[FOM] First-order arithmetical truth

Hartley Slater slaterbh at cyllene.uwa.edu.au
Mon Oct 16 21:46:29 EDT 2006

Arnon Avron appreciates there is a problem of some magnitude, for 
people who cannot grasp what the standard model of Arithmetic is:

>If you do not see or understand what are the (real) natural numbers,
>then ... what can you understand
>and see? It seems to me that practically nothing.

And indeed the problem, more generally, is what one does, and does 
not know if one does not understand semantics, as Andrej Bauer says:

>I am not denying either the usefulness of semantics (as a tool) nor
>any possible philosophical status to it. It may be that mathematics is
>"really" about objects and truths that can only be understood in terms
>of models, rather than in terms of deduction. I just don't know what
>those objects and truths are (perhaps yet).

But second order Arithmetic (as suggested by Stephen Pollard) gets 
one no closer, since all categoricity ensures is the uniform 
structure of all models, and not the distinctive details of any one 
model.   Hence one has not provided *Foundations for Mathematics*, if 
mathematics is first of all about the natural numbers  - that's the 
size of the problem.

Of course some people are drawn by the appeal of Structuralism, at 
this point - "let's forget about the natural numbers and concentrate 
on the general properties of omega sequences" may be recommended. 
And there is another, well-known, historical route, to the same 
proposal: at one time it was thought that certain omega sequences of 
sets were the natural numbers, until Benacceraf dissuaded people from 
that sort of close identification - and left the natural numbers 
themselves still undefined.

So Axiomatics doesn't do the trick, and neither does Set Theory, and 
the question remains: where are the Foundations of Mathematics to be 
found? (Naturally I have a paper on this which is available upon 
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

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