# [FOM] Logic/Syntax versus Arithmetic

Timothy Y. Chow tchow at math.princeton.edu
Wed Feb 26 20:44:54 EST 2020

```Alan Weir wrote:

> I discuss (in my book chapter six) the concept of equiform tokens,
> noting how complex and resistant to definition it is (a sound and an ink
> mark in some font can be equiform) but assume it is acceptable to take
> it as primitive for the purposes of syntax. Then with, for example, a
> bit of mereology, a 'tipe' can be taken to be the mereological sum of
> all (actual) tokens equiform to a given actual token. This gives us
> finitely many tipes but nothing like numbers because the tipes aren't
> structured in the right sort of way.

So what's wrong with this?  Introduce the concept of equicardinal tokens
(in effect, strings of symbols that contain the same number of symbols).
Note how complex and resistant to definition the word "equicardinal" is
but assume it is acceptable to take it as primitive.  A "natural number
tipe" is then a mereological sum of all actual tokens equicardinal to a
given actual token.  This gives us finite many natural number tipes.

Even if this particular plan has some subtle flaw---if you're going to
talk about wffs then somewhere along the way you're going to have to talk
about "strings of 3 symbols" and if you've made some sense of that then
you've made sense of the number 3.  Haven't you?

Tim
```