# [FOM] Logic/Syntax versus Arithmetic

Timothy Y. Chow tchow at math.princeton.edu
Mon Mar 2 20:31:54 EST 2020

```Alan Weir wrote:

> Hence I agree one can introduce cardinal numbers as mereological fusions
> of all wffs of the same length as a given one.

Ah, good!

> I have a number(!) of problems with this. Firstly, we have strong
> empirical reasons to believe that on this account there are only
> finitely many numbers whereas we can prove that there are infinitely
> many. This argument will not persuade strict finitists.
>
> Secondly, we need more than the numbers, we need to be able to
> characterise various relations and functions over them:- less than,
> addition, multiplication and so on.

Recall that my main contention is that arithmetic isn't any harder or
easier to make sense of than syntax.  I'm not sure if you agree, so let me
spell it out further.  You seem to have agreed that if we're content with
either finitely many numbers or finitely many wffs, then we can get to
that point nominalistically.  So no fundamental distinction between syntax
and arithmetic arises at this stage.

Now you point out that we might want more.  Maybe we want infinitely many
numbers, and maybe we want arithmetic relations and functions.  To this I
would respond, maybe we want infinitely many wffs, and maybe we want
syntactic relations and functions---concatenation, shorter than, and so
on.

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.  I haven't seen anything in your
account that would let me give a philosophically satisfactory development
of syntax that wouldn't also let me give a philosophically satisfactory
development of arithmetic (to an analogous degree).

Do you disagree?  Do you see a fundamental difference between syntax and
arithmetic, and if so, what?

Tim
```