Logic/syntax versus arithmetic

[Mikhail Katz]

Hi Tim and everybody,

I read Leng's piece and it left me a bit puzzled, as well.  Possibly
Leng's problem is with viewing 2+3=5 not as a metalanguage statement
but rather as a statement in the object language, with an apparent
implication of infinitary ontology behind it.  Perhaps logical rules
of inference are more defensible (according to Leng) because they are
surveyable at the metalanguage level.  Perhaps what Leng finds
objectionable is the mathematician's facile identification of
metalanguage counting numbers with "the" natural numbers (at the level
of the object language).

Mikhail Katz

On Thu, 13 Feb 2020, Timothy Y. Chow wrote:

> 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
>