Re: Current Issues in the Philosophy of Practice of Mathematics & Informatics

[Vaughan Pratt <pratt@cs.stanford.edu>]

While agreeing wholeheartedly with Robert, I would like to point a
finger at what I think of as the "monolithic mathematics mob", MMM.
These are the people who treat mathematics as a single theory.  Carl
Hewitt, with whom I shared an admin at MIT for a decade, has a proof of
the consistency of mathematics based on that premise at

https://docs.google.com/file/d/0B79uetkQ_hCKbkFpbFJQVFhvdU0/edit?usp=sharing

along with a little more that so far I've been unable to pin down.  But
I rather suspect that pretty much everyone who finds G??del's second
incompleteness theorem paradoxical shares Hewitt's view of mathematics
as a single theory.

I find the following difficulties with the MMM view.

1.  You can't have a theory without a language.  What is the language of
mathematics?  Judging by appearances it would seem to be a living thing
that grows in different directions following the many varied and
evolving interests of mathematicians, pure, applied, Arcturan, or whatever.

This leads to:

Principle 1.  There will never come a time when mathematicians have
settled on the language of mathematics.

2.  In the unlikely event that Principle 1 is violated, namely by
collecting every mathematical symbol that will ever be needed in
mathematics into a single language possessed of a single consistent
theory T, there is no reason to expect any such thing to be recursively
enumerable.  With the requisite assumptions this is G??del's first rather
than his second incompleteness theorem.

But this raises the imponderable question of whether mathematics is what
mathematicians know, or what they could ever know (by enumeration of
theorems), or the aforementioned theory T, which they can never know at
any future time t even as all the consequences-in-principle of whatever
finite axiomatization of T they have agreed on by time t.

Which leads to:

Principle 2.  There will never come a time when mathematicians have
settled on what constitutes mathematics.

G??del's first incompleteness theorem suffices for Principle 2.
Principle 1 is even more elementary.  Much as I appreciate G??del's
second incompleteness theorem, it seems to me that his first is all
that's needed to answer those who find the second paradoxical.

Vaughan Pratt

On 7/29/2015 6:56 AM, Robert Dawson wrote:
...
> However, the loop is not closed, and cannot be.  There are questions
> which are legitimate parts of mathematics/logic that cannot be answered
> internally.  I'm not talking about Goedel incompleteness here (though
> one might), but about why we do what we do.  If we want to say what
> constitutes mathematics worth doing - to say why Fermat's Last Theorem
> or the Riemann Hypothesis are more important that the (3n+1) problem or
> finding palindromic sequences in the decimal expansion of pi - we cannot
> do this by calculation and proof.  This is an example of a place where
> philosophy of mathematics can have a genuine connection.
>
> -Robert Dawson


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