[FOM] Foundations/Philosophy

Hartley Slater slaterbh at cyllene.uwa.edu.au
Tue Sep 23 21:12:19 EDT 2003

At 12:01 PM -0400 23/9/03, Harvey Friedman quotes me:

>  > Martin Davis has written to me:
>>>  I would encourage you to post explaining why you think that the
>>>  set-theoretic foundation for mathematics with which most advanced
>>>  mathematics textbooks begin, is no longer relevant.
>Slater did not take up the Davis challenge in what he wrote.

That was because I had posted explanations of the issue many times before.

>  > That was because I expressed to him my disappointment at what seemed
>>  to me a shallow interest in foundational issues amonst members on the
>>  list, despite it being 'FOM'.
>Whatever disappointment you might have is overshadowed by my disappointment
>in seeing such a statement.

I was disappointed that no-one took up my challenge to provide a 
non-rubbery foundation for Mathematics.   There had been a 
preparedness on the part of several correspondents (e.g. Lucas Wiman 
FOM Digest Vol 4 Issue 20, Arnon Avron, FOM Digest Vol 5 Issue 37) to 
accept my point that numbers are not sets, but also for some to argue 
that you could still say they were.  Thus Mark Steiner wrote (FOM 
Digest Vol 5 Issue 28): ' I think that core mathematicians would have 
little sympathy for the argument, however valid, that real numbers 
are not "really" Dedekind cuts. (I'm not taking sides here, just 
stating a sociological fact)', and  Matt Insall wrote (FOM Digest Vol 
5 Issue 17): 'Mathematicians do not use identity in the same way as 
philosophers...It seems to me that Slater is objecting to what 
Mathematicians commonly call 'an abuse of notation'.  This argument 
therefore seems silly and trivial'.  I replied to Insall (FOM Digest 
Vol 5 Issue 20): ' But if it was just a carefree matter...there would 
be no problem - no problem at all - about stating the rules at the 
bottom of the relevant mathematical practices.  Or is it all so 
rubbery as to be quite lawless?  Either he must confess there are no 
rules at the bottom of the relevant mathematical pracices, or he must 
now specify them, since they are evidently much needed, in the sober 
and rigorous study we call 'The Foundations of Mathematics' to 
augment Leibniz' Law'.

>The set theoretic foundations of mathematics is still the only workable
>universal foundations for mathematics that we have. Other candidates are
>either philosophically incoherent, or are naturally mutually interpretable
>with the set theoretic foundations.

I mentioned, in FOM Digest Vol 5 Issue 29, the paper by Agustin Rayo 
JSL 67.4 Dec 2002 1623-1638 'Frege's Unofficial Arithmetic'.  Maybe 
Harvey should have a try at interpreting that set-theoretically.

>The great power and scope of foundations is that a surprisingly varied array
>of fruitful, satisfying, illuminating, exciting, deep, subjects flow
>naturally and prolifically out of virtually any competent philosophical
>In contrast, the philosopher would normally act on points about "what
>numbers cannot be" by, say, continuing with "what numbers could be" or "what
>numbers really are". The philosopher will take sides on various related

Unfortunately, even if Harvey would like to call something 
'foundations of mathematics' it is only such if it is about numbers. 
Otherwise it is just recreational mathematics.
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html

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