[FOM] Concerning Ultrafinitism

[Hartley Slater]

Mirco Mannucci writes

>In my modest belief though, the single most important thing Nelson 
>ever wrote is not his
>magnificient mathematics, not even his predicative arithmetics (not 
>radical enough
>for ultrafinitism, by the way), but a small confession (available at
>http://www.math.princeton.edu/~nelson/papers/s.pdf), where he candidly tells
>the world when and where he lost his faith in N.

I have read this paper ('Syntax and Semantics'), and can readily see 
why it should be thought important.  Having found that the procedures 
Hilbert put in place cannot prove that 2+3=5, for instance, (because 
that would involve giving a semantics to the metamathematical syntax) 
the novel idea is to deny that there is such a semantically based 
fact, enabling one to continue guiltlessly with Hilbert's line of 
research into what are then regarded as the only proper facts - 
syntactic ones.  Gone is the thought that there might be a conclusive 
proof that 2+3=5 some other way, that might be a more appropriate 
object for the Foundations of Mathematics to locate and investigate.

So 'Hilbert's plane geometry is flawless' (Nelson, p3)??  Certainly 
it meticulously avoided any statement about lines and planes and was 
scrupulously concerned merely with the words 'lines' and 'planes', in 
a certain accurately deductive context.  But that shows that it was 
not in the same business as Euclid, not that it improved on that 
business.  At one time I thought that a use-mention confusion was 
endemic in foundational studies, when metamathical investigations 
about uninterpreted words were thought to be establishing, on a 
firmer basis, the older, mathematical truths, which involved words 
that were interpreted.   And maybe that was the case in the early 
part of the 20th century.  But the more modern trend, illustrated in 
this paper by Nelson, and some other messages to FOM recently, 
clearly involves, instead, just the total abnegation of use.
-- 
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) 6488 1246 (W), 9386 4812 (H)
Fax: (08) 6488 1057
Url: http://www.philosophy.uwa.edu.au/staff/slater