, well, we've gotten along perfectly well without trimming the
> newsgroups that I put on the thread originally so if you want to
> broadcast to your own selected groups I suggest you find another
> thread to do it on.
I'm not sure I agree with that, Lester.  We are each independent 
agents with complete control over which groups we choose to post 
on.  If I thought it would work, I would do it too.
-- 
I consider it quite possible that physics cannot be based
on the field concept, i. e., on continuous structures. In that
case nothing remains of my entire castle in the air,
gravitation theory included, [and of] the rest of modern physics.
        -- Albert Einstein in a 1954 letter to Michele Besso.
===
Subject: Re: Epistemology 201: The Science of Science
>[Newsgroups trimmed a bit.  Sorry if Tony's group was omitted.]
>> Yeah, well, we've gotten along perfectly well without trimming the
>> newsgroups that I put on the thread originally so if you want to
>> broadcast to your own selected groups I suggest you find another
>> thread to do it on.
>I'm not sure I agree with that, Lester.  We are each independent 
>agents with complete control over which groups we choose to post 
>on.  If I thought it would work, I would do it too.
I never suggested we weren't independent agents. In fact early on I
pointed out exactly that to Osher. It just strikes me as rude to jump
into a thread with 4000-5000 post history and try to redirect it.
===
Subject: Re: Epistemology 201: The Science of Science
Jesse F. Hughes said:
> [Newsgroups trimmed a bit.  Sorry if Tony's group was omitted.]
> robert j. kolker said:
>> 
>> 
>> Now, do you believe that there exists infinitely many naturals 
between
>> any pair of naturals or not? If not, how do you explain the 
conclusion you 
>> drew?
>> 
>> It depends on what is meant by between. Let R be any linear 
>> (transitive, non-symmetric and non-reflexive and trichotomy holds wrt 
R) 
>> ordering (dense or not). We say z is R-between x and y if and only if 
>> R(x,z) and R(z,y). There are many linear dense ordering on a countable 
>> set, hence many linear dense orderings on the integers.
>> 
>> Bob Kolker
>> 
>> 
> Have you been paying attention Bob? In the context of numbers, 
between means 
> greater than one and less than the other, in quantity. By this 
deifnition, does 
> the conclusion make any sense?
> Bull.  In this context, between meant greater[1] than one and 
less
> Daryl has been absolutely precise on this point.
> No one here has said that *according to the usual ordering*, there
> exists infinitely many naturals between any two naturals.  That is
> obviously false.
> Daryl constructed a particular (non-standard) ordering in which it is
> true.  What is so hard to understand about that?
> Look: A linear order on a set S is any relation R satisfying the
> following (for all x, y, z in S): 
>   (1) x R x
>   (2) x R y and y R z implies x R z
>   (3) x R y and y R x implies x = y
>   (4) x R y or y R x
> Daryl showed that there is a relation on N satisfying (1) - (4) and
> also the additional property that for all x, y such that x R y, there
> exist infinitely many z such that x R z and z R y.  There isn't a damn
> thing controversial about that claim.
> Unless you're too thick to realize that there are non-standard linear
> orders on N.
> Footnotes: 
> [1]  I hesitate to use the words greater and less here, since 
they
> might imply the standard order.  I mean nothing more than stand in
> the relation this way or that, loosely speaking.
Jesse, apparently you haven't been paying attention either. When you come to 

conclusions that make no sense, except in the case of an arbitrary ordering 

that makes no sense, do you take those conclusions as fact? The usual 
ordering of numbers is the one that makes them make sense. Numbers are 
naturally ordered by quantity, since that's what they represent.
As was pointed out, precision does not equate to accuracy. If your shots 
cluster tightly, great, you're precise. But, if they are hitting the tree 8 

feet to the left of the target, the gun is useless. If mathematicians 
conclude 
based on cardinality that there are as many integers as rationals, when in 
fact 
there are an infinite number of rationals for every integer (and in fact a 
larger infinity than the entire set of integers), then there is something 
wrong 
with cardinality, or at least the interpretation of it. As I've determined, 

the 
root flaw in the method is the assumption that one can rearrange the number 

system without violating the definition of its members. So, bull right 
back 
atcha!
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
 <ErXYd.1221$Ai3.17974@news20.bellglobal.com>
 <vcb3buz8x3u.fsf@beta19.sm.ltu.se>
 <lSXYd.1567$Ai3.23620@news20.bellglobal.com>
 <d11tnk$1lio$2@msunews.cl.msu.edu>
 <MPG.1c9f8785dcad079f9898e0@newsstand.cit.cornell.edu>
 <d14i8401dgn@drn.newsguy.com>
 <MPG.1c9faeb58f9b3c7d9898fd@newsstand.cit.cornell.edu>
 <d14qc002ftc@drn.newsguy.com>
 <MPG.1c9fc00f82a6e1f798990a@newsstand.cit.cornell.edu>
 <d14ubv02ti7@drn.newsguy.com>
 <MPG.1c9fd85594f28221989913@newsstand.cit.cornell.edu>
 <d156h50n8q@drn.newsguy.com>
 <MPG.1ca0d893c7fb49ba98991f@newsstand.cit.cornell.edu>
 <d175tf0l65@drn.newsguy.com>
 <MPG.1ca12a58f596fab6989940@newsstand.cit.cornell.edu>
 <39r2t1F6495bnU1@individual.net>
 <MPG.1ca2422c4a8ff2bc989954@newsstand.cit.cornell.edu>
 <87y8cnmlct.fsf@phiwumbda.org>
 <MPG.1ca2597563bc0d00989967@newsstand.cit.cornell.edu>
 Discussion, linux)
> Jesse F. Hughes said:
>> Bull.  In this context, between meant greater[1] than one and 
less
>> Daryl has been absolutely precise on this point.
>> No one here has said that *according to the usual ordering*, there
>> exists infinitely many naturals between any two naturals.  That is
>> obviously false.
>> Daryl constructed a particular (non-standard) ordering in which it is
>> true.  What is so hard to understand about that?
>> Look: A linear order on a set S is any relation R satisfying the
>> following (for all x, y, z in S): 
>>   (1) x R x
>>   (2) x R y and y R z implies x R z
>>   (3) x R y and y R x implies x = y
>>   (4) x R y or y R x
>> Daryl showed that there is a relation on N satisfying (1) - (4) and
>> also the additional property that for all x, y such that x R y, there
>> exist infinitely many z such that x R z and z R y.  There isn't a damn
>> thing controversial about that claim.
>> Unless you're too thick to realize that there are non-standard linear
>> orders on N.
>> Footnotes: 
>> [1]  I hesitate to use the words greater and less here, since 
they
>> might imply the standard order.  I mean nothing more than stand in
>> the relation this way or that, loosely speaking.
> Jesse, apparently you haven't been paying attention either. When you come 

to 
> conclusions that make no sense, except in the case of an arbitrary 
ordering 
> that makes no sense, do you take those conclusions as fact? The usual 
> ordering of numbers is the one that makes them make sense. Numbers are 
> naturally ordered by quantity, since that's what they represent.
Numbers do indeed have a natural ordering.  They also have other
orderings.  What Daryl has said applies to other orderings.
> As was pointed out, precision does not equate to accuracy. If your shots 
> cluster tightly, great, you're precise. But, if they are hitting the tree 

8 
> feet to the left of the target, the gun is useless. If mathematicians 
conclude 
> based on cardinality that there are as many integers as rationals, when in 

fact 
> there are an infinite number of rationals for every integer (and in fact a 

> larger infinity than the entire set of integers), then there is something 

wrong 
> with cardinality, or at least the interpretation of it. 
One might conclude that cardinality isn't the right concept for size
when infinite sets are involved -- one *might* conclude that, but I
can't imagine a better definition myself.  Cardinality is very well
motivated from a philosophical standpoint.
But even *if* one doubted that cardinality was the right
generalization of size for infinite sets, that has nothing to do with
your complaints about Daryl's obvious and uncontroversial result.
There is a relation R satisfying (1) - (4) such that it also satisfies
the density property I mention above.  His proof is simple and
obvious.  
> As I've determined, the root flaw in the method is the assumption
> that one can rearrange the number system without violating the
> definition of its members. So, bull right back atcha!
Yeah, well, the difference between my claim of bull and yours is
that the supporting text made sense in my case.  
Look, maybe you have trouble with the term order when it's a
non-standard order.  Call a relation R satisfying (1)-(4) a
gigglegoggle.  Do you agree or disagree that there exists a
gigglegoggle such that for all x and y where x R y, there exists a z
(and hence infinitely many z) such that x R z and z R y?
-- 
[Criticizing JSH's mathematics will result in] one of the worst debacles 
in
the history of the world.  It is foretold in most mythologies and 
religions.
And yes, you are the ones, the cursed ones, who destroy the world.  
    --James S. Harris reads from the Aztec Book of the Damned 
Mathematicians
===
Subject: Re: Epistemology 201: The Science of Science
Jesse F. Hughes said:
> Jesse F. Hughes said:
>> 
>> Bull.  In this context, between meant greater[1] than one and 
less
>> Daryl has been absolutely precise on this point.
>> 
>> No one here has said that *according to the usual ordering*, there
>> exists infinitely many naturals between any two naturals.  That is
>> obviously false.
>> 
>> Daryl constructed a particular (non-standard) ordering in which it is
>> true.  What is so hard to understand about that?
>> 
>> Look: A linear order on a set S is any relation R satisfying the
>> following (for all x, y, z in S): 
>>   (1) x R x
>>   (2) x R y and y R z implies x R z
>>   (3) x R y and y R x implies x = y
>>   (4) x R y or y R x
>> 
>> Daryl showed that there is a relation on N satisfying (1) - (4) and
>> also the additional property that for all x, y such that x R y, there
>> exist infinitely many z such that x R z and z R y.  There isn't a damn
>> thing controversial about that claim.
>> 
>> Unless you're too thick to realize that there are non-standard linear
>> orders on N.
>> 
>> 
>> Footnotes: 
>> [1]  I hesitate to use the words greater and less here, since 
they
>> might imply the standard order.  I mean nothing more than stand in
>> the relation this way or that, loosely speaking.
>> 
>> 
> Jesse, apparently you haven't been paying attention either. When you 
come to 
> conclusions that make no sense, except in the case of an arbitrary 
ordering 
> that makes no sense, do you take those conclusions as fact? The 
usual 
> ordering of numbers is the one that makes them make sense. Numbers are 
> naturally ordered by quantity, since that's what they represent.
> Numbers do indeed have a natural ordering.  They also have other
> orderings.  What Daryl has said applies to other orderings.
> As was pointed out, precision does not equate to accuracy. If your shots 
> cluster tightly, great, you're precise. But, if they are hitting the 
tree 8 
> feet to the left of the target, the gun is useless. If mathematicians 
conclude 
> based on cardinality that there are as many integers as rationals, when 
in fact 
> there are an infinite number of rationals for every integer (and in fact 
a 
> larger infinity than the entire set of integers), then there is 
something wrong 
> with cardinality, or at least the interpretation of it. 
> One might conclude that cardinality isn't the right concept for size
> when infinite sets are involved -- one *might* conclude that, but I
> can't imagine a better definition myself.  Cardinality is very well
> motivated from a philosophical standpoint.
> But even *if* one doubted that cardinality was the right
> generalization of size for infinite sets, that has nothing to do with
> your complaints about Daryl's obvious and uncontroversial result.
> There is a relation R satisfying (1) - (4) such that it also satisfies
> the density property I mention above.  His proof is simple and
> obvious.  
> As I've determined, the root flaw in the method is the assumption
> that one can rearrange the number system without violating the
> definition of its members. So, bull right back atcha!
> Yeah, well, the difference between my claim of bull and yours is
> that the supporting text made sense in my case.  
> Look, maybe you have trouble with the term order when it's a
> non-standard order.  Call a relation R satisfying (1)-(4) a
> gigglegoggle.  Do you agree or disagree that there exists a
> gigglegoggle such that for all x and y where x R y, there exists a z
> (and hence infinitely many z) such that x R z and z R y?
As I said, the conclusion that there are an infinite number of naturals 
between 
any two naturals is nonsensical to me. When that conclusion is drawn from an 

assumption that numbers can be placed between numbers when their values 
aren't 
between the values of the other numbers, then I credit that assumption with 

the 
error in the conclusion. So, like I said to Daryl, if you want to make that 

assumption, then sure, it is valid to draw that conclusion, but you can't 
claim 
the conclusion is sound unless you can justify the assumption. Since the 
conclusion makes no sense, and the assumption makes no sense, I reject the 
argument.
If one can create a 1-1 and onto relation between members of two sets of 
numbers, *in their natural quantitative order*, and create a bijection, then 

the two sets have exactly the same size. When you introduce reordering 
of 
numbers, then you confound the picture so that you can basically only 
distinguish between countable infinities and continuous infinities of 
different 
dimensions. So, while you cannot imagine a better method of distinguishing 
infinities, I can. Is that really bull?
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
 <lSXYd.1567$Ai3.23620@news20.bellglobal.com>
 <d11tnk$1lio$2@msunews.cl.msu.edu>
 <MPG.1c9f8785dcad079f9898e0@newsstand.cit.cornell.edu>
 <d14i8401dgn@drn.newsguy.com>
 <MPG.1c9faeb58f9b3c7d9898fd@newsstand.cit.cornell.edu>
 <d14qc002ftc@drn.newsguy.com>
 <MPG.1c9fc00f82a6e1f798990a@newsstand.cit.cornell.edu>
 <d14ubv02ti7@drn.newsguy.com>
 <MPG.1c9fd85594f28221989913@newsstand.cit.cornell.edu>
 <d156h50n8q@drn.newsguy.com>
 <MPG.1ca0d893c7fb49ba98991f@newsstand.cit.cornell.edu>
 <d175tf0l65@drn.newsguy.com>
 <MPG.1ca12a58f596fab6989940@newsstand.cit.cornell.edu>
 <39r2t1F6495bnU1@individual.net>
 <MPG.1ca2422c4a8ff2bc989954@newsstand.cit.cornell.edu>
 <87y8cnmlct.fsf@phiwumbda.org>
 <MPG.1ca2597563bc0d00989967@newsstand.cit.cornell.edu>
 <87sm2vmk1n.fsf@phiwumbda.org>
 <MPG.1ca363c9cd7e60e98996c@newsstand.cit.cornell.edu>
 Discussion, linux)
> Jesse F. Hughes said:
>> Look, maybe you have trouble with the term order when it's a
>> non-standard order.  Call a relation R satisfying (1)-(4) a
>> gigglegoggle.  Do you agree or disagree that there exists a
>> gigglegoggle such that for all x and y where x R y, there exists a z
>> (and hence infinitely many z) such that x R z and z R y?
> As I said, the conclusion that there are an infinite number of naturals 
between 
> any two naturals is nonsensical to me. When that conclusion is drawn from 

an 
> assumption that numbers can be placed between numbers when their values 
aren't 
> between the values of the other numbers, then I credit that assumption 
with the 
> error in the conclusion. So, like I said to Daryl, if you want to make 
that 
> assumption, then sure, it is valid to draw that conclusion, but you can't 

claim 
> the conclusion is sound unless you can justify the assumption. Since the 
> conclusion makes no sense, and the assumption makes no sense, I reject the 

> argument.
I really have no idea what you mean.  Let me ask again:
Do you agree or disagree that there is a relation R on N satisfying
the following conditions?
(1) x R x
(2) x R y and y R z implies x R z
(3) x R y and y R x implies x = y
(4) x R y or y R x
(5) x R y implies there is a z such that x R z and z R y
This is what Daryl has explicitly shown.  You seem to think it
involves moving numbers about, but it doesn't.  It's a simple fact
about the exist of a certain relation on N.
> If one can create a 1-1 and onto relation between members of two sets of 
> numbers, *in their natural quantitative order*, and create a bijection, 
then 
> the two sets have exactly the same size. When you introduce reordering 
of 
> numbers, then you confound the picture so that you can basically only 
> distinguish between countable infinities and continuous infinities of 
different 
> dimensions. So, while you cannot imagine a better method of distinguishing 

> infinities, I can. Is that really bull?
I don't see why order types are a particularly natural method of
judging the sizes of sets.  A good method of judging the sizes of
linearly ordered sets *as such*, perhaps.  But the set of natural
numbers is a set as well as a linear ordered set.  We can ask how it
compares to the set Q without reference to its order as well as how it
compares with respect to its order.  They're distinct questions and
they're both sensible questions.
-- 
[Y]ou never understood the real role of mathematicians.  The
position is one of great responsibility and power. [...] You people
have no concept of what it means to be an actual mathematician versus
pretending to be one, dreaming you understand.  -- James S. Harris
===
Subject: Re: Epistemology 201: The Science of Science
Jesse F. Hughes said:
> Jesse F. Hughes said:
>> 
>> Look, maybe you have trouble with the term order when it's a
>> non-standard order.  Call a relation R satisfying (1)-(4) a
>> gigglegoggle.  Do you agree or disagree that there exists a
>> gigglegoggle such that for all x and y where x R y, there exists a z
>> (and hence infinitely many z) such that x R z and z R y?
>> 
>> 
> As I said, the conclusion that there are an infinite number of naturals 
between 
> any two naturals is nonsensical to me. When that conclusion is drawn 
from an 
> assumption that numbers can be placed between numbers when their values 
aren't 
> between the values of the other numbers, then I credit that assumption 
with the 
> error in the conclusion. So, like I said to Daryl, if you want to make 
that 
> assumption, then sure, it is valid to draw that conclusion, but you 
can't claim 
> the conclusion is sound unless you can justify the assumption. Since the 
> conclusion makes no sense, and the assumption makes no sense, I reject 
the 
> argument.
> I really have no idea what you mean.  Let me ask again:
> Do you agree or disagree that there is a relation R on N satisfying
> the following conditions?
> (1) x R x
> (2) x R y and y R z implies x R z
> (3) x R y and y R x implies x = y
> (4) x R y or y R x
> (5) x R y implies there is a z such that x R z and z R y
> This is what Daryl has explicitly shown.  You seem to think it
> involves moving numbers about, but it doesn't.  It's a simple fact
> about the exist of a certain relation on N.
> If one can create a 1-1 and onto relation between members of two sets of 
> numbers, *in their natural quantitative order*, and create a bijection, 
then 
> the two sets have exactly the same size. When you introduce 
reordering of 
> numbers, then you confound the picture so that you can basically only 
> distinguish between countable infinities and continuous infinities of 
different 
> dimensions. So, while you cannot imagine a better method of 
distinguishing 
> infinities, I can. Is that really bull?
> I don't see why order types are a particularly natural method of
> judging the sizes of sets.  A good method of judging the sizes of
> linearly ordered sets *as such*, perhaps.  But the set of natural
> numbers is a set as well as a linear ordered set.  We can ask how it
> compares to the set Q without reference to its order as well as how it
> compares with respect to its order.  They're distinct questions and
> they're both sensible questions.
Daryl's example for infinite naturals between any two naturals involved 
an 
ordering which was entirely non-standard. That's why he was able to draw 
that 
conclusion. In the context of the ordering, the conclusion is valid, but the 

ordering is artificial, and the conclusion cannot be applied to natural 
numbers 
in their natural order. By any normal sense of the naturals, it is not at 
all 
true that there are an infinite number of naturals between any two naturals. 

It 
is only in the context of a number system which totally rearranged so as to 

be 
almost unrecognizable.
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
 <MPG.1c9f8785dcad079f9898e0@newsstand.cit.cornell.edu>
 <d14i8401dgn@drn.newsguy.com>
 <MPG.1c9faeb58f9b3c7d9898fd@newsstand.cit.cornell.edu>
 <d14qc002ftc@drn.newsguy.com>
 <MPG.1c9fc00f82a6e1f798990a@newsstand.cit.cornell.edu>
 <d14ubv02ti7@drn.newsguy.com>
 <MPG.1c9fd85594f28221989913@newsstand.cit.cornell.edu>
 <d156h50n8q@drn.newsguy.com>
 <MPG.1ca0d893c7fb49ba98991f@newsstand.cit.cornell.edu>
 <d175tf0l65@drn.newsguy.com>
 <MPG.1ca12a58f596fab6989940@newsstand.cit.cornell.edu>
 <39r2t1F6495bnU1@individual.net>
 <MPG.1ca2422c4a8ff2bc989954@newsstand.cit.cornell.edu>
 <87y8cnmlct.fsf@phiwumbda.org>
 <MPG.1ca2597563bc0d00989967@newsstand.cit.cornell.edu>
 <87sm2vmk1n.fsf@phiwumbda.org>
 <MPG.1ca363c9cd7e60e98996c@newsstand.cit.cornell.edu>
 <874qfatbwq.fsf@phiwumbda.org>
 <MPG.1ca3a2a885eb9fde989993@newsstand.cit.cornell.edu>
 Discussion, linux)
> Jesse F. Hughes said:
>> I don't see why order types are a particularly natural method of
>> judging the sizes of sets.  A good method of judging the sizes of
>> linearly ordered sets *as such*, perhaps.  But the set of natural
>> numbers is a set as well as a linear ordered set.  We can ask how it
>> compares to the set Q without reference to its order as well as how it
>> compares with respect to its order.  They're distinct questions and
>> they're both sensible questions.
> Daryl's example for infinite naturals between any two naturals 
involved an 
> ordering which was entirely non-standard. 
Duh.  Explicitly so.
> That's why he was able to draw that conclusion.
Duh.  He never claimed otherwise and he explicitly and repeatedly drew
attention to the non-standard nature of his ordering.
> In the context of the ordering, the conclusion is valid, but the
> ordering is artificial, and the conclusion cannot be applied to
> natural numbers in their natural order.
No kidding.  No one has claimed that it *does* apply to the naturals
in their standard order.
Anyway, nice to see that you're in total agreement with Daryl.  
-- 
Jesse F. Hughes
I think the burden is on those people who think he didn't have
weapons of mass destruction to tell the world where they are.
           -- White House spokesman Ari Fleischer
===
Subject: Re: Epistemology 201: The Science of Science
Tony says...
>As I said, the conclusion that there are an infinite number of
>naturals between any two naturals is nonsensical to me.
It's nonsensical to me, as well. As I said, nobody has drawn that
conclusion. You got that conclusion by misquoting what I have said,
by leaving out the most important part, namely, according to a 
particular ordering.
>When that conclusion is drawn from an 
>assumption that numbers can be placed between
>numbers when their values aren't between the
>values of the other numbers,
It's not an assumption if it is provably true. It is provably
true that there exists orderings on the naturals other than the
usual one.
Write out what you think the assumption is in mathematical
terms, and tell me why you think the assumption is false.
Here's what I believe about orderings:
    Definition: If S is a set, and R is a binary relation on
    S, then we say that R is a total ordering on S if
        1. For all x and y in S, either R(x,y) or R(y,x)
        2. For all x, y and z in S, R(x,y) and R(y,z) implies R(x,z).
    Definition: If R is a total ordering on set S, and x,y, and z
    are elements of S, then we say that y is between x and z according
    to that ordering if R(x,y) and R(y,z).
    Theorem: There exists a total ordering R on the naturals such
    that 2 is between 0 and 1 according to that ordering.
    Proof: Let R(x,y) == if (x=2 and y=1) then true else x < y
    By definition  of R, we find R(0,2) and R(2,1).
    By definition of between, 2 is between 0 and 1
    according to ordering R.
This is the ordering 0 2 1 3 4 5 ...
    Theorem: There exists a total ordering R on the naturals such
    that there are infinitely many naturals between 0 and 1 according
    to that ordering.
    Proof: Let R(x,y) == if x=y then false else if (x=1 or y=2) then true
                         else x < y
This is the ordering 0 2 3 4 5 6 ... 1
So exactly what am I assuming that you don't believe? Are you saying
that I can't write down the above orderings? I just did.
>then I credit that assumption with the 
>error in the conclusion.
If you would actually quote the *entire* conclusion,
then you would see that it isn't an error. It becomes
an error when you misquote it. That makes the error
due to *your* mistakes, not those of anyone else.
>So, like I said to Daryl, if you want to make that 
>assumption, then sure, it is valid to draw that conclusion,
>but you can't claim the conclusion is sound unless you can
>justify the assumption.
One way to justify an assumption is to prove it. That's been done.
>Since the conclusion makes no sense,
I agree that your conclusion makes no sense, but it wasn't *my*
conclusion. My conclusion included the phrase according to the
ordering, which is a critical point.
>and the assumption makes no sense, I reject the argument.
Why don't you reject the habit of misquoting conclusions?
>If one can create a 1-1 and onto relation between members of two sets of 
>numbers, *in their natural quantitative order*, and create a bijection, 
then 
>the two sets have exactly the same size.
You don't mean that. There exists a bijection between the
naturals and the even naturals that preserves the natural
quantitative ordering:
    n --> 2n
You really don't know what you mean, do you?
>When you introduce reordering of numbers, then you
>confound the picture
I would say that when you misquote people in such a way
as to change the meaning of what they are saying, you
confound the picture.
>so that you can basically only distinguish between countable
>infinities and continuous infinities of different 
>dimensions.
*Who* can only distinguish between countable infinities
and continuous infinities of different dimensions? I
can certainly distinguish between the rationals and
the naturals based on their order type, or based on
the fact that one is a subset of the other. I can
certainly distinguish between the naturals and the
even numbers based on the fact that one is a subset
of the other, and also based on the fact that the
relative frequency of one set is 1/2 the relative
frequency of the other set.
You seem to have a totalitarian mindset in which anything
that is not forbidden is mandatory. The fact that I *can*
compare two sets using cardinality doesn't imply
that I *must* compare them using cardinality.
>So, while you cannot imagine a better method of distinguishing 
>infinities, I can.
No, you can't. You haven't come up with any better method. There
are no distinctions that you have been able to make that are not
already well understood by conventional mathematics, which includes
the tools of cardinality, relative frequency, measure, subset
relationships, and order types.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Epistemology 201: The Science of Science
 <d11tnk$1lio$2@msunews.cl.msu.edu>
 <MPG.1c9f8785dcad079f9898e0@newsstand.cit.cornell.edu>
 <d14i8401dgn@drn.newsguy.com>
 <MPG.1c9faeb58f9b3c7d9898fd@newsstand.cit.cornell.edu>
 <d14qc002ftc@drn.newsguy.com>
 <MPG.1c9fc00f82a6e1f798990a@newsstand.cit.cornell.edu>
 <d14ubv02ti7@drn.newsguy.com>
 <MPG.1c9fd85594f28221989913@newsstand.cit.cornell.edu>
 <d156h50n8q@drn.newsguy.com>
 <MPG.1ca0d893c7fb49ba98991f@newsstand.cit.cornell.edu>
 <d175tf0l65@drn.newsguy.com>
 <MPG.1ca12a58f596fab6989940@newsstand.cit.cornell.edu>
 <39r2t1F6495bnU1@individual.net>
 <MPG.1ca2422c4a8ff2bc989954@newsstand.cit.cornell.edu>
 <87y8cnmlct.fsf@phiwumbda.org>
 <MPG.1ca2597563bc0d00989967@newsstand.cit.cornell.edu>
 <87sm2vmk1n.fsf@phiwumbda.org>
 <MPG.1ca363c9cd7e60e98996c@newsstand.cit.cornell.edu>
 <d1c9sq01hvg@drn.newsguy.com>
 Discussion, linux)
> Tony says...
>>As I said, the conclusion that there are an infinite number of
>>naturals between any two naturals is nonsensical to me.
> It's nonsensical to me, as well. As I said, nobody has drawn that
> conclusion. You got that conclusion by misquoting what I have said,
> by leaving out the most important part, namely, according to a 
> particular ordering.
>>When that conclusion is drawn from an 
>>assumption that numbers can be placed between
>>numbers when their values aren't between the
>>values of the other numbers,
> It's not an assumption if it is provably true. It is provably
> true that there exists orderings on the naturals other than the
> usual one.
> Write out what you think the assumption is in mathematical
> terms, and tell me why you think the assumption is false.
> Here's what I believe about orderings:
>     Definition: If S is a set, and R is a binary relation on
>     S, then we say that R is a total ordering on S if
>         1. For all x and y in S, either R(x,y) or R(y,x)
>         2. For all x, y and z in S, R(x,y) and R(y,z) implies
>            R(x,z).
I guess you usually want anti-symmetry, too: R(x,y) and R(y,x) implies
x = y.  Else, you have a total pre-order.
You don't need reflexivity, since that's provable by (1).  I've been
including it in my list, but it's not necessary.
-- 
Jesse F. Hughes
Ultimately, I can bring the entire mathematical establishment to its
knees...  Live in a fantasy world if you wish, but to me that's just
an expression of your intellectual inferiority. --James Harris
===
Subject: Re: Epistemology 201: The Science of Science
 <MPG.1c9f8785dcad079f9898e0@newsstand.cit.cornell.edu>
 <d14i8401dgn@drn.newsguy.com>
 <MPG.1c9faeb58f9b3c7d9898fd@newsstand.cit.cornell.edu>
 <d14qc002ftc@drn.newsguy.com>
 <MPG.1c9fc00f82a6e1f798990a@newsstand.cit.cornell.edu>
 <d14ubv02ti7@drn.newsguy.com>
 <MPG.1c9fd85594f28221989913@newsstand.cit.cornell.edu>
 <d156h50n8q@drn.newsguy.com>
 <MPG.1ca0d893c7fb49ba98991f@newsstand.cit.cornell.edu>
 <d175tf0l65@drn.newsguy.com>
 <MPG.1ca12a58f596fab6989940@newsstand.cit.cornell.edu>
 <39r2t1F6495bnU1@individual.net>
 <MPG.1ca2422c4a8ff2bc989954@newsstand.cit.cornell.edu>
 <87y8cnmlct.fsf@phiwumbda.org>
 <MPG.1ca2597563bc0d00989967@newsstand.cit.cornell.edu>
 <87sm2vmk1n.fsf@phiwumbda.org>
 <MPG.1ca363c9cd7e60e98996c@newsstand.cit.cornell.edu>
 <d1c9sq01hvg@drn.newsguy.com> <87zmx2rs0r.fsf@phiwumbda.org>
 Discussion, linux)
>> Tony says...
>As I said, the conclusion that there are an infinite number of
>naturals between any two naturals is nonsensical to me.
>> It's nonsensical to me, as well. As I said, nobody has drawn that
>> conclusion. You got that conclusion by misquoting what I have said,
>> by leaving out the most important part, namely, according to a 
>> particular ordering.
>When that conclusion is drawn from an 
>assumption that numbers can be placed between
>numbers when their values aren't between the
>values of the other numbers,
>> It's not an assumption if it is provably true. It is provably
>> true that there exists orderings on the naturals other than the
>> usual one.
>> Write out what you think the assumption is in mathematical
>> terms, and tell me why you think the assumption is false.
>> Here's what I believe about orderings:
>>     Definition: If S is a set, and R is a binary relation on
>>     S, then we say that R is a total ordering on S if
>>         1. For all x and y in S, either R(x,y) or R(y,x)
>>         2. For all x, y and z in S, R(x,y) and R(y,z) implies
>>            R(x,z).
> I guess you usually want anti-symmetry, too: R(x,y) and R(y,x) implies
> x = y.  Else, you have a total pre-order.
Oops!  You're using a slightly different definition in the rest of
your post, namely one in which R(x,y) implies NOT R(y,x).
Nothing controversial, of course.  Just the difference between <= and
<.
-- 
What if [...] these people HATE mathematics itself, but possibly hang
out here to prove to themselves that there's nothing really to it and
live in the fantasy that they can conquer mathematics itself? 
You know, like arsonists who become firefighters. --James S Harris
===
Subject: Re: Epistemology 201: The Science of Science
Daryl McCullough said:
> Tony says...
>As I said, the conclusion that there are an infinite number of
>naturals between any two naturals is nonsensical to me.
> It's nonsensical to me, as well. As I said, nobody has drawn that
> conclusion. You got that conclusion by misquoting what I have said,
> by leaving out the most important part, namely, according to a 
> particular ordering.
The ordering was the premise, and the conclusion about naturals is 
nonsensical, 
because the premise is nonsensical, because numbers come after smaller 
numbers 
and before larger numbers, by their very nature. In other words, once you 
rearrange the numbers, it is no longer the natural number system. What if 
you 
reordered the numbers from the beginning with Peano's axioms and said 3 came 

before 2? Wouldn't that just constitute switching the meanings of the two 
symbols? Wouldn't 3 mean this many ** and 2 mean this many ***?
>When that conclusion is drawn from an 
>assumption that numbers can be placed between
>numbers when their values aren't between the
>values of the other numbers,
> It's not an assumption if it is provably true. It is provably
> true that there exists orderings on the naturals other than the
> usual one.
Is it provably true that the meaning that numbers derive from their order 
remains intact when the order is changed? I imagine it's provably false.
> Write out what you think the assumption is in mathematical
> terms, and tell me why you think the assumption is false.
> Here's what I believe about orderings:
>     Definition: If S is a set, and R is a binary relation on
>     S, then we say that R is a total ordering on S if
>         1. For all x and y in S, either R(x,y) or R(y,x)
>         2. For all x, y and z in S, R(x,y) and R(y,z) implies R(x,z).
>     Definition: If R is a total ordering on set S, and x,y, and z
>     are elements of S, then we say that y is between x and z according
>     to that ordering if R(x,y) and R(y,z).
>     Theorem: There exists a total ordering R on the naturals such
>     that 2 is between 0 and 1 according to that ordering.
>     Proof: Let R(x,y) == if (x=2 and y=1) then true else x < y
>     By definition  of R, we find R(0,2) and R(2,1).
>     By definition of between, 2 is between 0 and 1
>     according to ordering R.
> This is the ordering 0 2 1 3 4 5 ...
>     Theorem: There exists a total ordering R on the naturals such
>     that there are infinitely many naturals between 0 and 1 according
>     to that ordering.
>     Proof: Let R(x,y) == if x=y then false else if (x=1 or y=2) then true
>                          else x < y
> This is the ordering 0 2 3 4 5 6 ... 1
> So exactly what am I assuming that you don't believe? Are you saying
> that I can't write down the above orderings? I just did.
You can create any orderings you like, and draw conclusions based on those 
orderings. You cannot apply those conslusions generally. If you have a use 
for 
such a number system and such a fact concerning it, by all means derive your 

conclusions. But conclusions drawn from such ordering cannot be applied to 
the 
number system in its natural order. There are infinitely many more rationals 

than integers, in the real world, despite what ordering the rationals into a 

traversible matrix may imply in the Garden of Cantor.
>then I credit that assumption with the 
>error in the conclusion.
> If you would actually quote the *entire* conclusion,
> then you would see that it isn't an error. It becomes
> an error when you misquote it. That makes the error
> due to *your* mistakes, not those of anyone else.
I left out your premise.
>So, like I said to Daryl, if you want to make that 
>assumption, then sure, it is valid to draw that conclusion,
>but you can't claim the conclusion is sound unless you can
>justify the assumption.
> One way to justify an assumption is to prove it. That's been done.
So, you proved that 15 comes before 2 and 6? Well, you also proved 3<2, so 
what 
do I expect?
>Since the conclusion makes no sense,
> I agree that your conclusion makes no sense, but it wasn't *my*
> conclusion. My conclusion included the phrase according to the
> ordering, which is a critical point.
That phrase was an assertion, along with the validity of the method of 
bijection.
>and the assumption makes no sense, I reject the argument.
> Why don't you reject the habit of misquoting conclusions?
>If one can create a 1-1 and onto relation between members of two sets of 
>numbers, *in their natural quantitative order*, and create a bijection, 
then 
>the two sets have exactly the same size.
> You don't mean that. There exists a bijection between the
> naturals and the even naturals that preserves the natural
> quantitative ordering:
>     n --> 2n
Then why does 4 appear before 3 does? Perhaps the definition needs 
refinement. 
It's a work in progess. I never claimed to have a full system worked out. 
That 
doesn't invalidate my objections.
> You really don't know what you mean, do you?
>When you introduce reordering of numbers, then you
>confound the picture
> I would say that when you misquote people in such a way
> as to change the meaning of what they are saying, you
> confound the picture.
Boo hoo, you separated my premise frmm my conclusion to indetify the flaw in 

my 
argument. Come on Daryl, you know that given A then B is the same as 
if A is 
true then B is true, so your statement is equivalent to if there is an 
ordering R, then there are infinitely many naturals between any two 
naturals. 
The ordering serves as a starting assertion and the statement about naturals 

is 
the conclusion drawn. In the context of the premise the conclusion is valid, 

but conclusions drawn on artifical orderings cannot be applied to naturally 

ordered number systems. The fact is, there are infinitely many more 
rationals 
than integers, and the fact is that there are not really in infinite number 

of 
naturals between any two naturals. It is possible to draw any conclusion you 

want, validly, by using unsound assertions, but if you want to draw sound 
consclusions you have to start with real, true assertions.
>so that you can basically only distinguish between countable
>infinities and continuous infinities of different 
>dimensions.
> *Who* can only distinguish between countable infinities
> and continuous infinities of different dimensions? I
> can certainly distinguish between the rationals and
> the naturals based on their order type, or based on
> the fact that one is a subset of the other. I can
> certainly distinguish between the naturals and the
> even numbers based on the fact that one is a subset
> of the other, and also based on the fact that the
> relative frequency of one set is 1/2 the relative
> frequency of the other set.
> You seem to have a totalitarian mindset in which anything
> that is not forbidden is mandatory. The fact that I *can*
> compare two sets using cardinality doesn't imply
> that I *must* compare them using cardinality.
I am only saying that cardinality does not capture these distinctions when 
it 
introduces artificial orderings. The other methods you mentioned for 
catching 
those distinction, if you will notice, involve no reordering of the number 
system. That's why they are able to make those distinctions.
>So, while you cannot imagine a better method of distinguishing 
>infinities, I can.
> No, you can't. You haven't come up with any better method. There
> are no distinctions that you have been able to make that are not
> already well understood by conventional mathematics, which includes
> the tools of cardinality, relative frequency, measure, subset
> relationships, and order types.
I have successfully pointed out flaws in Cantor's method and their root 
source. 
I have not yet completed the method which will replace cardinality and 
unseat 
Cantor. I certainly have demonstrated that I can imagine a better method 
that 
draws sounder conclusions.
> --
> Daryl McCullough
> Ithaca, NY
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
> As I said, the conclusion that there are an infinite number of naturals 
between 
> any two naturals is nonsensical to me.
If you are talking about the usual linear ordering, you are quite 
correct. If you are talking about -any- linear ordering you are quite 
wrong. There are linear orderings of the naturals which are dense and 
there are linear orders of the naturals which  are not dense.
The important thing is that the ordering in question must be specified 
or defined first, then statements can be made about it.
Bob Kolker
===
Subject: Re: Epistemology 201: The Science of Science
robert j. kolker said:
> 
> As I said, the conclusion that there are an infinite number of naturals 
between 
> any two naturals is nonsensical to me.
> If you are talking about the usual linear ordering, you are quite 
> correct. If you are talking about -any- linear ordering you are quite 
> wrong. There are linear orderings of the naturals which are dense and 
> there are linear orders of the naturals which  are not dense.
> The important thing is that the ordering in question must be specified 
> or defined first, then statements can be made about it.
> Bob Kolker
See, Bob? You say the same thing over and over, refuse to acknowledge the 
problems I have pointed out, and insist on drawing false conclusions from 
false 
premises. You accuse me of not understanding you, but your responses sound 
like 
I have said what? instead of addressing particulars, because you dson't 
acknowledge them. Ho hum. 
The important thing is that the order of numbers not contradict their 
values, 
or the conclusion you draw from that order will contradict sense.
-- 
Smiles,
Tony
===
Subject: Distinct linear orderings on Z
> Jesse, apparently you haven't been paying attention either. When you come 

to 
> conclusions that make no sense,
If the ordering is defined it makes sense. It may not make the sense you 
want to make, but it makes sense. The set of distinct linear orderings 
on Z  (the integers) has infinte cardinality.
Now here is a serious question: (therefore not addressed to Tony or 
Albert). How many (in the sense of cardinality) distinct linear 
orderings are there on Z. I say two linear orderings L1 and L2 are 
distinct if L1 cannot be gotten from L2 by an order isomorphism with the 
ground set Z.
Bob Kolker
===
Subject: Re: Distinct linear orderings on Z
robert j. kolker said:
> 
> Jesse, apparently you haven't been paying attention either. When you 
come to 
> conclusions that make no sense,
> If the ordering is defined it makes sense. It may not make the sense you 
> want to make, but it makes sense. The set of distinct linear orderings 
> on Z  (the integers) has infinte cardinality.
> Now here is a serious question: (therefore not addressed to Tony or 
> Albert). How many (in the sense of cardinality) distinct linear 
> orderings are there on Z. I say two linear orderings L1 and L2 are 
> distinct if L1 cannot be gotten from L2 by an order isomorphism with the 
> ground set Z.
> Bob Kolker
Ooohh Ooohh!! Aleph-1! Aleph-1!!!
-- 
Smiles,
Tony
===
Subject: Re: Distinct linear orderings on Z
> Now here is a serious question: (therefore not addressed to Tony or 
> Albert). How many (in the sense of cardinality) distinct linear 
> orderings are there on Z. I say two linear orderings L1 and L2 are 
> distinct if L1 cannot be gotten from L2 by an order isomorphism with 
> the ground set Z. 
I think the answer is aleph1 (the first uncountable ordinal).  There are 
certainly aleph1 nonisomprophic well-orderings (for aleph1 is the set of 
all countable ordinals).  Isn't any linear ordering of  Z  a pasting of 
a reverse well-ordering to a well-ordering?  Or can one show otherwise 
that there are at most aleph1 linear orderings?
-- 
Stephen J. Herschkorn                        sjherschko@netscape.net
===
Subject: Re: Epistemology 201: The Science of Science
> Have you been paying attention Bob? In the context of numbers, between 
means 
> greater than one and less than the other, in quantity. By this deifnition, 

does 
> the conclusion make any sense?
Greater with respect to what linear ordering? There are many linear 
orderings. If you mean the ordering induced by the Peano successor 
function there between an integer and its successor there is no integer.
Bob Kolker
===
Subject: Re: Epistemology 201: The Science of Science
robert j. kolker said:
> 
> Have you been paying attention Bob? In the context of numbers, 
between means 
> greater than one and less than the other, in quantity. By this 
deifnition, does 
> the conclusion make any sense?
> Greater with respect to what linear ordering? There are many linear 
> orderings. If you mean the ordering induced by the Peano successor 
> function there between an integer and its successor there is no integer.
> Bob Kolker
And what other ordering of the naturals would you prefer? Does any other 
than 
the inherent ordering make sense? Clearly, not. Therefore the conclusion 
that 
there are infinitely many naturals between any pair of naturals is clearly 
wrong. Therefore there is clearly a flaw in the method. Therefore, if I were 

you, I would not try to prove anything to ME using that method, because I 
won't 
be fooled, now that I've identified the root problem with it. If I were you, 

I 
would further stop accepting it as true, just because it hasn't been voted 
out 
of office yet.
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
> And what other ordering of the naturals would you prefer? Does any other 
than 
> the inherent ordering make sense? 
Any well defined linear ordering makes as much sense (or as little 
sense) as any other well defined linear ordering.
> Clearly, not. Therefore the conclusion that 
> there are infinitely many naturals between any pair of naturals is clearly 

> wrong. 
This is the case wrt the normal ordering, i.e. the ordering induced by 
the successor function. But it is NOT generally true of all linear 
orderings on Z. Consider  a mapping g from the set of integers Z into 
the set of rationals such that g is one to one onto. Such functions 
exist. Now define R(x,y) where x,y in Z to mean g(x) < g(y). It is 
trivial to show that R is a linear ordering on Z. Now let x, y in Z be 
distinct integers. Assume without loss of generality that g(x) < g(y) in 
the set of Rationals. Then there exist a rational number w such that 
g(x) < w < g(y).  By definition of R, R(x, g-inverse(w)) and
R(g-inverse(w), y). So in the R ordering g-inverse (w) is between 
(i.e. with respect to the linear ordering R) x and y. In particular this 
is true for x = 1 an y = 2. So there are linear orderings on the 
integers such that there is an integer between 1 and 2. So to say 
there is no integer between an integer and its successor one must 
specify the linear ordering to which between pertains.
Bob Kolker
===
Subject: Re: Epistemology 201: The Science of Science
robert j. kolker said:
> 
> And what other ordering of the naturals would you prefer? Does any other 
than 
> the inherent ordering make sense? 
> Any well defined linear ordering makes as much sense (or as little 
> sense) as any other well defined linear ordering.
> Clearly, not. Therefore the conclusion that 
> there are infinitely many naturals between any pair of naturals is 
clearly 
> wrong. 
> This is the case wrt the normal ordering, i.e. the ordering induced by 
> the successor function. But it is NOT generally true of all linear 
> orderings on Z. Consider  a mapping g from the set of integers Z into 
> the set of rationals such that g is one to one onto. Such functions 
> exist. Now define R(x,y) where x,y in Z to mean g(x) < g(y). It is 
> trivial to show that R is a linear ordering on Z. Now let x, y in Z be 
> distinct integers. Assume without loss of generality that g(x) < g(y) in 
> the set of Rationals. Then there exist a rational number w such that 
> g(x) < w < g(y).  By definition of R, R(x, g-inverse(w)) and
> R(g-inverse(w), y). So in the R ordering g-inverse (w) is between 
> (i.e. with respect to the linear ordering R) x and y. In particular this 
> is true for x = 1 an y = 2. So there are linear orderings on the 
> integers such that there is an integer between 1 and 2. So to say 
> there is no integer between an integer and its successor one must 
> specify the linear ordering to which between pertains.
> Bob Kolker
Yet another example of number theory obfuscation. Very good. 1 and 2 have 
lost 
their meanings. Good job. Have a cookie.
You might want to consider if you could possibly get all the 
counterintuitive results you get without reordering numbers out of 
sequence. 
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
aeo6 says...
>Yet another example of number theory obfuscation. Very good. 1 and 2 have 
lost 
>their meanings. Good job. Have a cookie.
Are you saying that if I write down
    3 1 4
that suddenly 3 has lost its meaning? *I* still remember what it
means, even if you don't. With a little practice, you can learn
to recite your telephone number, or the digits of pi, even when
the digits are out of order. The digits don't lose their meaning.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Epistemology 201: The Science of Science
> aeo6 says...
>>Yet another example of number theory obfuscation. Very good. 1 and 2 have 

lost 
>>their meanings. Good job. Have a cookie.
> Are you saying that if I write down
>     3 1 4
> that suddenly 3 has lost its meaning? *I* still remember what it
> means, even if you don't. With a little practice, you can learn
> to recite your telephone number, or the digits of pi, even when
> the digits are out of order. The digits don't lose their meaning.
A telephone number is not a number.  It is a character string 
composed of numerals.
-- 
I consider it quite possible that physics cannot be based
on the field concept, i. e., on continuous structures. In that
case nothing remains of my entire castle in the air,
gravitation theory included, [and of] the rest of modern physics.
        -- Albert Einstein in a 1954 letter to Michele Besso.
===
Subject: Re: Epistemology 201: The Science of Science
:> aeo6 says...
:>>Yet another example of number theory obfuscation. Very good. 1 and 2 have 

lost 
:>>their meanings. Good job. Have a cookie.
:> Are you saying that if I write down
:>     3 1 4
:> that suddenly 3 has lost its meaning? *I* still remember what it
:> means, even if you don't. With a little practice, you can learn
:> to recite your telephone number, or the digits of pi, even when
:> the digits are out of order. The digits don't lose their meaning.
: A telephone number is not a number.  It is a character string 
: composed of numerals.
An ordering is not a number either.  Once again you miss
the point.  
Stephen
===
Subject: Re: Epistemology 201: The Science of Science
stephen@nomail.com said:
> :> aeo6 says...
> :> 
> :> 
> :>>Yet another example of number theory obfuscation. Very good. 1 and 2 
have lost 
> :>>their meanings. Good job. Have a cookie.
> :> 
> :> 
> :> Are you saying that if I write down
> :> 
> :>     3 1 4
> :> 
> :> that suddenly 3 has lost its meaning? *I* still remember what it
> :> means, even if you don't. With a little practice, you can learn
> :> to recite your telephone number, or the digits of pi, even when
> :> the digits are out of order. The digits don't lose their meaning.
> : A telephone number is not a number.  It is a character string 
> : composed of numerals.
> An ordering is not a number either.  Once again you miss
> the point.  
> Stephen
It's not hard to miss the point of a statement that has no connection to 
anything that anyone has said. I think Albert hit the nail on the head by 
calling it a phone number. I never said you can't write digits in any order. 

It 
was a pointless question. It's you who miss the point of the entire 
discussion. 
Who ever said an ordering was a number? Take a vitamin.
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
Daryl McCullough said:
> aeo6 says...
>Yet another example of number theory obfuscation. Very good. 1 and 2 have 
lost 
>their meanings. Good job. Have a cookie.
> Are you saying that if I write down
>     3 1 4
> that suddenly 3 has lost its meaning? *I* still remember what it
> means, even if you don't. With a little practice, you can learn
> to recite your telephone number, or the digits of pi, even when
> the digits are out of order. The digits don't lose their meaning.
> --
> Daryl McCullough
> Ithaca, NY
If you are saying 3 comes before 1, which immediately precedes 4 in a number 

system, then you are changing the meaning of the numerals. Of course there 
are 
an infinite number of uses for placing numbers in all sorts of orders. I am 

talking specifically about rearranging the number system, but I thought I 
made 
that clear already. Pay attention.
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
Tony says...
>Daryl McCullough said:
>> Are you saying that if I write down
>>     3 1 4
>> that suddenly 3 has lost its meaning? *I* still remember what it
>> means, even if you don't. With a little practice, you can learn
>> to recite your telephone number, or the digits of pi, even when
>> the digits are out of order. The digits don't lose their meaning.
>If you are saying 3 comes before 1, which immediately precedes 4 in a 
number 
>system, then you are changing the meaning of the numerals.
Who said anything about a number system? I am talking about orderings
of the natural number system. Sometimes, we write 3 before 1; for example,
when we are writing out the decimal expansion of pi. That doesn't mean
that 3 has lost its meaning.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Epistemology 201: The Science of Science
Daryl McCullough said:
> Tony says...
>Daryl McCullough said:
>> Are you saying that if I write down
>> 
>>     3 1 4
>> 
>> that suddenly 3 has lost its meaning? *I* still remember what it
>> means, even if you don't. With a little practice, you can learn
>> to recite your telephone number, or the digits of pi, even when
>> the digits are out of order. The digits don't lose their meaning.
> 
>If you are saying 3 comes before 1, which immediately precedes 4 in a 
number 
>system, then you are changing the meaning of the numerals.
> Who said anything about a number system? I am talking about orderings
> of the natural number system. Sometimes, we write 3 before 1; for 
example,
> when we are writing out the decimal expansion of pi. That doesn't mean
> that 3 has lost its meaning.
> --
> Daryl McCullough
> Ithaca, NY
That is entirely unrelated to the reordering of the number system that I was 

cautioning against. The use of digits to form numbers is a matter of a 
digital 
system for expressing numbers as numerals. If you write 3.1415, that's a 
number 
between 3.1 and 3.2. Do you think I am saying that all 1's should come 
before 
all 2's? I have never made any such stupid assertion. My statement was 
specifically in the context of cardinality and concerning arbitrary 
reorderings 
of the number system in general to achieve a mapping function. And people 
accuse me of being confused.......
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
Tony says...
>Daryl McCullough said:
>> Who said anything about a number system? I am talking about orderings
>> of the natural number system. Sometimes, we write 3 before 1; for 
example,
>> when we are writing out the decimal expansion of pi. That doesn't mean
>> that 3 has lost its meaning.
>That is entirely unrelated to the reordering of the number system that I 
was 
>cautioning against.
Nobody talked about reordering the number system. You are confused.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Epistemology 201: The Science of Science
Daryl McCullough said:
> Tony says...
>Daryl McCullough said:
>> Who said anything about a number system? I am talking about orderings
>> of the natural number system. Sometimes, we write 3 before 1; for 
example,
>> when we are writing out the decimal expansion of pi. That doesn't mean
>> that 3 has lost its meaning.
> 
>That is entirely unrelated to the reordering of the number system that I 
was 
>cautioning against.
> Nobody talked about reordering the number system. You are confused.
> --
> Daryl McCullough
> Ithaca, NY
So, sticking 15 between 2 and 5 isn't reordering the number system? Counting 

through the multiples of two before getting to 9 isn't reordering the number 

system? What are all these orderings performed in bijections, then? I am not 

confused. 
followed immediately by, I am talking about orderings of the natural 
number 
system. I guess the answer to the question would be you, based on the 
statement that immediately follows it. And now you say, Nobody talked 
about 
reordering the number system. You are confused. So, what were you talking 
about immediately before? It's all right there, so don't accuse me of 
misquoting you. I am not the confused one.
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
Tony says...
>Daryl McCullough said:
>> Nobody talked about reordering the number system. You are confused.
>So, sticking 15 between 2 and 5 isn't reordering the number system?
Sticking 15 between 2 and 5 sounds as if someone is picking numbers
up and moving them around. That isn't possible, since numbers are
abstractions, not physical objects. Instead, what people have done
is to define a new ordering on the naturals. Defining an binary
relation on numbers doesn't do anything to the numbers.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Epistemology 201: The Science of Science
> Daryl McCullough said:
>>aeo6 says...
>What I object to is the use of numbers in orderings that are not 
consistent, 
>using a technique that is not proven, to derive results that contradict 
more 
>basic notions of set theory
>>Well, nobody is using numbers in orderings that are not consistent, and
>>nobody is deriving results that contradict more basic notions of set 
theory.
> If the ordering of quantities is inconsistent with the quantities, then 
the 
> ordering is inconsistent. If a set contains all the elements of another 
set, 
> plus some more, then it has more. These are pretty basic concepts that I 
> shouldn't have to repeat.
>>I already told you, the result is not there are an infinite number
>>of naturals between any two naturals, the result is that for the
>>particular ordering that I defined, there are infinitely many
>>naturals between any two naturals.
>Right, but do you believe the conclusion based on that ordering?
>>The conclusion is: There exists an ordering R(x,y) on naturals
>>such that between any two naturals x and y, there exists infinitely
>>many naturals z such that R(x,z) and R(z,y).
>>Yes, I believe that. I constructed such an ordering.
>Is it true that there are infinitely many naturals between
>any two naturals, given the normal meaning of between for
>natural numbers?
>>Of course not.
>If not, then  you should examine the train of thought that led to
>that conclusion
>>Nothing led to that conclusion. You are misquoting the conclusion.
>>I came to the conclusion that
>>There exists an ordering R(x,y) on naturals
>>such that between any two naturals x and y,
>>there exists infinitely many naturals z such
>>that R(x,z) and R(z,y).
> You are not being honest. Here is your actual quote:
> I didn't say that. I said that for that particular *ordering* on the
> naturals, there exists infinitely many naturals between
> any pair of naturals. It is a property of the ordering, not the
> underlying set.
> Now, do you believe that there exists infinitely many naturals between
> any pair of naturals or not? If not, how do you explain the conclusion 
you 
> drew?
Read what you quoted, Tony. It's all there. And stop smoking up - it 
addles your brain.
[...]
===
Subject: Re: Epistemology 201: The Science of Science
Wolf Kirchmeir said:
<snip Now, do you believe that there exists infinitely many naturals 
between
> any pair of naturals or not? If not, how do you explain the conclusion 
you 
> drew?
> Read what you quoted, Tony. It's all there. And stop smoking up - it 
> addles your brain.
> [...]
Pay attention, don't be insulting, and answer the question, as I have 
already 
pointed out the flaw in your orderings, and apparently need to rub your nose 

in 
it and admit that the conclusion is false to get anywhere. Read carefully 
and 
answer the question, oh self-addler.
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
aeo6 says...
>Daryl McCullough said:
>> aeo6 says...
>What I object to is the use of numbers in orderings that are not 
consistent,
>using a technique that is not proven, to derive results that contradict 
more
>>basic notions of set theory
>> Well, nobody is using numbers in orderings that are not consistent, and
>> nobody is deriving results that contradict more basic notions of set 
theory.
>If the ordering of quantities is inconsistent with the quantities
What is that phrase supposed to mean? An ordering is a binary relation.
Are you trying to say that you think that there is only one possible
binary relation on the naturals?
>then the ordering is inconsistent.
What does it mean to say that an ordering is inconsistent? You're
not making any sense.
>If a set contains all the elements of another set, 
>plus some more, then it has more.
It has more according to the subset relation on sets. Nobody
argues with that.
>These are pretty basic concepts that I shouldn't have to repeat.
Nobody is arguing about the basic concepts:
   1. If every element of A is also an element of B, but not
   vice-versa, then A is a proper subset of B.
   2. The set of reals in [0,1] has half the measure of the set
   of reals in [0,2].
   3. The relative frequency of even naturals is half the relative
   frequency of all naturals.
   4. The order type of the rationals is larger than the order
   type of the naturals. (That is, the naturals can be embedded
   in the rationals in an order-preserving way, but not vice-versa).
The part that people disagree with is that you want to use the
phrase has more elements for all 4 cases. That's just being
perverse. The technical definitions of subset, measure,
relative frequency, order embedding, and cardinality
are different. No definition captures everything that you
might want to say about the relationship between sets.
You are arguing about words, and insisting that people use
your terminology, which glosses over at least 5 different
distinctions.
>> There exists an ordering R(x,y) on naturals
>> such that between any two naturals x and y,
>> there exists infinitely many naturals z such
>> that R(x,z) and R(z,y).
>You are not being honest. Here is your actual quote:
>I didn't say that. I said that for that particular *ordering* on the
>naturals, there exists infinitely many naturals between
>any pair of naturals. It is a property of the ordering, not the
>underlying set.
How is that different? In both cases, I'm saying that the
betweeness relationship is a property of the *ordering*,
and whether or not there are infinitely many naturals
between any two naturals depends on what *ordering* you
are using.
>Now, do you believe that there exists infinitely many naturals between
>any pair of naturals or not?
Why do you keep leaving out the important phrase for the particular
ordering? That is the *key* phrase, and you keep deleting it.
Betweeness is a property of the ordering.
>If not, how do you explain the conclusion you drew?
The conclusion is that for that particular *ordering* on the
naturals, there exists infinitely many naturals between
any pair of naturals. That's a true conclusion. It is *provably*
true. If you are rejecting it, then you are being inconsistent.
>> Or, what's actually the case, you misquoted the conclusion.
>Or not.
You did misquote it. I specifically said for that particular
ordering, and you left that phrase out.
>>This is yet another example of how artificial orderings that
>>violate the natural order of real numbers can lead to erroneous
>>results.
>> The only erroneous results are due to you.
>Try again.
Okay. The only erroneous results are due to you.
>> My conclusion wasn't wrong---yours was.
>Read your quoted conclusion above, again.
You misquoted it by leaving out the phrase for the
particular ordering I defined.
>> Nobody disagrees with the fact that, with the usual ordering
>> on the rationals, there are infinitely many rationals between
>> any two integers.
>Are there infinitely many integers between any two rationals?
For which ordering? Betweeness depends on the ordering. For
the *usual* ordering, there are infinitely many rationals
between any two naturals, and not vice-versa.
>No? What does that tell you about the relative numbers integers 
>and rationals?
Nothing. It tells something about the *order* type of the rationals.
>>To draw a conclusion so drastically different from the blatantly
>>obvious,
>> The only conclusion is that there exists a bijection between
>> the rationals and the integers. That's provably the case.
>More bijection genuflection. What does this case prove?
It's just a fact. A fact that you have trouble with, for some reason.
>Certainly nothing about the relative numbers of integers and rationals.
There are several *different* relationship between the set of
integers and the set of rationals:
    1. The order type of the rationals is larger than the order
    type of the integers.
    2. The integers are a subset of the rationals.
    3. The cardinality of the integers is the same as the cardinality
    of the rationals.
For you to say there are more rationals than there are integers is
*ambiguous*. It might mean in the sense of order type, it might mean
in the sense of subsets, it might mean in the sense of cardinality.
>>using a technique that is assumed to work because it
>>looks neat,
>> That's not true.
>Why else is it accepted?
It's accepted for the same reasons any good theory is accepted:
It is rigorous, it produces interesting results, it produces results
that are useful in a wide variety of fields.
>>but is not supported by any empirical evidence or
>>application to reality,
>> That's not true.
>What empirical evidence can you offer?
Cardinality theory is a critical part of measure theory, which
in turn is a critical ingredient to probability theory. Probability
theory has been very successful in a wide range of practical
applications.
>> What conclusion? The concusion is that there is a bijection
>> between the integers and the rationals. You disagree with that
>> conclusion?
>No, just with the conclusion that there exists infinitely many naturals 
>between any pair of naturals
Nobody made that conclusion. The conclusion was that 
    For the particular binary relation R(x,y) that I defined,
    there are infinitely many naturals between any two naturals.
Leaving out the phrase for the particular binary relation is
leaving out the important part. Betweenness is a property of *orderings*.
>> I don't see how the word entitled applies here. You are almost
>> completely ignorant of the subject that you are criticizing, and
>> your criticism doesn't amount to a hill of beans. The complaints
>> you have are almost all complaints about *word* choice, which is
>> about the silliest basis for criticizing mathematics.
>That is entirely untrue.
Then what *mathematical* complaint do you have? Express it
in the language of mathematics. If you are going to use a
phrase such as has more elements, then give a *mathematical*
definition of how you are interpreting that phrase. I offered
4 different interpretations of has more elements:
    1. Has a higher relative frequency.
    2. Has a larger cardinality.
    3. Has a larger order type.
    4. Is a superset.
(Each of these have precise definitions, if you are unclear about
them.)
If you want to use yet another notion of has more elements,
then give a mathematical definition. You can't use an informal
phrase in mathematics without giving a mathematical definition.
>see that you have not been reading too carefully, as I've been making a 
clear 
>case that the conclusions drawn from bijection are overdrawn. Do you now 
agree 
>that the conclusion you drew, which I quoted, is incorrect?
You have consistently quoted me incorrectly. In particular, I have
consistently said that the word betweenness in a phrase such as
      There are infinitely many elements z between x and y
is a property of the *ordering*, not a property of the underlying
sets. When you leave off the specification of the ordering, you
are changing the meaning of the phrase.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Epistemology 201: The Science of Science
Daryl McCullough said:
> aeo6 says...
>Daryl McCullough said:
>> aeo6 says...
>> 
>What I object to is the use of numbers in orderings that are not 
consistent,
>using a technique that is not proven, to derive results that contradict 
more
>>basic notions of set theory
>> 
>> Well, nobody is using numbers in orderings that are not consistent, 
and
>> nobody is deriving results that contradict more basic notions of set 
theory.
>If the ordering of quantities is inconsistent with the quantities
> What is that phrase supposed to mean? An ordering is a binary relation.
> Are you trying to say that you think that there is only one possible
> binary relation on the naturals?
>then the ordering is inconsistent.
> What does it mean to say that an ordering is inconsistent? You're
> not making any sense.
There is one natural ordering of numbers in general, and that's where 
smaller 
quantitative values precede larger values. Any other ordering violates the 
members' identities as numbers.
>If a set contains all the elements of another set, 
>plus some more, then it has more.
> It has more according to the subset relation on sets. Nobody
> argues with that.
Except Mr. Cardinality, but he's half blind.
>These are pretty basic concepts that I shouldn't have to repeat.
> Nobody is arguing about the basic concepts:
>    1. If every element of A is also an element of B, but not
>    vice-versa, then A is a proper subset of B.
>    2. The set of reals in [0,1] has half the measure of the set
>    of reals in [0,2].
>    3. The relative frequency of even naturals is half the relative
>    frequency of all naturals.
>    4. The order type of the rationals is larger than the order
>    type of the naturals. (That is, the naturals can be embedded
>    in the rationals in an order-preserving way, but not vice-versa).
> The part that people disagree with is that you want to use the
> phrase has more elements for all 4 cases. That's just being
> perverse. The technical definitions of subset, measure,
> relative frequency, order embedding, and cardinality
> are different. No definition captures everything that you
> might want to say about the relationship between sets.
> You are arguing about words, and insisting that people use
> your terminology, which glosses over at least 5 different
> distinctions.
See below.
>> There exists an ordering R(x,y) on naturals
>> such that between any two naturals x and y,
>> there exists infinitely many naturals z such
>> that R(x,z) and R(z,y).
>You are not being honest. Here is your actual quote:
>I didn't say that. I said that for that particular *ordering* on the
>naturals, there exists infinitely many naturals between
>any pair of naturals. It is a property of the ordering, not the
>underlying set.
> How is that different? In both cases, I'm saying that the
> betweeness relationship is a property of the *ordering*,
> and whether or not there are infinitely many naturals
> between any two naturals depends on what *ordering* you
> are using.
You denied saying there were infinite many naturals between any two 
naturals, 
so I showed you your quote and asked you to defend it. If you saw no 
difference 
between the two, then why did you accuse me of misquoting and rephrase it? 
Is 
there something wrong with the way you phrased it the first time? This is 
exactly what I object to. That there are an infinite number of naturals 
between 
any two naturals is rotten hogwash used to make diaper tea. You can't really 

believe that statement, can you?
>Now, do you believe that there exists infinitely many naturals between
>any pair of naturals or not?
> Why do you keep leaving out the important phrase for the particular
> ordering? That is the *key* phrase, and you keep deleting it.
I delete it because it is the flaw in your argument. I am asking if, in any 

real sense, in your mind, regardless of any ordering, it seems at all 
plausible 
that there are infinitely many naturals between any pair of naturals, using 

a 
sensible consistent definition for between. What naturals are more than 
2 and 
less than 3? The conclusion is wrong, therefore there is a problem with the 

method, or some assumption applied to it. The problem is in the reordering 
of 
inherently ordered elements, while using the inherent orders as well. This 
is 
some typical makings of your standard paradox. Paradoxes are all resolvable. 

they all boil down to mutually contradictory terms where the contradiction 
is 
not obvious. Incidentally, all paradoxes are counterintuitive, aren't they?
> Betweeness is a property of the ordering.
>If not, how do you explain the conclusion you drew?
> The conclusion is that for that particular *ordering* on the
> naturals, there exists infinitely many naturals between
> any pair of naturals. That's a true conclusion. It is *provably*
> true. If you are rejecting it, then you are being inconsistent.
You can prove it based on Cantor's theorems, but are they provable? 
Somewhere 
in the mix is the assumption that you can change the order of numbers and 
preserve their identities as numbers. This is false, and needs to be 
eliminated 
if any kind of consistency between these methods is to be achieved.
>> Or, what's actually the case, you misquoted the conclusion.
>Or not.
> You did misquote it. I specifically said for that particular
> ordering, and you left that phrase out.
That's not how you corrected it. You changed the language so it didn't say 
there are an infinite number of naturals between any pair of naturals, which 

was dishonest. I left out the ordering aspect because it's not relevant to 
what 
I am asking. Do you actually, not in the context of any ordering but in 
general, see that conclusion as being correct? I have asked this several 
times, 
and you refuse to answer. Can you answer the question directly? I can.
>>This is yet another example of how artificial orderings that
>>violate the natural order of real numbers can lead to erroneous
>>results.
>> 
>> The only erroneous results are due to you.
>Try again.
> Okay. The only erroneous results are due to you.
>> My conclusion wasn't wrong---yours was.
>Read your quoted conclusion above, again.
> You misquoted it by leaving out the phrase for the
> particular ordering I defined.
Explained twice already above.
>> Nobody disagrees with the fact that, with the usual ordering
>> on the rationals, there are infinitely many rationals between
>> any two integers.
>Are there infinitely many integers between any two rationals?
> For which ordering? Betweeness depends on the ordering. For
> the *usual* ordering, there are infinitely many rationals
> between any two naturals, and not vice-versa.
Correct. For the regular, non-ridiculous, non-meaning-violating order.
>No? What does that tell you about the relative numbers integers 
>and rationals?
> Nothing. It tells something about the *order* type of the rationals.
Which indicates there are more rationals than integers, which means they 
comprise a larger set.
>>To draw a conclusion so drastically different from the blatantly
>>obvious,
>> 
>> The only conclusion is that there exists a bijection between
>> the rationals and the integers. That's provably the case.
>More bijection genuflection. What does this case prove?
> It's just a fact. A fact that you have trouble with, for some reason.
What does the bijection prove? What inference can you draw from it, besides 

that, according to the method you have used, it is the conclusion you have 
drawn? The bijection does not mean the sets are of equal size. It means that 

the cardinality measure as defined does not distinguish between them. Please 

show a proof of the validity of equating set size with cardinality for 
infinite 
sets, which doesn't circularly fall back on the definition of cardinality as 

the size of the set.
>Certainly nothing about the relative numbers of integers and rationals.
> There are several *different* relationship between the set of
> integers and the set of rationals:
>     1. The order type of the rationals is larger than the order
>     type of the integers.
>     2. The integers are a subset of the rationals.
>     3. The cardinality of the integers is the same as the cardinality
>     of the rationals.
> For you to say there are more rationals than there are integers is
> *ambiguous*. It might mean in the sense of order type, it might mean
> in the sense of subsets, it might mean in the sense of cardinality.
I don't think it's ambiguous at all. I think it is the case in fact, and any 

method that fails to detect that fails in that respect. It's not a matter of 

opinion. It's not sometimes true. It's true universally. If the 
interpretation 
of cardinality is that it's not true, then the interpretation is wrong.
I think it is ambiguous to have several different ways of answering the same 

question, and getting different answers depending on how you ask it. The 
question, in your mind, has no definite answer. If that's not ambiguity, 
then I 
don't know what is.
>>using a technique that is assumed to work because it
>>looks neat,
>> 
>> That's not true.
>Why else is it accepted?
> It's accepted for the same reasons any good theory is accepted:
> It is rigorous, it produces interesting results, it produces results
> that are useful in a wide variety of fields.
What is the application of the fact that there are as many integers as 
rationals in some abstract sense, operationally?
>>but is not supported by any empirical evidence or
>>application to reality,
>> 
>> That's not true.
>What empirical evidence can you offer?
> Cardinality theory is a critical part of measure theory, which
> in turn is a critical ingredient to probability theory. Probability
> theory has been very successful in a wide range of practical
> applications.
In dealing with infinite sets of rationals vs. integers?
>> What conclusion? The concusion is that there is a bijection
>> between the integers and the rationals. You disagree with that
>> conclusion?
>No, just with the conclusion that there exists infinitely many naturals 
>between any pair of naturals
> Nobody made that conclusion. The conclusion was that 
>     For the particular binary relation R(x,y) that I defined,
>     there are infinitely many naturals between any two naturals.
I stated the conclusion correctly. The first part is an assertion upon which 

the conclusion is based. This is basic logic, to wit:
 the particular binary relation R(x,y)
 -> 
 there are infinitely many naturals between any two naturals
 in the form A->B (A implies B)
Now, if A->B and B is false, then what can you say about A? This is easy, 
and 
what I have been pointing out all along. It seems like I have to iterate my 

way 
through the infinite set of ways of saying the same thing to get anywhere 
with 
this. A is false, the ordering is self-contradictory, and the conclusions 
drawn 
from a self-contradictory construct are unsound at best.
> Leaving out the phrase for the particular binary relation is
> leaving out the important part. Betweenness is a property of *orderings*.
Numbers are already inherently ordered.
>> I don't see how the word entitled applies here. You are almost
>> completely ignorant of the subject that you are criticizing, and
>> your criticism doesn't amount to a hill of beans. The complaints
>> you have are almost all complaints about *word* choice, which is
>> about the silliest basis for criticizing mathematics.
>That is entirely untrue.
> Then what *mathematical* complaint do you have? Express it
> in the language of mathematics. If you are going to use a
> phrase such as has more elements, then give a *mathematical*
> definition of how you are interpreting that phrase. I offered
> 4 different interpretations of has more elements:
>     1. Has a higher relative frequency.
>     2. Has a larger cardinality.
>     3. Has a larger order type.
>     4. Is a superset.
> (Each of these have precise definitions, if you are unclear about
> them.)
> If you want to use yet another notion of has more elements,
> then give a mathematical definition. You can't use an informal
> phrase in mathematics without giving a mathematical definition.
>see that you have not been reading too carefully, as I've been making a 
clear 
>case that the conclusions drawn from bijection are overdrawn. Do you now 
agree 
>that the conclusion you drew, which I quoted, is incorrect?
> You have consistently quoted me incorrectly. In particular, I have
> consistently said that the word betweenness in a phrase such as
>       There are infinitely many elements z between x and y
> is a property of the *ordering*, not a property of the underlying
> sets. When you leave off the specification of the ordering, you
> are changing the meaning of the phrase.
For the last time, the ordering is the flaw. I can't say this any more 
clearly 
than I already have. If you believe that there are in fact infinitely many 
naturals between any two naturals, by the normal definition of between, 
then 
we should just drop it, because there's no place to go from there.
> --
> Daryl McCullough
> Ithaca, NY
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
> Daryl McCullough said:
[...]
>>Why do you keep leaving out the important phrase for the particular
>>ordering? That is the *key* phrase, and you keep deleting it.
> I delete it because it is the flaw in your argument.
No, you deklete it becasue if you you leave it oin you would realise 
that the question you pose in the next sentence is unanswerable. For a 
expositon on unanswerable questions, go to Augutsine - he's been dead. 
lo these many centuries, but he realsied that abdly formulated question 
couldn't be answerd. One of his
> I am asking if, in any 
> real sense, in your mind, regardless of any ordering, it seems at all 
plausible 
> that there are infinitely many naturals between any pair of naturals, 
using a 
> sensible consistent definition for between. 
This is a meaningless question. Ordering is a way of defining what it 
means for an object to be between two other objects. You're also begging 
the question by use of that weasel word sensible, by which you clearly 
intyend acceptable to Tony. IOW, if the definition implies conclusions 
you disagree with, the definition can't have been sensible. That's 
been the logical form of your claim all along, in fact - you don't agree 
with X, so X must be nonsense.
Attempting to analyse your notion of natural order, I infer that you 
think that the counting sequence is the only way to arrange the natural 
numbers. Maybe you assume that {1st, 2nd, 3rd, ...} is the same set as 
{1, 2, 3, ...}. These are different: that is, theorems that are true 
about the cardinal numbers aren't true about the natural numbers, and 
vice versa. Is that not obvious?
> What naturals are more than 2 and 
> less than 3?
More than and less than isn't the same as to one side of and 
to 
other side of w/ ref. to the number row. As a matter of fact, you can 
deduce that 3 > 2  and  2 < 3 from a specific ordering of the natural 
numbers, and not from other orderings.
> The conclusion is wrong, therefore there is a problem with the 
> method, or some assumption applied to it. The problem is in the reordering 

of
> inherently ordered elements, while using the inherent orders as well.
There are no inherently ordered elements. (I suspect that this belief 
arises from our experience of time.) All orderings are constructed. This 
is obvious in the case of alphabetical order, not obvious in the case of 
  the natural numbers or cardinal numbers. Counting Order is just one 
way to order them. But occasionally, even in ordinary life, this order 
is violated, and may even be ignored. Eg, a selection rule might be, 
Pick the even numbered ones, then the odd numbered ones. Or, Pick the 
large ones, then the middle sized ones, then the leftovers, which 
_ignores_ counting order.
> This is 
> some typical makings of your standard paradox. Paradoxes are all 
resolvable. 
> they all boil down to mutually contradictory terms where the contradiction 

is 
> not obvious. Incidentally, all paradoxes are counterintuitive, aren't 
they?
[...]
Well, depends what you mean by counter-intuitive. Some of the most 
famous are thoroughly inutuitive, such as Zeno's paradox that the 
Achilles will never catch up to the Tortoise. Zeno showed that reasoning 
based on intuitions could lead you all the way into absurdity. IMO, 
paradoxes arise when people rely on incompatible intuitions, so 
paradoxes are all intuitive in this sense. They are counter-intuitive in 
the sense that they set intuitions against each other -- ie, they reveal 
that the inutitons are inconsistent.
===
Subject: Re: Epistemology 201: The Science of Science
aeo6 says...
>There is one natural ordering of numbers in general, and that's where 
smaller 
>quantitative values precede larger values. Any other ordering violates the 

>members' identities as numbers.
How does it violate the member's identities as numbers? You
are saying that as soon as I write down an ordering
such as
    R(x,y) == (x+y is even and x > y) or (x+y is odd and x < y)
that I'm *violating* the natural numbers? So I'm not allowed
to define such a relation? That doesn't make a bit of sense.
 
>>If a set contains all the elements of another set, 
>>plus some more, then it has more.
>> It has more according to the subset relation on sets. Nobody
>> argues with that.
>Except Mr. Cardinality, but he's half blind.
Who do you think Mr. Cardinality is? Nobody said that
cardinality is the only relationship between sets. Mathematics
uses cardinality, subset, measure theory, order types, relative
frequency as appropriate.
>>These are pretty basic concepts that I shouldn't have to repeat.
>> Nobody is arguing about the basic concepts:
>>    1. If every element of A is also an element of B, but not
>>    vice-versa, then A is a proper subset of B.
>>    2. The set of reals in [0,1] has half the measure of the set
>>    of reals in [0,2].
>>    3. The relative frequency of even naturals is half the relative
>>    frequency of all naturals.
>>    4. The order type of the rationals is larger than the order
>>    type of the naturals. (That is, the naturals can be embedded
>>    in the rationals in an order-preserving way, but not vice-versa).
>> The part that people disagree with is that you want to use the
>> phrase has more elements for all 4 cases. That's just being
>> perverse. The technical definitions of subset, measure,
>> relative frequency, order embedding, and cardinality
>> are different. No definition captures everything that you
>> might want to say about the relationship between sets.
>> You are arguing about words, and insisting that people use
>> your terminology, which glosses over at least 5 different
>> distinctions.
>See below.
> There exists an ordering R(x,y) on naturals
> such that between any two naturals x and y,
> there exists infinitely many naturals z such
> that R(x,z) and R(z,y).
>>You are not being honest. Here is your actual quote:
I didn't say that. I said that for that particular *ordering* on the
>>naturals, there exists infinitely many naturals between
>>any pair of naturals. It is a property of the ordering, not the
>>underlying set.
>> How is that different? In both cases, I'm saying that the
>> betweeness relationship is a property of the *ordering*,
>> and whether or not there are infinitely many naturals
>> between any two naturals depends on what *ordering* you
>> are using.
>You denied saying there were infinite many naturals between
>any two naturals, so I showed you your quote and asked you
>to defend it.
Leaving out the phrase for that particular ordering changes
the meaning of the quote. So you are not quoting accurately.
>If you saw no difference 
>between the two, then why did you accuse me of misquoting and rephrase
>it?
Because you misquoted it. You left out the qualifying phrase
for that particular ordering. 
>Is there something wrong with the way you phrased it the first time?
No, there is something wrong with the way *you* phrased it, namely,
you left out the phrase for that particular ordering.
>This is exactly what I object to.
If you object to misquoting people, then don't do it.
>That there are an infinite number of naturals between 
>any two naturals is rotten hogwash used to make diaper tea.
>You can't really believe that statement, can you?
I never made that statement. I made the statement
  for that particular *ordering* on the
  naturals, there exists infinitely many naturals between
  any pair of naturals. It is a property of the ordering,
  not the underlying set.
You keep deleting the crucial phrase for that particular ordering,
which completely changes the meaning.
>>Now, do you believe that there exists infinitely many naturals 
between
>>any pair of naturals or not?
>> Why do you keep leaving out the important phrase for the particular
>> ordering? That is the *key* phrase, and you keep deleting it.
>I delete it because it is the flaw in your argument.
How does misquoting me point out a flaw in my argument?
>I am asking if, in any real sense, in your mind, regardless of
>any ordering, it seems at all plausible that there are infinitely
>many naturals between any pair of naturals, using a 
>sensible consistent definition for between.
There *is* no meaning of between that does not depend on
a choice of ordering. Between inherently involves ordering.
>The conclusion is wrong,
The conclusion
    For the particular ordering that I defined on the
    naturals, there exists infinitely many naturals between
    any pair of naturals. It is a property of the ordering,
    not the underlying set.
is not wrong. For that notion of ordering, it is true that
there are infinitely many naturals between any pair of naturals.
>therefore there is a problem with the 
>method, or some assumption applied to it.
There is a problem when misquote a claim so as to change the meaning.
>The problem is in the reordering of 
>inherently ordered elements,
There is no problem, other than the fact that you continually
misquote what I say.
>while using the inherent orders as well. This is 
>some typical makings of your standard paradox.
There is no paradox. There is no problem, other than your
sloppiness.
>> Betweeness is a property of the ordering.
>>If not, how do you explain the conclusion you drew?
>> The conclusion is that for that particular *ordering* on the
>> naturals, there exists infinitely many naturals between
>> any pair of naturals. That's a true conclusion. It is *provably*
>> true. If you are rejecting it, then you are being inconsistent.
>You can prove it based on Cantor's theorems, but are they provable?
Yes. That's why they are called theorems.
>Somewhere in the mix is the assumption that you can change the
>order of numbers and preserve their identities as numbers.
>This is false
I didn't *change* the ordering of the naturals, I defined a *different*
ordering on the naturals. The fact that I define a new binary relation
on the naturals doesn't do *anything* to the naturals.
You have a totalitarian view of mathematics, in which anything
that is not forbidden is mandatory. The fact that I choose to
use a *different* ordering on the naturals for some purpose does
not imply that I have eliminated the usual ordering.
>> You did misquote it. I specifically said for that particular
>> ordering, and you left that phrase out.
>That's not how you corrected it. You changed the language so it didn't say 

>there are an infinite number of naturals between any pair of naturals, 
which 
>was dishonest.
      There exists an ordering R(x,y) on naturals
      such that between any two naturals x and y,
      there exists infinitely many naturals z such
      that R(x,z) and R(z,y).
But z is between x and y (according to ordering R) means the
same thing as R(x,z) and R(z,y). That's what between means.
>I left out the ordering aspect because it's not relevant to what 
>I am asking.
If you leave out the ordering, then between doesn't mean
*anything*. Between only has meaning for an ordering. So
by leaving out the ordering aspect, you completely changed
the meaning of the sentence.
>Do you actually, not in the context of any ordering but in 
>general, see that conclusion as being correct?
Between doesn't mean anything except in the context of an ordering.
>I have asked this several times, 
>and you refuse to answer. Can you answer the question directly? I can.
You ask a nonsense question, and you get mad because I don't answer it.
You can't ask someone to disregard ordering, and then *also* ask him
to say how many elements are between 1 and 2. Between only makes sense
relative to an ordering.
>> You misquoted it by leaving out the phrase for the
>> particular ordering I defined.
>Explained twice already above.
By leaving out the phrase for the particular ordering I defined,
you completely changed the meaning.
>>Are there infinitely many integers between any two rationals?
>> For which ordering? Betweeness depends on the ordering. For
>> the *usual* ordering, there are infinitely many rationals
>> between any two naturals, and not vice-versa.
>Correct. For the regular, non-ridiculous, non-meaning-violating order.
The phrases non-ridiculous and non-meaning-violating add nothing
*mathematically* to the statement. The fact is between is relative
to an ordering, and whether there are infinitely many elements z between
x and y is a fact about an *ordering*, not about the underlying set.
>>No? What does that tell you about the relative numbers integers 
>>and rationals?
>> Nothing. It tells something about the *order* type of the rationals.
>Which indicates there are more rationals than integers, which means they 
>comprise a larger set.
No, it doesn't. It indicates that the order type of the rationals
under the usual ordering is larger than the order type of the naturals
under the usual ordering.
>To draw a conclusion so drastically different from the blatantly
>obvious,
> 
> The only conclusion is that there exists a bijection between
> the rationals and the integers. That's provably the case.
More bijection genuflection. What does this case prove?
>> It's just a fact. A fact that you have trouble with, for some reason.
>What does the bijection prove?
It doesn't prove anything. It is just a fact about rationals and naturals.
There are a number of things we can say about the naturals and the 
rationals:
    1. The set of naturals is a proper subset of the set of rationals.
    2. The usual order type of the naturals is smaller than the usual
    order type of the rationals.
    3. The cardinality of the naturals is equal to the cardinality of
    the rationals.
These three facts together tell more about the relationship between
the rationals and the naturals than any single fact separately. The
phrase There are more rationals than naturals doesn't say anything,
since it is ambiguous whether it means 1,2, 3 or something else.
>What inference can you draw from it, besides 
>that, according to the method you have used, it is the conclusion you have 

>drawn? The bijection does not mean the sets are of equal size.
That's because there are about 5 different definitions of size.
Cardinality is one. Measure is another. Order type is another. Subset
is another. Relative frequency is another.
>It means that the cardinality measure as defined does not
>distinguish between them.
That's right. Cardinality makes some distinctions, measure theory
makes other distinctions, the subset relationship makes other distinctions,
relative frequency makes other distinctions, order type makes yet other
distinctions. Why do you want to pick on cardinality? It doesn't make
all possible distinctions, but no notion of size can possibly make all
possible distinctions.
>Please show a proof of the validity of equating set size with
>cardinality for infinite sets which doesn't circularly fall
>back on the definition of cardinality as the size of the set.
What does it mean to prove that a definition is valid?
>>Certainly nothing about the relative numbers of integers and rationals.
>> There are several *different* relationship between the set of
>> integers and the set of rationals:
>>     1. The order type of the rationals is larger than the order
>>     type of the integers.
>>     2. The integers are a subset of the rationals.
>>     3. The cardinality of the integers is the same as the cardinality
>>     of the rationals.
>> For you to say there are more rationals than there are integers is
>> *ambiguous*. It might mean in the sense of order type, it might mean
>> in the sense of subsets, it might mean in the sense of cardinality.
>I don't think it's ambiguous at all.
I gave you three different common interpretations of the phrase
set A has more elements than set B. That makes it ambiguous.
>It's not a matter of opinion.
No, it's a matter of definition. Whether set A has more elements
than set B depends on how you define has more elements.
>I think it is ambiguous to have several different ways of answering the 
same 
>question,
If the question is a *mathematical* question, then there is a single
answer for it. But a question involving *intuitive* notions of size
is not a mathematical question until you give a mathematical definition
of what you mean by size.
>> It's accepted for the same reasons any good theory is accepted:
>> It is rigorous, it produces interesting results, it produces results
>> that are useful in a wide variety of fields.
>What is the application of the fact that there are as many integers as 
>rationals in some abstract sense, operationally?
In computer science, the fact that every countable set can be coded
as an integer is an important fact. It allows us to iterate over
rationals, lists, ordered pairs, etc. as easily as we loop over integers.
>but is not supported by any empirical evidence or
>application to reality,
> 
> That's not true.
What empirical evidence can you offer?
>> Cardinality theory is a critical part of measure theory, which
>> in turn is a critical ingredient to probability theory. Probability
>> theory has been very successful in a wide range of practical
>> applications.
>In dealing with infinite sets of rationals vs. integers?
> What conclusion? The concusion is that there is a bijection
> between the integers and the rationals. You disagree with that
> conclusion?
No, just with the conclusion that there exists infinitely many 
naturals 
>>between any pair of naturals
>> Nobody made that conclusion. The conclusion was that 
>>     For the particular binary relation R(x,y) that I defined,
>>     there are infinitely many naturals between any two naturals.
>I stated the conclusion correctly.
No. You didn't. Leaving out the context for the particular
binary relation R(x,y) completely changes the meaning.
>> You have consistently quoted me incorrectly. In particular, I have
>> consistently said that the word betweenness in a phrase such as
>>       There are infinitely many elements z between x and y
>> is a property of the *ordering*, not a property of the underlying
>> sets. When you leave off the specification of the ordering, you
>> are changing the meaning of the phrase.
>For the last time, the ordering is the flaw.
For the last time, no it's not.
>If you believe that there are in fact infinitely many 
>naturals between any two naturals
I have said that I *don't* believe that. You have to specify
according to what ordering.
>by the normal definition of between
I've said many times that according to the usual ordering on
rationals, there are infinitely many rationals between any
pair of naturals. That is a fact about the ordering, not about
the size of the underlying sets.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Epistemology 201: The Science of Science
 <d0j6n8$fdl$5@msunews.cl.msu.edu>
 <ErXYd.1221$Ai3.17974@news20.bellglobal.com>
 <vcb3buz8x3u.fsf@beta19.sm.ltu.se>
 <lSXYd.1567$Ai3.23620@news20.bellglobal.com>
 <d11tnk$1lio$2@msunews.cl.msu.edu>
 <MPG.1c9f8785dcad079f9898e0@newsstand.cit.cornell.edu>
 <d14i8401dgn@drn.newsguy.com>
 <MPG.1c9faeb58f9b3c7d9898fd@newsstand.cit.cornell.edu>
 <d14qc002ftc@drn.newsguy.com>
 <MPG.1c9fc00f82a6e1f798990a@newsstand.cit.cornell.edu>
 <d14ubv02ti7@drn.newsguy.com>
 <MPG.1c9fd85594f28221989913@newsstand.cit.cornell.edu>
 <d156h50n8q@drn.newsguy.com>
 <MPG.1ca0d893c7fb49ba98991f@newsstand.cit.cornell.edu>
 <d175tf0l65@drn.newsguy.com>
 <MPG.1ca12a58f596fab6989940@newsstand.cit.cornell.edu>
 <d19kd502cur@drn.newsguy.com>
 <MPG.1ca241b7ba7a89dd989953@newsstand.cit.cornell.edu>
 <d1a39r0tn5@drn.newsguy.com>
 Discussion, linux)
> aeo6 says...
>>If you saw no difference 
>>between the two, then why did you accuse me of misquoting and rephrase
>>it?
> Because you misquoted it. You left out the qualifying phrase
> for that particular ordering. 
I think that Tony fails to see the equivalence between the following
two statements.  Maybe not, but why else complain about your
rephrasing?
There exists an ordering R(x,y) on naturals such that between any two
naturals x and y, there exists infinitely many naturals z such that
R(x,z) and R(z,y).
I didn't say that. I said that for that particular *ordering* on the
naturals, there exists infinitely many naturals between any pair of
naturals. It is a property of the ordering, not the underlying set.
-- 
Jesse F. Hughes
Besides, discoverers are too proud to kiss butt.  Indiana Jones would
never kiss some academic's ass to get published, and neither will I.
  --James Harris
===
Subject: Re: Epistemology 201: The Science of Science
: aeo6 says...
:>What is the application of the fact that there are as many integers as 
:>rationals in some abstract sense, operationally?
: In computer science, the fact that every countable set can be coded
: as an integer is an important fact. It allows us to iterate over
: rationals, lists, ordered pairs, etc. as easily as we loop over integers.
I have already pointed out that the fact each rational number
is computable is an application of the fact that there are not
more rationals than integers.  For each rational number there clearly 
exists a Turing Machine that computes it, and each Turing Machine
can be encoded as an integer.
I have yet to hear what the application of the fact that there are 
not as many integers as rationals is.  
Stephen
===
Subject: Re: Epistemology 201: The Science of Science
stephen@nomail.com said:
> : aeo6 says...
> :>What is the application of the fact that there are as many integers as 
> :>rationals in some abstract sense, operationally?
> : In computer science, the fact that every countable set can be coded
> : as an integer is an important fact. It allows us to iterate over
> : rationals, lists, ordered pairs, etc. as easily as we loop over 
integers.
> I have already pointed out that the fact each rational number
> is computable is an application of the fact that there are not
> more rationals than integers.  For each rational number there clearly 
> exists a Turing Machine that computes it, and each Turing Machine
> can be encoded as an integer.
> I have yet to hear what the application of the fact that there are 
> not as many integers as rationals is.  
> Stephen
Sure, you can translate any countable grammar into any other if you want. 
That's fine. It doesn't explain the flaw in the idea that for any integer x 

there are an infinite number of rationals between it and the next integer 
x+1,  
namely x*n+m/n for for m=1 to oo and n=m+1 to oo, and that if this is true 
of 
every integer, it is true for integers in general. The set of rationals 
should 
be considered to have the integral of the size of the integers, not the same 

size, that is, if the size of the integers is x, then the size of the 
rationals 
is x^2/2, and the size of the set of reals should be considered to be the 
infinitieth integral of x. You may consider this exercise an application of 

the 
theory to the reality of the real numbers.
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
>: aeo6 says...
>:>What is the application of the fact that there are as many integers as 
>:>rationals in some abstract sense, operationally?
>: In computer science, the fact that every countable set can be coded
>: as an integer is an important fact. It allows us to iterate over
>: rationals, lists, ordered pairs, etc. as easily as we loop over 
integers.
>I have already pointed out that the fact each rational number
>is computable is an application of the fact that there are not
>more rationals than integers.  For each rational number there clearly 
>exists a Turing Machine that computes it, and each Turing Machine
>can be encoded as an integer.
>I have yet to hear what the application of the fact that there are 
>not as many integers as rationals is.  
Well for that matter we're all waiting with bated breath to hear what
the application of sets and subsets of equal undefined cardinality is?
===
Subject: Re: Epistemology 201: The Science of Science
 <MPG.1c965cd95cdcbb1198989c@newsstand.cit.cornell.edu>
 <d0j6n8$fdl$5@msunews.cl.msu.edu>
 <ErXYd.1221$Ai3.17974@news20.bellglobal.com>
 <vcb3buz8x3u.fsf@beta19.sm.ltu.se>
 <lSXYd.1567$Ai3.23620@news20.bellglobal.com>
 <d11tnk$1lio$2@msunews.cl.msu.edu>
 <MPG.1c9f8785dcad079f9898e0@newsstand.cit.cornell.edu>
 <d14i8401dgn@drn.newsguy.com>
 <MPG.1c9faeb58f9b3c7d9898fd@newsstand.cit.cornell.edu>
 <d14qc002ftc@drn.newsguy.com>
 <MPG.1c9fc00f82a6e1f798990a@newsstand.cit.cornell.edu>
 <d14ubv02ti7@drn.newsguy.com>
 <MPG.1c9fd85594f28221989913@newsstand.cit.cornell.edu>
 <d156h50n8q@drn.newsguy.com>
 <MPG.1ca0d893c7fb49ba98991f@newsstand.cit.cornell.edu>
 <d175tf0l65@drn.newsguy.com>
 <MPG.1ca12a58f596fab6989940@newsstand.cit.cornell.edu>
 <d19kd502cur@drn.newsguy.com>
 <MPG.1ca241b7ba7a89dd989953@newsstand.cit.cornell.edu>
 Discussion, linux)
>> What does it mean to say that an ordering is inconsistent? You're
>> not making any sense.
> There is one natural ordering of numbers in general, and that's where 
smaller 
> quantitative values precede larger values. Any other ordering violates the 

> members' identities as numbers.
Violates the members' identities as numbers?  Is that a misdemeanor
or a felony?  Is Daryl gonna get nicked for that one?
This is nonsense.  A relation R is a linear order of N iff for every
n,m,p in N, the following hold:
  (1) n R n
  (2) n R m and m R p implies n R p
  (3) n R m and m R n implies n = m
  (4) n R m or m R n
There are gobs of such relations on N.  Oodles even.  I'd say there
were infinitely many, but certain small brains in this discussion
might object.  The existence of such relations don't violate the
members' identities as numbers, their sensibilities, chastity or
anything else.  It's just a particular kind of relation, nothing
scarier than a subset of N x N satisfying certain nice properties.
-- 
Jesse F. Hughes
But nothing's being Dr. Ullrich is a particular case of something's
being such that nothing is it: (Ex)~(Ey)(y = x) 
  -- John Correy on the failings of first order logic
===
Subject: Re: Epistemology 201: The Science of Science
      <zjqYd.1717$v4.35393@news2.nokia.com> 
      <39eq3eF614usvU2@individual.net> 
      <nP0Zd.40295$k4.786746@news1.nokia.com> 
      <4234a632.28350469@netnews.att.net> 
      <39jq7jF61e1pvU2@individual.net> 
      <9x3Zd.10663$Fy.1366@okepread04> 
      <MPG.1c9f8bc68ca9436b9898e4@newsstand.cit.cornell.edu> 
      <39lvtbF63mmkhU2@individual.net> 
      <MWnZd.10709$Fy.3082@okepread04> 
      <39mo82F63lffvU2@individual.net> 
      <T%rZd.10724$Fy.2626@okepread04> 
      <39n5fvF62s52iU2@individual.net> 
      <VfBZd.11441$Fy.1067@okepread04> 
      <d171oa$8iu$1@msunews.cl.msu.edu> 
      <KrJZd.29$45.6305@news.uchicago.edu>
In <KrJZd.29$45.6305@news.uchicago.edu>, on 03/15/2005
   at 10:36 PM, mmeron@cars3.uchicago.edu said:
>How about a fly?
That's a rather arthropromorphic concept. The arthropods, and indeed
all of the metazoans, are outnumbered and outweighed by the bacteria.
-- 
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: Epistemology 201: The Science of Science
> That's a rather arthropromorphic concept. The arthropods, and indeed
> all of the metazoans, are outnumbered and outweighed by the bacteria.
God is a germ?
Bob Kolker
===
Subject: Re: Epistemology 201: The Science of Science
>> That's a rather arthropromorphic concept. The arthropods, and indeed
>> all of the metazoans, are outnumbered and outweighed by the bacteria.
>God is a germ?
I've often speculated that based on interbreedable numbers homo s. is
the most successful species in the history of the earth.
===
Subject: Re: Epistemology 201: The Science of Science
>That's a rather arthropromorphic concept. The arthropods, and indeed
>all of the metazoans, are outnumbered and outweighed by the bacteria.
>>God is a germ?
> I've often speculated that based on interbreedable numbers homo s. is
> the most successful species in the history of the earth.
What do you mean by interbreedable numbers? And don't speculate, go 
out there and count 'em. Or read the papers of people who have counted 'em.
===
Subject: Re: Epistemology 201: The Science of Science
>>That's a rather arthropromorphic concept. The arthropods, and indeed
>>all of the metazoans, are outnumbered and outweighed by the bacteria.
>God is a germ?
>> I've often speculated that based on interbreedable numbers homo s. is
>> the most successful species in the history of the earth.
>What do you mean by interbreedable numbers? And don't speculate, go 
>out there and count 'em. Or read the papers of people who have counted 
'em.
Or count the people who have interbred. Wolf, you have a rather nasty
habit of telling people what to do. If you can't make anything of my
statement, go research the conept for yourself for a change. Then
maybe you could explain what you don't understand instead of whining.
===
Subject: Re: Epistemology 201: The Science of Science
Lester Zick said:
>> 
>> 
>> 
>> 
>>That's a rather arthropromorphic concept. The arthropods, and indeed
>>all of the metazoans, are outnumbered and outweighed by the bacteria.
>>God is a germ?
>> 
>> 
>> I've often speculated that based on interbreedable numbers homo s. is
>> the most successful species in the history of the earth.
>> 
>What do you mean by interbreedable numbers? And don't speculate, go 
>out there and count 'em. Or read the papers of people who have counted 
'em.
> Or count the people who have interbred. Wolf, you have a rather nasty
> habit of telling people what to do. If you can't make anything of my
> statement, go research the conept for yourself for a change. Then
> maybe you could explain what you don't understand instead of whining.
And we were so close to becoming a ring species. (sigh)
-- 
Smiles,
Tony
===
Subject: Re: Epistemology 201: The Science of Science
>Lester Zick said:
> 
> 
> 
> 
>That's a rather arthropromorphic concept. The arthropods, and indeed
>all of the metazoans, are outnumbered and outweighed by the 
bacteria.
>>God is a germ?
> 
> 
> I've often speculated that based on interbreedable numbers homo s. is
> the most successful species in the history of the earth.
> 
What do you mean by interbreedable numbers? And don't speculate, go 
>>out there and count 'em. Or read the papers of people who have counted 
'em.
>> Or count the people who have interbred. Wolf, you have a rather nasty
>> habit of telling people what to do. If you can't make anything of my
>> statement, go research the conept for yourself for a change. Then
>> maybe you could explain what you don't understand instead of whining.
>And we were so close to becoming a ring species. (sigh)
Ring species?
===
Subject: Re: Epistemology 201: The Science of Science
>> And a species that can predict that event and model an escape stands a 
>> chance of surviving even that.  Catastrophes will continue erasing the 
>Two chances: slim and none.
>> blackboard until something gets the right answer.
>It is human reproductive behaviour that extends our stay (as a species) 
>not how smart we are. The people who are doing the most reproduction 
>generally are the least bright of our species. The lower the 
>intelligence the greater the number of children (on average).
No.  You are confusing today's demographics with what has happened
over the past, long-term.
/BAH
Subtract a hundred and four for e-mail.
===
Subject: Re: Epistemology 201: The Science of Science
>> People keep confusing cause and effect.  Evolution is a description
>> of changes we see that has happened.   It is a process; people
>> have assigned a noun to it.  It is not a cause; it isn't an effect.
>> It is a summary of all effects observed so far.  You can do
                                                       ^not
>> x, and have an evolution immediately pop out.
>Evolution = Variation + Natural Selection. Selection is on the basis of 
>reproductive fitness. Those species that do not reproduce well become 
>extinct. Those the reproduce in great numbers survive barring a 
>catastrophe like the comet that hit earth about 65,000,000 ybp
This occurs over a long period of time (where long is dependent
on life span counts).  My point is that the theory is not
about one event but a summary of many events after they have 
happened.  Now take a look at what the Creationists assume.
They think that somebody gave an incantation and Poof! there
was a fully grown animal.  Read the arguments that have been
given to you here.
/BAH
Subtract a hundred and four for e-mail.
===
Subject: A short question about Banach algebra.
let B(N,C)=l^{infinity}(N). Then for me it easily follows with norm
defined by sup_n|x_n|, l^{infinity} is a banach space.
I think Banach algebra is built by giving product as
(x_n)*(y_n):=(x_n*y_n) in banach space because banach space is already
complete normed vector space. So I think now I have a ring structure on
l^{infinity}.
If somebody uses the term 'unital' it means the space has identity?
What is the unit element in l^{infinity}?
===
Subject: Re: A short question about Banach algebra.
>let B(N,C)=l^{infinity}(N). Then for me it easily follows with norm
>defined by sup_n|x_n|, l^{infinity} is a banach space.
>I think Banach algebra is built by giving product as
>(x_n)*(y_n):=(x_n*y_n) in banach space because banach space is already
>complete normed vector space. So I think now I have a ring structure on
>l^{infinity}.
>If somebody uses the term 'unital' it means the space has identity?
Yes
>What is the unit element in l^{infinity}?
Well, what do you think x_n should be if x_n*y_n = y_n for all y?
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
===
Subject: Re: A short question about Banach algebra.
   <d1aj3r$r4i$1@nntp.itservices.ubc.ca>
:) It should be x_n = 1 i.e. constant sequence ?
===
Subject: Flies in the ointment.
The recent sci.math mudfest with the character Jason contained a
number of references to some [perhaps different person] Gabriel,
in which it was suggest the this Gabriel person had exhibited a
similar lack of propriety on some newsgroup, presumably (at least
presumed by me), this occurred on sci.math.
I have been unable to find any reference to (John?) Gabriel, and don't
actually recall him. As a result, I'm somewhat curious about seeing
pointers to his, er, contributions.
Dale.
===
Subject: REPOST: Re: Flies in the ointment.
[Randy Poe]
...
> I'm beginning to suspect that all original John Gabriel
> posts have been deleted. Check out this exchange from
> the numerical analysis group, where John Gabriel
> appears only in quoted text, but a character named
> Jason Wells tells him he is a genius who has revolutionized
> calculus:
> http://tinyurl.com/3rryy
LOL!  Jason Wells offering sage counsel to John Gabriel:
    You have a very abrupt style of responding to objections - if I
    may, you may try posing your responses as questions rather than
    'attacks' as has been interpreted by most.
Uh huh.
I also stumbled into this amazing scam on Wikipedia:
    John Gabriel's Nth root algorithm
    http://en.wikipedia.org/wiki/John_Gabriel%27s_Nth_root_algorithm
Nothing wrong with the algorithm, but it's just a clumsily-stated direct 
application of Newton's method to finding a zero of f(y) = y^n-x (iterate y 

<- y - f(y)/f'(y)).  This certainly doesn't go under the name of John 
Gabriel in any numerical analysis circles I've run in <heh>. 
===
Subject: Re: REPOST: Re: Flies in the ointment.
   <reposted.GZ6dnfoY873DcaXfRVn-gg@adelphia.com [Randy Poe]
> ...
> I'm beginning to suspect that all original John Gabriel
> posts have been deleted. Check out this exchange from
> the numerical analysis group, where John Gabriel
> appears only in quoted text, but a character named
> Jason Wells tells him he is a genius who has revolutionized
> calculus:
> http://tinyurl.com/3rryy
> LOL!  Jason Wells offering sage counsel to John Gabriel:
>     You have a very abrupt style of responding to objections - if I
>     may, you may try posing your responses as questions rather than
>     'attacks' as has been interpreted by most.
> Uh huh.
> I also stumbled into this amazing scam on Wikipedia:
>     John Gabriel's Nth root algorithm
>     http://en.wikipedia.org/wiki/John_Gabriel%27s_Nth_root_algorithm
> Nothing wrong with the algorithm,
  The Gabriel algorithm says the iterations are defined by
       s_(i+1) = ((s_i)^(x/(n - 1)) + (n - 1) s_i)) / n,
which seems a little silly.  The object is to compute the n-th root
of x.  As can be seen above, in order to compute the n-th root
of x, you will need to compute the (n-1)st root of (s_i)^x.  But if
you have a way of computing (n - 1)st roots [for arbitray integer n,
presumably] in the first place, why do you need the algorithm ?
A straightforward implementation of Newton's method to solve
        y^n = x
is simpler and requires only that you can compute integer powers,
rather than roots.
  Nora B.
but it's just a clumsily-stated direct
> application of Newton's method to finding a zero of f(y) = y^n-x
(iterate y
> <- y - f(y)/f'(y)).  This certainly doesn't go under the name of
John 
> Gabriel in any numerical analysis circles I've run in <heh>.
===
Subject: Re: REPOST: Re: Flies in the ointment.
...
[Tim Peters]
>> I also stumbled into this amazing scam on Wikipedia:
>>     John Gabriel's Nth root algorithm
>>     http://en.wikipedia.org/wiki/John_Gabriel%27s_Nth_root_algorithm
>> Nothing wrong with the algorithm,
[Nora Baron]
>  The Gabriel algorithm says the iterations are defined by
>       s_(i+1) = ((s_i)^(x/(n - 1)) + (n - 1) s_i)) / n,
> which seems a little silly.
Maybe this is browser-dependent.  When I look at that page, it's clearly 
s_i/(x^(n-1)) instead.  Which is why I said what I said <wink>.
If you look at the history of the page, the original version was ASCII art, 

and not open to this misinterpretation (well, not in a fixed-width font 
anyway); this is cut-and-paste from page's first revision:
               x
           ---------                      means x divided by
               (n-1)                      s ^(n-1)
   s     =    s       +  (n-1)*s           i
    i+1        i                i
           ----------------------    (2)
                        n
[...]
> A straightforward implementation of Newton's method to solve
>        y^n = x
> is simpler and requires only that you can compute integer powers,
> rather than roots.
That's what I said:
>> but it's just a clumsily-stated direct application of Newton's method
>> to finding a zero of f(y) = y^n-x (iterate y <- y - f(y)/f'(y)).
I think there's nothing here but a misunderstanding. 
===
Subject: Re: REPOST: Re: Flies in the ointment.
>>[Randy Poe]
>>...
>I'm beginning to suspect that all original John Gabriel
>posts have been deleted. Check out this exchange from
>the numerical analysis group, where John Gabriel
>appears only in quoted text, but a character named
>Jason Wells tells him he is a genius who has revolutionized
>calculus:
>http://tinyurl.com/3rryy
>>LOL!  Jason Wells offering sage counsel to John Gabriel:
>>    You have a very abrupt style of responding to objections - if I
>>    may, you may try posing your responses as questions rather than
>>    'attacks' as has been interpreted by most.
>>Uh huh.
>>I also stumbled into this amazing scam on Wikipedia:
>>    John Gabriel's Nth root algorithm
>>    http://en.wikipedia.org/wiki/John_Gabriel%27s_Nth_root_algorithm
>>Nothing wrong with the algorithm,
>   The Gabriel algorithm says the iterations are defined by
>        s_(i+1) = ((s_i)^(x/(n - 1)) + (n - 1) s_i)) / n,
> which seems a little silly.  The object is to compute the n-th root
> of x.  As can be seen above, in order to compute the n-th root
> of x, you will need to compute the (n-1)st root of (s_i)^x.  But if
> you have a way of computing (n - 1)st roots [for arbitray integer n,
> presumably] in the first place, why do you need the algorithm ?
> A straightforward implementation of Newton's method to solve
>         y^n = x
> is simpler and requires only that you can compute integer powers,
> rather than roots.
>   Nora B.
I think the problem is with the crappy formatting.
Newton's method would give this:
        s_(i+1) = ( x/ s_i^(n-1)  + (n-1) s_i ) / n
That is, in the Wikipedia entry, the first term in the numerator for
s_(i+1) should read x / s_i^(n-1), not s_i^(x/(n-1)).
As such, it would be a correct reading of Newton's method.
However, for Gabriel to have appropriated it as his own is ludicrous.
Dale
===
Subject: Re: REPOST: Re: Flies in the ointment.
   <reposted.GZ6dnfoY873DcaXfRVn-gg@adelphia.com>
   <4Al_d.11477$C47.9472@newssvr14.news.prodigy.com>[Randy Poe]
>>...
>I'm beginning to suspect that all original John Gabriel
>posts have been deleted. Check out this exchange from
>the numerical analysis group, where John Gabriel
>appears only in quoted text, but a character named
>Jason Wells tells him he is a genius who has revolutionized
>calculus:
>http://tinyurl.com/3rryy
LOL!  Jason Wells offering sage counsel to John Gabriel:
    You have a very abrupt style of responding to objections - if I
>>    may, you may try posing your responses as questions rather than
>>    'attacks' as has been interpreted by most.
Uh huh.
I also stumbled into this amazing scam on Wikipedia:
    John Gabriel's Nth root algorithm
>>
http://en.wikipedia.org/wiki/John_Gabriel%27s_Nth_root_algorithm
Nothing wrong with the algorithm,
>   The Gabriel algorithm says the iterations are defined by
>        s_(i+1) = ((s_i)^(x/(n - 1)) + (n - 1) s_i)) / n,
> which seems a little silly.  The object is to compute the n-th root
> of x.  As can be seen above, in order to compute the n-th root
> of x, you will need to compute the (n-1)st root of (s_i)^x.  But if
> you have a way of computing (n - 1)st roots [for arbitray integer
n,
> presumably] in the first place, why do you need the algorithm ?
> A straightforward implementation of Newton's method to solve
>         y^n = x
> is simpler and requires only that you can compute integer powers,
> rather than roots.
>   Nora B.
> I think the problem is with the crappy formatting.
> Newton's method would give this:
>       s_(i+1) = ( x/ s_i^(n-1)  + (n-1) s_i ) / n
> That is, in the Wikipedia entry, the first term in the numerator for
> s_(i+1) should read x / s_i^(n-1), not s_i^(x/(n-1)).
> As such, it would be a correct reading of Newton's method.
  You're right - that makes a LOT more sense.
> However, for Gabriel to have appropriated it as his own is ludicrous.
  You're even more right on this.  Wikipedia needs better editing.
  Nora B.
> Dale
===
Subject: Re: REPOST: Re: Flies in the ointment.
Nora Baron a .8ecrit :
>>[Randy Poe]
>>...
>I'm beginning to suspect that all original John Gabriel
>posts have been deleted. Check out this exchange from
>the numerical analysis group, where John Gabriel
>appears only in quoted text, but a character named
>Jason Wells tells him he is a genius who has revolutionized
>calculus:
>http://tinyurl.com/3rryy
>>LOL!  Jason Wells offering sage counsel to John Gabriel:
>>   You have a very abrupt style of responding to objections - if I
>>   may, you may try posing your responses as questions rather than
>>   'attacks' as has been interpreted by most.
>>Uh huh.
>>I also stumbled into this amazing scam on Wikipedia:
>>   John Gabriel's Nth root algorithm
> http://en.wikipedia.org/wiki/John_Gabriel%27s_Nth_root_algorithm
>>Nothing wrong with the algorithm,
>  The Gabriel algorithm says the iterations are defined by
>       s_(i+1) = ((s_i)^(x/(n - 1)) + (n - 1) s_i)) / n,
>which seems a little silly.  The object is to compute the n-th root
>of x.  As can be seen above, in order to compute the n-th root
>of x, you will need to compute the (n-1)st root of (s_i)^x.  But if
>you have a way of computing (n - 1)st roots [for arbitray integer
> n,
>presumably] in the first place, why do you need the algorithm ?
>A straightforward implementation of Newton's method to solve
>        y^n = x
>is simpler and requires only that you can compute integer powers,
>rather than roots.
>  Nora B.
>>I think the problem is with the crappy formatting.
>>Newton's method would give this:
>>      s_(i+1) = ( x/ s_i^(n-1)  + (n-1) s_i ) / n
>>That is, in the Wikipedia entry, the first term in the numerator for
>>s_(i+1) should read x / s_i^(n-1), not s_i^(x/(n-1)).
>>As such, it would be a correct reading of Newton's method.
>   You're right - that makes a LOT more sense.
>>However, for Gabriel to have appropriated it as his own is ludicrous.
>   You're even more right on this.  Wikipedia needs better editing.
Not really possible: I edited it , and it was reverted back to its 
original misleading (to say the least) version in less than one minute.
More and more JSH's like.
>   Nora B.
>>Dale
===
Subject: Re: REPOST: Re: Flies in the ointment.
   <reposted.GZ6dnfoY873DcaXfRVn-gg@adelphia.com>
   <4Al_d.11477$C47.9472@newssvr14.news.prodigy.com>
   <423a8bf5$0$25019$8fcfb975@news.wanadoo.fr Not really possible: I edited it , and it was reverted back to its
> original misleading (to say the least) version in less than one
minute.
> More and more JSH's like.
Looks like it was reverted by someone called Inter, who is probably an
admin on wikipedia. Reverting back in less than minute is most likely
an automated process!
===
Subject: Re: REPOST: Re: Flies in the ointment.
   <reposted.GZ6dnfoY873DcaXfRVn-gg@adelphia.com>
   <4Al_d.11477$C47.9472@newssvr14.news.prodigy.com>
   <423a8bf5$0$25019$8fcfb975@news.wanadoo.fr Nora Baron a .8ecrit :
[Randy Poe]
>>...
>I'm beginning to suspect that all original John Gabriel
>posts have been deleted. Check out this exchange from
>the numerical analysis group, where John Gabriel
>appears only in quoted text, but a character named
>Jason Wells tells him he is a genius who has revolutionized
>calculus:
>http://tinyurl.com/3rryy
>>LOL!  Jason Wells offering sage counsel to John Gabriel:
>>   You have a very abrupt style of responding to objections - if
I
>>   may, you may try posing your responses as questions rather
than
>>   'attacks' as has been interpreted by most.
>>Uh huh.
>>I also stumbled into this amazing scam on Wikipedia:
>>   John Gabriel's Nth root algorithm
> http://en.wikipedia.org/wiki/John Gabriel%27s Nth root algorithm
>>Nothing wrong with the algorithm,
>>  The Gabriel algorithm says the iterations are defined by
>>       s (i+1) = ((s i)^(x/(n - 1)) + (n - 1) s i)) / n,
>>which seems a little silly.  The object is to compute the n-th
root
>of x.  As can be seen above, in order to compute the n-th root
>of x, you will need to compute the (n-1)st root of (s i)^x.  But
if
>you have a way of computing (n - 1)st roots [for arbitray integer
> n,
>presumably] in the first place, why do you need the algorithm ?
>A straightforward implementation of Newton's method to solve
>>        y^n = x
>>is simpler and requires only that you can compute integer powers,
>rather than roots.
>>  Nora B.
>>I think the problem is with the crappy formatting.
>>Newton's method would give this:
    s (i+1) = ( x/ s i^(n-1)  + (n-1) s i ) / n
That is, in the Wikipedia entry, the first term in the numerator
for
>>s (i+1) should read x / s i^(n-1), not s i^(x/(n-1)).
As such, it would be a correct reading of Newton's method.
>   You're right - that makes a LOT more sense.
>>However, for Gabriel to have appropriated it as his own is
ludicrous.
>   You're even more right on this.  Wikipedia needs better editing.
> Not really possible: I edited it , and it was reverted back to its
> original misleading (to say the least) version in less than one
minute.
> More and more JSH's like.
Probably Gabriel aka Jason is monitoring this thread
I've been reading Wikipedia. Of particular interest are
Now though this is clearly a (badly-formatted) version
of the Newton-Raphson method, Gabriel is trying to claim
it is original research. Wikipedia rules prohibit publishing
original research.
Wikipedia overview: http://en.wikipedia.org/wiki/Wikipedia
(see Policies, Disputes, and Vandalism).
Vandalism mechanism:
http://en.wikipedia.org/wiki/Wikipedia:Dealing with vandalism
http://en.wikipedia.org/wiki/Wikipedia:Votes for deletion
mentions the Newton-Raphson algorithm. I'm going to write
an n-th root version of this and link back to the square
root page.
Square roots:
http://en.wikipedia.org/wiki/Square root
over the line from vanity to vandalism, and Wikipedia
has established mechanisms for dealing with that,
ultimately resulting in a blocked user.
Let's fight the good fight, but it may take a few
weeks to sort out. I'm still learning my way through
the Wiki world.
            - Randy
===
Subject: Re: REPOST: Re: Flies in the ointment.
   <reposted.GZ6dnfoY873DcaXfRVn-gg@adelphia.com>
   <4Al_d.11477$C47.9472@newssvr14.news.prodigy.com>
   <423a8bf5$0$25019$8fcfb975@news.wanadoo.fr mentions the Newton-Raphson algorithm. I'm going to write
> an n-th root version of this and link back to the square
> root page.
Done! It's rough, could be reorganized and tightened up
a bit, and could stand to be expanded. But now there's
http://en.wikipedia.org/wiki/N-th_root_algorithm
Suggested expansions:
    - Methods for picking initial guesses
    - Termination rules
    - Examples
    - The alternate derivation from fixed-point iteration
(which I'm not familiar with).
         - Randy
===
Subject: Re: REPOST: Re: Flies in the ointment.
   <reposted.GZ6dnfoY873DcaXfRVn-gg@adelphia.com>
   <4Al_d.11477$C47.9472@newssvr14.news.prodigy.com However, for Gabriel to have appropriated it as his own is
ludicrous.
>   You're even more right on this.  Wikipedia needs better editing.
>   Nora B.
Wikipedia is open-source, but it's supposed to be
self-correcting. I notice that somebody has made a
comment that it's Newton's method. Think I'll take a
and see what happens.
         Randy
===
Subject: Re: REPOST: Re: Flies in the ointment.
[Randy Poe]
> Wikipedia is open-source, but it's supposed to be
> self-correcting. I notice that somebody has made a
> comment that it's Newton's method.
Given that the comment is a nearly verbatim repeat of what I posted here 
last night, my guess is that either Jason Wells added that comment, or that 

I did <wink>.
> and see what happens.
on the Newton-Raphson method instead, and give this as an interesting 
example.  The Numerical Analysis section of Wikipedia doesn't appear to 
===
Subject: Re: REPOST: Re: Flies in the ointment.
   <reposted.GZ6dnfoY873DcaXfRVn-gg@adelphia.com>
   <4Al_d.11477$C47.9472@newssvr14.news.prodigy.com>
   <86KdnaQ5TImBhaffRVn-vQ@comcast.com [Randy Poe]
> Wikipedia is open-source, but it's supposed to be
> self-correcting. I notice that somebody has made a
> comment that it's Newton's method.
> Given that the comment is a nearly verbatim repeat of what I posted
here
> last night, my guess is that either Jason Wells added that comment,
or that
> I did <wink>.
> and see what happens.
> If you have the time (I sure don't), it would be great to write an
> on the Newton-Raphson method instead, and give this as an interesting
> example.  The Numerical Analysis section of Wikipedia doesn't
appear to
I had that thought, but just discovered there's a very nice
http://en.wikipedia.org/wiki/Newton%27s_method
to discuss n-th root-finding as a specific application of
I've never done any writing for Wikipedia so I'm not sure
what the process is, but I'm definitely going to take
a stab at it over the next day or two.
           Randy
===
Subject: Re: REPOST: Re: Flies in the ointment.
[Tim Peters, to Randy Poe]
>> If you have the time (I sure don't), it would be great to write an
>> interesting example.  The Numerical Analysis section of Wikipedia
[Randy Poe]
> I had that thought, but just discovered there's a very nice
> http://en.wikipedia.org/wiki/Newton%27s_method
So there is!  Cool.  I can't find it again now, but last time I looked I 
ended up on what claimed to be an alphabetized list of numerical analysis 
topics, and there was no relevant entry under N.  But what I found just 
now:
   http://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
doesn't have that problem (or form), and is intelligently organized.  Damn 
software bugs <wink>.
> to discuss n-th root-finding as a specific application of
Or even delete it.
> I've never done any writing for Wikipedia so I'm not sure
> what the process is, but I'm definitely going to take
> a stab at it over the next day or two.
I haven't either, and would also like to.  Won't have sufficient free time 
for more than a week, so the glory is all yours (and the money, the 
groupies, everything!).  If you can tolerate a hint from an old Open Source 

hand, write something _you'd_ like to read.  Open Source works so long as 
people contribute work they love.  In that sense, John Gabriel's n'th-root 
contribution was likely appropriate after all <wink>. 
===
Subject: Re: REPOST: Re: Flies in the ointment.
> I notice that somebody has made a
> comment that it's Newton's method. Think I'll take a
> and see what happens.
Please see also my comment that it is a simple 
fixed-point iteration obtainable without Newton's 
method.
--r.e.s.
===
Subject: Re: REPOST: Re: Flies in the ointment.
[Randy Poe]
>> I notice that somebody has made a
>> comment that it's Newton's method. Think I'll take a
>> and see what happens.
[r.e.s.]
> Please see also my comment that it is a simple fixed-point iteration 
> obtainable without Newton's method.
Don't you think that's a stretch, though?  I do, for two reasons:
1. General.  A zero r of a function f(x) is always a fixed point of a
   Newton-method iteration (well, assuming f'(r) is defined).  Deriving
   the same formula from a fixed-point perspective may be an exercise
   in cleverness, but if you can get the same thing directly from
   applying Newton's method, the latter is preferable because entirely
   straightforward.
2. Specific.  You can find the n'th-root Newton method all over the
   web, in its weighted-average and raw forms.  Heck, it's frequently
   given as a programming assignment in intro numerical analysis
   courses.  In these contexts, it's always presented as an application
   of Newton-Raphson (proof:  I looked at a lot of course handouts
   Google could find <wink>).
===
Subject: Re: REPOST: Re: Flies in the ointment.
        reply-type=response
> [r.e.s.]
>> Please see also my comment that it is a simple fixed-point iteration 
>> obtainable without Newton's method.
> Don't you think that's a stretch, though?  I do, for two reasons:
> 1. General.  A zero r of a function f(x) is always a fixed point of a
>   Newton-method iteration (well, assuming f'(r) is defined).  Deriving
>   the same formula from a fixed-point perspective may be an exercise
>   in cleverness, but if you can get the same thing directly from
>   applying Newton's method, the latter is preferable because entirely
>   straightforward.
> 2. Specific.  You can find the n'th-root Newton method all over the
>   web, in its weighted-average and raw forms.  Heck, it's frequently
>   given as a programming assignment in intro numerical analysis
>   courses.  In these contexts, it's always presented as an application
>   of Newton-Raphson (proof:  I looked at a lot of course handouts
>   Google could find <wink>).
I want to add one more comment to my previous reply on those
two points.  The following is a well-known method intended to 
improve convergence of fixed-point iterations (or to produce 
convergence where there otherwise is none):
If y = h(y), then y = y + p*(h(y)-y), where p is arbitrary. 
This is sometimes called a dampening or averaging 
method, with p in [0,1];  note that y = p*h(y) + (1-p)*y.  
By adjusting p, the iteration y <- y + p*(h(y)-y) can often 
be made to converge in cases where y <- h(y) does not.
In the present case of y^n = x, the natural first attempt
at a fixed-point iteration is y = x/y^(n-1) = h(y), and the 
averaging method then gives the revised iteration 
y <- p*x/y^(n-1) + (1-p)*y
which is the same algorithm as obtained by Newton-Raphson 
if one takes p = 1/n (although other values of p also work).
This method is not so well-known as Newton-Raphson, to be 
sure, but I recall it from a numerical analysis course back 
in the '60s.  It seems remotely possible that the wikipedia
page on the so-called John Gabriel's method may have 
originated along these lines.
===
Subject: Re: REPOST: Re: Flies in the ointment.
        reply-type=response
> [Randy Poe]
> I notice that somebody has made a
> comment that it's Newton's method. Think I'll take a
> and see what happens.
> [r.e.s.]
>> Please see also my comment that it is a simple fixed-point iteration 
>> obtainable without Newton's method.
> Don't you think that's a stretch, though?  
Not a stretch, really, and I do think it's worth noting that this 
algorithm has a simple derivation that does not require differentiation. 
(It's not that the differentiation is difficult.)  
> I do, for two reasons:
> 1. General.  A zero r of a function f(x) is always a fixed point of a
>   Newton-method iteration (well, assuming f'(r) is defined).  Deriving
>   the same formula from a fixed-point perspective may be an exercise
>   in cleverness, but if you can get the same thing directly from
>   applying Newton's method, the latter is preferable because entirely
>   straightforward.
Of course Newton's method *is* a fixed-point iteration, but the
point is that it's as straightforward to derive the algorithm --
as posted at the wikipedia site -- in three lines of algebra as it 
is to manipulate the equation from Newton's method.  Both require 
about the same not-so-clever manipulations. 
> 2. Specific.  You can find the n'th-root Newton method all over the
>   web, in its weighted-average and raw forms.  Heck, it's frequently
>   given as a programming assignment in intro numerical analysis
>   courses.  In these contexts, it's always presented as an application
>   of Newton-Raphson (proof:  I looked at a lot of course handouts
>   Google could find <wink>).
True, but it's nice also to bring in the connection to fixed-point
iterations (and also a fixed-point theorem relating to convergence). 
===
Subject: Re: REPOST: Re: Flies in the ointment.
> Wikipedia is open-source, but it's supposed to be
> self-correcting. I notice that somebody has made a
> comment that it's Newton's method. Think I'll take a
> and see what happens.
possible. So much for self-correction.
===
Subject: REPOST: Re: Flies in the ointment.
> The recent sci.math mudfest with the character Jason contained a
> number of references to some [perhaps different person] Gabriel,
> in which it was suggest the this Gabriel person had exhibited a
> similar lack of propriety on some newsgroup, presumably (at least
> presumed by me), this occurred on sci.math.
> I have been unable to find any reference to (John?) Gabriel, and
don't
> actually recall him. As a result, I'm somewhat curious about seeing
> pointers to his, er, contributions.
I can't find it (in Google) either, yet I do remember
something about average tangents.
I'm beginning to suspect that all original John Gabriel
posts have been deleted. Check out this exchange from
the numerical analysis group, where John Gabriel
appears only in quoted text, but a character named
Jason Wells tells him he is a genius who has revolutionized
calculus:
http://tinyurl.com/3rryy
         - Randy
===
Subject: Re: Flies in the ointment.
        reply-type=response
> I also stumbled into this amazing scam on Wikipedia:
>    John Gabriel's Nth root algorithm
>    http://en.wikipedia.org/wiki/John_Gabriel%27s_Nth_root_algorithm
> Nothing wrong with the algorithm, but it's just a clumsily-stated direct 
> application of Newton's method to finding a zero of f(y) = y^n-x (iterate 

> y <- y - f(y)/f'(y)).  This certainly doesn't go under the name of John 
> Gabriel in any numerical analysis circles I've run in <heh>.
Surprisingly, it's cited at
http://www.cs.princeton.edu/introcs/96optimization/
which seems not to recognize it as an application of Newton's Method.
It is somewhat interesting that the recurrence relation has been
written in the form of an *average* (shades of an average tangent?).
This post is even more interesting ...
http://mathforge.net/index.jsp?page=seeReplies&messageNum=635
  There are infinitely many numbers between 0.999.. and 1
  [...] My name is JOhn Gabriel and I am on a crusade [...]
  Why not call 0.999... what it really is - an irrational
  like 1/3, pi, e or sqrt(2). 
===
Subject: Re: Flies in the ointment.
[Tim Peters]
...
>> I also stumbled into this amazing scam on Wikipedia:
>>    John Gabriel's Nth root algorithm
>>    http://en.wikipedia.org/wiki/John_Gabriel%27s_Nth_root_algorithm
>> Nothing wrong with the algorithm, but it's just a clumsily-stated
>> direct application of Newton's method to finding a zero of f(y) = y^n-x
>> (iterate y <- y - f(y)/f'(y)).  This certainly doesn't go under the
>> name of John Gabriel in any numerical analysis circles I've run
>> in <heh>.
[r.e.s.]
> Surprisingly, it's cited at
> http://www.cs.princeton.edu/introcs/96optimization/
Brrrrrr.
> which seems not to recognize it as an application of Newton's Method.
> It is somewhat interesting that the recurrence relation has been
> written in the form of an *average* (shades of an average tangent?).
Well, it's a weighted averge, and it's not unusual to write it like that; 
e.g.,
http://www.gnu.org/software/gmp/manual/html_node/Nth-Root-Algorithm.html
It takes a bit of algebraic manipulation to get it into that form, so I'm 
not surprised if a teacher who understands Newton-Raphson but hasn't 
actually applied it to n-th roots in anger didn't recognize the 
weighted-average form.  Still, to give a reference to a web page that can't 

do better than characterize its convergence as much faster is pretty 
lame 
even for a professor <wink>.
> This post is even more interesting ...
> http://mathforge.net/index.jsp?page=seeReplies&messageNum=635
>  There are infinitely many numbers between 0.999.. and 1
>  [...] My name is JOhn Gabriel and I am on a crusade [...]
>  Why not call 0.999... what it really is - an irrational
>  like 1/3, pi, e or sqrt(2).
Excellent!  That link just took me to the top of a large page, so I started 

skimming down.  When I got to:
    There are infinitely many numbers between 0.999.. and 1
    By definition, 0.999.. is less than 1.
    ... 1000 100 10 Units . 1/10 1/100 1/1000 1 . 0 0 0 0 . 9 9 9 ...
    Look at the above carefully! Now most mathematicians are
    unfortunately fools and will pull out all stops to cover their
    folly.
my first thought was ah, so Jason Wells posted here too.  Giving 1/3 as 
an 
example of an irrational just nailed it.  LOL -- maybe they're just 
identical twins.  I _would_ like to believe this kind of insanity is 
confined to rare unfortunate families <wink>. 
===
Subject: Re: Flies in the ointment.
        reply-type=response
> [Tim Peters]
> ...
> I also stumbled into this amazing scam on Wikipedia:
>    John Gabriel's Nth root algorithm
>    http://en.wikipedia.org/wiki/John_Gabriel%27s_Nth_root_algorithm
> Nothing wrong with the algorithm, but it's just a clumsily-stated
> direct application of Newton's method to finding a zero of f(y) = y^n-x
> (iterate y <- y - f(y)/f'(y)).  This certainly doesn't go under the
> name of John Gabriel in any numerical analysis circles I've run
> in <heh>.
> [r.e.s.]
>> Surprisingly, it's cited at
>> http://www.cs.princeton.edu/introcs/96optimization/
> Brrrrrr.
>> which seems not to recognize it as an application of Newton's Method.
>> It is somewhat interesting that the recurrence relation has been
>> written in the form of an *average* (shades of an average tangent?).
> Well, it's a weighted averge, and it's not unusual to write it like that; 

> e.g.,
> http://www.gnu.org/software/gmp/manual/html_node/Nth-Root-Algorithm.html
> It takes a bit of algebraic manipulation to get it into that form, so I'm 

> not surprised if a teacher who understands Newton-Raphson but hasn't 
> actually applied it to n-th roots in anger didn't recognize the 
> weighted-average form.  Still, to give a reference to a web page that 
> can't do better than characterize its convergence as much faster is 
> pretty lame even for a professor <wink>.
Possibly, at the link I cited above, the algorithm is to be seen
as a fixed point iteration, rather than an application of Newton's
method -- y = x^n can be manipulated into the fixed-point equation
y = (x/y^(n-1) + (n-1)y) / n  (with appropriate assumptions).
===
Subject: Re: Flies in the ointment.
        reply-type=response
> Possibly, at the link I cited above, the algorithm is to be seen
> as a fixed point iteration, rather than an application of Newton's
> method -- y = x^n can be manipulated into the fixed-point equation
                
   I meant  y = x^(1/n)  of course.
> y = (x/y^(n-1) + (n-1)y) / n  (with appropriate assumptions).
===
Subject: Re: Flies in the ointment.
> The recent sci.math mudfest with the character Jason contained a
> number of references to some [perhaps different person] Gabriel,
> in which it was suggest the this Gabriel person had exhibited a
> similar lack of propriety on some newsgroup, presumably (at least
> presumed by me), this occurred on sci.math.
> I have been unable to find any reference to (John?) Gabriel, and
> don't
> actually recall him. As a result, I'm somewhat curious about seeing
> pointers to his, er, contributions.
> I can't find it (in Google) either, yet I do remember
> something about average tangents.
Back in an older thread, Larry Hammick posted the following
link to some original John Gabriel material:
http://www.cut-the-knot.org/htdocs/dcforum/DCForumID6/403.shtml
However, it seems to have been cleaned up since the time of
Larry's posting, so that the infantile tantrum to which Larry
alluded is no longer present there.  I can say it certainly
*was* there a month ago, and it did look uncannily similar to
Jason Wells' writing style.  Of course this may prove nothing
more than that crank math of a certain kind attracts similar
personalities.
===
Subject: Re: Flies in the ointment.
Russell a .8ecrit :
>The recent sci.math mudfest with the character Jason contained a
>number of references to some [perhaps different person] Gabriel,
>in which it was suggest the this Gabriel person had exhibited a
>similar lack of propriety on some newsgroup, presumably (at least
>presumed by me), this occurred on sci.math.
>I have been unable to find any reference to (John?) Gabriel, and
>>don't
>actually recall him. As a result, I'm somewhat curious about seeing
>pointers to his, er, contributions.
>>I can't find it (in Google) either, yet I do remember
>>something about average tangents.
> Back in an older thread, Larry Hammick posted the following
> link to some original John Gabriel material:
> http://www.cut-the-knot.org/htdocs/dcforum/DCForumID6/403.shtml
> However, it seems to have been cleaned up since the time of
> Larry's posting, so that the infantile tantrum to which Larry
> alluded is no longer present there. 
But what is left is enough for any reader (except perhaps Jason) to 
make up his/her  mind. It sems, besides, that the guy behind this 
(Gabriel?) was also vandalizing the site...
  I can say it certainly
> *was* there a month ago, and it did look uncannily similar to
> Jason Wells' writing style.  Of course this may prove nothing
> more than that crank math of a certain kind attracts similar
> personalities.
===
Subject: Re: Flies in the ointment.
> The recent sci.math mudfest with the character Jason contained
a
> number of references to some [perhaps different person]
Gabriel,
> in which it was suggest the this Gabriel person had exhibited a
> similar lack of propriety on some newsgroup, presumably (at least
> presumed by me), this occurred on sci.math.
 I have been unable to find any reference to (John?) Gabriel, and
> don't
> actually recall him. As a result, I'm somewhat curious about
seeing
> pointers to his, er, contributions.
> I can't find it (in Google) either, yet I do remember
> something about average tangents.
> Back in an older thread, Larry Hammick posted the following
> link to some original John Gabriel material:
> http://www.cut-the-knot.org/htdocs/dcforum/DCForumID6/403.shtml
> However, it seems to have been cleaned up since the time of
> Larry's posting, so that the infantile tantrum to which Larry
> alluded is no longer present there.  I can say it certainly
> *was* there a month ago, and it did look uncannily similar to
> Jason Wells' writing style.  Of course this may prove nothing
> more than that crank math of a certain kind attracts similar
> personalities.
So, probably he'll be back with another sock puppet. If
so, I'll respond impassively to the mathematical arguments
and not make any accusations of puppetry.
         - Randy
===
Subject: Re: lyapunov function for a computer networking problem
> Unfortunately, (mu/2, mu/2) is on the line x1 + x2 = mu, and you 
haven't
> correctly specified what happens on that line.  Until you do, nothing 
further
> can be said about it.
> 
> I agree with you. So, please, what's the continuous-time version of:
> x_1(k+delta) = x_1(k) + delta * I(x_1(k) + x_2(k) <= mu) - beta *
> delta * x_1(k) * I(x_1(k) + x_2(k) > mu)
> x_2(k+delta) = x_2(k) + delta * I(x_1(k) + x_2(k) <= mu) - beta *
> delta * x_2(k) * I(x_1(k) + x_2(k) > mu)
> considering a very small delta?
> Here, 0 < beta < 1,  mu > 0, delta > 0  and k > 0.
> I(x < mu) is and indicator function, which is equal to 1 if x < mu,
> and 0 otherwise.
There doesn't necessarily have to be a continuous-time version.
In this case, let's see what happens when your system comes 
close to the line x_1 + x_2 = mu.  It's simpler to change 
variables: take
y_1 = x_1 + x_2 - mu
y_2 = x_1 - x_2
Thus your system becomes 
y_1(k + delta) = y_1(k) + 2 * delta  when y_1(k) <= 0 
               = y_1(k) - beta * delta *(y_1(k) + mu) when y_1(k) > 0
y_2(k + delta) = y_2(k) when y_1(k) <= 0
               = (1 - beta * delta) * y_2(k) when y_1(k) > 0
So y_1 will increase at a constant rate until it crosses the line y_1 = 0,
then oscillate back and forth across that line.  Since over the long run
it won't change much after that, and the steps forward are of size 2*delta
while the steps backward are of size approximately beta*mu*delta, 
on the average the fraction of time you spend (after crossing y_1 = 0)
with y_1 > 0  will be 2/(2 + beta*mu), and the fraction of time with y_1 < 
0
will be beta*mu/(2+beta*mu).
Thus I would say the appropriate continuous-time version would have
d/dt y_1 = 2 for y_1 < 0
         = 0 for y_1 = 0 
         = -beta*(y_1 + mu) for y_1 > 0
d/dt y_2 = 0 for y_1 < 0
         = -2*beta/(2 + beta*mu) * y_2 for y_1 = 0
         = -beta*y_2 for y_1 > 0
(these being only one-sided derivatives at the time you hit y_1 = 0).
And for this system it's easy to see that y_1 = y_2 = 0 (i.e. 
x_1 = x_2 = mu/2) is the unique fixed point, and is asymptotically
stable.
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
===
Subject: Re: lyapunov function for a computer networking problem
> Thus I would say the appropriate continuous-time version would have
> d/dt y_1 = 2 for y_1 < 0
>          = 0 for y_1 = 0 
>          = -beta*(y_1 + mu) for y_1 > 0
> d/dt y_2 = 0 for y_1 < 0
>          = -2*beta/(2 + beta*mu) * y_2 for y_1 = 0
>          = -beta*y_2 for y_1 > 0
> (these being only one-sided derivatives at the time you hit y_1 = 0).
d/dt y_2 = -2*beta/(2 + beta*mu) * y_2 for y_1 = 0
?  
I wasn't able to figure out what was the motivation of this equation. 
I understand that this equation needs to have the form -k*y2.  You
suggest
- (approx. average fraction of time y1 > 0)*beta*y2 . Why?
Maybe because we are trying to make the convergence time of the
continuous-time system equal to the convergence time of the
discrete-time system.  But how is it accomplished?  Indeed, I made
some simulations, and the convergence time of both systems is approx.
the same.  But again, I wasn't able to figure out why.
Also, one philosophical fuzzy question: why the discrete time system
is so simple, and the continuous time system is a bit intricate? 
Would it be possible to find a continuous-time system which also needs
just two conditions, and not three, in order to characterize the
desired behaviour?  Continuous random variables, for instance, have
the property that P{X=x}=0.  Would it be possible to use something
like that in order to simplify the continuous-time equations above?
Daniel Sadoc
===
Subject: Re: lyapunov function for a computer networking problem
>> Thus I would say the appropriate continuous-time version would have
 
>> d/dt y_1 = 2 for y_1 < 0
>>          = 0 for y_1 = 0 
>>          = -beta*(y_1 + mu) for y_1 > 0
 
>> d/dt y_2 = 0 for y_1 < 0
>>          = -2*beta/(2 + beta*mu) * y_2 for y_1 = 0
>>          = -beta*y_2 for y_1 > 0
 
>> (these being only one-sided derivatives at the time you hit y_1 = 0).
 
>d/dt y_2 = -2*beta/(2 + beta*mu) * y_2 for y_1 = 0
>?  
>I wasn't able to figure out what was the motivation of this equation. 
>I understand that this equation needs to have the form -k*y2.  You
>suggest
>- (approx. average fraction of time y1 > 0)*beta*y2 . Why?
Think of it this way.  In the discrete system, look at what happens
in N steps after the first time it crosses y1 = 0.  Of these steps,
say k are forward (with y1 increasing by 2*delta) and the other N-k 
backward (with y1 decreasing by approximately beta*mu*delta).  Since 
the system must stay within one step of y1=0, 
2*delta*k - beta*mu*delta*(N-k) is nearly 0, 
so k is approximately beta*mu/(2+beta*mu) * N, and N-k is
approximately 2/(2+beta*mu)*N.  
Now in the k forward steps y2 doesn't change, but in each of the 
N-k backward steps it changes by -beta*delta*y2, for an average
of 2/(2+beta*mu)*beta*delta*y2 per step.  So in the continuous-time
version, you should see 
d/dt y2 = lim_{delta -> 0} (2/(2+beta*mu)*beta*delta*y2)/delta
        = 2*beta/(2+beta*mu)*y2
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
===
Subject: advice for proving things about the tensor product of vectors?
    I've been through just about every algebra book in the university
library and they're all the same:  they have a paragraph, *MAYBE* a
page, about the tensor product of vectors, while having whole chapters
about the tensor product of vector _spaces_, linear transformations
etc.  I'm frantic because I have a homework assignment to prove a ton
of little properties about tensor product of vectors; the problems look
simple but there's just nothing to say.  You open your mouth to start
saying a proof, and it hits you-- you can't say a thing, because
everything you know is about the tensor product of _spaces_ or
transformations.  The canonical concrete construction of the tensor
product of vectors is quite worthless (I'm talking about the quotient
version here, where you include an n-tuple into the free module and add
(take quotient map) a rather complicated mess).
     One book put forth a bunch of little properties about the things,
and the proof went like:  this is immediate.  Er, no, it's not
immediate at all!  And to make matters worse, the problems are general
in the sense that I'm asked to prove things about tensor product of
many (arbitrary, not necessarily finite) vectors.  What little any
texts have to say about the matter at all, is almost entirely
restricted to the case of 2 vectors!
     So if anyone can help - even if just giving references to books
that actually offer insight - it would be greatly valued.
===
Subject: Re: advice for proving things about the tensor product of vectors?
Post a problem,maybe somebody will do it and you will get the
idea.Tensor Products are characterized by a universal property and you
need to use it in proofs.
===
Subject: Re: advice for proving things about the tensor product of vectors?
> Post a problem,maybe somebody will do it and you will get the
> idea.Tensor Products are characterized by a universal property and
you
> need to use it in proofs.
Here is a problem then.  I've struggled quite a lot with this.  I use
(x) to denote tensor product.
Given zin V (x) W, and given that z has two representations of the
form
z=sum_{i=1}^r (v_i) (x) (w_i),
z=sum_{i=1}^s (v'_i) (x) (w'_i),
where {v_i}, {w_i}, {v'_i} and {w'_i} are linearly indep. (just with
themselves, not necessarily with eachother, ie perhaps some v'_i is a
combination of the v_i)... then prove r=s.
All I can think of doing is induction.  If a term in the 2nd sum is a
lin. comb. of the terms in the 1st, then (inducting on, say, r+s) we
can say it is strictly a multiple of one particular term from the 1st,
and then playing around with it we can conclude it in fact equals a
term from the 1st, so we can subtract this term from both sides and
finish by induction.  This is all assuming we can prove the claim for
the special case when one of r or s is 1-  which I haven't actually
managed to do, that being itself rather hard.  Much harder is the case
if nothing in the 2nd sum is a lin. comb. of the terms in the first.
If this is the case, we can move terms from one side to the other
without breaking the hypothesis.  And likewise we can add a copy of any
term to both sides.  By doing the former, we can reduce the difference
between r and s by 2, assuming it's >2 to begin with, and by doing the
latter, we can reduce it by 1, assuming it's >1 to begin with, so we
can basically reduce everything to the case when the difference between
r and s is 1.  Then we're at a dead-end because there's just nothing we
can say.
Anyway, it seems there should be an easier solution anyway- my method
is leading toward a 2-page proof!
===
Subject: Re: advice for proving things about the tensor product of vectors?
> Post a problem,maybe somebody will do it and you will get the
> idea.Tensor Products are characterized by a universal property and
> you
> need to use it in proofs.
> Here is a problem then.  I've struggled quite a lot with this.  I use
> (x) to denote tensor product.
> Given zin V (x) W, and given that z has two representations of the
> form
> z=sum_{i=1}^r (v_i) (x) (w_i),
> z=sum_{i=1}^s (v'_i) (x) (w'_i),
> where {v_i}, {w_i}, {v'_i} and {w'_i} are linearly indep. (just with
> themselves, not necessarily with eachother, ie perhaps some v'_i is a
> combination of the v_i)... then prove r=s.
> All I can think of doing is induction.  If a term in the 2nd sum is a
> lin. comb. of the terms in the 1st, then (inducting on, say, r+s) we
> can say it is strictly a multiple of one particular term from the
1st,
> and then playing around with it we can conclude it in fact equals a
> term from the 1st, so we can subtract this term from both sides and
> finish by induction.  This is all assuming we can prove the claim for
> the special case when one of r or s is 1-  which I haven't actually
> managed to do, that being itself rather hard.  Much harder is the
case
> if nothing in the 2nd sum is a lin. comb. of the terms in the first.
> If this is the case, we can move terms from one side to the other
> without breaking the hypothesis.  And likewise we can add a copy of
any
> term to both sides.  By doing the former, we can reduce the
difference
> between r and s by 2, assuming it's >2 to begin with, and by doing
the
> latter, we can reduce it by 1, assuming it's >1 to begin with, so we
> can basically reduce everything to the case when the difference
between
> r and s is 1.  Then we're at a dead-end because there's just nothing
we
> can say.
> Anyway, it seems there should be an easier solution anyway- my method
> is leading toward a 2-page proof!
You have to use the characterizing property of tensor products,namely
the lifting of bilinear maps to linear maps.Still,it takes some
experience to see how to use it in this problem.
To do the problem one proves that every v_i is a linear combination of
the v'_i 's and the same argument (or symmetry )shows that every v'_i
is a linear combination of the v_i's.This means the spans of the v_i's
and v'i's are the same and the dimension of this span is r=s since both
sequences are basis for the span.
For this ,fix a positive integer k, at most r,and construct a linear
functional f on W with f(w_k)=1 and f(w_i)=0 for all i not = k.(Just
extend the w_i's to a basis for W and let f(w)= the kth coordinate of
the expansion of w as a finite linear combination of the extended
w_i's.
Now let G(v,w)= f(w)v ,G is a bilinear function from VxW into V ,and
thus by the so called universal property of tensor products ,there is a
linear function g from V(x)W into V with g(v(x)w)= f(w)v for all v,w.
 g(sum(v_i(x)w_i))= g( sum(v'_i(x)w'_i)) implies that
sum (f(w_i)v_i)= sum f(w'_i)v'_i) which gives by simplifying the left
curious -What text are you using for tensor products ?
===
Subject: Rationalize Expression
http://mygate.mailgate.org/mynews/sci/sci.math/47752daae0951dfb038a7d3d15467
0
a3.61944%40mygate.mailgate.org
How would you rationalize [1/2^(1/2)] + [1/3^(1/3)] + 1/5^(1/5)]?
-- 
===
Subject: Re: Rationalize Expression
> How would you rationalize [1/2^(1/2)] + [1/3^(1/3)] + 1/5^(1/5)]?
I'm not sure what rationalize means in this context.
If it means to combine into a single term with no
fractional powers in the denominator, then look for
a common denominator which has no such powers, i.e.
put each term over the common denominator 2*3*5.
For instance, multiply numerator and denominator of
the second term by 2*3^(2/3)*5. Something similar
can be done for the other two terms.
           - Randy
===
Subject: Re: Rationalize Expression
http://mygate.mailgate.org/mynews/sci/sci.math/023140da1f3347a0036134f1527cd
e
82.61944%40mygate.mailgate.org
> How would you rationalize [1/2^(1/2)] + [1/3^(1/3)] + 1/5^(1/5)]?
> I'm not sure what rationalize means in this context.
> If it means to combine into a single term with no
> fractional powers in the denominator, then look for
> a common denominator which has no such powers, i.e.
> put each term over the common denominator 2*3*5.
> For instance, multiply numerator and denominator of
> the second term by 2*3^(2/3)*5. Something similar
> can be done for the other two terms.
>            - Randy
Silly me.
I meant 1 / [2^(1/2) + 3^(1/3) + 5^(1/5)]
That is, find an x such that x * [2^(1/2) + 3^(1/3) + 5^(1/5)] is an
integer.
-- 
===
Subject: Re: Rationalize Expression
   <023140da1f3347a0036134f1527cde82.61944@mygate.mailgate.org>
Here's the easy way to do it.
Let a = 2^(1/2),  b = 3^(1/3),  c = 5^(1/5)
The the answer will be a polynomial in a,b,c
P(a,b,c) = sum A_n,m,p a^n b^m c^p
with  0<=n<2,  0<=m<3, 0<=p<5
So set (a+b+c)*P(a,b,c) = 1, and it will lead you to a system of linear
equations with rational coefficients. Thus each of the A_m,n,p will be
rational.
Note that Rusin's answer introduced irrational quantities like
cos(pi/5), which suggests that either his answer is wrong or more
likely that it is simply not in simplest form.
===
Subject: Re: Rationalize Expression
|>I meant 1 / [2^(1/2) + 3^(1/3) + 5^(1/5)]
|>That is, find an x such that x * [2^(1/2) + 3^(1/3) + 5^(1/5)] is an
|>integer.
Maple 9.5 says:
> rationalize(1 / (2^(1/2) + 3^(1/3) + 5^(1/5)));
-1/867818606*(-3^(1/3)-5^(1/5)+2^(1/2))*(2*3^(2/3)+3*3^(2/3)*5^(2/5)+3*3^(1/

3)
-2*3^(1/3)*5^(3/5)+4*3^(1/3)*5^(1/5)+4-4*5^(2/5)+5^(4/5)-6*5^(1/5))
*(-1676849-2931551*5^(1/5)+3730879*5^(2/5)-995123*5^(3/5)+3557707*5^(4/5))
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
===
Subject: Re: Rationalize Expression
<023140da1f3347a0036134f1527cde82.61944@mygate.mailgate.org>,
> I meant 1 / [2^(1/2) + 3^(1/3) + 5^(1/5)]
> That is, find an x such that x * [2^(1/2) + 3^(1/3) + 5^(1/5)] is an
> integer.
[meaning, a non-zero integer, and presumably meaning x expressed 
without fractions, in some sense, so as to rule out wiseguy answers 
like x = 73 / [2^(1/2) + 3^(1/3) + 5^(1/5)]]
The product of the conjugates of 2^(1/2) + 3^(1/3) + 5^(1/5) will 
be an integer. The conjugates of 2^(1/2) + 3^(1/3) + 5^(1/5) are 
the numbers a + b + c, where a = plus or minus 2^(1/2), 
b = w 3^(1/3), c = z 5^(1/5), where w runs through the three roots 
of unity, and z through the five fifth roots of unity.
-- 
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: Rationalize Expression
>> That is, find an x such that x * [2^(1/2) + 3^(1/3) + 5^(1/5)] is an
>> integer.
>[meaning, a non-zero integer, and presumably meaning x expressed 
>without fractions, in some sense, so as to rule out wiseguy answers 
>like x = 73 / [2^(1/2) + 3^(1/3) + 5^(1/5)]]
But Gerry, the right answer IS  x = 867818606 / [2^(1/2) + 3^(1/3) + 
5^(1/5)]],
for exactly the reasons you gave.  You could, of course, express it as an
integer polynomial in  a=2^(1/2), b=3^(1/3), c=5^(1/5), d=cos(pi/5), and
e=cos(2pi/5), but you would not want to do so because then it would be 
-21665049+34735946*a-36130691*b+86604619*c+30341150*e-30341150*d-27594638*a*

c*
e+12260718*b*c*e+61422896*b^2*c*e-12201798*c^4*b*e-3774890*c^4*b^2*e-2041555

2*
b*c^2*e+81991118*a*c^2*e+36571890*b^2*c^2*e+43466136*a*c^3*e+7217050*c^3*b*e

+
2142042*c^3*b^2*e-6159832*c^4*a*e+25669360*c^4*a*b*e+5029878*c^4*b^2*a*e+
278949*c^4*a*b-15793112*c^4*b^2*a-15333920*a*b*d-152670*a*b^2*d+27594638*a*c

*d
-12260718*b*c*d-61422896*b^2*c*d-81991118*a*c^2*d+20415552*b*c^2*d-36571890*

b^
2*c^2*d-43466136*a*c^3*d-7217050*c^3*b*d-2142042*c^3*b^2*d+12201798*c^4*b*d+

6159832*c^4*a*d+3774890*c^4*b^2*d+15333920*a*b*e+152670*a*b^2*e-12683047*a*c

^2
*b^2-6111030*a*c^3*b-162336*a*c^3*b^2+40367536*a*c-40446855*a*c*b^2+10769044

*a
*c^2*b+61575566*a*b*c*d-25236964*a*b*c-21147843*a*b+18356878*a*b^2-19219693*

b*
c+6880086*b^2*c+6750956*a*c^2+18486008*b*c^2+29677811*b^2*c^2+18794077*c^3*b

^2
-116613*c^4*b^2-104582022*c*d+18584095*a*c^3-29353139*c^3*b+12792147*c^4*b-
6681117*c^4*a+146915910*a*d-18569110*b*d+40483310*b^2*d+18569110*b*e-8910565

4*
c^2*d+16307708*c^3*d+58555770*c^4*d-40483310*b^2*e+104582022*c*e+89105654*c^

2*
e-16307708*c^3*e-58555770*c^4*e-146915910*a*e+100113403*b^2-2318395*c^2-
6263948*c^3+28795241*c^4-14111296*a*c*b^2*d+50683186*a*c^2*b*d+20503672*a*c^

2*
b^2*d-18361630*a*c^3*b*d+21894470*a*c^3*b^2*d-25669360*c^4*a*b*d-5029878*c^4

*b
^2*a*d-61575566*a*b*c*e+14111296*a*c*b^2*e-50683186*a*c^2*b*e-20503672*a*c^2

*b
^2*e+18361630*a*c^3*b*e-21894470*a*c^3*b^2*e
and nobody wants to see that!
dave
===
Subject: Re: Rationalize Expression
> 
> find an x such that x * [2^(1/2) + 3^(1/3) + 5^(1/5)] is an integer.
>> [meaning, a non-zero integer, and presumably meaning x expressed 
>> without fractions, in some sense, so as to rule out wiseguy answers 
>> like x = 73 / [2^(1/2) + 3^(1/3) + 5^(1/5)]]
> But Gerry, the right answer IS  x = 867818606/[2^(1/2) + 3^(1/3) + 
5^(1/5)]],
> for exactly the reasons you gave.  You could, of course, express it as an
> integer polynomial in  a=2^(1/2), b=3^(1/3), c=5^(1/5), d=cos(pi/5), and
> e=cos(2pi/5), but you would not want to do so because then it would be 
>-21665049+34735946*a-36130691*b+86604619*c+30341150*e-30341150*d-27594638*a

*c*
>e+12260718*b*c*e+61422896*b^2*c*e-12201798*c^4*b*e-3774890*c^4*b^2*e-204155

52*
>b*c^2*e+81991118*a*c^2*e+36571890*b^2*c^2*e+43466136*a*c^3*e+7217050*c^3*b*

e+
>2142042*c^3*b^2*e-6159832*c^4*a*e+25669360*c^4*a*b*e+5029878*c^4*b^2*a*e+
>278949*c^4*a*b-15793112*c^4*b^2*a-15333920*a*b*d-152670*a*b^2*d+27594638*a*

c*d
>-12260718*b*c*d-61422896*b^2*c*d-81991118*a*c^2*d+20415552*b*c^2*d-36571890

*b^
>2*c^2*d-43466136*a*c^3*d-7217050*c^3*b*d-2142042*c^3*b^2*d+12201798*c^4*b*d

+
>6159832*c^4*a*d+3774890*c^4*b^2*d+15333920*a*b*e+152670*a*b^2*e-12683047*a*

c^2
>*b^2-6111030*a*c^3*b-162336*a*c^3*b^2+40367536*a*c-40446855*a*c*b^2+1076904

4*a
>*c^2*b+61575566*a*b*c*d-25236964*a*b*c-21147843*a*b+18356878*a*b^2-19219693

*b*
>c+6880086*b^2*c+6750956*a*c^2+18486008*b*c^2+29677811*b^2*c^2+18794077*c^3*

b^2
>-116613*c^4*b^2-104582022*c*d+18584095*a*c^3-29353139*c^3*b+12792147*c^4*b-

>6681117*c^4*a+146915910*a*d-18569110*b*d+40483310*b^2*d+18569110*b*e-891056

54*
>c^2*d+16307708*c^3*d+58555770*c^4*d-40483310*b^2*e+104582022*c*e+89105654*c

^2*
>e-16307708*c^3*e-58555770*c^4*e-146915910*a*e+100113403*b^2-2318395*c^2-
>6263948*c^3+28795241*c^4-14111296*a*c*b^2*d+50683186*a*c^2*b*d+20503672*a*c

^2*
>b^2*d-18361630*a*c^3*b*d+21894470*a*c^3*b^2*d-25669360*c^4*a*b*d-5029878*c^

4*b
>^2*a*d-61575566*a*b*c*e+14111296*a*c*b^2*e-50683186*a*c^2*b*e-20503672*a*c^

2*b
>^2*e+18361630*a*c^3*b*e-21894470*a*c^3*b^2*e
> and nobody wants to see that!
It's  q p(u+v,w)  more simply, where  u = 2^(1/2), v = 3^(1/3), w = 5^(1/5)
              2                                      2
    (1770832 v + 18893046 v + 58073126 - u(18308051 v + 12555731 v + 
3601201))
q = 
--------------------------------------------------------------------------
                                   867818606
and   p(x,y) = (x^5+y^5)/(x+y) [= x^4 - y x^3 + y^2 x^2 - y^3 x + y^4 ]
It is tedious but straightforward to derive the above expression, namely:
 x^5+5 = x^5+w^5 = (x+w)p(x,w) => 1/(x+w) = p(x,w)/(x^5+5). Here x = u+v,
 x^5+5 = (u+v)^5+5 = (20u+3) v^2 + (15u+20) v + 4u+65
                              2         2       2             2
              1             (b  - a c) v  + (3 a  - b c) v + c  - 3 a b
Now    ---------------  =   -------------------------------------------
           2                           3      3    3          
        a v  + b v + c              9 a  + 3 b  + c  - 9 a b c
For  a,b,c = 20u+3,15u+20,4u+65  this specializes to
          (712 u - 495) v^2  + (695 u - 1007) v + 815 u - 2277
          ----------------------------------------------------
                              10147 u - 32768
Rationalizing the denominator yields the polynomial q(u,v) specified above.
Compare with Maple's result posted by Robert Israel:
 -(u-v-w)*(2vv+3vvww+3v-2vw^3+4vw+4-4ww+w^4-6w)
         *(-1676849-2931551w+3730879w^2-995123w^3+3557707w^4)/867818606
--Bill Dubuque
===
Subject: Re: Rationalize Expression
> Rationalizing the denominator yields the polynomial q(u,v)
Which expressed succinctly is not too painful to look at
(as opposed to the 19 line behemoth in D.R.'s prior post)
  1/(u+v+w) =  (10147 u + 32768)
              * ((712 u - 495) v^2 + (695 u - 1007) v + 815 u - 2277)
              * w^4 P((u+v)/w) / 867818606 
where    u = 2^(1/2), v = 3^(1/3), w = 5^(1/5)
      P(x) = (x^5+1)/(x+1) = x^4 - x^3 + x^2 + 1
I doubt it can be expressed much simpler than that.
--Bill Dubuque
===
Subject: Re: Rationalize Expression
> Compare with Maple's result posted by Robert Israel:
>                4        3       2       2                    2
>  (W + V - U) (W  - 2 V W  + (3 V  - 4) W  + (4 V - 6) W + 2 V  + 3 V + 4)
>                4           3            2
>    * (3557707 W  - 995123 W  + 3730879 W  - 2931551 W - 
1676849)/867818606
>  where  U = 2^(1/2), V = 3^(1/3), W = 5^(1/5)
This comes from rationalizing in the order: U, V, W  (vs. my  W, V, U )
namely  1/(T+U) = (T-U)/(T^2-2),  here  T = V+W
hence   1/(T^2-2) = 1/(V^2 + 2W V + W^2-2)
                             2         2       2             2
               1           (b  - a c) V  + (3 a  - b c) V + c  - 3 a b
Now     --------------  =  -------------------------------------------
           2                          3      3    3
        a V  + b V + c             9 a  + 3 b  + c  - 9 a b c
which for  a = 1, b = 2W, c = W^2-2  specializes to
        4        3       2       2                    2
       W  - 2 V W  + (3 V  - 4) W  + (4 V - 6) W + 2 V  + 3 V + 4
       ----------------------------------------------------------
                         4      3       2
                    - 6 W  + 6 W  + 12 W  + 41 W + 1
which times  T-U = W+V-U  with rationalized denominator is Maple's result.
--Bill Dubuque
===
Subject: Re: Rationalize Expression
http://mygate.mailgate.org/mynews/sci/sci.math/b341a29a87aa40960b32ec380d9e9
d
72.61944%40mygate.mailgate.org
> I meant 1 / [2^(1/2) + 3^(1/3) + 5^(1/5)]
> That is, find an x such that x * [2^(1/2) + 3^(1/3) + 5^(1/5)] is an
> integer.
As x would be a rather complicated expression, how would you find the
value of the denominator?
-- 
===
Subject: Re: What are numbers?
> A quantity is simply a property of a set. By not refering to the set, but
> instead its property of `size' or `total' 
The size of a set is its cardinality.  I know of two approaches (I bet
there are more):
(i) Take cardinality as an undefined primitive.  This is an approach due
to Tarski that is taken in Suppes  Axiomatic Set Theory (a recommended
read, btw).
(ii) Take the cardinality of a set to be itself a particular kind of
set, viz. an initial ordinal, where an ordinal is a particular kind of
set, viz. one ordered in a particular way by the is an element of
relation.
> I don't have to worry about
> members being and not being members of themselves.
> So I think of it this way.
> Quantity refers to a property of a set (size or totalness).
> Number refers to the names given to each distinct quantity (i.e. where
> quantities are the same they share the same name, different quantities
> have different names).
Worry: are there uncountably many quantities?  If so, how do you get to
name them all if name has its usual meaning?
> Numerals are the individual names refering to distinct quantities.
> I am not trying to construct the set of numbers here (although I seem to
> be getting close to it). By construction, I would do as you suggest, and
> start with sets. I'm just trying to provide a decent conception. I can't
> see how concieving of the natural numbers in this way leads to a problem
If all you want is natural numbers then go for Peano's axioms.  They
come in two varieties:
(i) First order: nice and simple but not categorical.
(ii) Second order: categorical, but sets rear their ugly heads!
> later. So far, I haven't seen any problems.
In that case you should proceed!  My worry: how _far_ can you proceed
before problems appear?
===
Subject: Re: What are numbers?
> I claim that you _can't_ get integers, rationals, reals, and complex
> numbers with *only* logic[*] you need a bit of set theory as well, and
> Ok, let me stupidly suggest that I can. Why can't I get integers by
> pairing natural numbers? Why can't I get the rationals by paring
> integers. Why can't I get reals through Dedekind cuts?
> Jackson
Ok, let the pair (n, m) be the integer n - m.  So -1, for example is all
of these
   (1, 2), (2, 3), (3, 4), ...
The usual thing is to lump them all together into a _set_ called an
equivalence class.  Ditto rationals as pairs of integers.
Clearly _sets_ of rational numbers are involved in the definition of
Dedekind cuts.
So you go beyond logic and use set theory.
If you want to expand the definition of logic so that it _includes_ set
theory, so be it.  But many people think that the theory of the
predicate is an element of goes beyond logic.
===
Subject: Re: What are numbers?
> Ok, let the pair (n, m) be the integer n - m.  So -1, for example is 
all
> of these
>    (1, 2), (2, 3), (3, 4), ...
> The usual thing is to lump them all together into a _set_ called an
> equivalence class.  Ditto rationals as pairs of integers.
> Clearly _sets_ of rational numbers are involved in the definition of
> Dedekind cuts.
> So you go beyond logic and use set theory.
> If you want to expand the definition of logic so that it _includes_ set
> theory, so be it.  But many people think that the theory of the
> predicate is an element of goes beyond logic.
Sure, its convenient to use set theory, but I don't have to use much of 
it--I'm not sure I need to use any of it. None of these things rests on 
the axioms. I can simply describe sets as elements catagorized by 
properties. I don't have to use is an element of because I can simply 
say has property P. I don't need to use compliments, replacement, 
choice etc. I don't know how far I can go before I will indeed have to 
make use of set theory, but as far as I can tell, the interest in basing 
numbers on set theory is simply to construct them from nothing--but this 
is where Godel showed that sets have content and thus are something.
Jackson
===
Subject: Re: What are numbers?
> Ok, let the pair (n, m) be the integer n - m.  So -1, for example is
> all
> of these
>    (1, 2), (2, 3), (3, 4), ...
> The usual thing is to lump them all together into a _set_ called an
> equivalence class.  Ditto rationals as pairs of integers.
> Clearly _sets_ of rational numbers are involved in the definition of
> Dedekind cuts.
> So you go beyond logic and use set theory.
> If you want to expand the definition of logic so that it _includes_ set
> theory, so be it.  But many people think that the theory of the
> predicate is an element of goes beyond logic.
> Sure, its convenient to use set theory, but I don't have to use much of
> it--I'm not sure I need to use any of it. None of these things rests on
> the axioms. I can simply describe sets as elements catagorized by
> properties. I don't have to use is an element of because I can simply
> say has property P. 
Eek!  No you can't!
Consider the property of not applying to itself.  It's a perfectly
reasonable property, after all the property of being red is not itself
red.  Call the property of not applying to itself R and ask yourself:
does R apply to R?  Frege had this problem.  You can Google for
Russell's paradox.
> I don't need to use compliments, replacement,
> choice etc. 
It depends what you're going to do with them once you've got them.  If
you're going to define the real numbers for use in analysis then you'll
need choice.  Choice isn't just there so that set theorists can play
with it, it has _real_ uses.
> I don't know how far I can go before I will indeed have to
> make use of set theory, but as far as I can tell, the interest in basing
> numbers on set theory 
Two things one might do:
(i) Define natural numbers as sets of a particular kind, say finite von
Neumann ordinals, and define all other numbers in terms of them.
(ii) Take natural numbers as undefined (as Peano did), and define all
other numbers in terms of them.
Clearly approach (i) uses set theory, but it seems clear to me that (ii)
does as well because the defining of all other numbers uses set theory
(as I think I indicated).
Btw, Zermelo-Fraenkel set theory with the axiom of infinity replaced by
its negation is equivalent to the theory of natural numbers.
If you want a theory of numbers without set theory consider these two
possibilities:
(i) Use lambda calculus to define the natural numbers.
(ii) Start with the reals as is done in beginning analysis texts.  This
won't eliminate set theory, because categorical axioms for the real
numbers quantify over sets of real numbers, but the intrusion will be
minimal.
> is simply to construct them from nothing--but this
> is where Godel showed that sets have content and thus are something.
> Jackson
===
Subject: Re: What are numbers?
      <140320051515000566%anniel@nym.alias.net.invalid> 
03/14/2005
   at 02:17 PM, quantanglement@gmail.com said:
>How is that? Why is the integer 17 different from the rational 17?
>etc.
That depends on what you mean by number. For instance, if you start
with the axioms for R and define C in terms of ordered pairs, then
clearly 17 and (17,0) are different. Similarly, if you start with the
integers and define rational numbers as equivalence classes of ordered
pairs then 17 and {(17,1)} are different. If you start with axioms for
C then all of those are the same.
-- 
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: What are numbers?
> Aren't numbers just names for quantities?
>> What is wrong with saying that numbers simply refer to their
> specific quantities?
> If you want to develop a theory of numbers from scratch, I think
>> you will find it more productive to start with a theory of sets, and
>> then define the natural numbers in terms of the far simpler notions
>> of first and next (like Peano's Axioms). Or start with a set
>> theory, like ZFC, in which these notions are already, in a sense,
>> built in. My own preference is the former.
>> Dan
>> Download DC Proof 1.0 at http://www.dcproof.com
> A quantity is simply a property of a set. By not refering to the set,
> but instead its property of `size' or `total' I don't have to worry
> about members being and not being members of themselves.
> So I think of it this way.
> Quantity refers to a property of a set (size or totalness).
> Number refers to the names given to each distinct quantity (i.e.
> where quantities are the same they share the same name, different
> quantities have different names).
> Numerals are the individual names refering to distinct quantities.
> I am not trying to construct the set of numbers here (although I seem
> to be getting close to it). By construction, I would do as you
> suggest, and start with sets. I'm just trying to provide a decent
> conception. I can't see how concieving of the natural numbers in this
> way leads to a problem later. So far, I haven't seen any problems.
>> I think you will encounter some problems when you try to do proofs by
>> induction.
>> Dan
>> Download DC Proof 1.0 at http://www.dcproof.com
> How So?
Suppose we have, for some predicate P:
P(1)
For all k in N, P(k) => P(k+1)
Prove: For all k in N, P(k)
Can you do this in your system?
As an amateur mathematician, I have struggled with such problems for many 
years -- my DC Proof program is the result.
Dan
===
Subject: Re: What are numbers?
> Suppose we have, for some predicate P:
> P(1)
> For all k in N, P(k) => P(k+1)
> Prove: For all k in N, P(k)
> Can you do this in your system?
Why not?
P(1) is some property.
For every numeral in the set of numbers,
if Property 1 applies to the quantity which the numeral k refers to then 
it  applies to the next largest quantity which is named k+1.
> As an amateur mathematician, I have struggled with such problems for
> many years -- my DC Proof program is the result.
> Dan
===
Subject: Re: What are numbers?
>> Suppose we have, for some predicate P:
>> P(1)
>> For all k in N, P(k) => P(k+1)
>> Prove: For all k in N, P(k)
>> Can you do this in your system?
> Why not?
> P(1) is some property.
> For every numeral in the set of numbers,
> if Property 1 applies to the quantity which the numeral k refers to then
> it  applies to the next largest quantity which is named k+1.
Is this an axiom in your system? Or is it a proof? You really need to 
formalize all the axioms and rules of inference for your proposed system. 
You might want to have a look at my DC Proof program to get some ideas. It 
is a line-by-line proof checker with what I think is very nice user 
interface.
Dan
Download DC Proof 1.0 at http://dcproof.com
===
Subject: Radioactive decay
I am not sure how to set up this equation in order to solve this
problem. Can someone help me or guide me in the direction that will
Question: If the half-life of a radioactive substance is 20 days, how
long will it take for 99 percent of the substance to decay?
===
Subject: Re: Radioactive decay
> I am not sure how to set up this equation in order to solve this
> problem. Can someone help me or guide me in the direction that will
> Question: If the half-life of a radioactive substance is 20 days, how
> long will it take for 99 percent of the substance to decay?
A(t) = A0.e^-rt
A0/2 = A0.e^-(r.20 day)
0.01 A0 = A0.e^-rt
1/2 = e^-(20r day)
2 = e^(20r day)
20 r day = log 2
r = (ln 2)/20 per day
0.01 = e^-rt
100 = e^rt
ln 100 = rt
t = (ln 100) / ((ln 2)/20 per day)
        = 20(ln 100)/(ln 2) days
        = 20(log_2 100) days
===
Subject: Re: Radioactive decay
If I'm not mistaken, here is the gist of the solution:
Find the const (k) of decay...Take logs of both side:
k=k/.Solve[1/2==E^(-20*k),k]
Log[2]/20
or...
N[k]
0.03465735902799726
Plug back k into the equation and solve for t (time)
t=t/.Solve[1/100==E^((-t)*k),t]
{(20*Log[100])/Log[2]}
N[t]
{132.87712379549453}
I think the answer is about 132 days.
-- 
Dana
>I am not sure how to set up this equation in order to solve this
> problem. Can someone help me or guide me in the direction that will
> Question: If the half-life of a radioactive substance is 20 days, how
> long will it take for 99 percent of the substance to decay?
===
Subject: Re: Radioactive decay
>I am not sure how to set up this equation in order to solve this
>problem. Can someone help me or guide me in the direction that will
>Question: If the half-life of a radioactive substance is 20 days, how
>long will it take for 99 percent of the substance to decay?
Solve (1/2)^x = 0.01, or 2^x = 100. Then t (in days) = 20x.
Thomas
===
Subject: integer sums of unit fractions
        3QLpj-NoP*NzsIC,boYU]bQ]H'<P%JD/,.J>y<#4ga3$21:
I'm interested in ways of partitioning an integer n>1 into a sum of a 
finite sequence of unit fractions, in such a way that no nontrivial 
subsequence has an integer sum.
For example, 2 = 1/2 + 1/3 + 1/3 + 1/4 + 1/7 + 1/7 + 1/7 + 1/7 + 1/84
but no subsequence of these fractions adds to one.
Is it possible to find examples with larger values of n?
Or with no two fractions equal to each other?
-- 
David Eppstein
Computer Science Dept., Univ. of California, Irvine
http://www.ics.uci.edu/~eppstein/
===
Subject: Re: integer sums of unit fractions
> Or with no two fractions equal to each other?
David:
The sum = 2 for these fractions:
1/2 1/3 1/4 1/5 1/7 1/9 1/10 1/11 1/13 1/14 1/24 1/26 1/55 1/88 1/99 1/273
1/2 1/3 1/4 1/5 1/7 1/9 1/10 1/11 1/14 1/16 1/21 1/24 1/55 1/88 1/99 1/112
1/2 1/3 1/4 1/5 1/7 1/9 1/10 1/11 1/13 1/14 1/24 1/33 1/44 1/55 1/99 1/2184
1/2 1/3 1/4 1/5 1/7 1/9 1/10 1/11 1/13 1/14 1/26 1/33 1/44 1/55 1/99 1/273
1/2 1/3 1/4 1/5 1/7 1/9 1/10 1/11 1/14 1/16 1/21 1/33 1/44 1/55 1/99 1/112
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/19 1/33 1/44 1/55 1/78 1/99 1/41496
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/19 1/33 1/44 1/55 1/84 1/99 1/1064
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/21 1/33 1/44 1/55 1/57 1/99 1/3192
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/21 1/33 1/44 1/55 1/58 1/99 1/1624
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/21 1/33 1/44 1/55 1/63 1/99 1/504
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/21 1/33 1/44 1/55 1/64 1/99 1/448
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/21 1/33 1/44 1/55 1/70 1/99 1/280
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/21 1/33 1/44 1/55 1/84 1/99 1/168
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/23 1/33 1/44 1/46 1/55 1/99 1/3864
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/26 1/33 1/39 1/44 1/55 1/99 1/728
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/28 1/33 1/34 1/44 1/55 1/99 1/2856
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/28 1/33 1/35 1/44 1/55 1/99 1/840
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/28 1/33 1/44 1/48 1/55 1/99 1/112
1/2 1/3 1/4 1/5 1/7 1/8 1/9 1/10 1/11 1/28 1/33 1/44 1/55 1/56 1/84 1/99
The sum of 3 appears to be much harder.  I found that a minimum of 12
terms was necessary for 2 in my search range.
Randall
===
Subject: Re: integer sums of unit fractions
> I'm interested in ways of partitioning an integer n>1 into a sum of a 
> finite sequence of unit fractions, in such a way that no nontrivial 
> subsequence has an integer sum.
> For example, 2 = 1/2 + 1/3 + 1/3 + 1/4 + 1/7 + 1/7 + 1/7 + 1/7 + 1/84
> but no subsequence of these fractions adds to one.
> Is it possible to find examples with larger values of n?
> Or with no two fractions equal to each other?
What happens when you try the greedy algorithm? e.g., 
3 = 1/2 + 1/3 + 1/3 + 1/4 + 1/5 + 1/5 + 1/5 + 1/5 + a few 1/7 
+ whatever it takes to get to 3? 
Similarly for the question with no repeated fractions.
-- 
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: integer sums of unit fractions
>> I'm interested in ways of partitioning an integer n>1 into a sum of a 
>> finite sequence of unit fractions, in such a way that no nontrivial 
>> subsequence has an integer sum.
 
>> For example, 2 = 1/2 + 1/3 + 1/3 + 1/4 + 1/7 + 1/7 + 1/7 + 1/7 + 1/84
>> but no subsequence of these fractions adds to one.
 
>> Is it possible to find examples with larger values of n?
>> Or with no two fractions equal to each other?
>What happens when you try the greedy algorithm? e.g., 
>3 = 1/2 + 1/3 + 1/3 + 1/4 + 1/5 + 1/5 + 1/5 + 1/5 + a few 1/7 
>+ whatever it takes to get to 3? 
3=1/2+1/3+1/3+1/4+1/5+1/5+1/5+1/5+1/7+1/7+1/7+1/7+1/7+1/16+1/153+1/85680
4=1/2+1/3+1/3+1/4+1/5+1/5+1/5+1/5+1/7+1/7+1/7+1/7+1/7+1/7+1/8+1/9+1/9
  +1/11+1/11+1/11+1/11+1/11+1/11+1/31+1/797+1/1065130+1/1326334812264
>Similarly for the question with no repeated fractions.
2=1/2+1/3+1/4+1/5+1/7+1/8+1/9+1/10+1/11+1/12+1/16+1/1047+1/1138151
   +1/3145940416080
   
It gets hard for larger integers (at least the way I tried to implement   
it), as you have a lot of subset sums to look at.
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel
University of British Columbia            Vancouver, BC, Canada
===
Subject: Re: integer sums of unit fractions
        3QLpj-NoP*NzsIC,boYU]bQ]H'<P%JD/,.J>y<#4ga3$21:
unit fractions sums to at most n, then one can partition the sequence 
into n+1 subsequences each of which sums to at most one.  [Bar-Noy, 
Ladner, and Tamir, Windows scheduling as restricted version of bin 
packing, SODA'04 pp. 217-226.]  These examples show that this can't 
always be improved to a partition into n subsequences.  Apparently the 
computational complexity of distinguishing sequences that require n+1 
subsequences from those that can be partitioned into n sequences is open.
-- 
David Eppstein
Computer Science Dept., Univ. of California, Irvine
http://www.ics.uci.edu/~eppstein/
===
Subject: Re: integer sums of unit fractions
        3QLpj-NoP*NzsIC,boYU]bQ]H'<P%JD/,.J>y<#4ga3$21:
> I'm interested in ways of partitioning an integer n>1 into a sum of a 
> finite sequence of unit fractions, in such a way that no nontrivial 
> subsequence has an integer sum.
> 
> For example, 2 = 1/2 + 1/3 + 1/3 + 1/4 + 1/7 + 1/7 + 1/7 + 1/7 + 1/84
> but no subsequence of these fractions adds to one.
> 
> Is it possible to find examples with larger values of n?
> Or with no two fractions equal to each other?
> What happens when you try the greedy algorithm? e.g., 
> 3 = 1/2 + 1/3 + 1/3 + 1/4 + 1/5 + 1/5 + 1/5 + 1/5 + a few 1/7 
> + whatever it takes to get to 3? 
Greedy meaning that it takes the largest fraction for which the total is 
at most n and that does not create a subsequence with integer sum?  Even 
much simpler greedy Egyptian fraction algorithms are hard to analyze.
-- 
David Eppstein
Computer Science Dept., Univ. of California, Irvine
http://www.ics.uci.edu/~eppstein/
===
Subject: Re: integer sums of unit fractions
> I'm interested in ways of partitioning an integer n>1 into a sum of a 
> finite sequence of unit fractions, in such a way that no nontrivial 
> subsequence has an integer sum.
> 
> For example, 2 = 1/2 + 1/3 + 1/3 + 1/4 + 1/7 + 1/7 + 1/7 + 1/7 + 1/84
> but no subsequence of these fractions adds to one.
> 
> Is it possible to find examples with larger values of n?
> Or with no two fractions equal to each other?
> 
> What happens when you try the greedy algorithm? e.g., 
> 3 = 1/2 + 1/3 + 1/3 + 1/4 + 1/5 + 1/5 + 1/5 + 1/5 + a few 1/7 
> + whatever it takes to get to 3? 
> Greedy meaning that it takes the largest fraction for which the total is 
> at most n and that does not create a subsequence with integer sum?  
Yes. 
> Even much simpler greedy Egyptian fraction algorithms are hard to 
analyze.
OK, but from your original question, you don't even know whether 
it can be done for n = 3; is it really that hard to analyze my 
suggestion in that case? 
I grant that my suggestion is unlikely to be of much use for large n.
-- 
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: integer sums of unit fractions
        3QLpj-NoP*NzsIC,boYU]bQ]H'<P%JD/,.J>y<#4ga3$21:
> OK, but from your original question, you don't even know whether 
> it can be done for n = 3; is it really that hard to analyze my 
> suggestion in that case? 
I found the n=2 example I posted by hand.  I'd be tempted to implement a 
greedy algorithm to see how well it works for larger n but it seems 
Robert Israel and Randall have already done so.
-- 
David Eppstein
Computer Science Dept., Univ. of California, Irvine
http://www.ics.uci.edu/~eppstein/
===
Subject: random model
Hallo,
I have a bit of an awkward question for you. I'm toying around with a GA 
and
I would like to test it by applying several models. Unfourtunately the
search is rather fast and I have to enter each time a new model by hand. So
I was thinking if there are algorithms present that could generate a
continuous model automatically, so I can feed it to the GA?
The model should incorporate all the standard operators and allow for an
undefined amount of parameters. Can this be done?
I would appreciate any links or pointers on how to procede from here.
===
Subject: Re: random model
> Hallo,
> I have a bit of an awkward question for you. I'm toying around with a GA 
> and
> I would like to test it by applying several models. Unfourtunately the
> search is rather fast and I have to enter each time a new model by hand. 
> So
> I was thinking if there are algorithms present that could generate a
> continuous model automatically, so I can feed it to the GA?
> The model should incorporate all the standard operators and allow for an
> undefined amount of parameters. Can this be done?
> I would appreciate any links or pointers on how to procede from here.
It's not clear, at least to me, what you are modeling. However, if I'm 
guessing correctly, you're asking about an encoding, a model, over which the 

genetic algorithm works. For instance, one model of human populations would 

include heights, weights, and skin color; you'd use the GA to come up with 
the best combination of those characteristics. Another model might include 
heights, weights, skin color, and waist measurement. Then again, I'm not 
sure how standard operators fit into this, because the operators I'm 
thinking of are inherent in the GA itself, whatever you're modeling; e.g. 
the mutation operator is often a part of the GA itself, not the thing you're 

modeling.
Could you be more specific about your problem, perhaps giving an example of 

the models you're using? 
lppl