[FOM] SOL vs. ZFC
Timothy Y. Chow
tchow at alum.mit.edu
Tue Jun 4 17:07:37 EDT 2013
Joe Shipman wrote:
> I am motivated by the question "how much of mathematics is logic in
> disguise?". If second order logic in the standard formalism is really
> "logic", then logic can express almost all of the mathematics anyone
> cares about, and the question reduces to which axioms can plausibly be
> described as "logical".
[...]
> but Logicism gets pretty far and I wonder why it seems to be unpopular.
A large part of the answer is historical. The original efforts to carry
out the logicist program (e.g., by Dedekind and Frege) ran into
difficulties because of the paradoxes. Hilbert also didn't like logicism.
It doesn't take much to make a philosophy unpopular.
One could ask why more people haven't tried to revive logicism. I think
part of the answer is that many people see it as a somewhat pointless
semantic debate, over what you consider to be "logic."
I imagine that only a minority of FOM readers take Alain Badiou seriously,
but for those who do, I'll remark that the only way I've been able to make
sense of Badiou's philosophy is to hypothesize that he regards much of set
theory as "logic" and that since logic applies universally, we should
expect set-theoretic concepts such as forcing extensions to show up in the
real world, and not just in mathematics papers. So I personally think of
Badiou as a kind of logicist, though I doubt he would describe himself in
those terms.
Tim
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