FOM: Re: Standards of mathematical rigour (and paradigms of Thomas Kuhn)

Charles Silver csilver at sophia.smith.edu
Sun Oct 25 09:50:26 EST 1998

On Sun, 25 Oct 1998, Vladimir Sazonov wrote:

> This is reply to a posting of Charles Silver from 20 Oct 1998.

> Thank you very much for recalling me the ideas of Thomas Kuhn on
> Structure of Scientific Revolutions.

You're welcome.  I thought you might have been alluding to them
earlier.

> It seems that at present practically everybody agreed with the
> framework based on set theory so that no problems with the
> rigour arise in everyday mathematical practice.  Those who
> knows formal rules of predicate calculus may have somewhat
> different, detailed standards. But this is a paradise only for
> those who do not try to look outside this framework or who is
> irrelevant to length of (imaginary?) proofs and to uncontrolled
> using implicit abbreviating mechanisms in the proofs.

I find your finitistic viewpoint very interesting, but there are a
couple of things I don't understand.  You seem to accept first-order
logic, which I think indicates that you accept the relation of logical
consequence in first-order logic.  Normally, this relation is defined
set-theoretically, and I believe--please correct me if I am wrong--that to
define logical consequence requires the existence of an infinite set. (If
logical consequence is to be defined some other way, I think you will
still need an infinite bunch of things.)  In fact, I think it is also the
case--again, please correct me if I am wrong--that you'll need to say
something about *all subsets* of an infinite set.  I am not claiming in
this latter case that you will strictly need a *transfinite* set, but I
believe you will still need to give an account of *all subsets* of an
infinite set (which may be difficult without postulating a transfinite
set).

In short, I can't see how you can speak intelligibly of
first-order logic within a finitistic framework.  Could you please explain
how you avoid the problems I've raised?  If I'm mistaken in supposing that
an infinite set is required, could you please explain where I went wrong?

Charlie Silver