From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: ZFC etc. (was Re: Report on Philosophies of Physicists)
Organization: Boise State University
Date: Mon, 28 Sep 1992 14:50:25 GMT

In article <1992Sep25.201622.40255@Cookie.secapl.com> frank@Cookie.secapl.com (Frank Adams) writes:
>In article <TORKEL.92Sep25094806@isis.sics.se> torkel@sics.se (Torkel Franzen) writes:
>>  What you say about the consistency of propositional logic can also
>>be said about, say, the theorem "every formula in propositional logic
>>has a disjunctive normal form". If this is false, our minds are
>>seriously flawed. However, it never occurs to anybody to try to read
>>any assertion that our minds are not flawed into a proof of this
>>latter theorem.
>>
>>  In sharp contrast, you find, again and again, reflections along the
>>following lines: how can we trust the proof that propositional logic
>>is consistent, if what we are putting in question is precisely whether
>>our thinking is consistent? This makes sense if you are
>>trying to interpret the statement proved - i.e. "propositional logic
>>is consistent" - as some sort of assurance that our thinking is ok,
>>our minds unflawed, etc. But it is nothing of the sort. Such assurances
>>must be sought outside mathematics. What is proved is just this:
>>propositional logic is consistent. The statement is an ordinary
>>mathematical statement, the proof is an ordinary mathematical proof.
>>There is no more point in agonizing over this particular statement and
>>proof than there is in agonizing over the theorem that every formula
>>has a disjunctive normal form.
>
>The point is that in proving the consistency of propositional logic, we
>are using reasoning techniques which include all the techniques built
>in to propositional logic.  Now, the main concern I have seen with this
>kind of statement is to wonder what the point is; we don't believe the
>proof unless we already believe what we are trying to prove.
>
>As for the other problem, let us step back for a moment to the point
>where neither of the theorems in question had been proved.  In this
>case, their status is very different.  If we were to have proved that
>some formulae don't have a disjunctive normal form, this would have
>been an interesting fact.  If we had proved that propositional logic
>was inconsistent, it would have been a catastrophe -- there was
>something wrong with our reasoning process.
>
>(In fact, just this kind of catastrophe occurred with naive set theory
>-- set theory with the axiom "for any property P, there is a set S of
>all sets having property P.  This did invalidate all, or even very
>much, of the reasoning process of mathematics up to that time;

No, it did not.  The effect on existing mathematics of the paradoxes
of set theory was zero.  It forced the program of founding mathematics
on arithmetic instead of geometry, and thus ultimately on set theory,
which was not very old at that point, to take a different tack.

 but it
>was nevertheless probably the most event in the history of
>mathematics.)

Is the word "important" missing here?  It was not the most important
event in the history of math by a long shot.  It was _an_ important
event, to be sure.  It was the most important event in the history of
the philosophical program for mathematics described above.  I am a
mathematical logician; I work with this stuff.

>
>The argument is not "we've proved this is true; if it turned out to be
>false it would be a real problem", but "we've assumed this is true (and
>incidently we've proved it); if it turned out to be false it would be a
>real problem".


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
Newsgroups: sci.math