VS: Logic/syntax versus arithmetic
Timothy Y. Chow
tchow at math.princeton.edu
Sun Feb 16 16:42:13 EST 2020
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.
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