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Feb 15, 2010, 3:28:45 PM2/15/10

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Health Warning:

Not only have I not thought much about maths for the last year or

so, and not posted here since last June, but this is the sort of

topic which always seems to cause a thread in sci.math to converge

rapidly to a cycle of abuse containing much use of the c___k word.

But it's been worrying me intermittently for a few days now, and I

don't know where would be a more appropriate forum to ask about it.

(Suggestions welcomed!)

The Actual Point:

In the UK, last Wednesday, BBC2 transmitted a television programme

in the scientific documentary series Horizon. Here is the episode

description in DigiGuide:

``To Infinity and Beyond.

Series exploring topical scientific issues. By our third year, most

of us will have learned to count. Once we know how, it seems as if

there would be nothing to stop us counting forever. But, while

infinity might seem like an perfectly innocent idea, keep counting

and you enter a paradoxical world where nothing is as it seems.

Older than time, bigger than the universe and stranger than fiction.

This is the story of infinity.

Copyright (c) GipsyMedia Limited.''

I was only half-watching the programme (Horizon is often annoyingly

dumbed-down - and I was doing something else at the same time), but

what grabbed my attention was an interview with Doron Zeilberger,

in which he claimed not only that there is a largest natural number

(I'd come across mention of Zeilberger's ultrafinitism before), but

that if you add 1 to this largest natural number, the result is 0.

(Pause for the reader to catch his or her breath, and re-read that

paragraph, to make sure that that was indeed what I said he said.)

This forced me to wonder again, not whether this is true or false

(I hold unremarkably to the orthodox view that there is no largest

natural number), but how, and indeed whether, anyone could possibly

seriously believe in such a proposition.

A visit to Zeilberger's website confirmed my vague memory that he

has a reputation for April Fool's Day mathematical pranks:

http://www.math.rutgers.edu/~zeilberg/

Homepage of Doron Zeilberger

But it also seemed to suggest that this was not one of them.

For confirmation (or not), see, for example:

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/enquiry.html

An Enquiry Concerning Human (and Computer!) [Mathematical] Understanding

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/real.html

"Real" Analysis is a Degenerate Case of Discrete Analysis

http://www.math.rutgers.edu/~zeilberg/OPINIONS.html

[many brief articles, controversial and/or humorous]

And, of course, he has done some remarkable mathematics, so he is

no fool; but am I, nevertheless, being made an April Fool of by him?

Although I'm scarcely able to concentrate on mathematics these days,

this thing has been intermittently worrying (and intriguing) me, and

I've been thinking of e-mailing him with at least a question or two.

But I don't want to be a pest (Chorus: "So why are you posting this

to sci.math, then?"), so I thought I would ask here first, to see if,

perhaps because of having been away from mathematics yet again, I am

just failing to see some obvious evidence that this is only a joke.

If it is not a joke, then I am struggling to understand: how he would

define the concept of 'finite'; and whether, if Z is his finite set

of all natural numbers, and if, for example, he were to construct the

Gaussian integers G = Z + Zi (or just the Cartesian product Z x Z), he

would regard G as finite; and if it is finite, then how many elements

does it contain; or, can some finite sets not have a cardinal number?

(I'm not worrying so much about such questions as: how the real line

could be hZ_p for some small real h and big prime p; or why he seems

to want his set of natural numbers to be a finite field; or exactly

how he would, or would not, define p itself, if p - 1 is the largest

natural number he was talking about in the documentary. It doesn't

seem impossible that he has some sort of reasonable answers to these

questions already worked out; but my worries seem a bit more basic.)

--

Angus Rodgers

(formerly, twi...@bigfoot.com;

alas, Bigfoot has gone tits-up)

Feb 15, 2010, 4:30:52 PM2/15/10

to

Try it on a 32-bit unsigned integer. You add 1 to the largest number

in that domain and, voila, you get zero as a result. Is that the April

Fool's?

in that domain and, voila, you get zero as a result. Is that the April

Fool's?

Feb 15, 2010, 5:25:26 PM2/15/10

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Huh? Zeilberger does indeed refer to his integer arithmetic as

taking place on some "giant computer in the sky" (or words to

that effect), but obviously it can't be a binary computer, if

the modulus is prime. I don't see your point (or joke, if that's

what it is). I'm sorry if I'm being rude, but it looks as if you

didn't understand what I wrote. I already mentioned Z_p.

--

Angus Rodgers

Feb 16, 2010, 2:19:08 AM2/16/10

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I can't understand the last part... A zillion marbles in a bag,

we had one marble to the bag, and then there are no marbles (?)

> (Pause for the reader to catch his or her breath, and re-read that

> paragraph, to make sure that that was indeed what I said he said.)

>

> This forced me to wonder again, not whether this is true or false

> (I hold unremarkably to the orthodox view that there is no largest

> natural number), but how, and indeed whether, anyone could possibly

> seriously believe in such a proposition.

>

> A visit to Zeilberger's website confirmed my vague memory that he

> has a reputation for April Fool's Day mathematical pranks:

>

> http://www.math.rutgers.edu/~zeilberg/

> Homepage of Doron Zeilberger

>

> But it also seemed to suggest that this was not one of them.

> For confirmation (or not), see, for example:

>

> http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/enquiry.html

> An Enquiry Concerning Human (and Computer!) [Mathematical] Understanding

[...]

In the first paragraph on page 20 of the essay, he says among other things

that the mathematical universe is the same as the physical universe,

that our unique universe is finite and that everything is computation.

Also, near the beginning he gives the impression of being fond of (or admiring)

skeptics, including Hume and others.

I think he probably is a finistic Platonist (whatever that means), but

it appears to me a bit of mysticism to say that the largest natural number,

plus one, is equal to zero.

David Bernier

Feb 16, 2010, 2:28:23 AM2/16/10

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David Bernier <davi...@videotron.ca> writes:

> I think he probably is a finistic Platonist (whatever that means), but

> it appears to me a bit of mysticism to say that the largest natural

> number, plus one, is equal to zero.

Taking the view that statements about naturals should be understood with

reference to the physical (computational?) universe (in some idealised

sense), and assuming we take it to follow on this view that there is a

largest natural, what to make of the successor of the largest natural is

a matter decided essentially by stipulation. We may decree it's zero,

that it's undefined, that applying the successor function to the largest

natural leaves it unperturbed, or pretty much anything that strikes our

fancy, anything we find convenient; just as in ordinary mathematics

whether zero is a natural or not is just a matter of stipulation.

(Lest there be any confusion, lest any innocent mind be led astray, let

us note here that it is not at all a necessary component of

ultra-finitism or ultra-intuitionism that there be a largest

natural. Coming clean, I must also admit that I didn't bother to consult

Zeilberger's essays before composing this reply.)

--

Aatu Koskensilta (aatu.kos...@uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"

- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Feb 16, 2010, 2:49:54 AM2/16/10

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On Feb 16, 3:28 am, Angus Rodgers <twirl...@yahoo.co.uk> wrote:

> ... Doron Zeilberger,

> in which he claimed not only that there is a largest natural number

> (I'd come across mention of Zeilberger's ultrafinitism before), but

> that if you add 1 to this largest natural number, the result is 0.

> ... Doron Zeilberger,

> in which he claimed not only that there is a largest natural number

> (I'd come across mention of Zeilberger's ultrafinitism before), but

> that if you add 1 to this largest natural number, the result is 0.

I don't see the problem. Ramanujan once wrote

1+2+3+4+... = -1/12

I think you'll agree the left side is positive and large.

Moreover if we multiply by 12, I think I can prove the result

is larger than any natural number you can name:

12*(1+2+3+4+...) = -1

So this is the largest natural number; if you add 1 you

get 0. Q.E.D.

I don't think you want to argue with Ramanujan about large

natural numbers. He knew about 1729, a number so large

that many languages can't even express it.

Sam (posting from jda's account)

Feb 16, 2010, 4:07:13 AM2/16/10

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On Mon, 15 Feb 2010 23:49:54 -0800 (PST), James Dow Allen

<jdall...@yahoo.com> wrote:

<jdall...@yahoo.com> wrote:

>I don't see the problem.

That must be nice.

--

Angus Rodgers

Feb 16, 2010, 4:25:24 AM2/16/10

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On Tue, 16 Feb 2010 02:19:08 -0500, David Bernier

<davi...@videotron.ca> wrote:

<davi...@videotron.ca> wrote:

>I can't understand the last part... A zillion marbles in a bag,

>we had one marble to the bag, and then there are no marbles (?)

Yes. One wonders what his numbers actually do for a living, and

what part they play in counting, or in the definition of what it

means for a set to be finite, which is so important to him.

It would seem, on the face of it, that his universe contains sets

'larger' than his set of natural numbers (such as the set Z x Z

that I mentioned), and a finite 'successor function' could easily

be defined on such a set (e.g., for the set Z x Z, we could take

(m, n)' = (m, n'), except when n' = 0, and then (m, n)' = (m', 0)),

and then it's not easy to see why this larger cycle of entities,

whose Platonic existence he presumably accepts, would not function

just as well in the role of 'natural numbers' as the cycle Z that

he started with. Of course, he has Z = Z_p, where p is (in some

sense!) a very large prime (never mind that it is equal to 0 in

his system!), so his Z is an integral domain, whereas the larger

system would not be. In the larger system, it would presumably

work out that p^2 = 0 - but I can't see how that's any worse than

p = 0! Nor can I see how he is able to get to choose that his Z

is an integral domain (therefore also a field) in the first place.

--

Angus Rodgers

Feb 16, 2010, 4:30:07 AM2/16/10

to

> alas, Bigfoot has gone tits-up)

We already discussed some stuff about Doron Zeibelger in the forum

once, some years ago, since it is a hook from which some cranks here

love to hang in some discussions.

I considere Doron the only mathematician I know who is not only a

crank with regards to the subject of infinity and surroundings but in

fact a not-so-nice one, as one can read in the following well-known

piece by him:

http://www.math.rutgers.edu/~zeilberg/Opinion68.html

As expected, he bases his case on computers and stuff. Pretty lame

imo, but of course that's his right. Calling others fools and

belittling them is already too much.

It is highly advisable to read the feedbacks at the bottom of the

above site, which rise some interesting points.

He also acknowledges and thankx (!!!) WM for something, and he also

receives a positive feedback from A. Zenkin from whom I read something

some years ago but I can't now remember what, though I clearly

remember that he was on the edge of crankhood, at least.

Tonio

Feb 16, 2010, 4:42:53 AM2/16/10

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Tonico <Toni...@yahoo.com> writes:

> He also acknowledges and thankx (!!!) WM for something, and he also

> receives a positive feedback from A. Zenkin from whom I read something

> some years ago but I can't now remember what, though I clearly

> remember that he was on the edge of crankhood, at least.

Zenkin's splendid contributions were discussed here a while back. He has

uncovered several fatal flaws in the diagonal argument, in particular

that it proves nothing, based on vaguely Wittgensteinian

reflections. Somewhat dreary and tedious, but great stuff nonetheless!

Feb 16, 2010, 4:44:06 AM2/16/10

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On Tue, 16 Feb 2010 09:28:23 +0200, Aatu Koskensilta

<aatu.kos...@uta.fi> wrote:

<aatu.kos...@uta.fi> wrote:

>David Bernier <davi...@videotron.ca> writes:

>

>> I think he probably is a finistic Platonist (whatever that means), but

>> it appears to me a bit of mysticism to say that the largest natural

>> number, plus one, is equal to zero.

>

>Taking the view that statements about naturals should be understood with

>reference to the physical (computational?) universe (in some idealised

>sense), and assuming we take it to follow on this view that there is a

>largest natural, what to make of the successor of the largest natural is

>a matter decided essentially by stipulation. We may decree it's zero,

>that it's undefined, that applying the successor function to the largest

>natural leaves it unperturbed, or pretty much anything that strikes our

>fancy, anything we find convenient; just as in ordinary mathematics

>whether zero is a natural or not is just a matter of stipulation.

If that's so, I can't understand how he can feel himself free to

'stipulate' any properties that he likes, when he is so insistent

that the way numbers are is the way that they really are in the

world.

I'm not saying you're wrong, just that I still can't imagine, on

this view, how to think like him. In support of your view, there

is the fact that his natural numbers conveniently form a field,

which presumably (or at least so it eventually occurred to me as

being likely) results from him preferring to avoid the possible

embarrassment or awkwardness of them failing to form an integral

domain, which in turn seems to imply that he is 'stipulating' his

Platonic numbers to have whatever properties he finds convenient

(or indeed, 'natural'). But I still can't get my head around it.

Perhaps what he's basically Platonist about is [some finitist

version] of set theory? The exact definition of what numbers

are, in any version of set theory, does always seem to involve

some sort of arbitrary stipulation, and perhaps he feels that

his arbitrary stipulation makes as much sense as any other?

I'm just guessing; I can never get very Platonically worked up

about set theory myself; so I still can't imagine what's going

on in his mind. But if it's something along these lines, then

at least the mystery is closer to the mystery of what people

usually take numbers to be when they take set theory as being

fundamental.

Me, I don't have any idea of what's fundamental, and I haven't

even been thinking about it recently, until this interview in

a television programme forced me to wonder what on Earth this

guy was thinking about!

>(Lest there be any confusion, lest any innocent mind be led astray, let

>us note here that it is not at all a necessary component of

>ultra-finitism or ultra-intuitionism that there be a largest

>natural. Coming clean, I must also admit that I didn't bother to consult

>Zeilberger's essays before composing this reply.)

He does mention that there are other forms of ultrafinitism (some

of which, at least, involve some sort of 'fading out' property).

--

Angus Rodgers

Feb 16, 2010, 4:56:42 AM2/16/10

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Angus Rodgers <twir...@yahoo.co.uk> writes:

> If that's so, I can't understand how he can feel himself free to

> 'stipulate' any properties that he likes, when he is so insistent that

> the way numbers are is the way that they really are in the world.

Whether there's a largest natural is of course not something we can

decide by stipulation -- no, such matters turn on the fundamental

physical nature of reality. Even so, there are many questions that do

not turn on physical fact, such as whether we count zero as a natural,

or what to say about the successor of the largest natural. We can say

it's zero, the largest natural itself, five, or whatever we want. This

is just a question of what convention to adopt.

> Perhaps what he's basically Platonist about is [some finitist version]

> of set theory? The exact definition of what numbers are, in any

> version of set theory, does always seem to involve some sort of

> arbitrary stipulation, and perhaps he feels that his arbitrary

> stipulation makes as much sense as any other?

There's no need to drag in any set theory. Putting ultra-finitism to one

side, we can ask what is, in the natural numbers, the predecessor of

zero? In some context it's convenient to say there is no such thing, the

predecessor function is not defined at zero, in others it's more

convenient to stipulate that zero is its own predecessor.

> He does mention that there are other forms of ultrafinitism (some

> of which, at least, involve some sort of 'fading out' property).

Yes, I hope to have something to say about that later on.

Feb 16, 2010, 5:07:54 AM2/16/10

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On Tue, 16 Feb 2010 01:30:07 -0800 (PST), Tonico <Toni...@yahoo.com> wrote:

>We already discussed some stuff about Doron Zeibelger in the forum

>once, some years ago, since it is a hook from which some cranks here

>love to hang in some discussions.

I'm sure they do, but let's hope we can avoid this becoming that

kind of thread. I only want to understand what's going on in the

guy's mind.

(I don't recall seeing any threads about DZ myself. In fact, I

once mentioned that it would be more interesting if he posted to

sci.math, instead of some of the abusive characters who do post

endlessly about their, shall we say, unorthodox opinions! Do you

have any references to such threads? Or were they just the usual

kind of c___k thread?)

>I considere Doron the only mathematician I know who is not only a

>crank with regards to the subject of infinity and surroundings

He's also unique in my (limited) experience. My curiosity as to

what he is thinking is unaffected by questions as to any of what

he is thinking is true in any sense. I just want to know how he

manages to think it! Or indeed, whether he really does think it,

or is joking.

>but in

>fact a not-so-nice one, as one can read in the following well-known

>piece by him:

>

>http://www.math.rutgers.edu/~zeilberg/Opinion68.html

>

>As expected, he bases his case on computers and stuff. Pretty lame

>imo, but of course that's his right. Calling others fools and

>belittling them is already too much.

I read that, but didn't get any impression of nastiness, just some

sort of tongue-in-cheek humour (which I don't get). Nope. Just

read it again, and I still don't get any impression of him being

like one of those sci.math c___ks. (Like the one who was literally

threatening me with an axe not long before I stopped posting here!)

>It is highly advisable to read the feedbacks at the bottom of the

>above site, which rise some interesting points.

>He also acknowledges and thankx (!!!) WM for something, and he also

>receives a positive feedback from A. Zenkin from whom I read something

>some years ago but I can't now remember what, though I clearly

>remember that he was on the edge of crankhood, at least.

How he is to be described - and indeed, even the question of the

truth-value of anything he says - doesn't bear on the mystery of

how he manages to think what he apparently does think (unless he

is joking). That's the objective problem for me.

There isn't any question of it being 'advisable' to take note of

any apparent murky associations with known c___ks (who may indeed

be nasty - I have never paid any attention to the WM threads in

sci.math, so I wouldn't know about him, although I remembered his

name when I saw the reference), unless there was some intention

of taking DZ's beliefs as authoritative in some way, thus risking

some terrible mental contagion. I'm just trying to resolve the

apparent paradox of how he is able to do some genuine mathematics

while also believing something which appears to me to be crazy or

a joke - and not just something about little green men or NASA

not landing men on the Moon, but something about the mathematics

he actually does with professional competence. Quite baffling!

--

Angus Rodgers

Feb 16, 2010, 5:15:53 AM2/16/10

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On Tue, 16 Feb 2010 11:56:42 +0200, Aatu Koskensilta

<aatu.kos...@uta.fi> wrote:

<aatu.kos...@uta.fi> wrote:

>Whether there's a largest natural is of course not something we can

>decide by stipulation -- no, such matters turn on the fundamental

>physical nature of reality. Even so, there are many questions that do

>not turn on physical fact, such as whether we count zero as a natural,

>or what to say about the successor of the largest natural. We can say

>it's zero, the largest natural itself, five, or whatever we want. This

>is just a question of what convention to adopt.

I see what you mean now.

However, this does seem to leave unexplained why he seems to believe

that the largest natural number is p - 1, for some very, very, very

large prime p. I'm not bothering so much about what sense he makes

of p when p = 0 in his system - I'll assume there is a way. The fact

that his largest natural is p - 1, however, does seem to suggest that

his system has built into it from the start the property that the

successor of the largest natural is 0.

--

Angus Rodgers

Feb 16, 2010, 5:17:08 AM2/16/10

to

Angus Rodgers <twir...@yahoo.co.uk> writes:

> I'm just trying to resolve the apparent paradox of how he is able to

> do some genuine mathematics while also believing something which

> appears to me to be crazy or a joke - and not just something about

> little green men or NASA not landing men on the Moon, but something

> about the mathematics he actually does with professional competence.

> Quite baffling!

A mathematician I met once proudly explained that mathematics is the

study of formal systems. When I queried about his work it turned out it

had absolutely nothing to do with formal theories but was concerned with

heavy going mathematics of algebraic this-or-that (of which I understood

nothing).

People are capable of believing, or at least professing to believe, the

most peculiar things without it affecting their work in the least, even

in areas that are seemingly directly implicated in these their

(professed) beliefs.

Feb 16, 2010, 5:18:23 AM2/16/10

to

Angus Rodgers <twir...@yahoo.co.uk> writes:

> The fact that his largest natural is p - 1, however, does seem to

> suggest that his system has built into it from the start the property

> that the successor of the largest natural is 0.

Quite possibly. As noted, I know nothing of his system.

Feb 16, 2010, 5:26:49 AM2/16/10

to

On Feb 16, 11:42 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:

> Tonico <Tonic...@yahoo.com> writes:

> > He also acknowledges and thankx (!!!) WM for something, and he also

> > receives a positive feedback from A. Zenkin from whom I read something

> > some years ago but I can't now remember what, though I clearly

> > remember that he was on the edge of crankhood, at least.

>

> Zenkin's splendid contributions were discussed here a while back. He has

> uncovered several fatal flaws in the diagonal argument,

> Tonico <Tonic...@yahoo.com> writes:

> > He also acknowledges and thankx (!!!) WM for something, and he also

> > receives a positive feedback from A. Zenkin from whom I read something

> > some years ago but I can't now remember what, though I clearly

> > remember that he was on the edge of crankhood, at least.

>

> Zenkin's splendid contributions were discussed here a while back. He has

> uncovered several fatal flaws in the diagonal argument,

"Flaws"? Within ZFC? Name one, just for the fun of it...;)

And it'd be interesting to get that paper(s) by Zenkin, if you have

some.

Tonio

in particular

> that it proves nothing, based on vaguely Wittgensteinian

> reflections. Somewhat dreary and tedious, but great stuff nonetheless!

>

> --

> Aatu Koskensilta (aatu.koskensi...@uta.fi)

>

> "Wovon man nicht sprechan kann, darüber muss man schweigen"

Feb 16, 2010, 5:37:58 AM2/16/10

to

Tonico <Toni...@yahoo.com> writes:

> "Flaws"? Within ZFC? Name one, just for the fun of it...;)

The main flaw is of course that the diagonal argument doesn't prove

anything.

> And it'd be interesting to get that paper(s) by Zenkin, if you have

> some.

From the abstract of Zenkin's

_LOGIC OF ACTUAL INFINITY AND G. CANTOR'S DIAGONAL PROOF OF THE

UNCOUNTABILITY OF THE CONTINUUM_

a little something to whet your appetite:

Since Cantor first constructed his set theory, two independent

approaches to infinity in mathematics have persisted: the Aristotle

approach, based on the axiom that "all infinite sets are potential,"

and Cantor's approach, based on the axiom that "all infinite sets

are actual." A detailed analysis of the "rule-governed" usage of

'actual infinity' reveals that Cantor's diagonal proof is based on

two hidden, but nonetheless necessary conditions never explicitly

mentioned but in fact algorithmically used both in Cantor's so

called "naive" set theory as well as modern "nonnaive" axiomatic set

the- ories. An examination of "rule-governed" usage of the first

necessary condition opens the way for a rigorous proof that in

reality Cantor's diagonal procedure proves nothing, and merely

reduces one problem, that associated with the un- countability of

real numbers (the continuum), to three new and additional

problems. The second necessary condition is simply a teleological

one possessing no real relation to mathematics.

Further analysis reveals that Cantor's Diagonal Method (CDM), being

the only procedure for distinguishing infinite sets on the basis of

their cardinalities, does not distinguish infinite from finite sets

just on the basis of the number of their elements (cardinality); the

results of CDM depend fatally upon the order of real numbers in the

sequences to which it is applied. Cantor's diagonal proof itself is

formally а "half" of the well-known "Liar" paradox but which can be

used to produce a new set-theoretical paradox of the "Liar" type.

- - -

The fact to be demonstrated is that ultimately Cantor's diagonal

proof engages us in an endless, potentially infinite, and quite

senseless paradoxical "game of two honest tricksters" (a new

set-theoretical paradox) which, as Wittgenstein alleged, "has no

relation to what is called a deduction in logic and mathematics."

- - -

Here it is argued that Cantor's proof does not in fact prove the

uncountability of the continuum, but rather proves something else

entirely, viz. Aristotle's Thesis (stated in its later canonized

Latin form): "Infinitum Actu Non Datur." In other words, it proves

that an actual infinity "is never permitted in mathematics" (Gauß),

or alternatively speaking, that in the words of Poincaré "there is

no actual infinity; Cantorians forgot that and fell into

contradictions. [... ] Later generations will regard set theory as a

disease from which one has recovered!"

The fabricated quote from Poincaré is a particularly nice touch

(although not an original contribution of Zenkin's).

--

Aatu Koskensilta (aatu.kos...@uta.fi)

Feb 16, 2010, 7:09:02 AM2/16/10

to

On Feb 16, 12:37 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:

> Tonico <Tonic...@yahoo.com> writes:

> > "Flaws"? Within ZFC? Name one, just for the fun of it...;)

>

> The main flaw is of course that the diagonal argument doesn't prove

> anything.

>

> Tonico <Tonic...@yahoo.com> writes:

> > "Flaws"? Within ZFC? Name one, just for the fun of it...;)

>

> The main flaw is of course that the diagonal argument doesn't prove

> anything.

>

That, of course, is your personal opinion. The diagonal argument as

used in Cantor Theorem's proof proves that the set of real numbers in

the interval [0,1] cannot be equinumerous to the natural numbers.

Nice. I don't know who "the cantorians" are, specially nowadays, but

the distinction between "potential and actual" mathematics as stressed

above, and in much lamer and nonsensical fashion by many cranks around

the house, isn't that clear to me: what do Zenkin, or you, or others,

believe that people that has no problem with the diagonal argument of

Cantor believe? I, for one, have no problem at all with that argument,

and it looks to me a rather simple and elegant way to prove that [0,1]

isn't countable. Fine, so what does that say about me with regards to

"potential and/or actual" infinity?

It'd be interesting as well to know to what "contradictions" does

Poincare refered to above...

Tonio

> Aatu Koskensilta (aatu.koskensi...@uta.fi)

Feb 16, 2010, 7:38:54 AM2/16/10

to

In the standard real analysis of today, the Intermediate Value Theorem and

the Mean Value Theorem are used a lot. For example, if f(x) is

a polynomial function of odd degree and real coefficients,

f: R -> R, then the IVT implies that f has at least one real root.

And the IVT rests upon the completeness of the reals.

The proof at Wikipedia uses the least upper bound property for reals:

< http://en.wikipedia.org/wiki/Intermediate_value_theorem >

Zenkin rejects cardinals above aleph_0, and also Cantor's Diagonal proof.

I'm not sure how those who reject the orthodox treatment of real numbers

would want to prove that f in IR[x] of odd degree has at least one real root ...

Go back to geometrical intuition? Construct root-finding algorithms?

I don't know.

Feb 16, 2010, 8:50:37 AM2/16/10

to

Tonico <Toni...@yahoo.com> writes:

> That, of course, is your personal opinion.

My dear Tonico, it was of course not my opinion but Zenkin's! My

objection to the highly suspect diagonal argument -- which, in this

context, we may take to be the very simple proof of the fact that |X| <

|P(X)|, rather than any messy stuff with reals and decimal expansions

|and whatnot -- is based on more abstruse and subtle considerations,

involving the problematic business of normalising the diagonal proof (it

gets a bit hairy, proof theoretically speaking, the sort of stuff that

is beyond the logical ken of pretty much any average mathematician;

hence the perfect spot to find something underhanded in the proof).

But Cantor's ghost is upon us, in a cardboard cut-out version, weeping

tears of despair and desolation. The ghost intones thusly:

The crowd was caught in a murderous cross-fire; hundreds more died in

the next few minutes, torn apart in a concentrated hail of cannon balls

and grapeshot.

Perhaps we'll find it in our hearts to mend our evil ways? Look into

your hearts. You will find, as I did, an angel of mercy, an angel crying

the unquenchable tears of Cantor's ghost, forever weeping over the evil

bile we spit at each other in news.

--

Aatu Koskensilta (aatu.kos...@uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"

Feb 16, 2010, 11:44:42 AM2/16/10

to

On Tue, 16 Feb 2010 07:38:54 -0500, David Bernier

<davi...@videotron.ca> wrote:

<davi...@videotron.ca> wrote:

>In the standard real analysis of today, the Intermediate Value Theorem and

>the Mean Value Theorem are used a lot. For example, if f(x) is

>a polynomial function of odd degree and real coefficients,

>f: R -> R, then the IVT implies that f has at least one real root.

>And the IVT rests upon the completeness of the reals.

>

>The proof at Wikipedia uses the least upper bound property for reals:

>< http://en.wikipedia.org/wiki/Intermediate_value_theorem >

>

>Zenkin rejects cardinals above aleph_0, and also Cantor's Diagonal proof.

>

>I'm not sure how those who reject the orthodox treatment of real numbers

>would want to prove that f in IR[x] of odd degree has at least one real root ...

>Go back to geometrical intuition? Construct root-finding algorithms?

I already seem to detect signs of the dreaded convergence towards

a discussion of the usual suspects and their heresies. Of course

I have no power to stop that, just because I started the thread,

but, unless either there is something new being said about other

mathematical heretics, or jokers (I haven't been around for a long

time, and I never followed all such threads - only occasionally the

JSH ones), or such a widened discussion sheds light on Zeilberger's

serious or humorous beliefs about the natural number system (by the

way, he seems to reject the entirety of mathematical analysis, root

and branch, as meaningless!), I would beg for the discussion not to

go too far in that direction.

On a more positive note, what I would like is to be helped to see

(a) how it is possible for anyone to think like Zeilberger at all,

and (b) how thinking like Zeilberger is nevertheless impossible for

me in particular, because it is incompatible with some principles I

hold to and he doesn't. If I can understand how even a professional

mathematician can believe something so fundamental, which I believe

to be false, I might better understand why I believe it to be false.

It might be a bit like coming to an understanding of the principle

of conservation of energy, by seeing in detail a failed attempt to

build a perpetual motion machine. That analogy might not work very

well - because I can't imagine coming to a better understanding of

field theory by following in detail some foolish person's attempt

to duplicate the cube! - but I think that is because it omits one

feature of this case, which is that what Zeilberger denies seems so

'obviously' true (unlike energy conservation, or the impossibility

of duplicating the cube by Euclidean means), that being able to

understand DZ's wacky point of view ought to help me to see beyond

the 'obvious', which is always a good thing - so long as you don't

completely lose sight of what is, in the end, still 'obvious'!

At least, that's why I think I'm interested, but any account of

reasons for being interested is less interesting than the thing

itself.

--

Angus Rodgers

Feb 16, 2010, 12:48:59 PM2/16/10

to

On Feb 16, 6:44 pm, Angus Rodgers <twirl...@yahoo.co.uk> wrote:

> On Tue, 16 Feb 2010 07:38:54 -0500, David Bernier

>

> On Tue, 16 Feb 2010 07:38:54 -0500, David Bernier

>

It was "obvious" that the heavier an object is the fastest it'll

freely fall towards the floor, specially after Mr. Aristotle

determined this, until Mr. Galileo proved otherwise.

That our intuition can get damaged by some facts in science in general

and in maths in particular is no news, though professional

mathematicians, used to this, could be more aware of the problem and

develop a "new mathematical intuition"...which, again, can be wrong

all the way.

I don't know why Doron believes what he does, and reading his stuff I

can't find some really compelling reason to that: perhaps a craving

for "originality", perhaps something in his personal educational

background, perhaps just a (mathematical or whatever) huntch. What I

find hard to digest from him is the patronizing and belittling

attitude he undertakes against those accepting "the paradise of fools"

that Cantors bestowed us. Let us not forget that not only the huge

majority of mathematicians, but at least a great deal (most of

them...?) of logicians have no problems with that paradise.

I'm gonna read some of Doron's writings once again...

Tonio

Feb 16, 2010, 3:42:43 PM2/16/10

to

On Tue, 16 Feb 2010 09:48:59 -0800 (PST), Tonico

<Toni...@yahoo.com> wrote:

<Toni...@yahoo.com> wrote:

>I don't know why Doron believes what he does, and reading his stuff I

>can't find some really compelling reason to that: perhaps a craving

>for "originality", perhaps something in his personal educational

>background, perhaps just a (mathematical or whatever) huntch. What I

>find hard to digest from him is the patronizing and belittling

>attitude he undertakes against those accepting "the paradise of fools"

>that Cantors bestowed us. Let us not forget that not only the huge

>majority of mathematicians, but at least a great deal (most of

>them...?) of logicians have no problems with that paradise.

>

>I'm gonna read some of Doron's writings once again...

Ditto.

The first statement of his philosophy that I read was this:

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/real.html

(extract, page 8)

``Myself, I don't care so much about the natural world. I am a

platonist, and I believe that integers, finite sets of finite

integers, and all finite combinatorial structures have an existence

their own, regardless of humans (or computers). I also believe that

symbols have an independent existence. What is completely

meaningless is any kind of infinite, actual or potential.

So I deny even the existence of the Peano axiom that every integer

has a successor. Eventually we would get an overflow error in the

big computer in the sky, and the sum and product [of] two integers

is well-defined only if the result is less than p, or if one

wishes, one can compute modulo p. Since p is so large, this is not

a practical problem, since the overflow in our earthly computers

comes so much sooner than the overflow errors in the big computer

in the sky.''

I haven't found anything that clarifies this. But he also keeps

saying things like this:

http://www.math.rutgers.edu/~zeilberg/Opinion69.html

``Since all knowable math is ipso facto trivial, why bother? So

only do /fun/ problems, that you really enjoy doing. It would be

a shame to waste our short lives doing "important" math, since

whatever /you/ can do, would be done, very soon (if not already)

faster and better (and more elegantly!) by computers. So we may

just as well enjoy our humble trivial work.''

This seems (at least to my depressed mind) to have an implicitly

depressive tone to it, and it might give some sort of clue to his

motivation, if not to his actual beliefs.

Some sort of inferiority complex in relation to the computer,

not only on his own behalf, but on behalf of humanity as a whole?

He seems to /enjoy/ being tied down to the finite, the concrete,

the physical, and the computable. I can't imagine enjoying that,

nor can I imagine understanding mathematics on such a basis, any

more than I can imagine understanding it on any other basis!

But it's what he actually believes that matters, not speculation

as to why he believes it.

Even if I'm going to disbelieve something, I like at least to have

a clear idea of what I'm disbelieving - and so far, I can't get my

head around his ideas at all.

There's another manifesto here:

http://www.math.rutgers.edu/~zeilberg/Opinion43.html

"It Is Time to Move On to NON-EUCLIDEAN MATHEMATICS"

And (in spite of the overall optimistic tone) there's that sad

refrain again:

"Let's face it, anything we humans can know for sure is trivial,

since we are creatures of such low complexity."

I haven't had a look at this yet (referenced from the above):

http://www.math.rutgers.edu/~zeilberg/GT.html

"Plane Geometry by Shalosh B. Ekhad XIV"

There's more about computers and programming here (I haven't read

this one right through yet):

http://www.math.rutgers.edu/~zeilberg/GT.html

"Don't Ask: What Can The Computer do for ME?,

But Rather: What CAN I do for the COMPUTER?"

That article contains some feedback, including his own reply to

Greg Kuperberg in sci.math.research, thus (edited down by me):

``

>Doron's argument for his thesis is a little crazy, but it's not

>completely crazy. You shouldn't interpret it literally,

>even though Doron himself might.

Of course, you should not interpret it literally, but neither

should you interpret anybody's text literally. As Derrida, Rorty

and several others have shown, we are slaves to our own final#

vocabularies and we always have hidden agendas, and our `objective

views' are just an instrument to bolster our ego, and to justify

to ourselves our miserable existence. Now that plain Racism and

Sexism is out of style, we cling to Human Chauvinism.

''

Derrida and Rorty? Oh dear. Oh dearie me.

That exchange (all of it, not just the bit I've quoted) seems

quite revealing, at least as to his motives and his general

philosophy, but it still leaves me in the dark as to how he can

actually believe what he believes about the natural numbers.

--

Angus Rodgers

Feb 16, 2010, 3:47:15 PM2/16/10

to

On Tue, 16 Feb 2010 20:42:43 +0000, I mistyped:

>http://www.math.rutgers.edu/~zeilberg/GT.html

>"Don't Ask: What Can The Computer do for ME?,

>But Rather: What CAN I do for the COMPUTER?"

Correction: that URL should be:

http://www.math.rutgers.edu/~zeilberg/Opinion36.html

--

Angus Rodgers

Feb 16, 2010, 4:08:06 PM2/16/10

to

On Tue, 16 Feb 2010 20:47:15 +0000, I wrote:

>>"Don't Ask: What Can The Computer do for ME?,

>>But Rather: What CAN I do for the COMPUTER?"

>

>Correction: that URL should be:

>

>http://www.math.rutgers.edu/~zeilberg/Opinion36.html

Now that I actually start to read through it, I see that it

even mentions the book title on which I based the title of

this thread: 'Surely You're Joking Mr. Feynman'!

I hadn't seen it, I promise!

--

Angus Rodgers

Feb 16, 2010, 4:22:03 PM2/16/10

to

Angus Rodgers wrote:

[...]

[...]

> On a more positive note, what I would like is to be helped to see

> (a) how it is possible for anyone to think like Zeilberger at all,

> and (b) how thinking like Zeilberger is nevertheless impossible for

> me in particular, because it is incompatible with some principles I

> hold to and he doesn't. If I can understand how even a professional

> mathematician can believe something so fundamental, which I believe

> to be false, I might better understand why I believe it to be false.

[...]

How can one know extensively about what exists in the physical

universe? There could be particles or things besides neutrinos, etc.

Generally speaking, aren't many philosophies close to systems of

beliefs? About (a), I'd say "computer-centric" ultrafinitism

is DZ's philosophy of mathematics, a kind of system of beliefs.

It seems to me that many philosophical views are very hard to

prove wrong, maybe because things can't be tested, etc.

I'd say that Zeilberger is quite settled in his finitistic views of

the universes of physics and maths.

I had a look at a PDF file of a talk he gave at a conference:

< http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf >

Feb 16, 2010, 5:24:36 PM2/16/10

to

On Tue, 16 Feb 2010 16:22:03 -0500, David Bernier

<davi...@videotron.ca> wrote:

<davi...@videotron.ca> wrote:

>I'd say that Zeilberger is quite settled in his finitistic views of

>the universes of physics and maths.

Yes. Although his /style/ is humorous and provocative, he seems

to be deadly serious about that. As with all other philosophies

of mathematics, I just don't understand it, but that's my problem!

--

Angus Rodgers

Feb 16, 2010, 5:49:55 PM2/16/10

to

In article <877hqdq...@dialatheia.truth.invalid>,

Aatu Koskensilta <aatu.kos...@uta.fi> wrote:

Aatu Koskensilta <aatu.kos...@uta.fi> wrote:

> People are capable of believing, or at least professing to believe, the

> most peculiar things without it affecting their work in the least, even

> in areas that are seemingly directly implicated in these their

> (professed) beliefs.

Indeed, an astronomer of my acquaintance firmly believes

that the universe was created about 6000 years ago, which

doesn't stop him from doing work that relies on the more

usual estimates. He is aware of the contradiction, and says

it's just one of those things he accepts; he's sure there's a way

to reconcile the two views, even if he himself hasn't come up

with one.

--

Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

Feb 16, 2010, 11:26:29 PM2/16/10

to

why is it that implied infinities are so bad, if

zeta(2) requires infinite terms to be 6/pi^2?

zeta(2) requires infinite terms to be 6/pi^2?

thus:

are the five eggs arrayed like a regular pentagon?

http://sites.google.com/site/tommy1729/home/eggs-problem

--Another Flower for Einstein:

http://www.21stcenturysciencetech.com/articles/spring01/Electrodynamiics.html

--les OEuvres!

http://wlym.com

--Stop Cheeny, Ricw & the ICC in Sudan;

no more Anglo-american quagmires!

http://larouchepub.com/pr/2010/100204rice

Feb 17, 2010, 2:34:24 AM2/17/10

to

David Bernier <davi...@videotron.ca> writes:

> It seems to me that many philosophical views are very hard to prove

> wrong, maybe because things can't be tested, etc.

In general the idea of proving a philosophical idea wrong is

silly. We're dealing rather with "a sort of persuasion" as the old

Witters put it. It's a matter of ideas, ways of thinking, various more

or less natural attitudes, and proof simply doesn't enter into it.

Feb 17, 2010, 5:55:41 AM2/17/10

to

Aatu Koskensilta wrote:

> David Bernier <davi...@videotron.ca> writes:

>

>> It seems to me that many philosophical views are very hard to prove

>> wrong, maybe because things can't be tested, etc.

>

> In general the idea of proving a philosophical idea wrong is

> silly. We're dealing rather with "a sort of persuasion" as the old

> Witters put it. It's a matter of ideas, ways of thinking, various more

> or less natural attitudes, and proof simply doesn't enter into it.

> David Bernier <davi...@videotron.ca> writes:

>

>> It seems to me that many philosophical views are very hard to prove

>> wrong, maybe because things can't be tested, etc.

>

> In general the idea of proving a philosophical idea wrong is

> silly. We're dealing rather with "a sort of persuasion" as the old

> Witters put it. It's a matter of ideas, ways of thinking, various more

> or less natural attitudes, and proof simply doesn't enter into it.

Yes, I think that's a good way of putting it. I've wondered at

times if there could be formal rules for debating (just as happens

on Usenet) that could be advantageous to a good debate.

For a two-person debate or discussion, some examples of rules or

guidelines would be:

(a) No ad-hominem attacks.

(b) To stay on topic (a bit of leeway is fine).

(c) Keep in mind that rhetoric isn't persuasive (at the end of the day?).

(d) Debaters can make one or more points per exchange.

(e) For each point raised by the other side, it's a good idea

to state what one's own view is, e.g.:

(i) Point 25a accepted.

(ii) The point is neither accepted nor challenged.

The point's meaning is clear. To be addressed later

by me (or should be).

(iii) Request for clarification on Point 25a: [to be completed].

(iv) [An opposition] My response to Point 25a is:

[statements] oppositions can also be called "counter-points".

(f) In addition to answering the other side's points or counterpoints,

one can make new points.

(g) When it's their turn, a debater can also ask a question addressed

to the other debater.

(h) A sensible reply to a question is for example:

(i) Give a reply.

(ii) Ask for the question to be clarified.

(iii) Accept the question as formulated and say something like:

"I'll think about it".

(I) Example of good question: What's your basis for saying [...] ?

[ If it's a scientific debate, one could ask what the source

of some unexplained statement is, e.g. for the statement

"Some birds of the same genus as penguins can fly." ]

(j) If a series of points/counterpoints goes on long enough,

and if both debaters feel it's worth it, they can enter

a sub-debate, labeled e.g. "Sub-debate on point

12b which was made by T. ."

(k) Alternatively, for long series of points and counterpoints,

if the debaters are tired of discussing that point,

they can mutually agree to close the debate on that point

to leave time for other points.

(l) It could be a good idea to propose conclusions and summaries

after a long enough period. To count, the conclusions and

summaries must be accepted by the two sides.

(m) Maybe think of following the Don't Repeat Yourself

principle, depending on circumstances.

Reference:

< http://en.wikipedia.org/wiki/Don%27t_repeat_yourself >

An example in a debate would be to cite one primary source

for a fact rather than come up with two separate secondary

sources, especially if both secondary sources rely on the

same primary source.

I'd be interested to hear about recommended ways to have

good debates.

David Bernier

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