FOM: second-order logic is a myth
csilver at sophia.smith.edu
Tue Mar 2 05:44:08 EST 1999
On Sun, 28 Feb 1999, Stephen G Simpson wrote:
> Let me concentrate instead on Shapiro's philosophical case.
> Boiled down to essentials, Shapiro's case for second-order logic seems
> to be as follows:
> 1. There is no sharp boundary between mathematics and logic.
> 2. In the present historical era, mathematicians standardly assume
> set-theoretic realism, including the existence of actual infinities
> and an absolute powerset operation applying to them.
> 3. Therefore, logicians ought to also assume these things.
> Against 3, I reject the notion that f.o.m. professionals ought to
> slavishly follow the current practice of `working mathematicians'.
> F.o.m. should take and does take a higher, broader, more universal
> perspective. As I have said many times here on the FOM list,
> f.o.m. addresses the place of mathematics in the totality of human
> knowledge. The Shapiro case leaves no way to do that. For instance,
> Shapiro says nothing about the obvious disconnect between Platonist
> realism and applied mathematics.
I didn't understand this paragraph very well. What "more
universal perspective" should we adopt? I didn't get the reference to
"the totality of human knowledge." I also did not fathom "the obvious
disconnect." Could you provide a few more words of explanation?
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