mm-1969
===
Subject: Re: final proof Einstein 1905 is crap.
>> Let's face it, it is beyond the intellectual capacity of moortel to
>> understand
>> how the sheer genius of Einstein hoodwinked the community for 100
>> years.
>> Recall that the light leaves A and reflects at B in
>> 2AB/(t'A-tA) = c to be a universal constant- the velocity of light
in
>> empty
>> space.
It's not just this which is a crap. The whole theory theory of relativity
is. Both its two postules and Lorentz transformation have been disputed by
reports from the Hubble. For example, the claim that the velocity of light
is
the upper limit of velocity is false. The galaxies are speeding outwards at
the rate of 10^20 km/sec. The basis of the postulate of relativity, that the
Cosmos is essentially empty
E. E. Escultura
===
Subject: Re: JSH: Simple proof
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id iB1DHxY29540;
Why don't you simply and briefly state the >>problematic<< property
of the ring of algebraic integers and/or the statement that you
want to prove.
Using standard mathematical terms and notions (for example from
commutative algebra) this should be possible in a few lines instead
of making a long story.
Why do we have to discuss things like >>what is a polynomial?<<
here? The notion of a polynomial is defined since a long time and
can be found in every introductory book on algebra.
If we consistently use the common definitions of mathematical
objects like polynomials we should rather quickly be able to
clarify the situation and avoid all the frustration that frequently
seems to culminate in personal attacks.
H
===
Subject: Putting FLT to Rest
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id iB1DI0q29566;
PUTTING FERMAT.89S LAST THEOREM TO REST
Now that counterexamples to FLT have been constructed it is appropriate to
put this false conjecture in perspective. Andrew Wiles made the most earnest
attempt to prove it. His strategy was quite simple: consider elliptic curves
on the surface, the latter supposedly defined by Fermat.89s equation for
n > 2, namely, x^n + y^n = z^n, and prove that they contain no point with
integer coordinates. A problem arises: curves and surfaces are topics in
analysis. This is minor, however, since Wiles could have devised suitable
analytical tools. He didn.89t. Instead, as number theorist, he resorted
to classical algebraic tools such as Galois theory and Hecke rings which are
vintage 19th Century. They are inadequate and uninformed by recent
developments. The more serious problem, however, as L. C. Young pointed out
in the thirties, is: all classical theories of curves and surfaces including
elliptic curves are flawed because they are not well-defined by their
functional representations. A c!
urve is well-defined by a pair of functions .9a its functional
representation and derivative. He developed this idea into the theory of
generalized curves in a series of papers from 1933 to 1937. Wiles was
unaware
of this development, a fatal error of omission that shut down his
ëproof..89 Moreover, elliptic curves were already known. He
merely tried to use them but without success. Therefore, Wiles did not open
up any area and I am not aware of any major paper that sprung up from his
work. For the benefit of the reader, we may define an elliptic curve as the
intersection of a surface and a plane and, for Wiles, a suitable family of
elliptic curves lie on planes perpendicular to and passing through integral
values on the z-axis. Every point on an elliptic curve satisfies both the
equation of the surface and the plane. It would have sufficed to prove that
no point on this curve has integral x- and y-coordinates and that this is so
for the other elliptic curves on Fermat.89s surface!
. However, this scheme failed because Wiles.89 tools lacked validity.
Like a curve, a surface is not well-defined by its functional representation
alone. Young fixed this error by well-defining a surface not only by a
system
of partial differential equations with suitable conditions along its
boundary
but also by associating with every element of area on the surface a directed
line segment perpendicular to it. This is called the jacobian. Young
developed this idea into the theory of generalized surfaces in a series of
papers from 1938 to 1954. Again, Wiles was not aware of it .9a
another fatal error since one cannot have a valid proof unless it is
justified by a valid theory. The most fundamental error Wiles committed,
however, is: he tried to tackle FLT without assessing the status of the
underlying fields .9a foundations, number theory and analysis.
Present mathematical reasoning is flawed. Number theory has no valid
axiomatization; therefore, no valid proof exists. Two of the a!
xioms of the real number system .9a the completeness and dichotomy
axioms .9a are false. Therefore, the real number system, the base
space of analysis, is ill-defined. Consequently, FLT as posed by Wiles in
the
context of curves and surfaces, which belong to analysis, is ill-defined.
The
remedy is the reconstruction of the real number system without these axioms
as well as upgrading of foundations and number theory that I undertook in
1998 (Nonlinear Studies, Vol. 5). The reconstruction upgrades number theory
by embedding the integers in the new real number system. After discarding
the
nonsense of the real number system the new real number system becomes finite
but unbounded, free from contradictions, enriched by the dark number d* and
unbounded number u* and has natural ordering, namely, the lexicographic
ordering. Wiles was also unaware of recent results especially my discovery
of
ambiguous sets (Nonlinear Analysis, Vol. 35, 2001) such as the set of points
in an elliptic curve!
. Scientific standards require that when an error is committed it is not
sufficient to present a correct alternative. The error must be analyzed and
criticized, which I did in a series of papers from 1996 to the present, and
the one who committed it must either refute the criticism and defend his
work
or accept it by default. Wiles did the latter. Finally, new mathematics
sprung up from my resolution of FLT among which are: the new real number
system, new nonstandard calculus, new arithmetic, dynamic modeling and
characterization of undecidable propositions, published in renowned
journals.
(For the interested reader visit the websites:
http://home.iprimus.com.au/pidro/ and
http://www.users.bigpond.com/pidro/home.htm)
===
Subject: Re: Putting FLT to Rest
Wow... You must be so bloody stupid...
===
Subject: Re: Putting FLT to Rest
So give us ONE counter example!
>PUTTING FERMAT.89S LAST THEOREM TO REST
>Now that counterexamples to FLT have been constructed it is appropriate to
put this false conjecture in perspective. Andrew Wiles made the most earnest
attempt to prove it. His strategy was quite simple: consider elliptic curves
on the surface, the latter supposedly defined by Fermat.89s equation for
n > 2, namely, x^n + y^n = z^n, and prove that they contain no point with
integer coordinates. A problem arises: curves and surfaces are topics in
analysis. This is minor, however, since Wiles could have devised suitable
analytical tools. He didn.89t. Instead, as number theorist, he resorted
to classical algebraic tools such as Galois theory and Hecke rings which are
vintage 19th Century. They are inadequate and uninformed by recent
developments. The more serious problem, however, as L. C. Young pointed out
in the thirties, is: all classical theories of curves and surfaces including
elliptic curves are flawed because they are not well-defined by their
functional representations. A c!
>urve is well-defined by a pair of functions .9a its functional
representation and derivative. He developed this idea into the theory of
generalized curves in a series of papers from 1933 to 1937. Wiles was
unaware
of this development, a fatal error of omission that shut down his
ëproof..89 Moreover, elliptic curves were already known. He
merely tried to use them but without success. Therefore, Wiles did not open
up any area and I am not aware of any major paper that sprung up from his
work. For the benefit of the reader, we may define an elliptic curve as the
intersection of a surface and a plane and, for Wiles, a suitable family of
elliptic curves lie on planes perpendicular to and passing through integral
values on the z-axis. Every point on an elliptic curve satisfies both the
equation of the surface and the plane. It would have sufficed to prove that
no point on this curve has integral x- and y-coordinates and that this is so
for the other elliptic curves on Fermat.89s surface!
>. However, this scheme failed because Wiles.89 tools lacked validity.
Like a curve, a surface is not well-defined by its functional representation
alone. Young fixed this error by well-defining a surface not only by a
system
of partial differential equations with suitable conditions along its
boundary
but also by associating with every element of area on the surface a directed
line segment perpendicular to it. This is called the jacobian. Young
developed this idea into the theory of generalized surfaces in a series of
papers from 1938 to 1954. Again, Wiles was not aware of it .9a
another fatal error since one cannot have a valid proof unless it is
justified by a valid theory. The most fundamental error Wiles committed,
however, is: he tried to tackle FLT without assessing the status of the
underlying fields .9a foundations, number theory and analysis.
Present mathematical reasoning is flawed. Number theory has no valid
axiomatization; therefore, no valid proof exists. Two of the a!
>xioms of the real number system .9a the completeness and
dichotomy axioms .9a are false. Therefore, the real number system,
the base space of analysis, is ill-defined. Consequently, FLT as posed by
Wiles in the context of curves and surfaces, which belong to analysis, is
ill-defined. The remedy is the reconstruction of the real number system
without these axioms as well as upgrading of foundations and number theory
that I undertook in 1998 (Nonlinear Studies, Vol. 5). The reconstruction
upgrades number theory by embedding the integers in the new real number
system. After discarding the nonsense of the real number system the new real
number system becomes finite but unbounded, free from contradictions,
enriched by the dark number d* and unbounded number u* and has natural
ordering, namely, the lexicographic ordering. Wiles was also unaware of
recent results especially my discovery of ambiguous sets (Nonlinear
Analysis,
Vol. 35, 2001) such as the set of points in an elliptic curve!
>. Scientific standards require that when an error is committed it is not
sufficient to present a correct alternative. The error must be analyzed and
criticized, which I did in a series of papers from 1996 to the present, and
the one who committed it must either refute the criticism and defend his
work
or accept it by default. Wiles did the latter. Finally, new mathematics
sprung up from my resolution of FLT among which are: the new real number
system, new nonstandard calculus, new arithmetic, dynamic modeling and
characterization of undecidable propositions, published in renowned
journals.
(For the interested reader visit the websites:
http://home.iprimus.com.au/pidro/ and
http://www.users.bigpond.com/pidro/home.htm)
===
Subject: Re: Cantor's diagonal proof wrong?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id iB1DHw529447;
>In sci.math, Shmuel (Seymour J.) Metz
>But it is still curious that it happens at 2^oo and not oo^oo.
>> Why? Is it curious that 3*3 is smaller than 3^3, or that 16*16 is
>> smaller than 2^16?
>I suspect that, if a set S has cardinality of at least card(N),
>then a bijective mapping can be found between the elements of 2^S
>and the elements of n^S, where n > 1 is an integer.
>However, I'd have to look.
Sure, that's correct; in fact, n can have cardinality anywhere
from 2 to 2^S. There's an injection 2^S --> n^S, and there's
an injection n --> 2^S (and so an injection
2^S --> (2^S)^S ~ 2^S);
now apply Schroeder-Bernstein.
Todd Trimble
===
Subject: Re: Counterexamples to FLT
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id iB1DHxd29519;
>congratulation.
> I'm using your proof in a short course,
>Implausibly Indeniable Proofs of a Really Old Problem. the assumption
>will
>be that these 4 or 5 or 6 elementary proofs taht I have found,
>mostly in the literature, are incorrect,
>either in execution or in concept (not XOR .-)
> that's all of the examples that I'm doing,
>thus far...
>second congratulation.
> I'm using the original formatting.
>> I am pleased to summarize the resolution of some issues in
>mathematics ...
>--Advice, 0.05; free, if wrong!
>http://tarpley.net/bush5.htm
E. E. Escultura
===
Subject: Vector invariants
I have 3 vectors A,B,C in the plane (with some origin O)
and like to construct invariants which are symmetric
(or antisymmetric) in A,B,C and independent of O,
by piling up cross and scalar products.
(Example: [AxB]*C. Not very helpful :-)
Same with 4 vectors. Any suggestions?
--
Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de
His-Ala-Sec-Lys-Glu Arg-Glu-Asp-Asp-Met-Ala-Asn-Asn
===
Subject: Re: Uncountable many reals without Cantor
> there are often threads in this group concerning
> the cardinality of the set of real numbers. Some
> persons seem to have strong objections against the
> Cantor Proof of the fact that the set of real
> numbers is not denumerable by the naturals.
> Cantor's Proof uses diagonalization. But there is
> a mesaure theoretic argument for the uncountability
> of the reals due to Borel which does not use this
> technique.
There are many alternative proofs, all of which presuppose the
consistency and truth of the definitions, in particular the
abstraction of actual infinity is present in any proof. Borel's
proof or any of the covering arguments is exactly that, showing that
the measure of computable reals is 0, based on the actual infinite
divisibility of the continuum. (Technically, that sounds such a nice
and acceptable thing, but *philosophically* I don't accept that the
extent of anything should be 0 if it exists. I will not argue that, I
don't believe that there is anybody here to argue over that point.)
So, I will say that none of this formalism solves the problems
pertinent in this implausible theory of the transfinite. The problems
are in the premises and concepts, not in the proofs per se.
But yes, we can fully expect several nonsensical consequences of such
faulty premises, which is indeed the case for continuum.
--
Eray Ozkural
===
Subject: Re: Uncountable many reals without Cantor
>there are often threads in this group concerning
>the cardinality of the set of real numbers. Some
>persons seem to have strong objections against the
>Cantor Proof of the fact that the set of real
>numbers is not denumerable by the naturals.
> People may have strong objections, but nobody
> has any _coherent_ objections - the people who
> object seem to be unable to follow very simple
> reasoning. Hence I doubt that they're going to
> be able to follow complicated chains of reasoning...
The point you are missing that it does not require any complicated
chains of reasoning. Cantor's first proof is perfectly adequate. Why
was a second proof needed at all? Did you ever think about that?
I think it was needed to hide the fact that the conclusion trivially
follows from the then accepted or refined definitions of Z and R.
There is no significant reasoning involved. It directly follows from
the basic concepts involved, and that's why the first proof can be
stated in a sentence or two, without resorting to any detailed
argument.
I hope you find this an interesting twist to the debate.
It's misled to think of it as a technical matter. It's a purely
philosophical issue. The diagonal argument is a short, and easy
argument. It's easier than 90% of the proofs I've studied. However,
it's just complex enough to make an appearance on the scene, unlike
the first argument.
It's a very strong claim to say that the people like Dr. Zenkin who
object to the diagonal proof are too stupid to follow such a simple
proof. Their objections are usually centered on the concept of actual
infinity sinisterly leaking in the proof, and they say, well this is
correct only if you accept the relevance of actual infinity in the
first place. They don't say that the proof is prima facie wrong. They
say the *concepts* are wrong.
That's an altogether different thing, which I think you can
understand.
--
Eray Ozkural
===
Subject: Re: Uncountable many reals without Cantor
> But seriously,
> the intuitionists and serious constructivists deny the uncountability
> of the reals
> No they don't.
See the replies, I think only the ones who are silly enough to admit
Platonism back in would admit the continuum
I had a talk with a so-called constructivist. He was repeating the
naive claims of Robin Chapman and Stephen Harris, that the TM is not a
physical model of computation, e.g. they don't exist in the world. He
said PCs are not TMs!!!!! I told him that a constructivist would
consider TMs to be physical models.
The poor chap was offended. A guy from Netherlands, he considered
himself to be a constructivist. He probably thought himself to be a
follower of Brouwer.
Now, if Brouwer was present in the discussion, he could teach a lesson
or two about constructivism to this chap. All that actual-infinity
tape talk is Platonist nonsense, and you would probably think that all
constructivists should be as naive as this poor chap.
It's clear that you know nothing about constructivism or why the
debate of the actual infinity is so central to these matters. As a
Platonist, you want to turn constructivism into the naive ideals of
Platonism.
But it is not. I consider true constructivism to reject actual
infinity talk at a fundamental level.
No candy for you,
--
Eray
===
Subject: Re: Uncountable many reals without Cantor
>> But seriously,
>> the intuitionists and serious constructivists deny the uncountability
>> of the reals
>> No they don't.
> See the replies, I think only the ones who are silly enough to admit
> Platonism back in would admit the continuum
> I had a talk with a so-called constructivist. He was repeating the
> naive claims of Robin Chapman and Stephen Harris, that the TM is not a
> physical model of computation, e.g. they don't exist in the world.
More dopey abuse from the egregious Ozkural. I certainly don't
equate physical model with existence in the world.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Uncountable many reals without Cantor
Discussion, linux)
> I had a talk with a so-called constructivist. He was repeating the
> naive claims of Robin Chapman and Stephen Harris, that the TM is not a
> physical model of computation, e.g. they don't exist in the world. He
> said PCs are not TMs!!!!! I told him that a constructivist would
> consider TMs to be physical models.
Of course, this is utterly stupid.
> The poor chap was offended.
Probably amused or slightly frightened.
--
Jesse F. Hughes
I'm not sure whether I'm not thinking clearly or clearly not
thinking. -- Ling Cheung, said on some day other than our wedding
day. Honest.
===
Subject: Re: Uncountable many reals without Cantor
> See the replies, I think only the ones who are silly enough to admit
> Platonism back in would admit the continuum
Like Brouwer?
===
Subject: Re: Uncountable many reals without Cantor
> See the replies, I think only the ones who are silly enough to admit
> Platonism back in would admit the continuum
> Like Brouwer?
You say that in Brouwer and other forthcoming constructivists, there
is actually abstraction of actual continuum. If that's right, then my
above statement is wrong-headed. But I don't think that's the case. If
you do admit the actual continuum, then you don't need any of the
detailed reworking of analysis that Bishop carried out.
I don't think you even understand what I refer to by admitting the
continuum.
--
Eray Ozkural
===
Subject: Re: Uncountable many reals without Cantor
> I don't think you even understand what I refer to by admitting the
> continuum.
The question is rather whether you understand what Brouwer is
saying.
===
Subject: Re: Uncountable many reals without Cantor
Discussion, linux)
>> See the replies, I think only the ones who are silly enough to admit
>> Platonism back in would admit the continuum
>> Like Brouwer?
> You say that in Brouwer and other forthcoming constructivists, there
> is actually abstraction of actual continuum. If that's right, then my
> above statement is wrong-headed. But I don't think that's the case. If
> you do admit the actual continuum, then you don't need any of the
> detailed reworking of analysis that Bishop carried out.
> I don't think you even understand what I refer to by admitting the
> continuum.
Now *that's* a strategy. Spout meaningless nonsense and when someone
responds as if it had meaning, call him on it.
--
It seems to me that some of you don't realize that [...] some day the
truth comes out. It doesn't matter if you're dead. I've made certain
that your name will live in infamy if it's known at all: Wiles, Ribet,
Granville, or anyone else from this generation. --JSH beyond the grave
===
Subject: Re: Uncountable many reals without Cantor
>>there are often threads in this group concerning
>>the cardinality of the set of real numbers. Some
>>persons seem to have strong objections against the
>>Cantor Proof of the fact that the set of real
>>numbers is not denumerable by the naturals.
>People may have strong objections, but nobody
>has any _coherent_ objections - the people who
>object seem to be unable to follow very simple
>reasoning. Hence I doubt that they're going to
>be able to follow complicated chains of reasoning...
>>Cantor's Proof uses diagonalization. But there is
>>a mesaure theoretic argument for the uncountability
>>of the reals due to Borel which does not use this
>>technique.
>>Let (a_i), i e {1,2,3,...} be a list of the reals in
>>the interval [0,1]. Let eps be any rational number
> 0.
>>Now consider a_1 in an interval of length eps/2, ...,
>>a_i in an interval of length eps/2^i. Since every
>>element of [0,1] is in some of the intervals, we
>>have
>>length([0,1]) <= eps/2 + eps/4 + ... + eps/2^i + ... = eps
>>for every rational eps > 0. A contradiction.
>I can imagine one of the objectors mentioned above
>_agreeing_ that this argument is right, because it's
>based on more familiar concepts. But I think the idea
>that it's actually simpler is bogus - if someone
>agrees to this but not to the diagonal argument I
>really don't think that he's understood all the details.
>This argument _is_ much more complicated, if you include
>the missing details. In particular you need a _proof_ of
>the intuitively reasonable fact that if [0,1] is contained
>in the union of countably many intervals I_n then
>(*) sum length(I_n) >= 1.
>How do you _prove_ that?
Assume the opposite, put the intervals end to end etc. This kind of
thing is proven in the beginning of any Real Variables text, e.g.
Royden. Where do you see a problem?
>(It really does require proof, you know. A _clever_
>objector to all this would point out that [0,1]
>is also the union of the closed intervals [x], for
>x in [0,1]. Note that the sum of the lengths of [x]
>for x in [0,1] is 0.
>Of course the reason this is not a contradiction
>is that (*) is not valid for uncountable unions.
>But (i) this shows at least that (*) for countable
>unions does require proof, and (ii) if we were
>insisting that there's no such thing as an uncountable
>set, maybe because we forgot to take our pills, then
>the explanation that (*) doesn't hold for uncountable
>unions doesn't work, and we conclude from this
>example that (*) is simply wrong! Suppose you do
>give a proof of (*) for countable unions - anyone
>stupid enough to be able to find flaws with the
>diagonal argument is going to have no problem
>finding flaws with that proof.
>Of course if our goal is to elicit _agreement_ instead
>of _understanding_ then the argument above is a good
>idea, because the objectors are going to be too dense
>to see the objections. But if the idea is actually
>to get someone to believe the reals are uncountable
>_for_ a valid _reason_ then the diagonal argument seems
>much better.)
>************************
>David C. Ullrich
===
Subject: Re: Uncountable many reals without Cantor
>>This argument _is_ much more complicated, if you include
>>the missing details. In particular you need a _proof_ of
>>the intuitively reasonable fact that if [0,1] is contained
>>in the union of countably many intervals I_n then
>>(*) sum length(I_n) >= 1.
>>How do you _prove_ that?
> Assume the opposite, put the intervals end to end etc. This kind of
> thing is proven in the beginning of any Real Variables text, e.g.
> Royden. Where do you see a problem?
Sounds like you want to use induction on the number of intervals.
Problem is, induction works only if the number of intervals is finite.
Hint: that's where compactness comes in.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: Uncountable many reals without Cantor
Dave Seaman says...
>(*) sum length(I_n) >= 1.
>How do you _prove_ that?
>> Assume the opposite, put the intervals end to end etc. This kind of
>> thing is proven in the beginning of any Real Variables text, e.g.
>> Royden. Where do you see a problem?
>Sounds like you want to use induction on the number of intervals.
>Problem is, induction works only if the number of intervals is finite.
>Hint: that's where compactness comes in.
Just for clarification of this comment.
Compactness for a topological space means the following: X is compact
if for any collection of open sets U_i whose union is X, there is a
finite subcollection whose union is also X. So a closed interval of
finite length is compact, but an open interval is not.
I was confused at first because I was misremembering the definition
of compactness. I remembered a definition along the lines of X is
compact if every Cauchy sequence converges. That doesn't directly
help much.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Uncountable many reals without Cantor
> Dave Seaman says...
>>(*) sum length(I_n) >= 1.
>>How do you _prove_ that?
> Assume the opposite, put the intervals end to end etc. This kind of
> thing is proven in the beginning of any Real Variables text, e.g.
> Royden. Where do you see a problem?
>>Sounds like you want to use induction on the number of intervals.
>>Problem is, induction works only if the number of intervals is finite.
>>Hint: that's where compactness comes in.
> Just for clarification of this comment.
> Compactness for a topological space means the following: X is compact
> if for any collection of open sets U_i whose union is X, there is a
> finite subcollection whose union is also X. So a closed interval of
> finite length is compact, but an open interval is not.
> I was confused at first because I was misremembering the definition
> of compactness. I remembered a definition along the lines of X is
> compact if every Cauchy sequence converges. That doesn't directly
> help much.
That property is called sequential compactness, and you're right that
it doesn't particularly help. The key here is the Heine-Borel theorem,
which says that closed, bounded intervals in R^n are compact.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: Uncountable many reals without Cantor
>> I was confused at first because I was misremembering the definition
>> of compactness. I remembered a definition along the lines of X is
>> compact if every Cauchy sequence converges. That doesn't directly
>> help much.
>That property is called sequential compactness, and you're right that
>it doesn't particularly help. The key here is the Heine-Borel theorem,
>which says that closed, bounded intervals in R^n are compact.
I don't know whether you were just being unclear but every Cauchy
sequence converges is certainly not the definition of a compact set,
it is the definition of a complete metric space.
The definition of sequential compactness is that every sequence in the
set has a subsequence that converges in the set.
--
I'm not interested in mathematics that might have anything
to do with reality. -- Russell Easterly, in sci.math
===
Subject: Re: Uncountable many reals without Cantor
Toni Lassila says...
>I don't know whether you were just being unclear but every Cauchy
>sequence converges is certainly not the definition of a compact set,
>it is the definition of a complete metric space.
Right. But there is a connection: every compact metric space is complete.
>The definition of sequential compactness is that every sequence in the
>set has a subsequence that converges in the set.
Right.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Uncountable many reals without Cantor
> I was confused at first because I was misremembering the definition
> of compactness. I remembered a definition along the lines of X is
> compact if every Cauchy sequence converges. That doesn't directly
> help much.
>>That property is called sequential compactness, and you're right that
>>it doesn't particularly help. The key here is the Heine-Borel theorem,
>>which says that closed, bounded intervals in R^n are compact.
> I don't know whether you were just being unclear but every Cauchy
> sequence converges is certainly not the definition of a compact set,
> it is the definition of a complete metric space.
> The definition of sequential compactness is that every sequence in the
> set has a subsequence that converges in the set.
Yes, sorry. I was being sloppy. In a metric space, compactness is
equivalent to sequential compactness, and both hold iff the space is
complete and totally bounded. But compactness is the property that's
needed in the measure theory proof, one way or another.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: Uncountable many reals without Cantor
>This argument _is_ much more complicated, if you include
>the missing details. In particular you need a _proof_ of
>the intuitively reasonable fact that if [0,1] is contained
>in the union of countably many intervals I_n then
>(*) sum length(I_n) >= 1.
>How do you _prove_ that?
>> Assume the opposite, put the intervals end to end etc. This kind of
>> thing is proven in the beginning of any Real Variables text, e.g.
>> Royden. Where do you see a problem?
>Sounds like you want to use induction on the number of intervals.
>Problem is, induction works only if the number of intervals is finite.
>Hint: that's where compactness comes in.
Amusing technicality:
In various places for various reasons we need to talk about
coverings by half-open intervals instead of open intervals.
Say [0,1) is the union of disjoint intervals [a_n, b_n).
Then sum(b_n - a_n) = 1. You can't prove that (directly)
by compactness, but you _can_ prove it by a simple
transfinite induction!
************************
David C. Ullrich
===
Subject: Re: Uncountable many reals without Cantor
>>This argument _is_ much more complicated, if you include
>>the missing details. In particular you need a _proof_ of
>>the intuitively reasonable fact that if [0,1] is contained
>>in the union of countably many intervals I_n then
>>(*) sum length(I_n) >= 1.
>>How do you _prove_ that?
> Assume the opposite, put the intervals end to end etc. This kind of
> thing is proven in the beginning of any Real Variables text, e.g.
> Royden. Where do you see a problem?
>>Sounds like you want to use induction on the number of intervals.
>>Problem is, induction works only if the number of intervals is finite.
>>Hint: that's where compactness comes in.
> Amusing technicality:
> In various places for various reasons we need to talk about
> coverings by half-open intervals instead of open intervals.
> Say [0,1) is the union of disjoint intervals [a_n, b_n).
> Then sum(b_n - a_n) = 1. You can't prove that (directly)
> by compactness, but you _can_ prove it by a simple
> transfinite induction!
That's not quite the approach I had in mind. I'm sure you are aware of
this argument, but I wonder why we need to talk about half-open
intervals to show that m([0,1]) = 1. What I meant was to show the outer
measure of [0,1] is >= 1 by considering open covers, extracting finite
subcovers, and using ordinary induction over the number of intervals in
the subcover. Then you can show the outer measure is less than 1+epsilon
for any epsilon > 0.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: Uncountable many reals without Cantor
>This argument _is_ much more complicated, if you include
>the missing details. In particular you need a _proof_ of
>the intuitively reasonable fact that if [0,1] is contained
>in the union of countably many intervals I_n then
(*) sum length(I_n) >= 1.
How do you _prove_ that?
>> Assume the opposite, put the intervals end to end etc. This kind of
>> thing is proven in the beginning of any Real Variables text, e.g.
>> Royden. Where do you see a problem?
>Sounds like you want to use induction on the number of intervals.
>Problem is, induction works only if the number of intervals is finite.
>Hint: that's where compactness comes in.
>> Amusing technicality:
>> In various places for various reasons we need to talk about
>> coverings by half-open intervals instead of open intervals.
>> Say [0,1) is the union of disjoint intervals [a_n, b_n).
>> Then sum(b_n - a_n) = 1. You can't prove that (directly)
>> by compactness, but you _can_ prove it by a simple
>> transfinite induction!
>That's not quite the approach I had in mind. I'm sure you are aware of
>this argument, but I wonder why we need to talk about half-open
>intervals to show that m([0,1]) = 1. What I meant was to show the outer
>measure of [0,1] is >= 1 by considering open covers, extracting finite
>subcovers, and using ordinary induction over the number of intervals in
>the subcover. Then you can show the outer measure is less than 1+epsilon
>for any epsilon > 0.
I understood all that - I wasn't disputing anything you said,
just pointing out a _similar_ result where it's curious that
one actually can use transfinite induction but not compactness
(as opposed to the previous result where one can use compactness
but not induction.)
************************
David C. Ullrich
===
Subject: Re: Uncountable many reals without Cantor
>>This argument _is_ much more complicated, if you include
>>the missing details. In particular you need a _proof_ of
>>the intuitively reasonable fact that if [0,1] is contained
>>in the union of countably many intervals I_n then
>>(*) sum length(I_n) >= 1.
>>How do you _prove_ that?
> Assume the opposite, put the intervals end to end etc. This kind of
> thing is proven in the beginning of any Real Variables text, e.g.
> Royden. Where do you see a problem?
>>Sounds like you want to use induction on the number of intervals.
>>Problem is, induction works only if the number of intervals is finite.
>>Hint: that's where compactness comes in.
> Amusing technicality:
> In various places for various reasons we need to talk about
> coverings by half-open intervals instead of open intervals.
> Say [0,1) is the union of disjoint intervals [a_n, b_n).
> Then sum(b_n - a_n) = 1. You can't prove that (directly)
> by compactness, but you _can_ prove it by a simple
> transfinite induction!
>>That's not quite the approach I had in mind. I'm sure you are aware of
>>this argument, but I wonder why we need to talk about half-open
>>intervals to show that m([0,1]) = 1. What I meant was to show the outer
>>measure of [0,1] is >= 1 by considering open covers, extracting finite
>>subcovers, and using ordinary induction over the number of intervals in
>>the subcover. Then you can show the outer measure is less than 1+epsilon
>>for any epsilon > 0.
> I understood all that - I wasn't disputing anything you said,
> just pointing out a _similar_ result where it's curious that
> one actually can use transfinite induction but not compactness
> (as opposed to the previous result where one can use compactness
> but not induction.)
Yeah, that's what I thought. I wouldn't have said anything if it hadn't
been for your choice of words in saying we need to talk about coverings
by half-open intervals. Didn't you recently chide someone else for
claiming that there is only one way to do something? (Or maybe two ways?)
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: Uncountable many reals without Cantor
>> I understood all that - I wasn't disputing anything you said,
>> just pointing out a _similar_ result where it's curious that
>> one actually can use transfinite induction but not compactness
>> (as opposed to the previous result where one can use compactness
>> but not induction.)
One other thought -- you may not be using compactness in that part of the
argument, but you do need some special property of the reals in order to
conclude that you only need to consider end-to-end placement of the
intervals in the first place. Consider covering the rationals in [0,1]
by collections of half-open intervals, for example.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: Uncountable many reals without Cantor
> I understood all that - I wasn't disputing anything you said,
> just pointing out a _similar_ result where it's curious that
> one actually can use transfinite induction but not compactness
> (as opposed to the previous result where one can use compactness
> but not induction.)
>One other thought -- you may not be using compactness in that part of the
>argument, but you do need some special property of the reals in order to
>conclude that you only need to consider end-to-end placement of the
>intervals in the first place.
??? I didn't say anything about end-to-end placement. (At least
I don't think I did, not _certain_ what you mean by the phrase).
If [0,1)is the union of countably many disjoint I_n = [a_n, b_n)
then the ordering of the I_n can be isomorphic to any
countable ordinal.
>Consider covering the rationals in [0,1]
>by collections of half-open intervals, for example.
************************
David C. Ullrich
===
Subject: Re: Uncountable many reals without Cantor
>> I understood all that - I wasn't disputing anything you said,
>> just pointing out a _similar_ result where it's curious that
>> one actually can use transfinite induction but not compactness
>> (as opposed to the previous result where one can use compactness
>> but not induction.)
>>One other thought -- you may not be using compactness in that part of the
>>argument, but you do need some special property of the reals in order to
>>conclude that you only need to consider end-to-end placement of the
>>intervals in the first place.
> ??? I didn't say anything about end-to-end placement. (At least
> I don't think I did, not _certain_ what you mean by the phrase).
> If [0,1)is the union of countably many disjoint I_n = [a_n, b_n)
> then the ordering of the I_n can be isomorphic to any
> countable ordinal.
Here's what you said:
> In various places for various reasons we need to talk about
> coverings by half-open intervals instead of open intervals.> Say [0,1)
is
> the union of disjoint intervals [a_n, b_n).> Then sum(b_n - a_n) = 1.
You
> can't prove that (directly)
> by compactness, but you _can_ prove it by a simple> transfinite
> induction!
For the record, it was someone else in the thread (J?rgen R., as the name
appears in my newsreader) who said Assume the opposite, put the
intervals end to end etc.. I don't know what end to end would mean
for open intervals, or for closed intervals, but I think end to end is
a very good description for a covering by disjoint half-open intervals,
such as you described.
>>Consider covering the rationals in [0,1]
>>by collections of half-open intervals, for example.
And how can you be sure that coverings by overlapping intervals won't
give you a lower estimate of the outer measure? That was the point of my
rational numbers example. I think you're going to need compactness or
completeness in some form.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: Uncountable many reals without Cantor
> I understood all that - I wasn't disputing anything you said,
> just pointing out a _similar_ result where it's curious that
> one actually can use transfinite induction but not compactness
> (as opposed to the previous result where one can use compactness
> but not induction.)
>One other thought -- you may not be using compactness in that part of
the
>argument, but you do need some special property of the reals in order to
>conclude that you only need to consider end-to-end placement of the
>intervals in the first place.
>> ??? I didn't say anything about end-to-end placement. (At least
>> I don't think I did, not _certain_ what you mean by the phrase).
>> If [0,1)is the union of countably many disjoint I_n = [a_n, b_n)
>> then the ordering of the I_n can be isomorphic to any
>> countable ordinal.
>Here's what you said:
>> In various places for various reasons we need to talk about
>> coverings by half-open intervals instead of open intervals.> Say
[0,1) is
>> the union of disjoint intervals [a_n, b_n).> Then sum(b_n - a_n) = 1.
You
>> can't prove that (directly)
>> by compactness, but you _can_ prove it by a simple> transfinite
>> induction!
>For the record, it was someone else in the thread (J?rgen R., as the name
>appears in my newsreader) who said Assume the opposite, put the
>intervals end to end etc.. I don't know what end to end would mean
>for open intervals, or for closed intervals, but I think end to end is
>a very good description for a covering by disjoint half-open intervals,
>such as you described.
>Consider covering the rationals in [0,1]
>by collections of half-open intervals, for example.
>And how can you be sure that coverings by overlapping intervals won't
>give you a lower estimate of the outer measure?
I didn't say that you could be sure of that. I really don't know
what you're going on about in all this - over and over you reply
to things that I didn't say. For the last time:
I thought the fact that that fact above can be proved by
transfinite induction might be interesting.
That was my one and only point.
>That was the point of my
>rational numbers example. I think you're going to need compactness or
>completeness in some form.
************************
David C. Ullrich
===
Subject: Re: Uncountable many reals without Cantor
>>Consider covering the rationals in [0,1]
>>by collections of half-open intervals, for example.
>>And how can you be sure that coverings by overlapping intervals won't
>>give you a lower estimate of the outer measure?
> I didn't say that you could be sure of that. I really don't know
> what you're going on about in all this - over and over you reply
> to things that I didn't say. For the last time:
> I thought the fact that that fact above can be proved by
> transfinite induction might be interesting.
Yes, it was interesting. I had to read it twice because I missed the
disjointness assumption on my first pass, and without disjointness I
couldn't see how to make the transfinite recursion work.
But you also suggested that compactness did not play a role in the
argument you were presenting. I disagree.
> That was my one and only point.
Then you weren't talking about a proof of uncountability of R without
Cantor, since you don't have a proof without addressing the disjointness
issue.
>>That was the point of my
>>rational numbers example. I think you're going to need compactness or
>>completeness in some form.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: Uncountable many reals without Cantor
>Consider covering the rationals in [0,1]
>by collections of half-open intervals, for example.
>And how can you be sure that coverings by overlapping intervals won't
>give you a lower estimate of the outer measure?
>> I didn't say that you could be sure of that. I really don't know
>> what you're going on about in all this - over and over you reply
>> to things that I didn't say. For the last time:
>> I thought the fact that that fact above can be proved by
>> transfinite induction might be interesting.
>Yes, it was interesting. I had to read it twice because I missed the
>disjointness assumption on my first pass, and without disjointness I
>couldn't see how to make the transfinite recursion work.
>But you also suggested that compactness did not play a role in the
>argument you were presenting. I disagree.
Where does compactness come in? You need _completeness_ to show
that the collection of intervals is well-ordered.
>> That was my one and only point.
>Then you weren't talking about a proof of uncountability of R without
>Cantor, since you don't have a proof without addressing the disjointness
>issue.
You're doing this on purpose, right? No, I wasn't talking about a
proof of uncountability of R. I never said that this little tidbit
had anything to do with that...
>That was the point of my
>rational numbers example. I think you're going to need compactness or
>completeness in some form.
************************
David C. Ullrich
===
Subject: Re: Uncountable many reals without Cantor
>>Consider covering the rationals in [0,1]
>>by collections of half-open intervals, for example.
>>And how can you be sure that coverings by overlapping intervals won't
>>give you a lower estimate of the outer measure?
> I didn't say that you could be sure of that. I really don't know
> what you're going on about in all this - over and over you reply
> to things that I didn't say. For the last time:
> I thought the fact that that fact above can be proved by
> transfinite induction might be interesting.
>>Yes, it was interesting. I had to read it twice because I missed the
>>disjointness assumption on my first pass, and without disjointness I
>>couldn't see how to make the transfinite recursion work.
>>But you also suggested that compactness did not play a role in the
>>argument you were presenting. I disagree.
> Where does compactness come in? You need _completeness_ to show
> that the collection of intervals is well-ordered.
You seem to have missed my point. If the intervals are not disjoint,
then they need not be well-ordered at all.
Besides, I said I thought you would need compactness or completeness in
some form to establish that the disjoint case is sufficient. I have not
seen any evidence to the contrary.
> That was my one and only point.
>>Then you weren't talking about a proof of uncountability of R without
>>Cantor, since you don't have a proof without addressing the disjointness
>>issue.
> You're doing this on purpose, right? No, I wasn't talking about a
> proof of uncountability of R. I never said that this little tidbit
> had anything to do with that...
So let's go back to the starting point of this subthread. It began when
I said:
>>One other thought -- you may not be using compactness in that part of the
>>argument, but you do need some special property of the reals in order to
>>conclude that you only need to consider end-to-end placement of the
>>intervals in the first place.
Would you like to start over and explain which part of that statement you
are disagreeing with? I explained what I meant by end-to-end placement,
and we need not revisit that. I specifically said that you don't need
compactness in that part of the argument, i.e., the transfinite
induction part.
And completeness is indeed a special property of the reals, at least in
the sense that it is not shared by the rationals, and therefore your
nitpicking about whether compactness or completeness constitutes the
required property is off target. I covered both possibilites from the
beginning. Which one you need depends on precisely what you are given
and what you are trying to prove, which is where we can't seem to agree.
>>That was the point of my
>>rational numbers example. I think you're going to need compactness or
>>completeness in some form.
You are behaving like the cranks you so often deride for shifting their
arguments whenever someone offers a critique of their statements. You
accuse me of bringing up things that you weren't talking about, but when
you look at my initial statement (quoted above) it should be clear that I
anticipated your objection and covered it even before you made it. If
you are going to claim that I had no business bringing up compactness in
a context that you weren't talking about, then I will counter that you
equally had no business bringing up transfinite induction, which I wasn't
talking about. My comment was every bit as relevant as yours was.
(And no, I didn't say you were a crank. The point is that you should
aspire to a higher standard of behavior.)
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: Uncountable many reals without Cantor
> Yes, it was interesting. I had to read it twice because I missed the
> disjointness assumption on my first pass, and without disjointness I
> couldn't see how to make the transfinite recursion work.
I meant transfinite induction, of course.
Hmm. I wonder if transfinite recursion would be useful in this program I'm
working on...
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: Uncountable many reals without Cantor
>there are often threads in this group concerning
>the cardinality of the set of real numbers. Some
>persons seem to have strong objections against the
>Cantor Proof of the fact that the set of real
>numbers is not denumerable by the naturals.
>>People may have strong objections, but nobody
>>has any _coherent_ objections - the people who
>>object seem to be unable to follow very simple
>>reasoning. Hence I doubt that they're going to
>>be able to follow complicated chains of reasoning...
>Cantor's Proof uses diagonalization. But there is
>a mesaure theoretic argument for the uncountability
>of the reals due to Borel which does not use this
>technique.
>Let (a_i), i e {1,2,3,...} be a list of the reals in
>the interval [0,1]. Let eps be any rational number
>> 0.
>Now consider a_1 in an interval of length eps/2, ...,
>a_i in an interval of length eps/2^i. Since every
>element of [0,1] is in some of the intervals, we
>have
>length([0,1]) <= eps/2 + eps/4 + ... + eps/2^i + ... = eps
>for every rational eps > 0. A contradiction.
>>I can imagine one of the objectors mentioned above
>>_agreeing_ that this argument is right, because it's
>>based on more familiar concepts. But I think the idea
>>that it's actually simpler is bogus - if someone
>>agrees to this but not to the diagonal argument I
>>really don't think that he's understood all the details.
>>This argument _is_ much more complicated, if you include
>>the missing details. In particular you need a _proof_ of
>>the intuitively reasonable fact that if [0,1] is contained
>>in the union of countably many intervals I_n then
>>(*) sum length(I_n) >= 1.
>>How do you _prove_ that?
>Assume the opposite, put the intervals end to end etc. This kind of
>thing is proven in the beginning of any Real Variables text, e.g.
>Royden.
Well of course this is proved in reals (although it's not so
clear to me that put the intervals end to end has much to
do with a proof - never mind that, not really relevant.)
>Where do you see a problem?
I don't see any problems with the validity of the proof.
I question the relevance in the present context because
when all the details are included it's much more complicated
than the diagonal argument. I mentioned a few other problems
in the next few paragraphs - again, they're not problems
with the proof, just problems with the propisition that
this is a way to attempt to convince the sort of idiot
who's not convinced by the diagonal argument:
>>(It really does require proof, you know. A _clever_
>>objector to all this would point out that [0,1]
>>is also the union of the closed intervals [x], for
>>x in [0,1]. Note that the sum of the lengths of [x]
>>for x in [0,1] is 0.
>>Of course the reason this is not a contradiction
>>is that (*) is not valid for uncountable unions.
>>But (i) this shows at least that (*) for countable
>>unions does require proof, and (ii) if we were
>>insisting that there's no such thing as an uncountable
>>set, maybe because we forgot to take our pills, then
>>the explanation that (*) doesn't hold for uncountable
>>unions doesn't work, and we conclude from this
>>example that (*) is simply wrong! Suppose you do
>>give a proof of (*) for countable unions - anyone
>>stupid enough to be able to find flaws with the
>>diagonal argument is going to have no problem
>>finding flaws with that proof.
>>Of course if our goal is to elicit _agreement_ instead
>>of _understanding_ then the argument above is a good
>>idea, because the objectors are going to be too dense
>>to see the objections. But if the idea is actually
>>to get someone to believe the reals are uncountable
>>_for_ a valid _reason_ then the diagonal argument seems
>>much better.)
>>************************
>>David C. Ullrich
************************
David C. Ullrich
===
Subject: Re: Uncountable many reals without Cantor
>>there are often threads in this group concerning
>>the cardinality of the set of real numbers. Some
>>persons seem to have strong objections against the
>>Cantor Proof of the fact that the set of real
>>numbers is not denumerable by the naturals.
>People may have strong objections, but nobody
>has any _coherent_ objections - the people who
>object seem to be unable to follow very simple
>reasoning. Hence I doubt that they're going to
>be able to follow complicated chains of reasoning...
>>Cantor's Proof uses diagonalization. But there is
>>a mesaure theoretic argument for the uncountability
>>of the reals due to Borel which does not use this
>>technique.
>>Let (a_i), i e {1,2,3,...} be a list of the reals in
>>the interval [0,1]. Let eps be any rational number
> 0.
>>Now consider a_1 in an interval of length eps/2, ...,
>>a_i in an interval of length eps/2^i. Since every
>>element of [0,1] is in some of the intervals, we
>>have
>>length([0,1]) <= eps/2 + eps/4 + ... + eps/2^i + ... = eps
>>for every rational eps > 0. A contradiction.
>I can imagine one of the objectors mentioned above
>_agreeing_ that this argument is right, because it's
>based on more familiar concepts. But I think the idea
>that it's actually simpler is bogus - if someone
>agrees to this but not to the diagonal argument I
>really don't think that he's understood all the details.
>This argument _is_ much more complicated, if you include
>the missing details. In particular you need a _proof_ of
>the intuitively reasonable fact that if [0,1] is contained
>in the union of countably many intervals I_n then
>(*) sum length(I_n) >= 1.
>How do you _prove_ that?
>>Assume the opposite, put the intervals end to end etc. This kind of
>>thing is proven in the beginning of any Real Variables text, e.g.
>>Royden.
>Well of course this is proved in reals (although it's not so
>clear to me that put the intervals end to end has much to
>do with a proof - never mind that, not really relevant.)
>>Where do you see a problem?
>I don't see any problems with the validity of the proof.
>I question the relevance in the present context because
>when all the details are included it's much more complicated
>than the diagonal argument.
OK, agreed.
===
Subject: Re: Uncountable many reals without Cantor
David C. Ullrich says...
>I don't see any problems with the validity of the proof.
>I question the relevance in the present context because
>when all the details are included it's much more complicated
>than the diagonal argument.
In a way, but the complexity of the measure-theoretic proof
is normal mathematics, while Cantor's diagonalization proof
seems like logic. Logic makes some people queasy, even good
mathematicians. I remember as an undergraduate asking a math
professor about Godel's theorem and he grimaced and said Ugh.
Mathematical logic! I'm not sure why people have that reaction,
but many people do.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Uncountable many reals without Cantor
What happens if you replace reals with rationals, irrationals,
algebraics, transcendentals, or other set dense in the reals?
Does it lead to a contradiction because there are functions bijecting
the rationals and integers?
What you have there is a restatement.
About the dart, basically the probability is between 1/2 and 1/3,
inclusive. That's based upon the deduction that the reals contain
exclusive rationals, algebraic irrationals, and transcendentals, and
that their union comprises the entire set of the reals, and that each
is dense in the reals. That's about the contiguous reals.
As usual, applying your argument to the natural/unit equivalency
function, your result does not hold in the related nonstandard real
numbers, and their related rational, irrational, algebraic and
transcendental real numbers.
I must admit that I don't know much about measure theory. For
example, where measure theory is useful for functions from reals to
reals with positive measure reflecting the geometric or analytical
results, there is a large disconnect in its consideration of an
infinite set with what is deemed to be zero measure. Between the
analog and discrete there is not an accessible experiment to validate
the results of measure theory where some infinite set has measure
zero, only in the abstract. So, I must investigate measure theory and
to promote my very own positions find either a contradiction or a
circularity, the contradiction invalidating that all infinitesimals
are zero and non-positive, and the circularity in that the assumption
that infinite sets are not equivalent in the definitions of measure
theory is the only reason that any non-empty set has measure zero.
Transfinite cardinality was bolted onto measure theory after the fact.
It's heavy, poisonous, and doesn't fit through doors.
That is to say: measure theory has utility in functions about which it
has a statement, and it is mute where its definitions are not
representative of the underlying number system.
Ullrich claims that logic is simple. He's often right. He's aware of
potential slight changes in very simple logic. Ullrich is proud that
I call him a hedgehog. Ullrich has stated a fallacy. Sarcasm is a
lie where the intent is to be truthful and the opposite. Is not your
model countable? Hodges' exposition contains flaws.
Consider the powerset and why the probabilities of one of its elements
containing either all or none are each 1 / 2^x, or for any other
particular element, where the sum from 1 to 2^x of 1/ 2^x = 1, and the
subtle distinction between singular and plural.
Are you unreasonably attached to the transfinite cardinals? Can you
use them to solve a problem? Would it be about transfinite cardinals?
If measure theory is redefined in terms of, yes, metrics, and not
transfinite cardinals, would anything change that was not nonsense
before?
Measure theory appears to be being brought into play to bulwark the
die agonal arguments. Why, what's wrong with them?
There is reason to believe that infinite sets are equivalent.
Ah yes, there is much in simple logical statements. F of x equals x
plus one. There are lots of rational numbers. It's an infinite set,
there's always one more.
Ross Finlayson
===
Subject: Re: Uncountable many reals without Cantor
>> But seriously,
>> the intuitionists and serious constructivists deny the uncountability
>> of the reals
> No they don't.
Some do. Brouwer certainly did. That's what led him to abandon
analysis.
Some of them even deny infinity. At that point you're veering right
out of mathematics into philosophy. But their point is that there is
in fact some largest number (size unknown, of course), because *in
fact* you're limited by the size of things you consider. Then you get
into arguments about what that largest number actually *is* (the
number, something else?). I personally think that sort of silliness
undermines their argument that there's a largest number -- once you
start thinking about the largest number, then just double it. But
that argument doesn't seem to affect them -- proving that they're
philosophers and not mathematicians.
Jon Miller
===
Subject: Re: Uncountable many reals without Cantor
|>> But seriously,
|>> the intuitionists and serious constructivists deny the uncountability
|>> of the reals
|> No they don't.
|
|Some do. Brouwer certainly did.
I don't think so. If he did, then it was an aberration, since
on other occasions he said that the integers and reals had
different cardinalities. Cantor's first proof of the uncountability
of the continuum is perfectly legitimate according to the
principles of intuitionist analysis.
None of the Bishop school deny the uncountability of the reals;
Bishop himself has a proof of it in his famous textbook of
constructive analysis. Quite to the contrary, the serious
constructivists understand that it's correct.
| That's what led him to abandon
|analysis.
Ridiculous. He spent much of his life reworking analysis on his terms.
|Some of them even deny infinity.
Name one. I don't think you know of any.
| At that point you're veering right
|out of mathematics into philosophy. But their point is that there is
|in fact some largest number (size unknown, of course), because *in
|fact* you're limited by the size of things you consider.
This sounds like a garbled description of the ultra-intuitionism
of Essenin-Volpin. But he does _not_ claim that there exists
a largest number. He just denies that such expressions as
10^12 necessarily denote natural numbers.
| Then you get
|into arguments about what that largest number actually *is* (the
|number, something else?). I personally think that sort of silliness
|undermines their argument that there's a largest number -- once you
|start thinking about the largest number, then just double it. But
|that argument doesn't seem to affect them -- proving that they're
|philosophers and not mathematicians.
I think you're doing a poor job of describing Essenin-Volpin's
point of view here. You're talking about him as though he had
followers, and I doubt you know of any. At least refrain from
trying to make his views sound worse than they are. I've read
some of his papers, and I don't believe he denies the existence
of infinity, either.
There's something about holding an unpopular point of view that
people seem to take as an invitation to start being extremely
sloppy about how the point of view is described. I don't know
whether you are doing so yourself, but if you are not then some
of your sources are, so be wary.
Keith Ramsay
===
Subject: Re: Uncountable many reals without Cantor
> Some do. Brouwer certainly did.
No he didn't. There's nothing intuitionistically problematic about
the reals being uncountable.
> That's what led him to abandon
> analysis.
No it wasn't. He didn't abandon analysis at all.
===
Subject: Re: Uncountable many reals without Cantor
>> Some do. Brouwer certainly did.
> No he didn't. There's nothing intuitionistically problematic about
>the reals being uncountable.
Well, in his inaugural address at the University of Amsterdam
(1912; I'm going by the 1913 English translation, Intuition
and Formalism, published in the Bulletin of the American
Mathematical Society, volume 20, p. 81), Brouwer did say that
the intuitionist doesn't recognize any infinite set of
cardinality greater than aleph-null. I'd quote him exactly--
of xerographic copying, I only have the right half of page 91,
from which I wish to quote. It says:
According to the statemen
aleph-null is the only infinit
recognize the existence.
which is surely a very bad haiku. But I do think I remember
his thrust correctly. Maybe he took it all back later?
(Or maybe you're being subtle, and being uncountable
is--intuitionistically--not the same as having an uncountable
cardinality?)
I *did* manage to copy all the words that I had gone in search
of (some of his comments on Kant), so I don't need to go back
to the library, and I'd rather not.
Lee Rudolph
===
Subject: Re: Uncountable many reals without Cantor
> Well, in his inaugural address at the University of Amsterdam
> (1912; I'm going by the 1913 English translation, Intuition
> and Formalism, published in the Bulletin of the American
> Mathematical Society, volume 20, p. 81), Brouwer did say that
> the intuitionist doesn't recognize any infinite set of
> cardinality greater than aleph-null.
In that address, Brouwer apparently only recognized lawlike choice
sequences. However, he does mention the possibility of admitting free
choice sequences, and notes that in such a case,
if, on the basis of the intuition of the linear continuum, [the
intuitionist] admits elementary series of free selections as
elements of construction, then each non-denumerable set constructed
by means of it contains a subset of the power of the continuum.
(A. S. Troelstra: Cholce sequences. Clarendo Press. Oxford 1977.)
sequences - a historical note (available on the web):
In 1914 Brouwer had changed his views on choice sequences as pointed
out by Troelstra in his book. In a review on Schoenflies and Hahns
book on set theory ( 139 - 144 ) Brouwer remarks in a footnote ( 140
):
Z. B. ist die Punktmenge: alle reellen Zahlen zwischen O und 1 mit
Ausnahme der endlichen Dualbruche, nur deshalb eine Wohlkonstruierte
Menge, weil die duale Entwicklung einer willkurlichen Zahl dieser
Menge eine Fundamentalreihe von endliche Gruppen von gleichen Ziffern
(abwechselnd O und l) liefert, so dass die Menge sich mittels einer
Fundamentalreihe von Auswahlen unter den endlichen Zahlen bestimmen
lasst. Dieser Schritt geht freilich weiter als mein romischer Vortrag
( 102 - 104 ), und auch weiter als die Borelschen Ausfuhrungen uber
wohlkonstruierte Mengen ( 827 - 878 ); er erscheint mir aber als eine
notwendige Konsequenz des Intuitionismus.
Brouwer's idea of a Menge was not, to be sure, that of Cantorian set
theory. But the proof that the Menge of real numbers between 0 and 1
is not countable is perfectly valid intuitionistically.
On a more basic note: recall that intuitionism does not recognize the
existence of infinite sets as completed totality. This does not
imply that the natural numbers form a finite totality from an
intuitionistic point of view. Similarly with the real numbers and
uncountability.
===
Subject: Re: Uncountable many reals without Cantor
for this information. Do you know why Brouwer changed his mind about
free choice sequences? And would law-like choice sequences be
similar to the requirement that constructions be recursive (a la
Markov(?)).
===
Subject: Re: Uncountable many reals without Cantor
> Do you know why Brouwer changed his mind about
> free choice sequences?
No, but I'm sure there are explanatory expositions in the literature.
> And would law-like choice sequences be
> similar to the requirement that constructions be recursive (a la
> Markov(?)).
A law-like sequence is one which is fully determined by a rule or
law, without any element of choice involved. It is not part of
the idea of a law-like sequence that it has to be recursive, though.
===
Subject: Levene's test & Two Independent Samples t-Test?
Someone told me that Levene's test is no longer used to test for equal
variances before performing the two independent samples t-test. This
is news to me, not that I'm a stat whiz but that every textbook that I
have still uses this test. If this is true could somebody point me to
a research paper or recent textbook or website, etc.?
===
Subject: Re: Escultura affair: publication scandal
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id iB1FkRA11431;
Reading this thread I got interested in what
E. Escultura is actually doing - and I got angry
reading his rather strange statements about FLT
and Wiles' proof.
In the >Zentralblatt f.9fr Mathematik< 9 publications
of E. Escultura are listed. For 5 of them the
Zentralblatt just mentions >not reviewed<, whatever
that means.
However I must admit that - looking at myself - I
wonder why persons like E. Escultura or J. Harris
provoke such strong reactions.
H
===
Subject: Re: Is this math test too easy?
> One problem is that the elementary and high school teachers
> have made it difficult for the students to understand the
> basics. This is continued by students demanding to be taught
> the same way in college. Most researchers are willing to teach
> students who want to understand, but not those who want to just
> learn the mechanics.
...
> No teacher who concentrates on memorization and routine can
> be even a fair teacher; if the curriculum calls for this,
> the curriculum prevents anyone from being this.
...
> If you are concentrating on teaching them how to perform
> arithmetic operations, you are not a good teacher, no matter
> how good a teacher you can be.
I've heard of a pedagogical trend in elementary schools which is to
-not- teach a particular algorithm for, say, addition, but rather let
the kid's discover their own algorithm (one that works of course!!)
by playing with the numbers (letting them discover tricks like
carrying or distributivity or whatever they dream up).
Is this a reasonable strategy for teaching understanding? Or is it
just a start? or is it the wrong way?
--
Mitch Harris
(remove q to reply)
===
Subject: Re: Is this math test too easy?
>> I was thinking f@x for function notation.
One can make a strong argument for xf instead. The
associativity of functions would then go in the same
order as the applications.
In my mathematical papers I
>> always start out with a few pages of definitions so that everyone
>> reading my paper knows exactly what I mean, I am aware that not
>> everyone does that.
But I suggest we drop the word definition. I prefer
characterization; there can be many characterizations
of something, but only one definition.
In my notes on topology, I give 14 equivalent characterizations
of topological space.
>Should we now argue that there should be only one definition for each term?
>I mean, why define continuity using epsilon-delta?
We definitely should not; even for one variable, intervals
are easier.
Wouldn't it be nicer if
>we just defined it using preimages of open sets?
One could, but this would be more difficult than the
existence of a neighborhood mapping into a neighborhood.
This is the one which carries over to uniform spaces,
for example.
>Forget about worrying about notation, we need to concentrate on defining
>our definitions first!
>And very first, we must come up with a nice, simple definition for
>number.
Definitely NOT. This, in my opinion, was one place where
the new math went wrong. There are different concepts
of number, and it would be a major mistake to use one
of them as THE idea of number.
We can say that the numbers form a system with certain
properties, and choose a sufficient number of those
properties to characterize the system.
My Merriam Webster Collegiate Dictionary lists 9 definitions for
>number, many of those have sub-definitions. If we can't even figure out
>what a number is, what value is there in trying use this thing we call a
>number?
So? That the integers or rational or real numbers can be
used in different ways does not change their properties.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Is this math test too easy?
>>Should we now argue that there should be only one definition for each
>>term? I mean, why define continuity using epsilon-delta?
> We definitely should not; even for one variable, intervals
> are easier.
> Wouldn't it be nicer if
>>we just defined it using preimages of open sets?
> One could, but this would be more difficult than the
> existence of a neighborhood mapping into a neighborhood.
> This is the one which carries over to uniform spaces,
> for example.
You did catch the part where I said I was using sarcasm to make a point? I
was trying to show that often times, having multiple ways to express
something, or multiple definitions actually make things easier.
- Tim
--
Timothy M. Brauch
NSF Fellow
Department of Mathematics
University of Louisville
email is:
news (dot) post (at) tbrauch (dot) com
===
Subject: Variable names Was: Re: Is this math test too easy?
.....................
>In Spanish, you would think, Necesito escribir una instruccin que haga
>algo MIENTRAS QUE una cierta condicin est satisfecha. Notice that the
>word while does not appear in the Spanish thought. The programmer
would
>then need to know the English term for mientras que.
The idea does appear; it is not a single word. It was
probably a mistake to use words for computer directions;
one reason for it, and for symbol grabbing (the use
of standard mathematical symbols in ways mathematicians
would never use them) was the reliance on overly limited
character sets. I know of a few languages which did try
to use physical subscripts and superscripts. At least
the producers of FORTRAN apologized for the necessity
of using ** for exponentiation.
As it appears here, I believe your Spanish spelling has
some errors.
The word for integer part in many computer languages is
entier.
>And then there is the whole giving variables meaningful names deal. When
>they piece together code available online, you get a nice jumble of English
>and Spanish.
Who cares about giving variables meaningful names? This
may or may not help in a particular application, and the
more characters typed, the greater chance for error. It
is difficult to read the program if a variable entitled
partial_vapor_pressure_of_methane is used instead of
one or two letters.
Before computers, variable names were almost always one
character, or one character with subscripts, which also
were restricted. One cannot understand variables unless
one can recognize that here, at least, Shakespeare was
correct.
>At least with math symbols, they are pretty much the same in every
>language. Without knowing Russian, I was able to follow a Russian linear
>algebra text for a proof, simply because the math notation was the same.
>The words between symbols, they were not as important because I could
>figure out how to get from step to the next.
Would it be any harder if Russian letters were used for
the variables? In fact, when I first looked at a Russian
and Greek letters for variables, and the same Latin
abbreviations for the trigonometric functions; Spanish
does not.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Is this math test too easy?
> The poster named The Ghost in the Machine has already responded at
> length. The gist of what the thread has been that computer programming
> languages eliminate the ambiguity often seen in common mathematical
> notation.
But this assumes you have a keen grasp of the English language. I lived in
Mexico for awhile and worked for the city government's IT department.
Trying to code in bilingual is not pretty. Computer language use words
like for and while and end and if among many other common
English
words. If you understand English, you can think to yourself, I need to
write an instruction that will do something WHILE some condition is
satisfied.
In Spanish, you would think, Necesito escribir una instrucci.97n que haga
algo MIENTRAS QUE una cierta condici.97n est.87 satisfecha. Notice that
the
word while does not appear in the Spanish thought. The programmer would
then need to know the English term for mientras que.
And then there is the whole giving variables meaningful names deal. When
they piece together code available online, you get a nice jumble of English
and Spanish.
At least with math symbols, they are pretty much the same in every
language. Without knowing Russian, I was able to follow a Russian linear
algebra text for a proof, simply because the math notation was the same.
The words between symbols, they were not as important because I could
figure out how to get from step to the next.
- Tim
--
Timothy M. Brauch
NSF Fellow
Department of Mathematics
University of Louisville
email is:
news (dot) post (at) tbrauch (dot) com
===
Subject: Re: Is this math test too easy?
> If sorting out the notation is your worst problem, you're in a better
> place than most working mathematicians.
> Maybe, but I hate math.
Which is doubtless a blessing for all concerned.
--
A.
===
Subject: Re: Is this math test too easy?
> Which is doubtless a blessing for all concerned.
Apart from myself, who would that concern?
Actually, to the extent that math might be applied to real-world
problems, it might be a disadvantage for the world as a whole, since
I've always had a high aptitude for math.
--
Transpose hotmail and mxsmanic in my e-mail address to reach me directly.
===
Subject: Re: Is this math test too easy?
> Which is doubtless a blessing for all concerned.
> Apart from myself, who would that concern?
> Actually, to the extent that math might be applied to real-world
> problems, it might be a disadvantage for the world as a whole, since
> I've always had a high aptitude for math.
How much math were you subjected to?
--
A.
===
Subject: Re: Is this math test too easy?
> How much math were you subjected to?
Enough to disgust me permanently.
--
Transpose hotmail and mxsmanic in my e-mail address to reach me directly.
===
Subject: Re: Is this math test too easy?
>> How much math were you subjected to?
>Enough to disgust me permanently.
How much non-computational math have you had an
opportunity to get into?
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Is this math test too easy?
> How much math were you subjected to?
> Enough to disgust me permanently.
Did you stick with it long enough to learn the Ham Sandwich Theorem?
Truly useful if you ever find yourself working as a sandwich builder in
a deli.
--
A.
===
Subject: Re: Is this math test too easy?
> Did you stick with it long enough to learn the Ham Sandwich Theorem?
No. I took nothing that I wasn't required to take, and I paid very
little attention to any of it.
The only occasions on which I ever studied math with any diligence were
occasions on which I needed it to solve some sort of problem, and in
those cases I taught myself only what I needed to know, and no more.
--
Transpose hotmail and mxsmanic in my e-mail address to reach me directly.
===
Subject: Re: Is this math test too easy?
>> Did you stick with it long enough to learn the Ham Sandwich Theorem?
>No. I took nothing that I wasn't required to take, and I paid very
>little attention to any of it.
>The only occasions on which I ever studied math with any diligence were
>occasions on which I needed it to solve some sort of problem, and in
>those cases I taught myself only what I needed to know, and no more.
There are those who use this educational philosophy, even
mathematicians. But how can you know what you need to
know, unless you know about its existence beforehand?
Even for practical purposes, that is not enough. If is
often wanted to evaluate distributions of the sum of
large numbers of random variables, and to make it easy,
they all have densities, possibly even the same one.
This problem only involves a large number of numerical
integrations as stated, and one does not even have to
know what a complex number is.
On the other hand, assuming there is a moment generating
function for the same real values for all of them, one
can compute the distribution of the sum by using complex
integration on the product of these. This can even work
VERY well, sometimes well enough that the low-order terms
of an appropriate expansion can give good results.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Is this math test too easy?
Originator: grubb@lola
>> Did you stick with it long enough to learn the Ham Sandwich Theorem?
>No. I took nothing that I wasn't required to take, and I paid very
>little attention to any of it.
>The only occasions on which I ever studied math with any diligence were
>occasions on which I needed it to solve some sort of problem, and in
>those cases I taught myself only what I needed to know, and no more.
And this is the basic point where you will differ from almost everyone
else here. The value of learning is not simply in applications. It
is not simply in whether it can be applied to someone's job. There
is a value in learning *for its own sake*. To minimize the amount you
learn is simply to remain ignorant. To suggest that others do the
same is to promote ignorance. Instead, we should be requiring
*more* from all our students. Yes, more math, more history, more
geography, more literature, more physics, etc. Learn as much as
possible!
--Dan Grubb
===
Subject: Re: Is this math test too easy?
> Did you stick with it long enough to learn the Ham Sandwich Theorem?
> No. I took nothing that I wasn't required to take, and I paid very
> little attention to any of it.
> The only occasions on which I ever studied math with any diligence were
> occasions on which I needed it to solve some sort of problem, and in
> those cases I taught myself only what I needed to know, and no more.
It shows.
===
Subject: Re: Is this math test too easy?
> Let's fix it rather than complain, please join in in making things
> better or at least stop complaining now that you've gotten people's
> attention.
> A new coding system is required that eliminates ambiguity. Any system
> will do as long as it's unambiguous. It could be based on the current
> system, as long as it is modified such that dx or f(x) or i
always
> mean the same thing.
I was thinking f@x for function notation. In my mathematical papers I
always start out with a few pages of definitions so that everyone
reading my paper knows exactly what I mean, I am aware that not
everyone does that. I'm torn between things that are easy to say an
type (like f@x) compared to things that might be easier to read, like
putting d@x inside a box to make it clear that it's a length.
> I'm concieving right now. To convince others I need support that what
> I'm proposing is going to be really better, so help me out please.
> I don't think most pure mathematicians want an unambiguous system,
I disagree, but I think this isn't helpful to discuss comapred to how
to actually fix things.
> since they aren't really interested in precision.
They aren't paid to write things better than is required for their
peers to understand, in general, but just because they are under
pressure to publish often and not rewarded for more clarity than is
required to be published doesn't mean that they didn't care.
> People who are interested in precision tend to take an
> interest in arithmetic and concrete applications of
> mathematics, not in pure mathematics.
I've never understood the distinction between pure and applied
mathematics. As a physicist I'm always interested if new mathematics
can allow me to do things in physics that I couldn't do or couldn't do
easily before. As a mathematician I'm interested in how applicable I
can make one and the same proof. Precision on arthmetic don't seem as
related to me as you make it, for instance describing geometrical
situations and movements, if you have to make predictions or do
reverse kinematics, something testable, then you want to be clear and
unambiguous, but arthmetic isn't quite enough, you need data
structures and models.
> Writing math symbolism to be consistent with coding would be an
> advantage too, since often you need to do math in a program.
> Yes. And it is notable that when complex equations are converted to
> programming code, all the ambiguity (necessarily) disappears.
Few programmers actually prove that there code does what they claim it
does. Do you think cleaning up mathematics to make it easier to do
that would encourage this?
> This is why I've often asked mathematicians to show me the software
> algorithm, rather than vague notation that doesn't really tell me what
> they are talking about.
Code is usually less than the original mathematics though. For
instance is someone uses a double, then the algorith might work for a
more precise mantessa, which would be more clear in the original math,
but ONLY if the original math was written more clearly.
> Let's fix things, I need specific criticism, not general stuff, give
> me specifics about things you found/find ambiguous in math please.
> Naming conventions are extremely ambiguous.
Such as?
> The juxtaposition of identifiers is equally ambiguous.
Such as?
> I was taught early in life that
> letters from the start of the Roman alphabet were constants, and letters
> from the end of the Roman alphabet were variables. I soon discovered,
> though, that this rule is not followed to any extent. Some letters
> are used preferentially for certain things, and others are taboo.
I think you already figured out that there isn't an existing fixed
rule on that.
> There
> are no clear rules, so you never really know what's a constant and what
> is a variable.
I could write Vx for a variable and Ix for a constant, would that
help?
[snip clear examples]
===
Subject: Re: Is this math test too easy?
> I was thinking f@x for function notation. In my mathematical papers I
> always start out with a few pages of definitions so that everyone
> reading my paper knows exactly what I mean, I am aware that not
> everyone does that.
Should we now argue that there should be only one definition for each term?
I mean, why define continuity using epsilon-delta? Wouldn't it be nicer if
we just defined it using preimages of open sets?
Forget about worrying about notation, we need to concentrate on defining
our definitions first!
And very first, we must come up with a nice, simple definition for
number. My Merriam Webster Collegiate Dictionary lists 9 definitions
for
number, many of those have sub-definitions. If we can't even figure out
what a number is, what value is there in trying use this thing we call a
number?
- Tim
Yes, this is a troll. I prefer to think of it as sarcasm to get a point
across, though.
--
Timothy M. Brauch
NSF Fellow
Department of Mathematics
University of Louisville
email is:
news (dot) post (at) tbrauch (dot) com
===
Subject: Re: Is this math test too easy?
> I've never understood the distinction between pure and applied
> mathematics.
Pure mathematics is mathematics for its own sake. Coming up with
various obscure proofs is a good example of this.
Applied mathematics is mathematics that serves only as a tool used to
solve problems.
> Few programmers actually prove that there code does what they claim it
> does. Do you think cleaning up mathematics to make it easier to do
> that would encourage this?
In most real-world situations, you don't need proofs. Proofs are
usually an academic exercise only.
> Such as?
Sometimes i is just any old variable, sometimes it's a specific number.
> Such as?
Sometimes f(x) means f times x, sometimes it doesn't.
> I think you already figured out that there isn't an existing fixed
> rule on that.
Yes, and that's one reason why my eyes glaze whenever I'm exposed to
pure mathematics.
> I could write Vx for a variable and Ix for a constant, would that
> help?
Anything with unambiguous rules would help.
--
Transpose hotmail and mxsmanic in my e-mail address to reach me directly.
===
Subject: Re: Is this math test too easy?
> Sometimes f(x) means f times x, sometimes it doesn't.
In context, it is almost always clear whether f(x) means f times x or
f
of x. If we remove conext, we end up with statements such as:
6 is *not* equal to 3! [1]
6 is equal to 3! [2]
Both are valid statements, given a certain context. If I am trying to
solve an equation, and end up with my final line being 6 = 3, then I might
scream out [1], six is not equal to three (with emotion, mind you). If
I
am trying to solve a problem of arranging 3 distinct items, I might
casually mention [2], six is equal to three factorial.
- Tim
--
Timothy M. Brauch
NSF Fellow
Department of Mathematics
University of Louisville
email is:
news (dot) post (at) tbrauch (dot) com
===
Subject: Re: Is this math test too easy?
> Both are valid statements, given a certain context.
If they require context, they are ambiguous.
--
Transpose hotmail and mxsmanic in my e-mail address to reach me directly.
===
Subject: Re: Is this math test too easy?
>> Both are valid statements, given a certain context.
> If they require context, they are ambiguous.
If there is no context, it isn't math. It is just applying an algorithm.
Show me the math in:
3 + 4 = 7
Computers can do algorithms. Most computers cannot solve the following
problem.
Johnny has three apples. Jane gives him four more. How many
apples does Johnny have now?
Which is math and which is an application of an algorithm?
- Tim
--
Timothy M. Brauch
NSF Fellow
Department of Mathematics
University of Louisville
email is:
news (dot) post (at) tbrauch (dot) com
===
Subject: Re: Is this math test too easy?
>> I've never understood the distinction between pure and applied
>> mathematics.
>Pure mathematics is mathematics for its own sake. Coming up with
>various obscure proofs is a good example of this.
>Applied mathematics is mathematics that serves only as a tool used to
>solve problems.
Applied mathematics is the application of pure mathematics.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Is this math test too easy?
> Applied mathematics is the application of pure mathematics.
Yes, that's what I said. When you do math without applying it, it's
pure mathematics--mathematics for its own sake.
--
Transpose hotmail and mxsmanic in my e-mail address to reach me directly.
===
Subject: Re: Is this math test too easy?
>> Applied mathematics is the application of pure mathematics.
>Yes, that's what I said. When you do math without applying it, it's
>pure mathematics--mathematics for its own sake.
Is there such a thing as applied mathematics? When one does
pure mathematics, will it ever get applied, and when? Famous
mathematicians have stated that parts of mathematics would
never be applied, and they were wrong.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Is this math test too easy?
> Is there such a thing as applied mathematics?
Sure. When I balance my checkbook, that's applied mathematics.
--
Transpose hotmail and mxsmanic in my e-mail address to reach me directly.
===
Subject: Re: Is this math test too easy?
> Most of my code nowadays is in C++, which is hardly
> unambiguously defined.
C++ was specifically designed to massively increase the degree to which
programs can seem superficially ambiguous. It's a solution looking for
a problem.
> The standards committee has made a
> number of clarifications of their intent since the finalization of
the
> standard.
I prefer to just write in C, without the ++. I haven't found much use
for the latter. C++ makes it far easier to make mistakes in coding than
ever was possible for ordinary C, and it provides no compensating
advantages.
> Machine code is simpler and closer to being precisely defined,
> but I think it would be naive to assume that CPUs were entirely
> bug-free.
Bugs and ambiguity are two different things. Machine code is always
completely unambiguous.
--
Transpose hotmail and mxsmanic in my e-mail address to reach me directly.
===
Subject: Re: Is this math test too easy?
> While I agree that this is true from the computer's perpsective (it will
> unambiguaousl do what it is programmed to do) it does not eliminate the
> ambiguity of informing another human what the computer has been
programmed
> to do.
While human beings can live on hamburgers, this does not eliminate the
fact that computers cannot.
> In all computer languages that I have used, from Z80 assembler to
> APL, it is possible to construct statemants that are opaque to human
> understanding.
Who constructs them? It cannot be a human being, since that would
require human understanding.
--
Transpose hotmail and mxsmanic in my e-mail address to reach me directly.
===
Subject: Re: Is this math test too easy?
> |Having a standard notation where (i,j) means {{0,i},{1,j}} is good
> |especially if (i,j,k) means {{0,i},{1,j},{2,k}}, where the pattern is
> |known and standard.
> If you represent ordered n-tuples this way you represent the
> ordered 4-tuples (1,2,3,0) and (4,0,1,2) both as the set
> {{0,1},{1,2},{2,3},{0,4}}, which is a bad thing.
Excellent point. I was considering the idea of reserving the (a,b,c)
notation for ordered pairs, I hadn't meant to define the specific
example above the same I did.
> One main problem with this kind of reform is the tradeoff
> between terseness and reuse. Eliminating the reuse would
> require a lot more verbosity than you might think. I've seen the
> notation (i,j) used to mean the ordered pair composed of i
> and j, the greatest common divisor of i and j, the ideal generated
> by i and j, the submodule generated by i and j, the open interval
> between i and j, the integers from i to j, an element of a K-group,
> the permutation sigma such that sigma(i)=j, sigma(j)=i,
> and sigma(k)=k if k<>i,j, and I'm sure a bunch of other things that
> just aren't coming to mind right now.
I think you might have missed my idea, that ambiguous notation can be
used in math, until the point that it decides to go mainstream in
which case it chooses a well-defined, not already in mainstream
useage. So many of those things don't need to be in secondary school,
and others can have more clear notation. For instance (i ... j) is
better to represent the integers from i to j than simply, (i, j). And
the open interval could be (< i, j <) making it clear that it depends
on an ording.
[snip examples]
> The point is that mathematicians have a really large number of
> concepts that need names and/or notations, and it would be more
> of an inconvenience than a help to separate out all of them so
> that they could be read out of context. This means that authors
> often have to start by defining their notation; better than the
> alternative.
> Keith Ramsay
Only ones to be taught to other than mathematicians need to be
standardized. It's like having a stable version of linux that
everyone uses as opposed to the next version still under construction.
The part underconstruction can be vague because only experts (or
people that understand that it's vague on purpose) should be using it.
I'm not trying to prevent repeated use for in-house mathematical
discussions, in fact I was thinking we should reserve some symbols for
in house discussions so that no one can reserve them to have public
fixed meanings.
===
Subject: Re: Is this math test too easy?
> Um, yes, so 30 would contain four bits for the 3 in the tens columns
and
> four bits for the 0 in the ones column, thus 30.
Yes, but they would not be a single byte, at least not in many earlier
systems (and in endless pocket calculators).
--
Transpose hotmail and mxsmanic in my e-mail address to reach me directly.
===
Subject: Re: Is this math test too easy?
> Um, yes, so 30 would contain four bits for the 3 in the tens
columns
and
> four bits for the 0 in the ones column, thus 30.
> Yes, but they would not be a single byte, at least not in many earlier
> systems (and in endless pocket calculators).
BCD pocket calculators?
> --
> Transpose hotmail and mxsmanic in my e-mail address to reach me directly.
===
Subject: Re: Is this math test too easy?
> BCD pocket calculators?
Most pocket calculators have traditionally used 4-bit, BCD-based
internal processors. I don't know what they use currently; I suppose
more advanced processors are so cheap today that it's more economical to
use a commodity 8-bit processor than a less common 4-bit processor.
--
Transpose hotmail and mxsmanic in my e-mail address to reach me directly.
===
Subject: Re: Cantor's diagonal proof wrong?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id iB1GAoW13882;
>>In sci.math, Shmuel (Seymour J.) Metz
>>But it is still curious that it happens at 2^oo and not oo^oo.
> Why? Is it curious that 3*3 is smaller than 3^3, or that 16*16 is
> smaller than 2^16?
>>I suspect that, if a set S has cardinality of at least card(N),
>>then a bijective mapping can be found between the elements of 2^S
>>and the elements of n^S, where n > 1 is an integer.
>>However, I'd have to look.
>Sure, that's correct; in fact, n can have cardinality anywhere
>from 2 to 2^S. There's an injection 2^S --> n^S, and there's
>an injection n --> 2^S (and so an injection
> 2^S --> (2^S)^S ~ 2^S);
>now apply Schroeder-Bernstein.
>Todd Trimble
Typo: displayed line should read n^S --> (2^S)^S ~ 2^S.
===
Subject: Re: Turing Machines and Physical Computation & TM 'S REVISIONISTS
> You've not yet read Copeland, EZzz ?
Indeed I have read, and I did not find his philosophy rigorous.
If you read the original post, you will see that the very existence of
hypercomputation requires infinite space (in the form of infinitely
big or infinitely small, doesn't matter)
Can you explain to me *why* physicists cannot store an infinite amount
of information on a fingertip, if hypercomputation is possible?
Please provide references pointing that such devices can be
constructed in the real world.
Oh, if you say that hypercomputation cannot be experimentally verified
or its existence cannot be proven by physicists, then I will say it
does not exist. Angels cannot be experimentally verified or proven,
either. And they doesn't exist, naturally.
Or will you say, I can imagine continuum, so it must exist. What a
great ontological argument! If you accept such sloppy argumentation,
then you will also accept Descartes's proof of substance dualism, and
the cosmological argument among other non-proofs.
Poulpes thinks of hypercomputation, therefore there is
hypercomputation! LOL You frenchmen ;)
--
Eray Ozkural
===
Subject: Re: Turing Machines and Physical Computation
>>lines made with real pencils and the pinpricks made by the point of a
>>real compass, mathematicians have always approximated the real world by
>>abstractions which are not physically realizable.
>Well, I don't know about that. Whose theory is this? I thought most
>mathematicians who expressed an opinion were generally some sort of
>Platonists, believing that effective mathematics is discovered, not
>invented, and that it is the real world which approximates the ideal.
> Both real and ideal worlds are discovered and invented. Neither is the
> other; they approximate each other. It's my theory.
In what sense is the real world invented? Do you mean we can invent
America, because we can build a big city like New York?
In my opinion, Real world does not approximate the ideal world in the
ordinary sense of the word, which can get fictional. But if you mean
creativity, and the ability to creatively shape our environment, I
might agree. It's unclear what you mean above.
--
Eray Ozkural
===
Subject: Re: Turing Machines and Physical Computation
>The putative difficulty in writing effective software is a more common
>and mundane version of the same thing, IMHO.
> That's actually a quite different problem.
> Computation is the manipulation of representations.
I am not sure if that's a good description.
This formal symbol manipulation idea got some otherwise ambitious
philosophers like Brian Cantwell Smith and gang quite confused.
Computers can work on representations, that's true. But it is not
necessary that what is being manipulated is representation.
--
Eray
===
Subject: Re: Turing Machines and Physical Computation
-----BEGIN PGP SIGNED MESSAGE-----
Hash: SHA1
>> Computation is the manipulation of representations.
>I am not sure if that's a good description.
>This formal symbol manipulation idea got some otherwise ambitious
>philosophers like Brian Cantwell Smith and gang quite confused.
>Computers can work on representations, that's true. But it is not
>necessary that what is being manipulated is representation.
It can be an abstract representation -- that is, one that doesn't
actually represent anything.
-------
The received view of AI is something like:
A system of sensors that generates representation in some fixed
manner.
A manipulation or transformation of these input representation into
output representations.
A system of effectors that, in fixed way, generates physical
actions (behaviors) from the output representations.
-----BEGIN PGP SIGNATURE-----
Version: GnuPG v1.3.91 (SunOS)
iD8DBQFBrOFrvmGe70vHPUMRAserAJ9Bk5z1K2lkqZ7ThW597I6umxD2jgCgmzad
+0Xgga+8Am5wYTUFVvDfBH8=
=lLCX
-----END PGP SIGNATURE-----
===
Subject: Re: Turing Machines and Physical Computation
> -----BEGIN PGP SIGNED MESSAGE-----
> Hash: SHA1
>> Computation is the manipulation of representations.
>I am not sure if that's a good description.
>This formal symbol manipulation idea got some otherwise ambitious
>philosophers like Brian Cantwell Smith and gang quite confused.
>Computers can work on representations, that's true. But it is not
>necessary that what is being manipulated is representation.
> It can be an abstract representation -- that is, one that doesn't
> actually represent anything.
> -------
> The received view of AI is something like:
> A system of sensors that generates representation in some fixed
> manner.
> A manipulation or transformation of these input representation into
> output representations.
> A system of effectors that, in fixed way, generates physical
> actions (behaviors) from the output representations.
We were talking about computation in general, not what perception or
action in the world requires.
--
Eray Ozkural
===
Subject: Re: Turing Machines and Physical Computation
>The putative difficulty in writing effective software is a more common
>and mundane version of the same thing, IMHO.
>>That's actually a quite different problem.
>>Computation is the manipulation of representations.
> I am not sure if that's a good description.
> This formal symbol manipulation idea got some otherwise ambitious
> philosophers like Brian Cantwell Smith and gang quite confused.
> Computers can work on representations, that's true. But it is not
> necessary that what is being manipulated is representation.
Can you give us an example of where a computer is *not* manipulating a
representation ?
patty
===
Subject: Re: Turing Machines and Physical Computation
The putative difficulty in writing effective software is a more common
>and mundane version of the same thing, IMHO.
>>That's actually a quite different problem.
>>Computation is the manipulation of representations.
> I am not sure if that's a good description.
This formal symbol manipulation idea got some otherwise ambitious
> philosophers like Brian Cantwell Smith and gang quite confused.
Computers can work on representations, that's true. But it is not
> necessary that what is being manipulated is representation.
Can you give us an example of where a computer is *not* manipulating a
> representation ?
Consider the generator of {0,1}*
--
Eray
===
Subject: Re: Turing Machines and Physical Computation
>The putative difficulty in writing effective software is a more common
>and mundane version of the same thing, IMHO.
>>That's actually a quite different problem.
>>Computation is the manipulation of representations.
> I am not sure if that's a good description.
> This formal symbol manipulation idea got some otherwise ambitious
> philosophers like Brian Cantwell Smith and gang quite confused.
> Computers can work on representations, that's true. But it is not
> necessary that what is being manipulated is representation.
> Can you give us an example of where a computer is *not* manipulating a
> representation ?
> patty
Patty, weren't you the one who suggested that representation is only
meaningful in the sense of representing something to an interpreting
observer? The machine goes through a sequence of changes in state.
Normally the people who program it think of some elements of those states
as
representing something... numbers, characters, cars, hurricanes, chess
pieces, whatever. But there may be ambiguity about what a particular
element represents, depending on ones point of view. There is nothing
about
for example the state of a memory location that makes it inherently a
representation of any particular type of thing. And if a computer were to
be executing some arbitrary (random, or of unknown origin) sequence of
operations, which was not a representation of anything particular to any
existing observer, I'm not sure in what sense you could say that it was
manipulating a representation. It is just going through a sequence of
causally dependent material state changes.
Of course, one could take the position that such a sequence is no longer a
computation. One could say that computing *means* manipulating
representations. Which might be a good thing, since it explains why we
are
uncomfortable saying that a projectile or a planet computes its
trajectory. We only ascribe computation to physical processes that we
take as representing other things. And with representation
dependendent
on an interpreting observer, all arguments about whether there really *are*
computations and representations in the brain or anything else become
moot... it all depends on whether you choose to so interpret them.
Bill
===
Subject: Re: Turing Machines and Physical Computation
>The putative difficulty in writing effective software is a more common
>and mundane version of the same thing, IMHO.
>>That's actually a quite different problem.
>>Computation is the manipulation of representations.
>I am not sure if that's a good description.
>This formal symbol manipulation idea got some otherwise ambitious
>philosophers like Brian Cantwell Smith and gang quite confused.
>Computers can work on representations, that's true. But it is not
>necessary that what is being manipulated is representation.
>>Can you give us an example of where a computer is *not* manipulating a
>>representation ?
>>patty
> Patty, weren't you the one who suggested that representation is only
> meaningful in the sense of representing something to an interpreting
> observer?
Yep that was me :) But i think the complete relationship should be a
four place predicate like: (represents x P x' P') where some agent (A)
interprets that x' in the process P' represents x in the process P. The
agent should also establish that there is some causative relationship
between x and x'; and that there is some measure in which x stands to P
as x' stands to P'. I think most natural language understandings of
represents can be coded in that form.
Let's say x is the number of dollars in a purchase transaction (P). x'
is that number in a computer memory, and P' might be a program
simulating the transaction, written by the programmer sitting as the
interpreting agent (A). All the slots in the predicate are filled, right?
But there is no reason that P' cannot *be* the interpretive agent (A).
Let say that x is a coffee cup, P is my desk, x' is the activity in my
skull directly caused by seeing the cup, and P' is other activity in my
head conditioned by my desk, and i am interpreting my perception of the
coffee cup to *be* the coffee cup. Same predicate, all the slots are
filled; but A == P', right?
I think the question is how can the running computer become the
interpretive agent; and when we solve that we might be closer to
creating AI. In other words: what is interpretive behavior?
> The machine goes through a sequence of changes in state.
> Normally the people who program it think of some elements of those states
as
> representing something... numbers, characters, cars, hurricanes, chess
> pieces, whatever. But there may be ambiguity about what a particular
> element represents, depending on ones point of view. There is nothing
about
> for example the state of a memory location that makes it inherently a
> representation of any particular type of thing. And if a computer were
to
> be executing some arbitrary (random, or of unknown origin) sequence of
> operations, which was not a representation of anything particular to any
> existing observer, I'm not sure in what sense you could say that it was
> manipulating a representation. It is just going through a sequence of
> causally dependent material state changes.
Well you are skirting very close to protoplasmic chauvinism. If the
computer is functioning without a programmer, then we have the same case
as me and my coffee cup. If it walks and talks like a duck, we should
call it a duck.
> Of course, one could take the position that such a sequence is no longer
a
> computation. One could say that computing *means* manipulating
> representations. Which might be a good thing, since it explains why we
are
> uncomfortable saying that a projectile or a planet computes its
> trajectory. We only ascribe computation to physical processes that
we
> take as representing other things. And with representation
dependendent
> on an interpreting observer, all arguments about whether there really
*are*
> computations and representations in the brain or anything else become
> moot... it all depends on whether you choose to so interpret them.
I don't understand this whole drive to make the word computation means
something other than just some kind of normitive activity. There is the
activity of a planet going around the sun. We can describe that
activity however, and we can interpret that activity to stand for
something else, and the latter represent the former. I fail to conceive
how the planet itself going around the sun could ever participate in the
interpretive behavior. What does bringing in this computation thingy
add to our predication of representations or help us understand
interpretive behavior? I don't get it.
patty
===
Subject: Re: Turing Machines and Physical Computation
<5uniq0trsetjjkql36v7vv8m4bhalsurd6@4ax.com>
<44ckq09b9mb3ehign5vac2lgi6fqet27ap@4ax.com>
The putative difficulty in writing
effective software is a more
>>common
>>and mundane version of the same thing, IMHO.
That's actually a quite different problem.
Computation is the manipulation of representations.
>>I am not sure if that's a good description.
>>This formal symbol manipulation idea got some otherwise ambitious
>>philosophers like Brian Cantwell Smith and gang quite confused.
>>Computers can work on representations, that's true. But it is not
>>necessary that what is being manipulated is representation.
>Can you give us an example of where a computer is *not* manipulating a
>representation ?
>patty
>> Patty, weren't you the one who suggested that representation is
>>only
>> meaningful in the sense of representing something to an interpreting
>> observer?
>Yep that was me :) But i think the complete relationship should be a
>four place predicate like: (represents x P x' P') where some agent (A)
>interprets that x' in the process P' represents x in the process P. The
>agent should also establish that there is some causative relationship
>between x and x'; and that there is some measure in which x stands to P
>as x' stands to P'. I think most natural language understandings of
>represents can be coded in that form.
>Let's say x is the number of dollars in a purchase transaction (P). x'
>is that number in a computer memory, and P' might be a program
>simulating the transaction, written by the programmer sitting as the
>interpreting agent (A). All the slots in the predicate are filled,
>right?
>But there is no reason that P' cannot *be* the interpretive agent (A).
>Let say that x is a coffee cup, P is my desk, x' is the activity in my
>skull directly caused by seeing the cup, and P' is other activity in
>my head conditioned by my desk, and i am interpreting my perception of
>the coffee cup to *be* the coffee cup. Same predicate, all the slots
>are filled; but A == P', right?
>I think the question is how can the running computer become the
>interpretive agent; and when we solve that we might be closer to
>creating AI. In other words: what is interpretive behavior?
>> The machine goes through a sequence of changes in state.
>> Normally the people who program it think of some elements of those states
as
>> representing something... numbers, characters, cars, hurricanes, chess
>> pieces, whatever. But there may be ambiguity about what a particular
>> element represents, depending on ones point of view. There is nothing
about
>> for example the state of a memory location that makes it inherently a
>> representation of any particular type of thing. And if a computer were
to
>> be executing some arbitrary (random, or of unknown origin) sequence of
>> operations, which was not a representation of anything particular to
any
>> existing observer, I'm not sure in what sense you could say that it was
>> manipulating a representation. It is just going through a sequence of
>> causally dependent material state changes.
>Well you are skirting very close to protoplasmic chauvinism. If the
>computer is functioning without a programmer, then we have the same
>case as me and my coffee cup. If it walks and talks like a duck, we
>should call it a duck.
>> Of course, one could take the position that such a sequence is no longer
a
>> computation. One could say that computing *means* manipulating
>> representations. Which might be a good thing, since it explains why we
are
>> uncomfortable saying that a projectile or a planet computes its
>> trajectory. We only ascribe computation to physical processes that
we
>> take as representing other things. And with representation
dependendent
>> on an interpreting observer, all arguments about whether there really
*are*
>> computations and representations in the brain or anything else become
>> moot... it all depends on whether you choose to so interpret them.
>I don't understand this whole drive to make the word computation
>means something other than just some kind of normitive activity. There
>is the activity of a planet going around the sun. We can describe that
>activity however, and we can interpret that activity to stand for
>something else, and the latter represent the former. I fail to
>conceive how the planet itself going around the sun could ever
>participate in the interpretive behavior. What does bringing in this
>computation thingy add to our predication of representations or help us
>understand interpretive behavior? I don't get it.
>patty
I predict that all you will see (are seeing) is a progressive (albeit
necessarily gradual) change in (natural) language where the
intensional idioms of propositional attitude etc are replaced by
extensional constructions.
As you can't see this, and how it's done (or why it's being done) you
keep arguing. That's just because you don't see what you are doing
yourself (cf. awareness of your own behaviour).
Learn a little more instead of arguing.
--
David Longley
http://www.longley.demon.co.uk
===
Subject: Re: Turing Machines and Physical Computation
>The putative difficulty in writing effective software is a more common
>and mundane version of the same thing, IMHO.
>>That's actually a quite different problem.
>>Computation is the manipulation of representations.
>I am not sure if that's a good description.
>This formal symbol manipulation idea got some otherwise ambitious
>philosophers like Brian Cantwell Smith and gang quite confused.
>Computers can work on representations, that's true. But it is not
>necessary that what is being manipulated is representation.
>>Can you give us an example of where a computer is *not* manipulating a
>>representation ?
>>patty
> Patty, weren't you the one who suggested that representation is only
> meaningful in the sense of representing something to an interpreting
> observer? The machine goes through a sequence of changes in state.
> Normally the people who program it think of some elements of those states
as
> representing something... numbers, characters, cars, hurricanes, chess
> pieces, whatever. But there may be ambiguity about what a particular
> element represents, depending on ones point of view. There is nothing
about
> for example the state of a memory location that makes it inherently a
> representation of any particular type of thing. And if a computer were
to
> be executing some arbitrary (random, or of unknown origin) sequence of
> operations, which was not a representation of anything particular to any
> existing observer, I'm not sure in what sense you could say that it was
> manipulating a representation. It is just going through a sequence of
> causally dependent material state changes.
You still may be left with the fact that it is manipulating symbols.
Symbols without semantic reference, but (ambiguous) symbols nonetheless.
I agree with the spirit of your post, though, and feel that people who
quibble with it don't really understand the guts of computers. They
think there's something magical going on, something more than physics
and materials science.
The question to me is whether the same applies to the human mind, where
most of the AI questions deserve to be aimed squarely at: what is it
that we insist is unique about our own processes? Just how did we see
fit to separate ourselves from the remainder of the animal kingdom, and
natural processes as a whole? Why do we concieve of our own mentality
as any different -- except in scale and physical substrate?
> Of course, one could take the position that such a sequence is no longer
a
> computation. One could say that computing *means* manipulating
> representations. Which might be a good thing, since it explains why we
are
> uncomfortable saying that a projectile or a planet computes its
> trajectory. We only ascribe computation to physical processes that
we
> take as representing other things. And with representation
dependendent
> on an interpreting observer, all arguments about whether there really
*are*
> computations and representations in the brain or anything else become
> moot... it all depends on whether you choose to so interpret them.
My little thought experiment is this: is is possible to concieve of a
computer existing independently of humans? For example, a pile of
sticks, pebbles on a beach, falling rocks, waves, etc. which are
computing autonomously (not derived from human engineering)? If not,
then I submit that the argument is couched in anthroporphism.
If so, but the answer is Only sentient species. For example, an
intelligent alien, non-human, species can concievably build bona fide
computing machinery. then I wonder whether there's some other
chauvenism at play there. And so, the question is already begged:
inorganic substrated intelligence cannot possibly exist independently of
biologically-derived mechanisms which provide the impetus.
So there's no point in trying to build AI: the decision is already made.
Indeed, the machinery has been damned. Came from a human or
intelligent species? Then it cannot possibly be, itself, intelligent.
I just don't like where that argument goes. It's based on some sort of
dualism to be sure.
For one thing, humans themselves -- as well as our machinery -- are
human derived. Both physically (egg+sperm) and
culturally/linguistically/intellectually.
===
Subject: Re: Turing Machines and Physical Computation
>We clearly disagree here. That may have something to do with why
>we disagree about computation and about mathematics.
I'm clearly making tendentious statements that could use a whole lot
more support than I'm giving them here. Discussed at great length, I
suspect we would agree more than it might seem. Wish I had time for
it, but am currently working a job at the end of a long commute.
>That's actually a quite different problem.
>Computation is the manipulation of representations. The difficulty
>in writing effective software, is because we don't start with
>representations. Rather, we start with a real world problem of some
>kind. Thus we must first find a way to reduce the problem to one of
>manipulating representations. Until we have done that, it is not a
>computational problem.
But I want to view computation ONLY as this wider problem, and
sporadically at least, so do you! And, I want to treat software
development as at least a rough equivalent to the issues of
computation.
>No, it is not that at all. It seems that the idea of software as the
>mechanical solving of problems (i.e. automation) has gone the way of
>the dodo. These days, software is all about writing GUI interfaces
>and other kinds of visual candy, so that we can keep people amused as
>they do the work that we are unable to automate.
>And perhaps automation has become less valuable, now that we can
>outsource the labor-intensive work to other places.
Gee I wish I said that ...
J.
===
Subject: Re: Turing Machines and Physical Computation
-----BEGIN PGP SIGNED MESSAGE-----
Hash: SHA1
>>Computation is the manipulation of representations. The difficulty
>>in writing effective software, is because we don't start with
>>representations. Rather, we start with a real world problem of some
>>kind. Thus we must first find a way to reduce the problem to one of
>>manipulating representations. Until we have done that, it is not a
>>computational problem.
>But I want to view computation ONLY as this wider problem, and
>sporadically at least, so do you!
The problem is far wider than you suspect -- too far off course to be
considered computation.
> And, I want to treat software
>development as at least a rough equivalent to the issues of
>computation.
There are many social issues that must be considered in software
development. It isn't just computation. As an example, consider the
social engineering methods used in distributing computer viruses.
-----BEGIN PGP SIGNATURE-----
Version: GnuPG v1.3.91 (SunOS)
iD8DBQFBrAVrvmGe70vHPUMRAtbyAJ41LAZDVpuA6YTjw9vI09LdaPJznACgyd8F
L7la3vQscJiR3MqegIY6kPw=
=IYCq
-----END PGP SIGNATURE-----
===
Subject: Re: What on earth! was Re: Turing Machines and Physical
Computation
> Abstract does not mean non-physical
> the Turing Machine is just another kind
> of automata, and its mechanism is firmly rooted in the physical world.
> Incuding its infinite tape no doubt
/me hands Robin an unbounded tape as a slot-in replacement
Phil
--
I used to have an interest in writing viral code and lost interest
quickly when Win95 came out. Hell how could any of us in the scene
write a more invasive program than Win95. It made us all obsolete.
-- Screaming Radish [NuKE] on alt.comp.virus.source.code
===
Subject: Re: What on earth! was Re: Turing Machines and Physical
Computation
>> Eray Are there integers with an infinite number of digits? Ozkural
>> Abstract does not mean non-physical
>> the Turing Machine is just another kind
>> of automata, and its mechanism is firmly rooted in the physical world.
>> Incuding its infinite tape no doubt
> /me hands Robin an unbounded tape as a slot-in replacement
Is that a potentially infinite tape? :-)
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: What on earth! was Re: Turing Machines and Physical
Computation
>> Eray Are there integers with an infinite number of digits? Ozkural
>> Abstract does not mean non-physical
>> the Turing Machine is just another kind
>> of automata, and its mechanism is firmly rooted in the physical
world.
>> Incuding its infinite tape no doubt
/me hands Robin an unbounded tape as a slot-in replacement
> Is that a potentially infinite tape? :-)
You'll find out eventually. Or maybe you won't.
Phil
--
I used to have an interest in writing viral code and lost interest
quickly when Win95 came out. Hell how could any of us in the scene
write a more invasive program than Win95. It made us all obsolete.
-- Screaming Radish [NuKE] on alt.comp.virus.source.code
===
Subject: Re: Turing Machines and Physical Computation
>ON THE COMPUTER METAPHOR
>'It has always bothered me that models of psychological
>processing were thought to be inspired by our understanding of
>the computer. The statement has always been false. Indeed, the
>architecture of the modern digital computer - the so-called Von
>Neumann architecture - was heavily influenced by people's (naive)
>view of how the mind operated. Perhaps I had better document
>this. Simply read the work on cybernetics and thought in the
>1940's and 1950's prior to the development of the digital
>computer. The group of workers included people from all
>disciplines: See the Macy Conferences on Cybernetics, or Her
>Majesty's Conference on Thought processes. Read the preface to
>Wiener's book on cybernetics. Everyone who was working together -
>engineers, physicists, mathematicians, psychologists,
>neuroscientists (not yet named) - consciously and deliberately
>claimed to be modelling brain processes.'
>Reflections on Cognition and Parallel Distributed Processing
>D.A. Norman
>(Ch 26, p534, Parallel Distributed Processing Volume 2)
>McClelland J and Rumelhart D 1986
>>I get that computer behavior and human behavior are essentially
>>different. Why they are so different is certainly what we came here to
>>discuss.
> Hi patty, every couple of weeks I click on a post totally at random to
> find the same silliness repeated, with 0.999 confidence.
> The problem with presenting a disembodied fragment is that it possibly
> implies what the fragmenter wants it to implie, but it hardly tells
> what the original writer actually meant. Follows are parts of the
> **REST** of Norman's story .... [caps are mine] ...
> .... here we are talking about a new form of COMPUTATION ...
> ... these MODELS are highly parallel, with thousands of elements
> INTERACTING primarily through actviation and inhibition ...
> ... each ELEMENT is highly interconnected with perhaps tens of
> thousands of connections ....
> ...these NEUROLOGICALLY INSPIRED COMPUTATIONAL PROCESSES pose new
> requirements on our understanding of computation, suggest novel
> THEORETICAL EXPLANATIONS of PSYCHOLOGICAL PHENOMENA, and suggest
> powerful new ARCHITECTURES for MACHINES OF THE FUTURE ....
> .... carry on a tradition that has long existed .... a tradition of
> BUILDING MODELS of NEUROLOGICAL PROCESSES ....
> [and later on .....]
> ... the whole point for the cognitive scientist is to UNDERSTAND
> COGNITION ... to do so, we insist upon explanation of the INTERNAL
> PROCESSING STRUCTURES THAT GIVE RISE TO COGNITIVE ACTIVITIES ....
> .... this is why we have spoken of REPRESENTATION, of MECHANISMS of
> memory and attention ...
Point taken :) Making collages of fragments of other people's writings
is art, not communication. What these collages refer to are mentalisms
in the head of the author of the collage.
patty
===
Subject: Re: Turing Machines and Physical Computation
<41a3ef28$0$576$b45e6eb0@senator-bedfellow.mit.edu>
<41a4a697$0$563$b45e6eb0@senator-bedfellow.mit.edu>
ON THE COMPUTER METAPHOR
>>'It has always bothered me that models of psychological
>>processing were thought to be inspired by our understanding of
>>the computer. The statement has always been false. Indeed, the
>>architecture of the modern digital computer - the so-called Von
>>Neumann architecture - was heavily influenced by people's (naive)
>>view of how the mind operated. Perhaps I had better document
>>this. Simply read the work on cybernetics and thought in the
>>1940's and 1950's prior to the development of the digital
>>computer. The group of workers included people from all
>>disciplines: See the Macy Conferences on Cybernetics, or Her
>>Majesty's Conference on Thought processes. Read the preface to
>>Wiener's book on cybernetics. Everyone who was working together -
>>engineers, physicists, mathematicians, psychologists,
>>neuroscientists (not yet named) - consciously and deliberately
>>claimed to be modelling brain processes.'
>>Reflections on Cognition and Parallel Distributed Processing
>>D.A. Norman
>>(Ch 26, p534, Parallel Distributed Processing Volume 2)
>>McClelland J and Rumelhart D 1986
>I get that computer behavior and human behavior are essentially
>different. Why they are so different is certainly what we came here
>to discuss.
>> Hi patty, every couple of weeks I click on a post totally at
>>random to
>> find the same silliness repeated, with 0.999 confidence.
>> The problem with presenting a disembodied fragment is that it
>>possibly
>> implies what the fragmenter wants it to implie, but it hardly tells
>> what the original writer actually meant. Follows are parts of the
>> **REST** of Norman's story .... [caps are mine] ...
>> .... here we are talking about a new form of COMPUTATION ... ...
>>these MODELS are highly parallel, with thousands of elements
>> INTERACTING primarily through actviation and inhibition ...
>> ... each ELEMENT is highly interconnected with perhaps tens of
>> thousands of connections ....
>> ...these NEUROLOGICALLY INSPIRED COMPUTATIONAL PROCESSES pose new
>> requirements on our understanding of computation, suggest novel
>> THEORETICAL EXPLANATIONS of PSYCHOLOGICAL PHENOMENA, and suggest
>> powerful new ARCHITECTURES for MACHINES OF THE FUTURE ....
>> .... carry on a tradition that has long existed .... a tradition of
>> BUILDING MODELS of NEUROLOGICAL PROCESSES ....
>> [and later on .....]
>> ... the whole point for the cognitive scientist is to UNDERSTAND
>> COGNITION ... to do so, we insist upon explanation of the INTERNAL
>> PROCESSING STRUCTURES THAT GIVE RISE TO COGNITIVE ACTIVITIES ....
>> .... this is why we have spoken of REPRESENTATION, of MECHANISMS of
>> memory and attention ...
>Point taken :) Making collages of fragments of other people's
>writings is art, not communication. What these collages refer to are
>mentalisms in the head of the author of the collage.
>patty
No, that is not the point you should take from this. That you do so just
makes you look as stupid as Michaels. What was it that Norman said about
the computer and the brain, and what has been shown about the
properties of so called ANNs apropos human and other animal behaviour?
What is it that they model do you think? (You haven't grasped how this
relates to natural stupidity)
The reason why you. Michaels etc don't pick up on any of this properly
is because you aren't trained in psychology. You don't have the context
or web if you like, to be able to make sense of it properly. The context
you have is that of an enthusiastic amateur with skills in other areas.
Those skills just don't equip you to read or talk in other fields and
that's the problem I have been illustrating. If you look to what many
people in AI or Cognitive Science get up to, you'll see this
repeated my many people. They commit the genetic fallacy. That is, they
may have skills in mathematics or programming, and they just assume they
have the ability or expertise in fields where in fact they have little
or none - in fact they have nothing more than pre-scientific folk
psychology. Why would Michaels or you presume otherwise one might ask
(one can ask this same question of many of his luminaries too).
Read the section of Fragments on transfer of training.
All you're actually doing here is reinforcing both your own and
Michaels' stupidity. The quote from Norman was simply to show what
people were doing back in the 40s and 50s. That they were doing that
doesn't mean that what they were doing was sound any more than Michaels'
or Norman's talk of ANNs being neurally inspired is sound! What these
models show is how we can model assumption violation, overfitting etc -
problems well known to those of us who use inferential statistics and
build models - hence construction, validation samples, reliability
measures, shrinkage etc. There is a subtle point being made about what
is being modelled which you are all missing.
See the two *other* sections of Fragments on a) ANNs *and* b)
hypothesis testing.
What is *your* behaviour illustrating?
--
David Longley
http://www.longley.demon.co.uk/Frag.htm
===
Subject: Re: The flux theory of gravitation
> Hi Folks,
> Since some of you have made references to the Flux Theory of
> Gravitation that I developed, IÍll offer some details.
Present
> mathematical physics uses conventional modeling to DESCRIBE nature by
> mathematical spaces, functions, equations and inequalities. Its main
> tool is computation. This is inadequate, the reason there are
> unsolved problems in mathematical physics such as finding the basic
> constituent of matter, the gravitational n-body problem and the
> structure of the electron. To overcome this difficulty I have
> introduced dynamic modeling that EXPLAINÍS nature, physical
systems
> and natural phenomena in terms of the laws of nature.
There is only one catch. Most of what Nature does or is is literally out
of our sight. We can only know it indirectly. Therefore we must rely on
hypotheses and inference, as opposed to direct knowledge. There only way
we can do that is by means of models grounded on physical hypotheses and
mathematical principles. Thus to EXPLAIN means to PREDICT and that is
what our theories do. The soundness of our theories rests on two
requirements.
1. The theories must be internally consistent (in the mathematical and
logical sense).
2. The predcitons must be empirically corroberated. Agreement with
experiment is the sine qua non of a sound theory.
If you think Nature can be deduced a priori from logically necessary
principles, you are in for dissapointment. Science is empirical, right
down to the basement level.
Bob Kolker
===
Subject: Re: The flux theory of gravitation
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id iB1DHw929472;
>> Since some of you have made references to the Flux Theory of
Gravitation
>> that I developed,
>They seem to be mathematicians. I requested
>them to define NUMBER a week ago but I have not seen their names
>ever since. Are they undergrads or mathematicians from antiquity?
>Some of my students might agree with the latter...
>Robert Israel israel@math.ubc.ca
>Department of Mathematics http://www.math.ubc.ca/~israel> J.E.:
>> >> Then you do not even know what the PoR is.
>> > >I'm sure you MEANT to say Principle of Relativity which means
that
>> >the laws of physics have the same form in all inertial frames of
>> >reference. It's a stupid principle which the modern theory makes
>> >unnecissary. It's a waste of time and just confuses people like
you.
>>
>> Except for the part about androsleaze being confused, I disagree and
>> I would say that your comment below is inconsistent with your comment
about
>> inertial frames.
>What IS there to be inconsistent. I don't USE the PoR. I use
>Minkowski geometry, so there are EVEN two things TO contradict each
>other, just ONE thing, Minkowksi geometry.
> [...]
>> >I was saying that Minkowksi geometry is based on set theory. SR is
>> >based on Minkowksi geometry.
>> Sure, but without the principle of relativity to connect that
geometry
>> to physics, all you have is minkowski geometry with no physical
meaning.
>> I could equally well say that any geometry is based on set theory, but
>> in order to decide which one applies to the universe, I also need a
>> connection to physical phenomena. That connection is the principle of
>> relativity, since what it requires is that physics be unchanged by an
>> infinitesimal displacement of the spacetime variables.
>I disagree wholeheartedly and completely. What you need is a mapping
>between physical events and minkowski geometry, and then a procedure
>for computing future observables in the geometry, which can then be
>mapped back to observations in reality. PoR is redundant, unecissary
>and confusing. Drop it. The symmetries of Minkowski geometry do all
>the work FOR you, you don't NEED to do anything else. Did Terrance's
>clock tick? Find the event. Did Stella see it? Trace a line to find
>out what event corresponds to Stella seeing it. What OTHER events
>that Stella saw did see observe it BETWEEN, that is ALL you need, the
>rest is fluff and if fluff confuses people, bag it. Body bag it.
>
>[snip disparagements against Androcles]
> But that won't obviously tell you whether the resulting theory of
> physics is one that our universe feels compelled to obey.
There simply is no way to THAT. The best you can hope is the do
experiments the distinguish between model/theory pairs that make
different predictions, and then everyone picks their favoriate
model/theory pair that DOES match all observations. Anything more is
impossible and so shouldn't be a standard.
> In Minkowski geometry, an inertial object's path is a straight line
> through the metric, and the properties of that line are the same
> regardless of whether or not such an object actually exists.
> The validity of the exercise depends on the idea that the existence of
> a real object, or the motion of a real object, has no effect on the
> lightbeam geometry.
> light along in their immediate vicinities (Fizeau), and even a
> reduction in lightspeeds (refractive index).
Do you not understand the concept of a vacuum or are you just joking?
> Special relativity does not claim validity for particulate media ...
> be a particulate medium of some sort. If you are an astronaut, your
> moving twin ought to be able to declare that the speed of light is
> reduced where it begins to encounter the glass of your helmet visor,
> and that that moving glass also produces an offset in the nearby speed
> of light in your direction of motion.
You can model light through non-vaccums in terms of straight lines in
Minkowski geometry too, you just need more than one line.
> So, while Minkowski geometry might well describe how the principle of
> relativity would apply to mathematical non-particulate observers
> exchanging signals, the real geometry of real particulate objects
> exchanging signals would seem to be different.
Um, does the phrase to within experimental erros mean anything to
you. We are talking about 10W signals verus 6W signals, that's a huge
difference, there is room for much errors, like extended bodies, not
quite instanteous turnaround, etc.
> Mechanims are in play
> that alter local lightspeeds around real-life observers, that do not
> exist in special relativity.
That's nonsense. SR theory doesn't make ANY predictions without a
material model. And if you make the RIGHT material model, then OF
COURSE you can models to effects of glass, non instant turn arounds,
extended bodies, etc. And obviously if you make a SIMPLER model that
ignores those things then it won't be 100% accurate. So what? No big
deal. Besides, right now I was discussing INTERNAL inconsistencies.
> Start with an empty perfect Minkowski
> metric and throw a rock through the region, and the rock ought to
> distort the metric and leave a sort of streak running through the
> block of spacetime, like an optical imperfection in a block of glass.
You are a big GR believer aren't you? Yes, things affects things, for
instance you can't have a laser be absorbed and emitted and yet be a
single wavelength, so all my equal separation stuff is impossible,
but GET THIS, for large amounts of time, the BIG effect 10 > 6 CAN be
observed EVEN dispite the simplicity of the model because we can
RESTRICT ourselves to the cases where the other effects are very
small, get it?
> relativity, then we need a theory that works consistently in
> curved-spacetime,
I was write you do believe in GR. Well get this, we can ALSO have a
background spacetime flat.
> and a metric for the description of inertial physics
> that is /not/ purely Minkowskian. Sure, we probably want to start with
> a flat background, but we need the physics played out against that
> flat background to incorporate localised velocity-dependent
> distortions.
Of EITHER a field OR the metric, I know which I'd prefer, ...
whichever has the eeasier mathematics of course.
> Q: Does SR cope robustly with this sort of velocity-dependent
> curvature?
> A: No, apparently not
Fallacy: Confusing models and theories.
> Q: Does GR currently incorporate these effects?
> A: Apparently no, not yet. Because people insist that it has to reduce
> to SR.
Interesting, but I've never EVER heard that before, so please
elaborate and/or provide references.
> Q: What research do we currently have on methods of amending SR and
> the Minkowski metric to incorporate velocity-dependent curvature?
> A: Apparently it can't be done, we seem to need a new type of metric
> and a new associated theory of how the deviations from the SR metric
> can also be relativistic.
I keep imagining everything you say about curvature being about
magnetism, you know that velocity dependant force. Rememeber how they
made a FIELD theory out of magnetism. Too bad we can't do the same
for gravity, wait! We CAN. Whew, you had me worried for a second.
> Q: What progress have we made so far on identifying these new
> relativistic rules and constructing the next-generation theory?
> A: Apparently none. Instead, we find it easier to insist that we know
> that spacetime is flat, and that SR/Minkowski is therefore known to be
> correct (even though experiment seems to say otherwise) ,and that
> these issues therefore do not need to be addressed.
More references to experimental verification of SR from a person who
can't distinguish a model from a theory. Go ahead, try and support
your claims.
> So, I think that we can play around with Minkowskian geometry to our
> heart's content, but what we discover still won't necessarily be real
> physics, because in real life, that's not how real matter seems to
> behave.
Yeah real electrons don't have the hyperfine corrections predicted by
AFT, how silly of me.
> Saying that these deviations form the idealised metric are small and
> don't need to be considered is not a legitimate response, IMO, that
> would be like saying that small deviations from energy conservation
> in a theory would be unimportant.
Saying that they have to be large enough to observe and predict IS a
requirement, since we can't MAKE a theory match your PERSONAL FEELINGS
about deviations.
> If there are localised
> velocity-dependent deviations from flat spacetime with real physical
> matter, we have to know how to deal with them relativistically.
The ONLY hard core proof of curvature (non-flatness) as opposed to
field interaction would be a non-euclidean topology. Do you have
evidence for that?
> With the Minkowski geometry generated by SR, we are using an idealised
> description of reality whose form we know breaks down when the
> idealisation is not perfect (usually a sign of a pathological theory),
> and the usual sanity test that we'd do in this situation -- exploring
> how the theory would come out if the idealisation was lost -- does not
> seem to have been done.
People have done it, their theories are more complicated, and frankly
since I've never heard of a necissity to USE them, I've never much
studied them.
> We can attempt to come up with a more realistic theory of relativity
> by discarding the Minkowski geometry and relationships and deriving a
> new theory of relativity from scratch -- but if we followed your
> suggestion of keeping the Minkowskian geometry and discarding the PoR
> as superfluous, then we'd appear to be cutting ourselves off from all
> hope of progress in this area.
Yeah, we'd never make a field theory based on observations, that
didn't happen over ten years ago where the flat Minkoski metric is
UNobservable and ONLY exists to MAKE THE MATH EASY. I must have
imagined that. I better alert Cambridge that their paper don't exist
I do wonder what those speakers were REALLY saying at those conflicts.
wouldn't have known. Either that, or you are WRONG and just attacking
a strawman because you don't understand the difference between a model
and a theory.
> IMO we'd then be locking ourselves into a theory that seems to be
> geometrically incompatible with real-life physics, and throwing away
> the key.
Or making math easy. Sheese, since I allow CURVES and ARBITRARY
lines, you could KEEP my model and use a different theory, and you
could keep the theory and use another model. Sheesh, the advantage of
the Minkowski geometry is to CLASSIFY and QUANTIFY certain lines in
certain ways. So it's just to make the math easy.
> IMO, if there's a possibility that we might have made a mistake in
> adopting SR so wholeheartedly, then we are obligated to check --
> either to put SR onto a new stronger footing, or to isolate the
> problem areas and the parts of physics that would have to be changed
> in the successor theory.
I won't stop you, if you want to do things that make the math harder
that don't affect any observations or experiments we have done or plan
to do, go for it. If it turns out we need it, I'll be grateful. But
please stop harassing me and making unfounded allegations about my
inability to add fields to a model to have more detail, because
you're wrong, and I can do it.
> Until that's done, we don't know for sure whether SR is deserves to be
> considered as hardcore physics, or whether its just a sidebranch that
> gives a Euclidean thumbnail sketch of some intrinsically non-Euclidean
> physics, and which provides a crude approximation of parts of the
> final theory, without being a full subset of it.
I have no idea why you mention Euclidean anything with reference to
Minkowksi based models. Do you not know what the words mean? Is this
a typo? Or did I just miss something?
> Minkowskian geometry may be self-consistent as a piece of abstract
> geometry, but that does not automatically make it physics.
Didn't say it did. But others were saying that certain SR predictions
that were predictions of an ELEMENTARY Minkowki model WERE
inconsistent, hence the motivation for this VERY SIMPLE model.
> It also
> does not automatically make it consistent with more general correct
> physics theory, in which some of the idealisations applied within
> Minkowski geometry might be incompatible with the more general
> principles being applied (eg GR says that localised energy
> concentrations warp spacetime, SR requires an arbitrarily-high
> metric -- one could argue that this should not be a legal situation,
> on principle, and that perhaps a full general theory ought not to
> reduce to SR, on principle).
Metric-fixation. You don't OBSERVE the metric, sorry to disappoint
you.
> Physics theory based upon Minkowski's metric may still turn out to be
> consistently right in some respects, but consistently wrong in
> others. To find out, we probably have to step outside the closed
> Euclidean mindset and see how the problem might look from outside.
No, to find out each side has to make models and theories and then
we have to compare to observations. I choose to make models and
theoires where the math is EASY for ME to do, if it's so hard that I
can't make either then I don't accomplish much of ANYTHING. So GO
AHEAD and do the math that is HARD for ME, and let's COMPARE to
observation. But saying A PRIORI how metric must be related to
observation in MY models and MY theories is CHEATING. Make your best
model, and compare to my best model, anything else is less than
honest.
===
Subject: Re: SR consistency is crap.
>>I'd rather teach QM from the start, and get special relativity as a
>>limiting case of that, and then Newtonian dynamics as a special case
>>of that.
>> How do you expect to get special relativity as a limiting case of QM?
>> David
>comes as a limiting case. Many of the so-called quantum corrections
>are actually present in classical SR, so in a sense QM has always been
>SR theory.
> When it comes to gravitational horizons, it seems that the NM energy
> relationships lead to a classical model that agrees (qualitatively,
> perhaps also quantitatively) with QM, whereas the SR relationships
> lead to GR's current description of the horizon as being inescapable,
> which is apparently not compatible with QM.
I have absolutely no idea what you are talking about. First, because
of time dilation I've never actually seen anything cross an event
horizon and I doubt can, so by Hawking radition the black hole
evaporates before you reach it. So the only information inside an
event horizon is from the initial collapse and that information is
sent out during the collapse.
Plus you can get gravitational theories in a flat SR metric, and the
fields can move at any speed relative to the background space, so I
don't get your concerns about that either.
> So in this sense, it does rather seem as though perhaps QM has always
> been /incompatible/ with SR theory.
Have you studied a serious SR based model of QM with gravity, there
are more than a few to pick from you know, and I don't know any
published ones that create problems like you mention.
> (SR's concept of how a lightspeed barrier works seems to be too
> clean to be compatible with QM, GR then inherits certain SR
> conventions and converts that clean lightspeed barrier into a
> clean event horizon, which makes the conflict more obvious --
> according to QM, information should be able to bleed outwards through
> the horizon, current GR says that it can't).
Current GR is a HORRIBLE place to do QM, so use a quantum
gravitational theory instead. There are quantum gravitational
theories that match all present data AND are even more restrictive
than GR since they restrict topologies.
===
Subject: Re: SR consistency is crap.
>> I took Physics 101 and Modern Phycsi 101 together during the same
>> semester as soon as I started undergraduate. The mentor protested
and
>> argued I should take Modern Physics (Relativity) after I take
>> Classical Mechanics, because that was a prerequisite. I protested,
I
>> argued it must be my choice not theirs. I won. I got a A- in
Classical
>> and an A+ in Modern.
>> Good for you. I didn't even realize I liked physics until Modern
>> Physics (which for us covered QM, solid state, Stat mech, etc. too)
I
>> just took it as an easy class since I was good at math.
>> Yesy, interesting to see that while it is an easy subject for some
>> people it is beyond comprehension for others.
>I think it's bad teaching. If I had a good background to easily
>understand a fixed way of teaching the subject, that doesn't make the
>subject inherently hard for others or easy for me, in means by
>background and the teaching lined up well.
> Funnily enough, I found that it was actually very easy to explain GR
> principles to complete physics newbies, with a suitable choice of
> words.
> You just say things like, increasing the strength of gravity in a
> region seems to make makes space seem more dense and time more
> rarefied, so that clocks there tick more slowly ... so if we want to
> create a map of the light-distances in a slice through the region, so
> that the distances in the map correspond to the distances in the
> region, we have to extrude the map in order ot be able to cram in the
> extra space (produce diagram of gravity-well, with a flourish).
I'd be interested in seeing that work in practise, you can really get
newbies to see that a parabola in space is a geodesic in spacetime?
> People seem to get that. You might get an occasional Gravity makes
> local time appear to go by slower? Really? Yup! Oh, Okay then
> But try to explain SR time dilation to the same person, and they'll
> usually get upset and insist that the thing is rubbish.
But how do you know you aren't using the wrong words? By thesis is
that SR is taught badly.
> So untrained people seem to be intuitively happier with supposedly
> advanced ideas like spacetime curvature than simple ones like
SR's
> Minkowski metric (I found). They seem to find most of the GR-ey
> principles easier to visualise and accept, stars bend light or
> gravity slows time or rotating stars pull stuff around with them
> seem to be easier concepts to take onboard than moving astronauts age
> slower than each other.
And there is an example of bad teaching. Astronauts don't age more
slowly than each other, you should discuss what the astronauts see,
and if they see each age slower then faster than themself, then who
aged more depends on what percentage of the observations were slower
versus faster.
> Maybe schools should teach the principles of general relativity first,
> and leave SR to the more advanced students. ;)
Maybe schools should teach things properly.
>> I think this is the way to go. Teach students everything together
as
>> competing theories and not relativity as an advance extension to
>> Newtonian Dynamics.
>> I'd rather teach QM from the start, and get special relativity as a
>> limiting case of that, and then Newtonian dynamics as a special case
>> of that.
>> Well, you are talking 1000 years from now when we will understand
>> better how quantum uncertainty gives rise to macrocosmic certainty.
>> For now, such approach can result only in confusion.
You seem to have strong opinions about things you obviously don't
>know. Bohm did a good job of explaining maro-certaintanty as a
>was decades ago. Teaching is always behind the cutting edge.
> I was taught at school that Newton's prism experiment proved that
> light was composed of seven colours, and we were made to memorise the
> seven colours and conduct the experiment ourselves and see the seven
> colours ourselves.
> Then we were given coloured filters to play with and were taught that
> white light was actually composed of three distinct colours, and we
> were shown how to conduct expeirments to prove /that/.
> No wonder the poor kids were a bit bewildered.
> I survived those classes by being arrogant enough to assume that if
> something sounded wrong it was probably bull.
Good, most things are oversimplifications to the point where they are
wrong.
> Hopefully anyone going through that syllabus with a real talent for
> physcs woudl have realised that thge problem was not with them but
> with the teaching, but I do sometimes wonder how many adepts leave
> physcs because they think,wrongly, that their misgivings about certain
> subjects are becuase they arenlt good at the subject.
> If the people taking up the subject are predominantly people who are
> more prepared to suspend disbelief than the norm, then that might not
> be good for the subject. I suppose the people who already have a
> grounding in physcs before they take the class (eg family background),
> or those who are pig-headed enough to believe that they are right and
> the system is wrong, may still get through.
You don't have to be simply pig-headed, you can just call your
teachers on anything that doesn't make sense and then disregard their
assertion if they don't back them up well enough.
> I do notice that a strangely high percentage of physics people seem to
> have physicists or teachers as parents or as elder siblings etc,
> Perhaps a support network makes it easier for one to survive
> introductory physics classes without having one's brain scrambled.
I took introductory physics by not caring about the subject.
>> But remember, you still have a problem. SR+GR are
>> axiomatic systems. No different from Euclidean geometry in that
>> respect. You gain understanding of the theories early on but also
you
>> raise the doubt in their foundations.
>> Huh? You want to avoid doubt? Science has doubt, predictions are
>> made, the predictions can be compared to data, they might pass or
>> fail.
>> Eventually, someone will lose.
>> Only theories can lose, not people, please explain more what you
mean.
>> I meant that popularization of SR, GR will turn out more questioning
>> and eventually abandoment. This is the fate of every theory that
>> becomes pupolarized.
I seriously doubt that. I think that when the correct SR theory is
>finally popularized the incorrect alleged SR theories will FINALLY
>be adandoned, and not a moment too soon, yeak!
> I think that perhaps part of why SR is such a bad subject is that
> there are probably a lot of pro-SR people strenuously insisting that
> what they were taught is right, even when it isn't.
> I think the Penrose/Terrell case illustrates this nicely, we had a
> situation where professional physcists had supposedly been pushing a
> wrong result for decades, even though it disagreed with the math,
> because they had been /taught/ that moving objects are seen to be
> contracted under SR.
> Social conditionaing overrode mathematics and geometry.
> Penrose was a mathematician who snuck his result out as a
> non-peer-revirewed letter in an obscure local journal, Terrell was an
> undergraduate who struggled for years to get his paper through peer
> review.
> So the matter was eventually tackled by a newbie and a mathematican,
> not by mature physics people.
> For some reason, these things always seem to end up being corrected by
> outsiders, the highly-trained mainstream don't seem to be willing or
> able to do it themselves.
I've considered that there might be large numbers of physicists that
think clearly about SR, but just assume that everyone else does too
and hence don't get invovled in debates about it. The irony is that
all of these physicists could compute what people SEE correctly, if
only they CARED enough to do so.
>> Those that are afraid of ending up with a loser keep SR+GR as
advance
>> subjects.
>> I'd love to have GR and SR as grade-school subjects, does that mean
>> I'm not afraid of losing? I don't know what you mean by afraid of
>> losing or lose, I do hope my predictions match the data,
that's
>> because if it's easy to be wrong, so why bother learning a theory if
>> it makes wrong predictions?
>> Remember what happened to Euclidean geometry?
>> What happened? It's still around!
>> As soos as
>> they started teaching it, thousands attempted to disprove the
Playfair
>> axiom (parallel lines never meet). The result was a
>> relative-consistency with spherical and hypoerbolic geometries.
>> Elemetary euclidean geometry was proved consistent, you can drop
>> relative-consistency and just say consistent if what you mean
is
>> that elementary hyperbolic geometry is as consistent as elementary
>> Euclidean geometry. And if you mean something other than
elementary,
>> then that useally means bringing in sets and once you've done that
>> then all become equiconsistent with set theory.
>> I agree. But I hope you recall that the consistency of Euclidean
>> geometry was under question for some time.
>Well, the same grounding to elementary euclidean geometry applies to
>elementary hyperbolic geometry, and I can do SR predictions with
>ELEMENTARY hyperbolic geometry. If you want not instantaneous
>accelerations, then I need FULL hyperbolic geometry, which is as
>consistent as FULL euclidean geometry.
> Mmmm, but the GR experience has hopefully taught everyone that
> mathematical consistency is not necessarily enough -- in physics, a
> structure also has to be appropriate.
> If basic Euclidean geometry is consistent, it doesn't neccessarily
> mean that it is appropriate or sufficient for describing lightbeam
> geometry in the presence of gravitational fields or accelerating
> masses (until you start adding dimensions), or even relativiely moving
> masses.
What was originally being debated was the internal consistency of SR
predictions. Therefore a constructed a slim version of SR without
frames or curves, just points and straight lines, which is KNOWN to be
consistent, and hence the internal consistency of the SR predictions
in my slim model should safe and garanteed.
And accelerating bodies can be handled to arbitrary precision. People
forget that the accelerating results of SR are KNOWN, and that THAT is
HOW we get the gravitational results in GR.
> So a theory or model can be completely consistent in its own
> (artificial) context, but physcally wrong when it comes to attempts to
> use it to model the behaviour of the real world.
> Mathematical theorems can appear to be rock-solid, but still be
> hopelessly wrong (or misunderstood) in the context of attempts to
> construct physical theory.
> Nuances of language can be incredibly important.
That's exactly what I'm doing, I'm contrasting a 4-d Newtonian model
that uses the manhattan metric with a minkowksi metric. Each is
internally consistent, but I'm asserting that the minkowski one would
match experiments, which I have GOTTEN to doing yet, because eleaticus
and Androcles are acting afraid of the model and I don't think you
guys are helping at all.
> BTW, This is one of the reasons why I'm still an SR sceptic
> (even though I count myself as a harcore relativist): apart from the
> fact that many or most of the experimental SR proofs seem to have been
> badly compromised, and that there still seems to me to be at least one
> major theoretical loophole that hasn't yet been dealt with (and which
> IMO should have been tackled decades ago), it's the sheer badness of
> most of the analysis.
> I don't honestly believe that physics people are usually this bad,
> without good reason.
How many people want to test SR for a living? Most good people do
that USES the results of SR as a necissary component rather than
testing SR as an end to itself.
> If SR was wrong, and the community was heroically struggling along
> with a reference theory that didn't work properly, then perhaps those
> analyses might be the best that could be achieved without exposing
> apparent conflicts with SR, and perhaps then we'd have a logical
> reason why the analyses seem to be so consistently compromised.
It's just not consistent, and there aren't problems with SR theory,
but there *might* be problems with that *some* people think is SR
theory, which as I'm saying is a problem for physics education to fix.
> If SR is /not/ a correct physcal theory, then perhaps these repeated
> screwups make some sort of sense and tell a story.
> Otherwise, if SR really is correct, I suppose the explanation would
> have to be that the whole community is just hopeless at these sorts of
> calculations, period.
These are fallacies that everyone screws up. Physicis education
literature discusses screw ups and how to avoid them in teachers as
well as students. For instance look at me, I'm not special in any
way, just a normal educator, but I don't make these erros you accuse
other people of. And there are tons of people like me. I do tend to
find frames bad news, for instance if one person hops frames, then the
distant events that were called *simultaneous* change in mid hop, so
clearly the notion of simultaneous is rather limited in it's use.
===
Subject: stochastically convergent sequence
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id iB1GqcQ17791;
(f_n) and (g_n) are stochastically convergent sequences of measurable
real functions, having limit functions f and g, respectively. Show that for
all a,b - real numbers, the sequence (af_n+bg_n) converges stochastically
to af+bg, and the sequences max(f_n,g_n), min(f_n,g_n) converge
stochastically to max(f,g), min(f,g), respectively.
===
Subject: density
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id iB1Gqc717797;
Let (W,A,m) be a measure space, T:W->W a mapping which together with its
inverse is an A-A-measurable bijection. Show that for every f which is
A-measurable, nonnegative, numerical function on W, the image measure T(fm)
has a density with respect to T(m), namely
f(T^-1).
===
Subject: shortest paths question
This question has me totally stuck.
Can anyone give me some advice please? Any sort of prodding on this
would be much much appreciated!!
Let G=(V,E) be a weighted directed graph with source vertex s, and let
G be initialized by INITIALIZE-SINGLE-SOURCE(G,s). Prove that if a
sequence of relaxation steps set pi(s) to a non-NIL value then G
contains a negative-weight graph.
For the unfamiliar,INITIALIZE-SINGLE-SOURCE is given by
INITIALIZE-SINGLE-SOURCE(G,s):
for each vertex v in V[G]
do d[v]=infty
pi[v]=NIL
d[s]=0
===
Subject: Re: This Week's Finds in Mathematical Physics (Week 209)
Hey John, how are you?
> The n-categories called n-groupoids magically know
> everything there is to know about homotopy theory,
That part I know...
> while those called
> n-categories with duals know everything there is to know about the
> topology of manifolds.
But how would this work? What are n-categories with duals anyway?
Squark.
===
Subject: Re: .99999... still=/= 1
> The 9's never stop repeating. It's like this. If you say, cat
repeately
> infinitely, catcatcat..., it never converges to dog.
> It just stays as cat.
When you (or your computer/calculator) successfully said cat (or
9) infinitely many times can you tell me the last decimal digit of
pi (since pi have infinitely many digits)?
0, 1, 2, 3, 4, 5, 6, 7, 8 or 9? (Hint: try with your favourite digit)
===
Subject: Re: .99999... still=/= 1
>.999... =/= 1
>>If so, then there must be a real number X such that 0.999... < X < 1.
What
>>is the decimal expansion of X?
> X = ..............999... = asymptote toward 1 never reaching it.
In what digit position does that differ from 0.999...?
===
Subject: Re: .99999... still=/= 1
>>.999... =/= 1
>If so, then there must be a real number X such that 0.999... < X < 1.
What
>is the decimal expansion of X?
>> X = ..............999... = asymptote toward 1 never reaching it.
>In what digit position does that differ from 0.999...?
It shows somewhere away from the starting point.
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
>.999... =/= 1
>>If so, then there must be a real number X such that 0.999... < X < 1.
What
>>is the decimal expansion of X?
> X = ..............999... = asymptote toward 1 never reaching it.
>>In what digit position does that differ from 0.999...?
>It shows somewhere away from the starting point.
Could you be more specific?
===
Subject: Re: .99999... still=/= 1
In sci.math, Josh Purinton
>>.999... =/= 1
>If so, then there must be a real number X such that 0.999... < X < 1.
What
>is the decimal expansion of X?
>> X = ..............999... = asymptote toward 1 never reaching it.
> In what digit position does that differ from 0.999...?
The next to last one, of course. :-) Namely, the Punultimate One....
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
===
Subject: Re: .99999... still=/= 1
> Again, I didn't use my calculator. I used the online math Partial
Sums
> LINK,
>>is just plain wrong. We know you are
>> I didn't write down anything. I pasted the solution from the Online
>Partial
>> Sums Convergence calculator from there to here.
>> incapable of mathematical
>>reasoning, but can you do simple arithmetic?
>> It's seems you are in an inaccurate loop of no logic. This shows you
>can't
>> handle logical reasoning.
>LOL
>Here is a simple exercise for you. The partial sums of SUM 3/10^{k}
>are defined by
>S_n := SUM_{k=1}^n 3/10^k.
>That is, S_1 = 0.3, and S_n = S_{n-1} + 3/10^n for n>1.
>Now, answer the following question:
>- is 3/10^n greater than 0?
>If the answer is yes, then surely you will agree that S_n > S_{n-1}
>(that is, the sequence is strictly INCREASING).
>Whatever tool you used gave you a sequence which was eventually
>CONSTANT. Thus what this tool output is GARBAGE. Stop using
>calculators or tools on the web, and start using your BRAIN.
>I gave you a definition of covergence, which is the one that is
>universally accepted. This definition does not require calculators, or
>rounding off, or any other nonsense. If you are able to understand
>this definition, apply it to the above sequence, and see it converges
>to 1/3. Otherwise, tell us what you don't understand about the
>definition, and we may be able to help you.
>Lasse
>---
Ok let's do things the old fashion way, we will fight it out with boxing
gloves ( humor).
1) Something that does converge doesn't mean equal to.
.999... =/= 1
2) go to number 1.
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
> In this debate as to whether or not, .999... = 1, so far it appears
that it
> converges using Partial Sums but to say .999... = 1 is incorrect
because a
> convergence value is an approximate value, not the exact value.
> But does .999... really converge at
> n->oo
> Lim SUM 9/10^n = 0
> It appears to converge because as n-->oo the ratio test shows
that,
> lim z_n+1/z_n = L
> If L < 1 the series converges
> If L > 1 the series diverges
> If L = 1 the test fails, that is no conclusion is possible.
Reread your textbook. First of all, if the ratio test fails,
you have learned nothing about the series. It may converge
or not. It just means this particular test was uninformative.
There are diverging series with L=1, and converging series
with L=1.
Second of all, z_n = 9/10^n for this series.
The ratio test gives you L = lim [0.9^(n+1) / 0.9^n] = lim (0.9) = 0.9.
L < 1. The series converges.
- Randy
===
Subject: Re: .99999... still=/= 1
Discussion, linux)
> Very good point. Let's look at that.
> x * 10 - 9 = x
> let x = .444...
> 4.444... - 9 =
> 4 + .444... - 9 = -5 + .444... = -4.555...
> -4.555... =/= .444...
> The proof is inconsistent. So this proof can't be used.
At this point, I'm pretty sure you're trolling for humor's sake. On
the off chance you're not, I'll respond anyway.
So you proved x = .444... isn't a solution for x * 10 - 9 = x. So
what? No one here claimed it was. Everyone here instead said that
(1) there is a unique solution to that equation and (2) that solution
is 1. It is also 0.999..., which is no contradiction at all, since
1 = 0.999... .
--
After years of arguing I realize that your intellects are too limited
to fully grasp my work. [...] Still, no matter how child-like your
minds are, [...] since you have language, [...] there's a chance that
I'll be able to find something that your minds can handle. --JSH
===
Subject: Re: .99999... still=/= 1
>>and after his calculations are done he shows it to be equal to
>>something else.
>How many solutions have the equation: x * 10 - 9 = x, where x is
>unknown?
>-1, 0, 1, 2, 3 or infinity many solutions?
>If you agree that there is only one solution then it follows that
there
>is no room for something else.
>Since you said that .999...* 10 - 9 = .999... then it follows that
the
>number denoted as the symbol .999... equals the number denoted as
the
>symbol 1.
> Very good point. Let's look at that.
> x * 10 - 9 = x
> let x = .444...
> 4.444... - 9 =
> 4 + .444... - 9 = -5 + .444... = -4.555...
> -4.555... =/= .444...
> The proof is inconsistent. So this proof can't be used.
Why you said that the point is Very good when you proved that it is
inconsistent?
===
Subject: Re: .99999... still=/= 1
> If the proof is correct it should work on a common scientific
calculator.
Untrue.
Calculators do approximate arithmetic when working with
real numbers. No calculator manufacturer has ever made any
claim otherwise.
Here's an exercise for you:
Do you agree 1 + x - 1 should be equal to x, no matter what
the value of x?
If so, do the following on your favorite calculator:
1. Enter 1.
2. Add 1e-30 (this is x).
3. Subtract 1.
4. Did you get x back?
- Randy
===
Subject: Re: .99999... still=/= 1
>> If the proof is correct it should work on a common scientific
>calculator.
>Untrue.
>Calculators do approximate arithmetic when working with
>real numbers. No calculator manufacturer has ever made any
>claim otherwise.
>Here's an exercise for you:
>Do you agree 1 + x - 1 should be equal to x, no matter what
>the value of x?
>If so, do the following on your favorite calculator:
>1. Enter 1.
>2. Add 1e-30 (this is x).
>3. Subtract 1.
>4. Did you get x back?
> - Randy
Yes. It brought me right back to 1e-30.
In fact, my calculator can do math at levels of about 1e+-~ 2000.
And the point I made was, repeating decimals can be observed within about
32
significant digits on my calculator, which will allow me to at least see
that
the series or sequence will continue to repeat a certain way. You can even
see
patterns of numbers repeating.
1/3 = .333..->32 digits
Well that's enought to let me know that,
.333... --> infinity
There is only a minor difference between,
.3333333333333333333333333333333333
on my calculator and,
.3333333333333333333326
Using Partial Sums Convergence.
The point is you can use the approximate convergence value in the
calculations whether it is a convergence test or a rounding off of numbers.
If a function diverges on my calculator, my calculator will show that it
can
handle the amount of digits in the calculation. Like for example
lim n! = divergent
n-->oo
My calculator would start approaching high number values until it goes
over,
1e+2000 values. This let's me know the value is divergent for a series like
that.
For convergence values where L < 1 the calculator will use a rounding off
significant digit. If it is an indeterminate, like with,
1/0
My calculator will say, can't divide by zero. So I know that in cases
where
L = 0 the calculator will say something like, can't evaluate function.
For values of L > 1 then the calculator just goes as far as it can with
the
calculations and then says something like overflow, or number too large to
calculate.
So you are incorrect, a calculator can be used to assist you in
determining
if a function in a series converges or diverges or is an indeterminate.
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
here are the first few partial sums:
1;
1+1/1 = 2;
1+1/1+1/1 = 4/3;
1+1/1+1/1+1/1 = 5/3 ... maybe that's the formula
for the *reciprocal* of e. then,
the formula for e is (1+1)/1+1)/1+1)/1+..., with extra (s
at the front of it.
or some thing. nota bene:
I left off the infinite string of zeros for real integers,
as opposed to REALs and INTEGERs;
don't do this on the Pentium Trillion!
>> If the proof is correct it should work on a common scientific
>calculator.
--Advice 0.05; free, if wrong, again!
http://tarpley.net/bush6.htm
===
Subject: Re: .99999... still=/= 1
mister Company is going to love this.
I just found an application of 0.9999...,
over and above OVERFLOW CONDITION or TENS COMPLIMENT.
unfortunately, it's not insurable because
it's a proof of Fermat's Last Theorem;
the one that began his hobby.
so, I'm just going to use it in a short course.
> don't do this on the Pentium Trillion!
--Advice, 0.05; free, if wrong!
http://tarpley.net/bush5.htm
===
Subject: Re: .99999... still=/= 1
>mister Company is going to love this.
> I just found an application of 0.9999...,
>over and above OVERFLOW CONDITION or TENS COMPLIMENT.
That's because,
m--> oo
Lim (1 - 1/10^m) = 1
divergent
> unfortunately, it's not insurable because
>it's a proof of Fermat's Last Theorem;
>the one that began his hobby.
> so, I'm just going to use it in a short course.
>> don't do this on the Pentium Trillion!
>--Advice, 0.05; free, if wrong!
>http://tarpley.net/bush5.htm
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
>mister Company is going to love this.
> I just found an application of 0.9999...,
>over and above OVERFLOW CONDITION or TENS COMPLIMENT.
> That's because,
> m--> oo
> Lim (1 - 1/10^m) = 1
> divergent
Convergent.
===
Subject: Re: .99999... still=/= 1
In sci.math, Richard Henry
:
>>mister Company is going to love this.
>> I just found an application of 0.9999...,
>>over and above OVERFLOW CONDITION or TENS COMPLIMENT.
>> That's because,
>> m--> oo
>> Lim (1 - 1/10^m) = 1
>> divergent
> Convergent.
Both. Be very very careful with your notation, guys! :-)
lim (1 - 10^m) = infinity
m-> +oo
lim (1 - 10^m) = 1
m-> -oo
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
===
Subject: Re: .99999... still=/= 1
Fine. Then do it for x=1e-2100. Let me know what you find.
- Randy
===
Subject: Re: .99999... still=/= 1
>Fine. Then do it for x=1e-2100. Let me know what you find.
>- Randy
Ok, it shows that the number is so small, the calculator considers it to be
0.
But this tells me something without it really showing that same number.
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
> Ok, it shows that the number is so small, the calculator considers it to
be 0.
> But this tells me something without it really showing that same number.
It proves that calculators do not do mathematics. 1 + x - 1 = x for all
x. If the calculator says otherwise, it is wrong.
Bob Kolker
===
Subject: Re: .99999... still=/= 1
>> Ok, it shows that the number is so small, the calculator considers it to
be
>> But this tells me something without it really showing that same number.
>It proves that calculators do not do mathematics. 1 + x - 1 = x for all
>x. If the calculator says otherwise, it is wrong.
>Bob Kolker
Correction for other post...
series n!
1 + 1 + (1*2) + (1*2*3) + ... + n!
n-->oo
Lim SUM (n!)
You could use the calculator to see if this diverges. If the calculator
say
number too large, then you know it diverged.
And what about this series on your calculator? Does it converge or
diverge?
1 - z^2/2! + z^4/4! - + ...
n-->oo
Lim Sum (-1)^n [ z^2n/(2n)!]
This just happens to be the cos (z) function on your calculator. So all
you
do is plug in high values for z, and see if it converges and it does.
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
> You could use the calculator to see if this diverges. If the
> calculator say
> number too large, then you know it diverged.
Try this one on your calculator,
(10^2000000) + (10^2000000)/2 + (10^2000000)/4 + (10^2000000)/8 +
+ (10^2000000)/16 + (10^2000000)/32 + ...
Tell me if your calculator says it diverges. Then I will tell you want
good ol' logic says happens.
- Tim
--
Timothy M. Brauch
NSF Fellow
Department of Mathematics
University of Louisville
email is:
news (dot) post (at) tbrauch (dot) com
===
Subject: Re: .99999... still=/= 1
>> You could use the calculator to see if this diverges. If the
>> calculator say
>> number too large, then you know it diverged.
>Try this one on your calculator,
>(10^2000000) + (10^2000000)/2 + (10^2000000)/4 + (10^2000000)/8 +
> + (10^2000000)/16 + (10^2000000)/32 + ...
>Tell me if your calculator says it diverges. Then I will tell you want
>good ol' logic says happens.
It diverges, I didn't need the calculator.
That series is of the form,
1 + 1/2 + 1/4 + 1/8+ ...
This diverges.
> - Tim
>--
>Timothy M. Brauch
>NSF Fellow
>Department of Mathematics
>University of Louisville
>email is:
>news (dot) post (at) tbrauch (dot) com
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
> It diverges, I didn't need the calculator.
> That series is of the form,
> 1 + 1/2 + 1/4 + 1/8+ ...
> This diverges.
You lame brain . The series adds up to 2.
You are truly one of the Incompetents of Great Note. You and JSH should
sing duet a capella.
Bob Kolker
===
Subject: Re: .99999... still=/= 1
>> You could use the calculator to see if this diverges. If the
>> calculator say
>> number too large, then you know it diverged.
>Try this one on your calculator,
>(10^2000000) + (10^2000000)/2 + (10^2000000)/4 + (10^2000000)/8 +
> + (10^2000000)/16 + (10^2000000)/32 + ...
>Tell me if your calculator says it diverges. Then I will tell you
want
>good ol' logic says happens.
> It diverges, I didn't need the calculator.
> That series is of the form,
> 1 + 1/2 + 1/4 + 1/8+ ...
> This diverges.
Converges. Limit = 2.
===
Subject: Re: .99999... still=/= 1
> n-->oo
> Lim SUM (n!)
SUM (n!) does not converge. The limit does not exist.
Bob Kolker
===
Subject: Re: .99999... still=/= 1
>> n-->oo
>> Lim SUM (n!)
>SUM (n!) does not converge. The limit does not exist.
>Bob Kolker
That's what I said before you snipped 90% of what I posted.
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
>> Ok, it shows that the number is so small, the calculator considers it to
be
>> But this tells me something without it really showing that same number.
>It proves that calculators do not do mathematics. 1 + x - 1 = x for all
>x. If the calculator says otherwise, it is wrong.
>Bob Kolker
But the calculator will determine if a function will produce a number
too
large, which is divergence. Or a number less than 1, which is convergence,
or
1, no conclusion possible.
If I have a series of the form,
n-->oo
Lim n! = other than zero = divergence
If I plug in high numbers of n! , the calculator goes into something like
1e200000 and then says something number too large. This is a sign of
divergence.
So the calculator does tell you something if you analyze it right.
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
but how can you tell if it applies
to a series that approches a limit, really slowly?
for instance,
the continued fraction for e;
you can start with any number, or within some range,
as I recall from the subject of conditionally converent series
-- together with unconditional ones, of course -- but
we can just use 1+1/1+1/1+..., which approximates the usual notation.
you have to subtract each partial quotient,
whatever is the proper term fot that, from e,
with a possible ten's compliment for when the larger is subrtracted
from the smaller. if that can be converted
into an laternating sum, then it's easier to handle; eh?
I think it is said,
e is hte most irrational number, because so slow in this way.
> patterns of numbers repeating.
> The point is you can use the approximate convergence value in the
> calculations whether it is a convergence test or a rounding off of
numbers.
> 1e+2000 values. This let's me know the value is divergent for a series
like
> So you are incorrect, a calculator can be used to assist you in
determining
> if a function in a series converges or diverges or is an indeterminate.
--Advice 0.05; Free, if wrong, again!
http://tarpley.net/bush6.htm
===
Subject: Re: .99999... still=/= 1
>but how can you tell if it applies
>to a series that approches a limit, really slowly?
Just find the function that describes the series and place large numbers
in
it. If the pattern shows it's greater than 1 then it diverges. If the
pattern
shows it's less than 1 then it converges and if the pattern shows something
moving closer to 1 then it is an indeterminate or the calculator will say
something like it can't perform that function.
> for instance,
>the continued fraction for e;
>you can start with any number, or within some range,
>as I recall from the subject of conditionally converent series
>-- together with unconditional ones, of course -- but
>we can just use 1+1/1+1/1+..., which approximates the usual notation.
> you have to subtract each partial quotient,
>whatever is the proper term fot that, from e,
>with a possible ten's compliment for when the larger is subrtracted
>from the smaller. if that can be converted
>into an laternating sum, then it's easier to handle; eh?
> I think it is said,
>e is hte most irrational number, because so slow in this way.
>> patterns of numbers repeating.
>> The point is you can use the approximate convergence value in the
>> calculations whether it is a convergence test or a rounding off of
numbers.
>> 1e+2000 values. This let's me know the
Correction: It can handle about 1e+200000 values.
value is divergent for a series like
>> So you are incorrect, a calculator can be used to assist you in
>determining
>> if a function in a series converges or diverges or is an indeterminate.
>--Advice 0.05; Free, if wrong, again!
>http://tarpley.net/bush6.htm
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
In sci.math, S. Enterprize Company
but how can you tell if it applies
>>to a series that approches a limit, really slowly?
> Just find the function that describes the series and place large
numbers in
> it.
Well, that's one way of doing it. A more rigorous method is to
play the epsilon-delta game. If one can prove that, for any
epsilon > 0, I can pick a delta > 0 such that, for all x within
delta of a except x = a, f(x) is within epsilon of L, then
lim (x->a) f(x) = L.
If one replaces delta with M and for all x within delta of a
with for all n > M, one can prove lim (n->+oo) f(n) = L.
I'm not sure what function describes the series .999... (since
it's a number, not a series) but one logical guess would be
that .999... is the limit as the number of digits increases
without bound of the series .9, .99, .999, ...; the terms of that
series can be represeted as 1 - 10^(-n) for arbitrary positive
integer n. For any epsilon > 0, I can pick M = ceil(-log_10(epsilon))
and get the desired result.
[rest snipped]
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
===
Subject: Re: .99999... still=/= 1
I don't care about the precision of your floating-point algorithm;
is it said to conform to the IEEE standard?
starting with a large number for the continued faction of e,
as far as I know, is not going to help it to converge ... but
I'd have to see how to do it, firstly.
well, never mind; you've found the ultimate expression
of never mind. to wit,
the calculator will say something!
> Just find the function that describes the series and place large
numbers in
> it. If the pattern shows it's greater than 1 then it diverges. If the
pattern
> shows it's less than 1 then it converges and if the pattern shows
something
> moving closer to 1 then it is an indeterminate or the calculator will say
> something like it can't perform that function.
--Advice, 0.05; free, if wrong!
http://tarpley.net/bush5.htm
===
Subject: Re: .99999... still=/= 1
>>and after his calculations are done he shows it to be equal to
>>something else.
>How many solutions have the equation: x * 10 - 9 = x, where x is
>unknown?
>-1, 0, 1, 2, 3 or infinity many solutions?
>If you agree that there is only one solution then it follows that
there
>is no room for something else.
>Since you said that .999...* 10 - 9 = .999... then it follows that
the
>number denoted as the symbol .999... equals the number denoted as
the
>symbol 1.
> Very good point. Let's look at that.
> x * 10 - 9 = x
> let x = .444...
> 4.444... - 9 =
> 4 + .444... - 9 = -5 + .444... = -4.555...
> -4.555... =/= .444...
> The proof is inconsistent. So this proof can't be used.
Why you said that the point is Very good when you proved that it is
inconsistent?
===
Subject: Re: .99999... still=/= 1
>and after his calculations are done he shows it to be equal to
>something else.
>>How many solutions have the equation: x * 10 - 9 = x, where x is
>>unknown?
>>-1, 0, 1, 2, 3 or infinity many solutions?
>>If you agree that there is only one solution then it follows that
>there
>>is no room for something else.
>>Since you said that .999...* 10 - 9 = .999... then it follows that
>the
>>number denoted as the symbol .999... equals the number denoted as
>the
>>symbol 1.
>> Very good point. Let's look at that.
>> x * 10 - 9 = x
>> let x = .444...
>> 4.444... - 9 =
>> 4 + .444... - 9 = -5 + .444... = -4.555...
>> -4.555... =/= .444...
>> The proof is inconsistent. So this proof can't be used.
>Why you said that the point is Very good when you proved that it is
>inconsistent?
I was complimenting you.
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
>> x = 0.9_
>> 10x = 9.9_
>> = 9 + x
>>If you substitue the value of x here,
>>9.999... = 9 + x
>>.999... = x
>> 9x = 9
>> x = 9/9
>> = 1
>I.e. 0.9_ = 1
>> And, let's see if your proof is consistent. A proof must be correct in
all
>>cases.
>> x = .555...
>>10 x = 5.555...
>>10x = 5 + x
>> 9 x = 5
>> x = 5/9 = .55555555555555555556
>> .5555555555555555556 =/= .555...
>Your calculation is wrong:
>Assume 5/9 = .55555555555555555556 then we should have
>1 = (9/5)*(5/9) = (9/5)* .55555555555555555556
>but (9/5)* .55555555555555555556 = 1.000000000000000000008 =/= 1,
>i.e a contradiction.
>In previous used notation 5/9 = .555... ; the limit of the sequence:
>{0.5, 0.55, 0.555, ...}, and your .55555555555555555556 is just an
>approximation of this limit. The digit 6 is due to round off errors of
your
>calculator.
Oh, since you refuse to admit you are in error, you blame my calculator.
Calculators don't make mistakes, you do.
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
> 9 * .999... = 8.99999999999999991
That is not correct, unless you are using some nonstandard meaning of
that notation. In ordinary mathematical usage, the middle part means
the limit of the series, the right part means an exact decimal
fraction, * means multiplication, and = means exact equality. Using
that meaning, the product on the left&middle does not equal the exact
decimal fraction on the right, so the = statement is false. If you mean
something different by any of those parts, please tell what you mean.
===
Subject: Re: .99999... still=/= 1
>>9 * .999... = 8.99999999999999991
> That is not correct, unless you are using some nonstandard meaning of
> that notation.
You are wasting your time with Enterprise. He either does not know the
difference between a mathematical proof and using a calculator or he is
perversly pretending not to know. You will got nowhere withhim. Do not
piss up a rope, you will only get wet.
Bob Kolker
===
Subject: Re: .99999... still=/= 1
>9 * .999... = 8.99999999999999991
>> That is not correct, unless you are using some nonstandard meaning of
>> that notation.
>You are wasting your time with Enterprise. He either does not know the
>difference between a mathematical proof and using a calculator or he is
>perversly pretending not to know. You will got nowhere withhim. Do not
>piss up a rope, you will only get wet.
A scientific calculator does approximate convergence with a certain
number
significant digits. I think my calculator
about 32 significant digits.
1/3 = .3333333333333333333333333333
1/3 = .333...
Partial Sum Convergence does something like this. It uses a certain number
of
finite significant digits, then it finds a converges value. This
convergence
value is a number that the series APPROACHES NOT EQUALS.
.999... can NEVER = 1
it can approach or converge to 1 however, but this is a rounding off method
used finitely.
.999... was to be evaluated infinitely NOT finitely.
>Bob Kolker
And,
I am using Advanced Engineering Mathematics. There are various types of
convergence tests. Here is one of them.
Theorem 1 ( Divergence )
If a series z_1 + z_2 + .... converges, then
lim z_m = 0
m-->oo
Hence if the series doesn't satisfy this condition, it diverges.
.999... = .9 + .09 + .... + ( 1 - 1/10^m)
Z_m = ( 1 - 1/10^m)
lim 1 - 1/10^m = 1
m -->oo
So,
.999.... doesn't converge to 1.
Again..., another thing people seem to be misunderstanding about
convergence
is, a series that does converge, will never EQUAL that convergence value.
It
only approachs that convergence value.
So even if .999... did converge,
.999... still=/= 1
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
> Theorem 1 ( Divergence )
> If a series z_1 + z_2 + .... converges, then
> lim z_m = 0
> m-->oo
> Hence if the series doesn't satisfy this condition, it diverges.
> .999... = .9 + .09 + .... + ( 1 - 1/10^m)
.999... = .9 + .09 + ... + 9/10^m + ...
> Z_m = ( 1 - 1/10^m)
z_m = 9/10^m
(Do you expect us to believe that you think .09 = 1 - 1/10^2 ?)
> lim 1 - 1/10^m = 1
> m -->oo
lim 9/10^m = 0
m -->oo
--
Daniel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
===
Subject: Re: .99999... still=/= 1
>> Theorem 1 ( Divergence )
>> If a series z_1 + z_2 + .... converges, then
>> lim z_m = 0
>> m-->oo
>> Hence if the series doesn't satisfy this condition, it diverges.
>> .999... = .9 + .09 + .... + ( 1 - 1/10^m)
>.999... = .9 + .09 + ... + 9/10^m + ...
>> Z_m = ( 1 - 1/10^m)
>z_m = 9/10^m
>(Do you expect us to believe that you think .09 = 1 - 1/10^2 ?)
>> lim 1 - 1/10^m = 1
>> m -->oo
> lim 9/10^m = 0
> m -->oo
Lim 9/10^m =/= Lim SUM 9/10^m
The form of this convergence test is not the SUM, but just Lim.
>--
>Daniel W. Johnson
>panoptes@iquest.net
>http://members.iquest.net/~panoptes/
>039 53 36 N / 086 11 55 W
SUM
.999... = .9 + .09 + .009 +... + 9/10^m
m-->oo
Lim SUM 9/10^m
is the same as,
m-->oo
Lim (1 - 1/10^m)
At m = 10
1 - 1/10^10 = .9999999999
At m = 100
1 - 1/100 =
.9999999999999999999999999999999999999999999999999999999999999999999999999
999999999999
At m = oo
1 - 1/10^m = .999...
Lim SUM 9/10^m and Lim (1 - 1/10^m)
are equivalent.
But, correct form of this test is,
m-->oo
Lim Z_m = 0 ,
then convergence, otherwise no convergence
m-->oo
Lim ( 1 - 1/10^m ) = 1
This means it doesn't converge.
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
S. Enterprize Company blithered:
> Lim SUM 9/10^m and Lim (1 - 1/10^m)
> are equivalent.
> But, correct form of this test is,
> m-->oo
> Lim Z_m = 0 ,
> then convergence, otherwise no convergence
Where the z_m are the TERMS (9/10^m), not the partial sums (1 - 1/10^m).
> m-->oo
> Lim ( 1 - 1/10^m ) = 1
> This means it doesn't converge.
--
Daniel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
===
Subject: Re: .99999... still=/= 1
> I am using Advanced Engineering Mathematics. There are various
> types of
> convergence tests. Here is one of them.
> Theorem 1 ( Divergence )
> If a series z_1 + z_2 + .... converges, then
> lim z_m = 0
> m-->oo
> Hence if the series doesn't satisfy this condition, it diverges.
> .999... = .9 + .09 + .... + ( 1 - 1/10^m)
^^^^^^^^^^^^^
Let's check that for the first few terms...
m = 1: 1 - 1/10 = 0.9
m = 2: 1 - 1/100 = 0.99
Now sum those and we end up with 1.89.
m = 3: 1 - 1/1000 = 0.999
for a sum of 2.889
m = 4: 1 - 1/10000 = 0.9999
for a sum of 3.8889
That equation is very much wrong.
- Tim
--
Timothy M. Brauch
NSF Fellow
Department of Mathematics
University of Louisville
email is:
news (dot) post (at) tbrauch (dot) com
===
Subject: Re: .99999... still=/= 1
> That equation is very much wrong.
The last term is 9/10^m which does go to zero as m increases.
Bob Kolker
===
Subject: Re: .99999... still=/= 1
>> I am using Advanced Engineering Mathematics. There are various
>> types of
>> convergence tests. Here is one of them.
>> Theorem 1 ( Divergence )
>> If a series z_1 + z_2 + .... converges, then
>> lim z_m = 0
>> m-->oo
>> Hence if the series doesn't satisfy this condition, it diverges.
>> .999... = .9 + .09 + .... + ( 1 - 1/10^m)
> ^^^^^^^^^^^^^
>Let's check that for the first few terms...
> m = 1: 1 - 1/10 = 0.9
> m = 2: 1 - 1/100 = 0.99
>Now sum those and we end up with 1.89.
> m = 3: 1 - 1/1000 = 0.999
>for a sum of 2.889
> m = 4: 1 - 1/10000 = 0.9999
>for a sum of 3.8889
You're right I would like to make the correction.
SUM
.999... = .9 + .09 +.... + 9/10^m
m-->oo
Lim SUM 9/10^m
But the Convergence Test I used was of the form,
Lim Z_m = 0 then convergence otherwise divergence
So this form has to be used.
.999... =
m-->oo
Lim ( 1 - 1/10^m) = 1
.999... doesn't converge to 1
>That equation is very much wrong.
> - Tim
>--
>Timothy M. Brauch
>NSF Fellow
>Department of Mathematics
>University of Louisville
>email is:
>news (dot) post (at) tbrauch (dot) com
Smart's Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: .99999... still=/= 1
> You're right I would like to make the correction.
> SUM
> .999... = .9 + .09 +.... + 9/10^m
> m-->oo
> Lim SUM 9/10^m
> But the Convergence Test I used was of the form,
> Lim Z_m = 0 then convergence otherwise divergence
> So this form has to be used.
You've got it wrong! The theorem you're thinking about says:
If SUM Z_m converges then Lim Z_m = 0, but the implication the other
way does not hold.
That is, if Lim Z_m = 0, you cannot conclude that SUM Z_m converges!
The harmonic series SUM 1/m is a classical counter example.
Anyway, the series above do converge since it is a geometric series
SUM 9/10^m, and 9/10^m < 1.
> .999... =
> m-->oo
> Lim ( 1 - 1/10^m) = 1
> .999... doesn't converge to 1
And now you're talking about sequences and not series! Your sequence
is:
p_m = {1 - 1/10^m}. Since p_m is monotonically increasing p_(m+1) >
p_m, and bounded 0 < p_m < 1 in R, it converges in R.
You've also calculated the limit of this sequence for us:
> Lim ( 1 - 1/10^m) = 1
Therefore 0.999... = 1 by your own calculation.
> Smart's Alt. Physics News Group
> http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1
> S. Enterprize (Science Journal)
> http://smart1234.s-enterprize.com/
===
Subject: Adleman-Pomerance-Rumely Primality Test
I'm looking for the full description of the Adleman-Pomerance-Rumely
Primality Test, I couldn't have found it anywhere in the net, so I
would be grateful for any description.
I have one more question, what is the fastest Primality Test for
numbers with approximately 1000-5000 binary digs with Primality
Certification.
Ps. Sorry for my weak English
Adam
===
Subject: Re: Adleman-Pomerance-Rumely Primality Test
> I'm looking for the full description of the Adleman-Pomerance-Rumely
> Primality Test, I couldn't have found it anywhere in the net, so I
> would be grateful for any description.
Crandall & Pomerance, /Prime Numbers, a Computational Perspective/
> I have one more question, what is the fastest Primality Test for
> numbers with approximately 1000-5000 binary digs with Primality
> Certification.
Marcel Martin's Primo.
> Ps. Sorry for my weak English
It's better than most English speakers' .
Phil
--
The gun is good. The penis is evil... Go forth and kill.
===
Subject: Re: Questions about math tutoring
>> However, in spite of these obvious reasons not to like tutoring as
>> a way of earning money, I'm thinking of offering some tutoring
>> again because my camel is hungry.
> Well, you can't buy much food for your camel (?!) on a couple
> hundred bucks a month.
> Probably a euphemism for cigarettes gobbling up his income.
Actually, I don't smoke. But I get the connection to camels. For what
it is worth, decades ago I gave up smoking by mathematical induction:
I realized that if I didn't smoke the next cigarette, I couldn't smoke
the one that came after it, and it seemed trivial to give up one cigarette.
Telling myself horror stories about my lungs and cancer didn't work,
since a few hundred more or less didn't make any difference from that
perspective. What worked was convincing myself it was easy to quit, by
the inductive argument.
So far no one has proposed a definite price that I should charge for
tutoring or for the high level tutoring I am calling consulting. So,
I guess it is up to me. I think I can charge $125/hour for tutoring in
coffee shops, more if I have to rent space specially to accomodate it,
and $500/hour for internet consulting. And if I do it in a coffee shop
that I normally go to anyway, I won't be greatly inconvienced by no-shows.
I don't think it will be too noisy, since if it was, I wouldn't use it
as a place to study.
Apart from that, I think all the major questions I posed have essentially
--
Allan Adler
* Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My actions
and
* comments do not reflect in any way on MIT. Also, I am nowhere near
Boston.
===
Subject: Re: Partition strangeness
>> Now define the jaggedness J of a tableaux to be the
>> difference between the longest and shortest row.
>> (In the above, 0 3 1 2 1 1 0.)
> What actually are you asking?
a) Why do exactly n partitions of n of jaggedness <=1
exist? (Clean formulation of the handwave proof.)
b) Is there already something known about partitions
with other jaggedness (other than the obvious J=0
corresponds to the number of divisors of n)?
c) Maybe it would be enlightening to extend my n<=5
table a bit by computer. I'm *very* good at pattern
spotting :-)
--
Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de
His-Ala-Sec-Lys-Glu Arg-Glu-Asp-Asp-Met-Ala-Asn-Asn
===
Subject: Re: Partition strangeness
> Now define the jaggedness J of a tableaux to be the
> difference between the longest and shortest row.
> (In the above, 0 3 1 2 1 1 0.)
>> What actually are you asking?
> a) Why do exactly n partitions of n of jaggedness <=1
> exist? (Clean formulation of the handwave proof.)
That's easy. Look at the conjugate partition. A partition
mu has jaggedness <= 1 iff its conjugate mu* has all parts,
save perhaps the smallest, equal. That is if mu* has largest
part k it has [n/k] parts equal to k, and perhaps an extra
part equal to n - [n/k]k (unless this is zero). So these mu*
are parametrized by k where 1 <= k <= n.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: ceil() mathematical formula
>>So can you give me the true function ?
>it might be noted, FWIW, that
> ceiling(x) = x + 1/2 + i/(2 pi) log( -exp(2 pi i x) )
> where log denotes the principal branch.
> So it seems that ceiling and related functions are elementary. ;-)
Maybe this is part of the joke, but isn't choosing the branch cheating
a little bit? Symbolically, you don't really have a particular branch,
you have all of them at once. So that ceiling(x) pops out as an
(arbitrary) artifact of the choice of branch. Is that right?
--
Mitch Harris
(remove q to reply)
===
Subject: Re: ceil() mathematical formula
>>So can you give me the true function ?
>it might be noted, FWIW, that
> ceiling(x) = x + 1/2 + i/(2 pi) log( -exp(2 pi i x) )
> where log denotes the principal branch.
> So it seems that ceiling and related functions are elementary. ;-)
> Maybe this is part of the joke, but isn't choosing the branch cheating
> a little bit?
Indeed. [Or you could say that it's the whole joke, and choosing the branch
is cheating a lot.]
> Symbolically, you don't really have a particular branch,
> you have all of them at once. So that ceiling(x) pops out as an
> (arbitrary) artifact of the choice of branch. Is that right?
Exactly.
This is a point I've made several times in the past in this newsgroup.
For example: Although the multivalued square root relation is certainly
algebraic, I seems to me that the principal-valued square root function
doesn't actually deserve to be called algebraic, technically. But
OTOH, I suspect you'd get some strange reactions if you were to call
it transcendental. Maybe we should meditate on that.
David Cantrell
===
Subject: Re: ceil() mathematical formula
> This is a point I've made several times in the past in this newsgroup.
> For example: Although the multivalued square root relation is certainly
> algebraic, I seems to me that the principal-valued square root function
> doesn't actually deserve to be called algebraic, technically. But
> OTOH, I suspect you'd get some strange reactions if you were to call
> it transcendental. Maybe we should meditate on that.
It wouldn't, I think, be too much of a stretch to call it
semi-algebraic (maybe it's no stretch at all; its *graph*
is certainly a semi-algebraic set in the technical sense).
Lee Rudolph
===
Subject: Re: ceil() mathematical formula
> This is a point I've made several times in the past in this newsgroup.
> For example: Although the multivalued square root relation is certainly
> algebraic, I seems to me that the principal-valued square root function
> doesn't actually deserve to be called algebraic, technically. But
> OTOH, I suspect you'd get some strange reactions if you were to call
> it transcendental. Maybe we should meditate on that.
It used to bug me that the sqrt function was considered part of the
transcendental unit in UCSD Pascal, but I don't suppose that's the
reason. They didn't have an algebraic unit.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: ceil() mathematical formula
>So can you give me the true function ?
>>it might be noted, FWIW, that
>> ceiling(x) = x + 1/2 + i/(2 pi) log( -exp(2 pi i x) )
>> where log denotes the principal branch.
>> So it seems that ceiling and related functions are elementary. ;-)
>> Maybe this is part of the joke, but isn't choosing the branch cheating
>> a little bit?
> Indeed. [Or you could say that it's the whole joke, and choosing the
> branch is cheating a lot.]
> Well, how about arcsin(sin x)? I mean, that is not so obviously
> steeped in complex branchism.
How about it? Well, for one thing, I don't see how to use that to get
something like ceiling(x). [But maybe I'm just overlooking something.]
Note that the sort of triangle wave given by arcsin(sin(x)) is
continuous, while of course ceiling(x) is not.
David Cantrell
===
Subject: Re: ceil() mathematical formula
<20050109113111.019$n5@newsreader.com> <34hmg8F4bneurU1@news.dfncis.de>
<20050111083330.299$vY@newsreader.com>
<20050111101626.165$aN@newsreader.com>
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi
$t^
VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
>So can you give me the true function ?
it might be noted, FWIW, that
ceiling(x) = x + 1/2 + i/(2 pi) log( -exp(2 pi i x) )
where log denotes the principal branch.
So it seems that ceiling and related functions are elementary. ;-)
Maybe this is part of the joke, but isn't choosing the branch
cheating
> a little bit?
>> Indeed. [Or you could say that it's the whole joke, and choosing the
>> branch is cheating a lot.]
>> Well, how about arcsin(sin x)? I mean, that is not so obviously
>> steeped in complex branchism.
> How about it? Well, for one thing, I don't see how to use that to
> get something like ceiling(x). [But maybe I'm just overlooking
> something.] Note that the sort of triangle wave given by
> arcsin(sin(x)) is continuous, while of course ceiling(x) is not.
Well, doh. Let's take arctan(tan(x)) then...
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: ceil() mathematical formula
>So can you give me the true function ?
it might be noted, FWIW, that
ceiling(x) = x + 1/2 + i/(2 pi) log( -exp(2 pi i x) )
where log denotes the principal branch.
So it seems that ceiling and related functions are elementary.
> ;-)
Maybe this is part of the joke, but isn't choosing the branch
> cheating a little bit?
>> Indeed. [Or you could say that it's the whole joke, and choosing the
>> branch is cheating a lot.]
>> Well, how about arcsin(sin x)? I mean, that is not so obviously
>> steeped in complex branchism.
> How about it? Well, for one thing, I don't see how to use that to
> get something like ceiling(x). [But maybe I'm just overlooking
> something.] Note that the sort of triangle wave given by
> arcsin(sin(x)) is continuous, while of course ceiling(x) is not.
> Well, doh. Let's take arctan(tan(x)) then...
Oh, I'd already considered using something like that before I posted my
expression using log. Here's the trouble: What do we do with things like
arctan(tan(pi/2))?
I did come up with a solution, but you might not like it.
Of course, I doubt that you'd mind using R*, the one-point extension of the
reals, for the range of the tangent function, thereby allowing us to say
tan(pi/2) = oo. But then what should we do with arctan(oo)?! [If we were
dealing with the _two_-point extension of reals, there would be no problem
with arctan. Standardly, arctan(-oo) = -pi/2 and arctan(+oo) = +pi/2. But
then there would be no reasonable way to have defined tan(pi/2).] Here's my
solution which you might not like. For arctan(unsigned oo), the values
-pi/2 and +pi/2 are equally reasonable. But our arctan is a
principal-valued function, and choosing what is to be the principal value
is necessarily arbitrary to some extent anyway. So let's extend arctan
to R* by just choosing to say arctan(unsigned oo) = +pi/2. That solves the
problem, and one can then express ceiling and related discontinuous
functions...
Please note that arctan is continuous on R (and also on the
two-point extension). But extending arctan to R* as I did above
makes it _discontinuous_ at oo.
I don't dislike this approach. However, I like the log approach better,
since it relies on a previously well established discontinuity.
David Cantrell
===
Subject: Re: FINALLY THE TRUMAN PROBE GOES PUBLIC
> stole it from?
I think that site is cute. All the paranoid schizophrenics can now have
their free form delusions confirmed by other paranoid schizophrenics.
===
Subject: Re: FINALLY THE TRUMAN PROBE GOES PUBLIC
---------------------------------------------s-o-s--------------------------
---------------
> stole it from?
> I think that site is cute. All the paranoid schizophrenics can now have
> their free form delusions confirmed by other paranoid schizophrenics.
hardy hah hah. I'm let down its not an actual forum.
But they've either copied my posts, or something is going on.
Forced telepathy satelites, that hear your thought from vibrations then
braodcast them?
That's my material to a tea. Same with the 'tortured by CIA for 2 years',
few people
saying that, using direct quotes from me.
They've got all the effects and tricks that the CIA can do down perfect,
there's no way
they made this up out of there head, but I've never posted these.
READING AND BROADCASTING THOUGHTS
FORCED MEMORY BLANKING
INDUCED ERRONOUS ACTIONS
INDUCED CHANGES TO HEARING
FORCED SPEECH
FORCED NUDGING OF ARM DURING DELICATE WORK
SLEEP PREVENTION
DROP IN YOUR TRACKS SLEEP INDUCEMENT
IRRESTIBLE GO HERE GO THERE COMMANDS
FREQUENT BRAKE AND ENTERS AT HOME
FORCED WAKING VISIONS / BODY MOTION
ARTIFICIAL TINNITUS
Welcome to my day. That's just the satelite, they control everything
plugged into 240V
and taped or live media is all controlled and synchronised. They also read
everyone
around you... never stops 3 years running of this political prisoner
nightmare.
Herc
===
Subject: Re: FINALLY THE TRUMAN PROBE GOES PUBLIC
<34hosoF4buvisU1@individual.net They've got all the effects and tricks that the CIA can do down
perfect, there's no way
> they made this up out of there head, but I've never posted these.
> READING AND BROADCASTING THOUGHTS
> FORCED MEMORY BLANKING
> INDUCED ERRONOUS ACTIONS
> INDUCED CHANGES TO HEARING
> FORCED SPEECH
> FORCED NUDGING OF ARM DURING DELICATE WORK
> SLEEP PREVENTION
> DROP IN YOUR TRACKS SLEEP INDUCEMENT
> IRRESTIBLE GO HERE GO THERE COMMANDS
> FREQUENT BRAKE AND ENTERS AT HOME
> FORCED WAKING VISIONS / BODY MOTION
> ARTIFICIAL TINNITUS
> Welcome to my day.
I think you'd notice the effects listed above would diminish if you cut
back on the booze.
> That's just the satelite, they control everything plugged into 240V
> and taped or live media is all controlled and synchronised. They
also read everyone
> around you... never stops 3 years running of this political prisoner
nightmare.
Sounds like a poor guy in the UK who believes MI5 are harrassing him
and that newsreaders are watching him in his home when they are live on
TV. They have a monitor next to the camera connected to hidden cameras
in his home, apparently.
Lazi
===
Subject: Re: FINALLY THE TRUMAN PROBE GOES PUBLIC
The brain would only give out modulated waves if natural selection so
programmed it. Because there was no need the brain does not do this.
The brain compares pattern (among other things) and does not do the
huge extra job of forming a signal to also broadcast the patterns.
I can slow my heart rate a bit with my thoughts. If my computer could
pick up my heartbeat, not too hard, I can also control my computer no
hands.
===
Subject: Re: algorithm for Modified Distribution Method / Transportation
Simplex Method/Transport Problem
> do anybody has any algorithm for the subject mentioned.
That is most easy. As the subject mentioned is an algorithm
for the subject mentioned, the algorithm is self referring.
Now as the self calling algorithm has no mentioned stop or cut off, it's
the eternal algorithm. So look in Knuth's three volume reference of
algorithms where there's a large variety of eternal algorithms.
Chose the one best fitted for your purposes. For a sole like you in such
a big rush, I recommend you use instant eternity, quick eternity or
eternity to go, which are designed for those that don't have enuf time
to wait for a short program to run.
===
Subject: Complex Variables: Deriving Lagrange's trigonometric identity.
I have managed to establish the identity for
1+z+z^2+...+z^n = (1- z^(n+1))/(1-z) where z not equal 1
but I am still unable to use it to derive the Lagrange's trigonometric
identity,
1+cosk + cos 2k + .... + cos nk = 1/2 + sin[(2n+1)k/2] /(2sin(k/2)
I tried using z=exp(ik), but somehow still unable to get to the right
hand side.
Help is much appreciated.
===
Subject: Re: Complex Variables: Deriving Lagrange's trigonometric identity.
> I have managed to establish the identity for
> 1+z+z^2+...+z^n = (1- z^(n+1))/(1-z) where z not equal 1
> but I am still unable to use it to derive the Lagrange's trigonometric
> identity,
> 1+cosk + cos 2k + .... + cos nk = 1/2 + sin[(2n+1)k/2] /(2sin(k/2)
cos k = (exp(i*k) + exp(-i*k))/2
Jose Carlos Santos
===
Subject: Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
>>I am only assuming mathematicians have *some* objective (i.e. effective)
>>procedure for deciding what methods to accept.
>>This is *not* a trivial assumption!
> Indeed, more is true: on the face of it, the assumption seems obviously
> false. Take, for example, There exists a strongly inaccessible
cardinal.
> Some mathematicians accept this as true; others don't.
But in what sense can any of them be said to *know* the answer?
I'm not even sure there even *is* an objective answer.
Before non-euclidean geometry was well understood, some mathematicians
accepted there exists a unique line through a given point parallel to a
given line, and some did not. Is there any sense in which one group was
right and the other wrong?
In fact, there is *another* unstated assumption: that mathematics is
*sound*. It is quite likely that mathematicians accept some false things,
even some contradictory things.
Ralph Hartley
===
Subject: Re: too much information!
)>>Someone else claimed that these random digits were
)>>complete crap and could be significantly compressed; let's see
)>>if your zip file reflects that.
)>>1 million decimal digits should be about 3.3 million bits;
)>>divide by 8 and get about 410,000 bytes.
415,241 bytes and change, if my calculator is accurate enough.
SaSW, Willem
--
Disclaimer: I am in no way responsible for any of the statements
made in the above text. For all I know I might be
drugged or something..
No I'm not paranoid. You all think I'm paranoid, don't you !
#EOT
===
Subject: Re: Who originated this proof of uncountability of reals?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id j0BMDa821882;
>Michael Orion http://mathfor
um.org/discuss/sci.math/m/670461/670461 Does anyone know who originated
the proof below for the
>> uncountability of the reals? I found it in Kuratowski,
>> Introduction to Set Theory and Topology, 1962 Pergamon.
>> It is different than Cantor's diagonalization proof.
>this. The other posts contain things you might also be
>interested in.
>The essence of Cantor's 1874 proof.
>http://mathfor
um.org/discuss/sci.math/m/388412/388425interval proof and Cantor's 1891 diagonalization proof.
>http://mathfor
um.org/discuss/sci.math/m/470447/470655Riemann integrable function has a dense set of continuity points,
>and whether this might have inspired Cantor's 1874 proof that the
>reals are uncountable.
>http://mathfor
um.org/discuss/sci.math/m/484926/485136
<344013F45nfv5U1@individual.net>
<344ggqF46mj3aU1@individual.net>
<345uvdF469qimU1@individual.net>
<3460t6F450klgU1@individual.net>
<346f8jF480pn6U1@individual.net what is the opposite of this proposition?
2 All coin sequences are computable to infinite length.
Herc
The only meaning I can think of for that proposition is All
> infinite
> coin sequences are computable. The opposite of that is Some
> infinite
> coin sequences are not computable.
> No, your mind has been programmed backwards you can't understand
> simple English
> so you can continue your charade of ignorace.
Well, really, there's no need to be rude. If you think you meant
> something else, just say what it is.
> either this is a lie or you can't think straight, as I see it.
What's a lie? Why don't you just answer the question?
> The only meaning I can think of for that proposition is All
> infinite
> all coin sequences are computable to infinite length,
> the thought
> all coin sequences are computable to unlimited length
> never occured?
> 2 All coin sequences are computable to infinite length.
If they're not computable to infinite length, what are they
> computable to?
Arbitrary finite segments of them are computable, but they
themselves
> are not.
> whats a finite segment? different to an infinite segment?
A finite segment of an infinite sequence is a segment consisting of the
first n digits, where n is some finite number. Yes, it's different to
an infinite segment.
> arbitrary segments are computable, ergo the sequence is computable to
arbitrary length.
If you want to put it that way, but this is different to saying the
entire sequence is computable.
> Herc
===
Subject: different of a number field
I have encountered two different definitions of the different of a
number field k. Hilbert defines it as follows. Let omega_1,..., omega_n
be a basis for the ring of integers of k as a Z-module. Let sigma range
over the nonidentity elements of the Galois group of the Galois closure
of k over Q, and let E_sigma be the ideal in the ring of integers of
the Galois closure of k generated by (omega_1-sigma(omega_1),
omega_2-sigma(omega_2), ... omega_n-sigma(omega_n)). Take the product
of all the E_sigmas, as sigma ranges over the nonidentity elements of
the Galois group of the Galois closure of k over Q, and take its
intersection with the ring of integers of k. This is the different of
the number field k.
Elsewhere I have seen it defined as follows. The different of k is the
inverse of the fractional ideal of k consisting of all a such that
trace(a.O_k) is contained in Z, where O_k is the ring of integers of k.
How do you prove these two definitions are equivalent?
===
Subject: Disjuncted miserations
The Axiom of Reducibility
Is supposed to better life for me
All tangles of a mystery
Fall down before equivalency
In the real world - a refreshing sight
But in the realm of intensions it loses its might
The Empty set: monochromically bright
Round circles, weak strengths - (oxy)moronic delight
As all of them empty the set in your sight.
The Reducibile Axiom cares not a whit
If plain-spoken falsehood maps to flibberty-git.
All numbers are elements - such is the same
As a falsehood more involved which enters your brain
All round square's motions 'round colors of time
Is the same as earlier, as ten pennies's are to dimes.
Such a usefull addition to the student's lodgic tree!
A true advertisement for [ ... ]
===
Subject: Re: phi
===
Subject: phi
Reply | Reply to Author | Forward | Print | Individual Message | Show
original | Report Abuse
Hi people! What is 0,618 or 1,618? I know it is phi but i am not sure
about
the deep meaning of it. It can be found in nature quite often. Of
course
every number relates to every other but can it be explained in an easy
way
like this?
----------------------------------------------------------------------------
---------
[Smart1234]
( It's not just found in nature, but also the sequence makes a copy of
it's self..., which is similar to DNA replication...)
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
( Just draw 1/4 a circle in each square and generate a spiral in the
Fibonaaci sequence)
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibrab.html
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat2.html
http://www.goldenmeangauge.co.uk/golden.htm
You can even hear music in the sequence... .
===
Subject: Re: Riemann Zeta Function Zeroes (not RH)
> Another argument, which calculates the values at negative odd integers.
> Gamma(s)zeta(s) = integral_0^infinity [1/(e^x - 1)] x^{s-1} dx
> for Re s > 1.
> I'm not sure if this is right, or even if it makes sense,
> one might suppose that there is a missing factor
> corresponding to the infinite place (or absolute valuation),
> and the so-called trivial zeros arise from this factor.
> When this factor is added one gets Riemann's Xsi(s),
> which is invariant under s -> 1-s and does not have the trivial zeros.
Just a little historical note: In that famous paper of his, Riemann stated
his conjecture in terms of Xi (all the zeros of Xi are real) and not
zeta
(all the zeros of zeta other than -2n are on the line |z-1|=|z| ). I
hope
to prove this conjecture, but it will have to wait until I finish another
blockbuster I'm working on ;)
===
Subject: Re: Integration in R^2
>Let D = [0,+oo[ x [0,1[ and f be a real-valued function (i) absolutely
>integrable on bounded measurable subsets of D. Assume that: (ii) lim
int(f,
>A_k x B_k) = 0 for every pair of ascending sequences: (A_k) and (B_k)
of
>measurable sets such that: U A_k = [0,+oo[ and U B_k = [0,1[. Must f
be
>absolutely integrable over D ?
Ok here's what I got (I think the answer is yes, but I'm not sure,
especially one point of my proof looks ridiculous and the problem
suggests
the answer is no).
Proof:
If f is not absolutely integrable over D, then by (ii) int(f+, D) = int(f-,
D) = +oo, where f+ = max(f,0), f- = max(-f,0). Now I set (for all k) B_k =
[0,1[ (<< that's the point of my proof that makes me suspicious).
The mapping: y->int(f+, [0,y[ x [0,1[) is continuous (since by (i) f+ is
absolutely continuous on bounded measurable subsets of D).
I set A_k = [0,k[ U C_k, where C_k is (measurable bounded, and f takes
negative values on C_k) such that:
int(f, C_k) + int(f, [0,k[) lies in [-3/2,-1/2] (such a C_k exists by the
previous paragraphs; note: int(f, [0,k[) is finite by (i)).
We will have U A_k = [0,+oo[, U B_k = [0,1[, but int(f, A_k x B_k) won't go
to 0 when k goes to +oo.
Where's the mistake ?
--
Julien Santini