Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)

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Date: 30 Oct 2004 00:03:08 -0700

examachine@gmail.com (Eray Ozkural exa) wrote in message news:<320e992a.0410271136.4479a210@posting.google.com>...
> Dear Chas,
>
> Thanks for your post.
>
> cbrown@cbrownsystems.com (Chas Brown) wrote in message news:<ba7dcfd7.0410161641.57a0b26e@posting.google.com>...
> > examachine@gmail.com (Eray Ozkural exa) wrote in message news:<320e992a.0410150808.6117de6f@posting.google.com>...
> > > "Plamen Petrov" <ppetrov@hotmail.com> wrote in message news:<2t70ugF1s991jU1@uni-berlin.de>...
> > > > Recently I had a private correspondence with Alexander Zenkin; among many
> > > > things I brought to his attention *this* thread discussing his publications
> > > > on Cantor.
> > > >
> > > > I am forwarding the following message on behalf of Dr. Zenkin.
> > > >
> > > > (You will find most of his comments below marked with "AZ" == "Alexander
> > > > Zenkin")
> > >
> > > Thanks, Plamen. I was almost convinced by Dr. Chapman that Dr. Zenkin
> > > had made an elementary mistake in the paper, such that we must be
> > > highly skeptical of it. Dr. Zenkin's explanations make it clear that
> > > the criticisms were not quite realistic.
> > >
> > > Regards,
> >
> > Good heavens! Assuming that you are serious in trying to understand
> > the mathematical basis for Cantor's argument, you could hardly do
> > worse than to accept his responses as addressing the criticisms that
> > have been leveled here.
> >
> > You should indeed be highly skeptical of this paper; and if you wish
> > to proceed with your studies of mathematical philosophy, you should
> > assure yourself of why this skepticism is justified.
>
> I wish to proceed with my studies of mathematical philosophy.
>

Excellent. It is an interesting field.

In my opinion, however, mathematical philosophy should not be the
study of, and exposition on, whatever one _as a philosopher_ thinks
mathematics is, but what one as a philosopher thinks about what
_mathematicians_ think mathematics is.

> The paper is a philosophical one. I cannot myself say I endorse each
> and every claim, but Zenkin makes criticism, and some of it I find of
> substance.
>
> Some people, like Rob, claimed elementary mistakes in the paper (like
> that Zenkin says there are only n n-bit bitstrings, rather than 2^n)
> which was not the case apparently. This would apparently make him a
> crank.

In that instance, I think it just makes Zenkin a poor "explainer" -
but his logical errors are still quite apparent.

> Anyway, it's a shame if people claimed elementary mistakes
> where there was none, only because "the guy said Cantor, he must be a
> crank".
>

If one wishes to study, and discourse upon, the operations of a dairy,
then it would practically fruitless to attempt this without first
understanding the needs of a dairy farm; and how dairy farmers go
about their activities.

Otherwise, the "philosophy of dairys" would resemble a guy at a bar,
who has never been on a farm, and knows little about a dairy either
first- or second-hand, saying "Y'know, they ought to feed those cows
chocolate, then they'd get chocolate milk!".

You should notice that no credible mathematician here respects the
_arguments_ made by Zenkin; not because he's a crank, but because they
fail the test of being about whatever it is that _mathematicians_
consider mathematics.

The reason why he's considered a crank is that, by professing to be a
philosopher of mathematics while displaying an ignorance of
mathematical logic, he displays the willful ignorance characteristic
of cranks.

He sits in a bar and tells the dairy farmer "Feed those cows
chocolate, I want chocolate milk!"

> There is one obvious place that Dr. Zenkin does not sound correct. The
> diagonal method is not the only means to establish a proof of Cantor's
> theorem that there are more reals than there are natural numbers.
> Cantor's first proof is an example for this, ...

Okay. So you know this, which is good. It is the work of a moment
(okay, maybe a bit more) for a _motivated_ person to discover this
fact, as you did. For a philosopher of mathematics, one would think
that this would be a "known fact".

But, as the saying goes, ignorance is no excuse before the law! If you
sent Zenkin an e-mail pointing out this gaping error in his
justification of his arguments, how should he respond? Shouldn't he
discard this line of argument as invalid? Do you think that you would
be the first person who had noticed this, and brought it to his
attention?

That's one reason to suspect him as a crank - when corrected, cranks
do not retract their arguments. And it beggars belief to imagine
someone trained in the history and philosophy of mathematics _not_
knowing the afore-mentioned fact. Conclusion: Zenkin is a likely
crank. I will examine his "arguments" below.

> ...and we can find other
> proofs which do not necessarily use infinitary reasoning.
>

More on "infinitary reasoning" below - but I cannot imagine a
reasonable definition of "infinitary reasoning" that applies to
Cantor's proof that the power set of a set cannot be put into
bijection with that set.

> He could elaborate his point by arguing that any such proof which does
> not use infinitary reasoning (which he argues is invalid), must
> necessarily use abstraction of actual infinity,

Okay, here's where you should really, _really_ try to understand what
is different about how mathematicians think about mathematics.

First, he introduces a term, "infinitary reasoning", _without defining
it_. He gives what he calls an _example_ of it (his failure to find a
counterexample); and he then attempts to makes an argument that
reasoning invalidates Cantor's arument.

This is like saying "consider the number 5, which is prime. Any number
which is like 5 I shall call 'fish-like'. Therefore, 2^10001 - 1 is
'fish-like', and therefore prime". I can't even evaluate this
argument, because I don't know if 2^10001 - 1 really is 'fish-like',
let alone verify that being 'fish-like' implies "being prime".

A mathematician would never accept Zenkin's argument as is, primarily
because there is no definition given for "infinitary reasoning" that
allows us to evaluate his (incorrect for other reasons) argument.

We can only _guess_ whether, for example, he considers the standard
argument proving that "there is no largest integer" also to be an
example of "infinitary reasoning". Maybe he thinks that it is, and
thefore the argument is false; and maybe he thinks it isn't - but
there's no way we can tell _from his argument_ because he hasn't been
specific enough in defining what he means by "infinitary reasoning".

>From the form of his argument, I can _guess_ that "using 'infinitary
reasoning' to prove proposition P" means that "this proof of
proposition P involves an unbounded number of steps". That would meet
the standard definition - a proof must have a _finite_ number of
steps.

But the next logical problem is, the proof he demonstrates as
requiring 'infinitary reasoning' is his own attempt to _refute_
Cantor's argument by showing the existence of a counterexample - so it
is _he_ who is using an infinite number of steps in a (failed, for
still other reasons) attempt to prove the existence of a
counterexample.

If he means to say that any proof of P, which can be disproven by a
proof T which is invalid because it contains an infinite number of
steps, is then not a valid proof, then he's not going to have a lot of
valid proofs sitting around. If a proof P is a _valid_ proof, then
_every_ "disproof" T _must_ be invalid for some reason - that's the
whole _point_ of "consistency".

But this all just shows Zenkin doesn't understand Cantor's argument in
the first place; nor mathematical concerns in general.

Even if we allowed his "infinitary reasoning" constructions to be
valid, it _still_ doesn't yield a counterexample. That's the whole
beauty of Cantor's proof - the existence of a counterexample will
_always_ be a contradiction, no matter how we construct it. So
Zenkin's "logic" is invalid for another reason - one unrelated to
whether or not one accepts the "existence" of an infinite set.

Since he gives no definition of "infinitary reasoning", and his
argument is certainly no "mathematical" argument, and really no
logical argument at all - it all boils down to just his thoughts about
what bothers him about his own lack of understanding of Cantor's
results.

As a "meta-mathematician", he is a man in a bar giving advice to a
dairy farmer.

If you want to understand mathematics as _mathematicians_ understand
it, you don't need to understand why Wiles' proof of Fermat's last
theorem is true (only a few probably could); but you _must_ get to the
point where you understand why _no_ mathematician worthy of the name
would accept Zenkin's arguments.

> which is a point that
> might be impossible to settle for two reasons:
> 1. It might turn out that we are unable to define what "abstraction
> of actual infinity" is precisely enough, because we do not really know
> what "abstraction" is,

It may turn out that you are unable to _agreee_ with how "abstraction
of actual infinity" is defined, but for a mathematician, there is a
perfectly satisfactory precise definition: It is equivalent to the
existence of the set defined by the axiom of infinity.

You may be thinking that "actual infinity" is "that which cannot be
completed, made complete", "something bigger than big", "the
unfathomable". That's fine - but then you're not talking about what
_mathematicians_ talk about when they talk about "actual infinity",
and you're therefore you're not doing _mathematical_ philosophy.

What a mathematician would find relevant is : given the above
definition, what would we do in the _absence_ of the axiom of
infinity? Which proofs still hold, and does the result correspond to
the set of problems that mathematicians are interested in? Does it
correspond to their intuitions? And, more philosophically, should we
then as rational beings accept the axiom of infinity or reject it?

Mathematicians who found the implications of Cantor's approach
distasteful to their intuitions did not cease doing math. They
eliminated those axioms and forms of logic which they found did not
correspond to their intuitions, but they still did what mathematicians
do - they investigated the logical consequences of their favorite
assumptions.

That is not what Zenkin is doing.

> or because "actual infinity" is illogical (and
> thus necessarily leads to paradoxes when trying to define its
> abstraction)

Without getting into Godel-ish complexities, if it could be "easily"
proven that the assumption of the axiom of infinity in the standard
set of axioms caused a contradiction (i.e. was inconsistent), believe
me, we'd have heard about it.

Just as dairy farmers work with cows every day, mathematicians work
with the logical implications of infinity all the time, and have done
so for years. From a position of humility, I'd think it would be
unlikely that I'd uncover one.

Is it possible? Sure, humans are fallible; but an awful lot of
pitching of hay, filling of feed troughs, and just plain milking was
done by set theoretic mathematicians in order to assure themselves
that these foundations lacked any contradicitions. Read a good book on
axiomatic set theory.

In any case, to accept that the axiom of infinity created
contradictions, I'd _still_ have to see it proven - and certainly,
nothing in Zenkin's paper proves this. If you want to be a
"meta-mathematician", you should learn to see why that is - why a
_mathematician_ would reject his paper as hopelessly flawed,
regardless of whether it has "resonance" to you as a philosopher.

> 2. One might argue that Cantor's first proof does not contain
> abstraction of actual infinity.
>

>From a mathematical view, one doesn't need the axiom of infinity to
verify the correctness of Cantor's proof that the power set of a set
cannot be put into bijection with that set. So that is _not_ why a
mathematician rejects Zenkin's arguments.

In the absence of the axiom of infinity, there is the problem of being
very specific about what you mean by "the naturals" and "the reals",
when your system lacks an axiom saying that the naturals actually _do_
exist "as a set", and without assuming that the naturals exist "as a
set" as part of your argument.

But that problem is not at all addressed by Zenkin.

> (I cannot say what I believe about 1, but 2 I disagree with)
>

On what basis? If you disagree with 2, but you don't know how you
would define "abstraction of actual infinity", how can you be possibly
be doing mathematical philosophy?

If you want to know "how can mathematicians describe the naturals, as
a set, without implicitly invoking the axiom of infinity?" then that
is a reasonable (and interesting) question; but you must expect that,
to understand the answer, you will need to follow a mathematical
argument.

And to learn how to follow a mathematical argument, you should assure
yourself that Zenkin's argument is logically, irreparably invalid.
Don't take my word for it; follow his argument and _prove_ it is
wrong.

> However, if he can settle the following metamathematical theorem:
>
> Any proof of Cantor's theorem either requires infinitary reasoning or
> abstraction of actual infinity. (Whatever it is!)
>
> Then, he would have put forward a much stronger argument.

That would be trying to "settle" the following theorem:

  Any proof of Cantor's theorem is invalid in the absence of the axiom
of infinity.

Then, he would have put forward a stronger, but equally false
argument.

As Barbie once said, "Math is hard". To study mathematical philosophy,
one must also study math. Study why Zenkin is wrong.

>
> Regards,

Cheers - Chas

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