# [FOM] Logic/Syntax versus Arithmetic

Timothy Y. Chow tchow at math.princeton.edu
Fri Mar 6 22:44:23 EST 2020

```Alan Weir wrote:

> Tim Chow wrote (FOM Vol 207 Issue 4.1): 'My hypothesis is that *if* you
> can come up with some kind of nominalistically satisfactory account of
> infinitely many wffs, and/or of syntactic operations, then you're going
> to be able to come up with a nominalistically satisfactory account of
> infinitely many numbers, and/or of arithmetic operations, essentially by
> mimicking whatever philosophical moves you make in the syntactic case.
[...]
> As to your hypothesis, yes I agree with the conditional and believe the
> antecedent too.
[...]
> The original worry (looking back to try to remember what that was!) was
> whether it made sense to challenge platonism in, say arithmetic, by
> appeal to logical consequence, read for example as derivability, a
> syntactic notion. For these logico-syntactic notions- proof, wff etc.-
> seem to be as abstract, and the syntactic theory of the pretty much the
> same power, e.g. entailing infinitely many distinct elements in a
> structure, as arithmetic. That's a good objection which
> nominalists/anti-platonists/formalists need to respond to, I agreed.

Thanks for your detailed reply.  I think that you and I are largely on the
same page (though I don't consider myself a nominalist).

As for "the original worry," let me quote from the opening paragraph of
Leng's article.  She quotes James Robert Brown as asking, "Is anyone
really agnostic about 2+3=5, and willing only to give assent to
PA->2+3=5?"  Leng affirms that she takes "precisely this attitude to
mathematical claims."  Leng's attitude remains mysterious to me.  The
question as phrased by Brown does not explicitly say that 2 is an abstract
object and PA is not, nor does it explicitly say that PA->2+3=5 is a
metalanguage statement while 2+3=5 is an object language statement.  It is
Leng, not Brown, who regards PA->2+3=5 as directly affirming some
nominalistically valid, non-abstract truth, yet for some reason does not
regard 2+3=5 as directly affirming some nominalistically valid,
non-abstract truth.  After all this discussion, I still can see no
justifiable reason for this double standard.

Tim
```