[vaillant]

> The particular thing that comes to mind repeatedly when reading this is the fact that more or less all of mathematics can be derived starting either from set theory or from logic theory.

I don't know if this is the actual foundation of mathematics though - we're seeing more advances in category theory, the Russell-Whitehead project of reducing mathematics to pure logic is generally considered a failure, and set theory's bogged down in issues of axioms in the wake of Cohen's proof of the indecidability of the continuum hypothesis. It's probably better to see these foundational projects as providing windows into the mathematical universe instead of being the actual substance of mathematics.

After all, we do mathematics without pure logic or sets all the time. Axioms are chosen for their elegance and ability to describe conceived mathematical concepts, not the other way around.

[solveit]

> The particular thing that comes to mind repeatedly when reading this is the fact that more or less all of mathematics can be derived starting either from set theory or from logic theory.

No, most of mathematics can be expressed in set theory. It's like saying every program can be written in C. It's more or less true, but the philosophical implications are overblown. That is, it's important that set theory and C are so powerful, but there's nothing[1] special about them in particular, we could just as well choose different foundations/Turing complete languages.

[1] Disclaimer: I am not a set theorist and I presume there's a reason set theorists study ZFC and its more powerful cousins so intensively.

[ogogmad]

> the Russell-Whitehead project of reducing mathematics to pure logic is generally considered a failure

sigh I find these discussions tedious, but oh well...

I've heard this before and it seems wrong and rooted in a misunderstanding. Can someone (not necessarily the person I'm replying to) explain why you believe this?

[thwayunion]

AFAICT among non-academics it's mostly rooted in pop sci story telling. Same genre as “Godel went insane because of his impossibility result” and nonsense like that.

Among academics (esp. mathematicians) this impression comes from the fact that if you look at almost any mathematics department, there aren't many people working in/on formal logic. But that's mostly because all of the mathematicians working on/in formal logic suffer the humiliation of sitting in the fancy new CS building with higher salaries and lower teaching loads ;)