VS: Logic/syntax versus arithmetic

[Timothy Y. Chow]

On Sun, 16 Feb 2020, Panu Raatikainen (TAU) wrote:
> Leng's view on truth in mathematics seems to amount to what has been 
> called "if-thenism" (or, sometimes, "deductivism").
> 
> It has some well-known problems:

Your objections seem to mostly be of the form, "Deductivism doesn't 
satisfactorily account for mathematical practice."  This is a reasonable 
objection if the proponent of deductivism is trying to account for 
mathematical practice.  I don't think that that is what Leng is trying to 
do, however.  What she's trying to argue is that, whatever account of 
mathematical practice---and its successful application to science---you 
might come up with, it should not demand a philosophical commitment to the 
existence of abstract objects.

The problem that I have with her strategy---which amounts to doing a 
search-and-replace in your theory of mathematics, substituting logic for 
arithmetic---is that it doesn't actually eliminate abstract objects.  It 
merely replaces one type of abstract object with another.

It's the same objection I have to Harty Field's "Science Without Numbers." 
He takes pains to eliminate conventional abstract mathematical entities, 
but in the end, his prescription for how to do science relies on other 
types of abstract entities.  The geometric entities preferred by Field 
seem just as abstract to me as numbers are, and I can't figure out the 
notion of abstractness that he's using to judge them to be non-abstract 
while judging numbers to be abstract.

Tim