Surely You're Joking, Mr. Zeilberger?

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Angus Rodgers

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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)

wht

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

Angus Rodgers

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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

David Bernier

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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

Aatu Koskensilta

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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

James Dow Allen

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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.

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)

Angus Rodgers

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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:

>I don't see the problem.

That must be nice.

--
Angus Rodgers

Angus Rodgers

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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:

>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

Tonico

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Feb 16, 2010, 4:30:07 AM2/16/10
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> (formerly, twir...@bigfoot.com;

> 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

Aatu Koskensilta

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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!

Angus Rodgers

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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:

>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

Aatu Koskensilta

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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.

Angus Rodgers

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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

Angus Rodgers

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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:

>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

Aatu Koskensilta

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

Aatu Koskensilta

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

Tonico

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


"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"

Aatu Koskensilta

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

Tonico

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


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)

David Bernier

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

Aatu Koskensilta

unread,
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"

Angus Rodgers

unread,
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:

>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

Tonico

unread,
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
>

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

Angus Rodgers

unread,
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:

>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

Angus Rodgers

unread,
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

Angus Rodgers

unread,
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

David Bernier

unread,
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 >

Angus Rodgers

unread,
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:

>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

Gerry Myerson

unread,
Feb 16, 2010, 5:49:55 PM2/16/10
to
In article <877hqdq...@dialatheia.truth.invalid>,
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)

spudnik

unread,
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?

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

Aatu Koskensilta

unread,
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.

David Bernier

unread,
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.

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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