Re: Anti-diagonalist page

On 3 Sep 2005 04:52:45 -0700, "William of Ockham"
<d3uckner@xxxxxxxxxxxxxx> wrote:

>I like the idea of a page on philosophers who have criticised the
>diagonal argument. But who would there be? Wittgenstein, of course.
>There is also Richard Arthur (Chair of the Department of Philosophy,
>McMaster University) who is, and Hartley Slater (professor of
>philosophy, UWA). Both are on record
>
>http://www.humanities.mcmaster.ca/~rarthur/papers/LeibCant.pdf
>
>http://www.philosophy.uwa.edu.au/staff/slater/publications/the_uniform_solution_of_the_paradoxes
>
>with papers presenting objections. Both are on apparently finitist
>lines. Arthur argues "it cannot automatically be assumed that any
>actual infinite
>will automatically have a corresponding infinite number, an infinite
>cardinality". And Slater argues as follows.
>
>--- Slater
>No justification for saying the natural numbers have a number can be
>given.

So Slater's an idiot - what's your point? (I don't think that any
mathematician has ever said that the natural numbers "have a number".
So that's a somewhat straw man. People certainly do say that they
have a cardinality, and people do maybe use the phrase "cardinal
number" as a synonym for "cardinal". But someone saying that no
justification can be given for saying that the natural numbers
have a cardinality is simply ignorant.

>For Dedekind defined infinite sets as those that could be put
>into one-one correlation with proper subsets of themselves, so the
>criteria for 'same number' bifurcate: if any two such infinite sets
>were numerable, then while, because of the correlation, their numbers
>would be the same, still, because there are items in the one not in the
>other, their numbers would be different. Hence such 'sets' are not
>numerable, and one-one correlation does not equate with equal
>numerosity, as Hume's Principle supposes. Cantor offered several proofs
>that there is no one-one correlation between the real numbers and the
>natural numbers, but ***only the presumption that there are infinite
>numbers can turn whatever impossibility there is here into a seeming
>demonstration that the number of the real numbers is greater than the
>number of natural numbers***. Skolem's Paradox brings this latter
>result into some doubt, but the facts about proper parts entirely
>defeat the presumption on which it is based (Slater [23]).
>---
>
>My ***emphasis***.

Thanks. Again, Slater is simply babbling here. There is no
_presumption_ regarding the existence of "infinite numbers"
(much more properly called "infinite cardinals") here.
The existence of such things is simply a matter of _definition_.

If he wanted to say that there's no such thing as infinite sets
in the first place that would be maybe different - we do in fact
have to assume that to get anywhere. One might argue in this
case that he was missing the meaning of the word "axiom"
instead of the meaning of the word "definition", but never
mind that, let's say if he were saying there's no such thing
as infinite sets he might not be spouting nothing but nonsense.

But that's certainly not what he's saying. If his point
were the nonexistence of infinite sets then he'd have
objections way before the theorem in question - he seems
to have no problem talking about the set of natural numbers,
etc. He seems to agree that there is a set of natural numbers,
that there is a set of reals, and that the proof that there
is no bijection between the two is correct. His objection
is to concluding that the number of reals is larger than
the number of naturals, and when he makes _that_ objection
in this context it's clear he simply doesn't know what
the word "definition" means. It's like he thinks that
the meaning of a technical term in mathematics is a priori
and forever exactly the same as its meaning in common
speech. Simply stupid.

His comment about Skolem's "paradox" is further evidence
that he simply does not understand the things he's
talking about.

Possibly I'm missing your point here. If the point to the
page is to demonstrate that it's possible for a professor
of philosophy to babble nonsense regarding things that he
simply knows nothing about then by all means go for it.
Not that I think that will be a big surprise to anyone.
But if your point is to demonstrate that there exist
coherent objections to the statement that the cardinality
of the reals is greater than the cardinality of the
natural numbers then you should find some examples that
are not so _obviously_ simply, let's say misinformed.

>Are there any other philosophers or mathematicians who have offered
>similar (or different) arguments against diagonalisation?

If you can name _one_ mathematician, other than one who objects
to the idea of infinite sets altogether, who has problems with
the statement that the cardinality of the reals is greater
than the cardinality of the natural numebers I'd really like
to know about that.

Oops, sounds like I'm suggesting that mathematicians
tend to understand mathematics better than non-mathematicians.
Oh well. Try to ignore that, and give me a reference to
the writings of such a mathematician.

************************

David C. Ullrich
.

Relevant Pages

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