# Logic/syntax versus arithmetic

Timothy Y. Chow tchow at math.princeton.edu
Thu Feb 13 13:30:11 EST 2020

```In a recent FOM post, I wrote:

> People feel comfortable with syntactical entities because of their
> familiarity with computers, so they are quick to regard "ZFC is
> consistent" as bivalent, but the farther one strays from syntax, the
> more leery they become.

https://cs.nyu.edu/pipermail/fom/2020-January/021969.html

I recently encountered an example of someone who is uneasy about saying
that "2 + 3 = 5" is true simpliciter, but who seems to have no qualms
about the truth of logical facts.  In her essay "In Defense of a Near
Absurdity" (reprinted in "The Best Writing on Mathematics 2019"), Mary
Leng explains that she is a nominalist, and balks at claims that numbers
exist or that "2 + 3 = 5" is true, unless these are understood in a
certain "Hilbertian" sense:

There is a notion of truth internal to mathematics according to which
to be true *mathematically* just is to be an axiom or a logical
consequence of accepted (minimally, logically possible---or *coherent*)
mathematical axioms, and to exist *mathematically* just is to be said
to exist in an accepted (minimally, logically possible) mathematical
theory.

My reading of this paragraph is that Leng regards statements such as
"Proposition P is a logical consequence of Axiom A" as simply true (or
false, as the case may be), without the need for further explanation of
the sense in which they are true, whereas the truth of "2 + 3 = 5" is
unclear unless understood as a facon de parler for a logical truth of the
form "2 + 3 = 5 is a theorem of first-order Peano Arithmetic".

As I said, this kind of comfort with logic/syntax (but discomfort with the
rest of mathematics, even arithmetic) seems to me to be rather common
nowadays.  On the other hand, I find Leng's view puzzling.  A nominalist
presumably balks at abstract entities, and an "axiom" is surely an
abstract entity, and "logical consequence" (or its syntactic counterpart,
provability) is surely an abstract relation.  So I don't see why a
nominalist should be any more comfortable with logic than with
mathematics.

Tim
```