[FOM] SOL vs. ZFC

[Timothy Y. Chow]

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