[FOM] First-order arithmetical truth
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
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