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.
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.
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?
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 ;)
You say this as if it contradicts some point that the author is trying to make or some key assumption that he holds.
Yet on the contrary, the author himself makes a similar point in the article:
> But the way we’ve modeled mathematics here has been much more about what statements can be derived (or entailed) than about any kind of abstract notion of what statements can be “tagged as true”. In other words, we’ve been more concerned with “structurally deriving” that “” than in saying that “1 + 1 = 2 is true”.
That is, math is not necessarily about truth or usefulness in some (meta)physical sense, it is about entailment from certain axioms via formal rules.
In physics there are real tests and relations that have meaning so it makes sense to ask if String Theory is correct or useful. In mathematics there are complex structures built from axioms and sometimes these structures can be related to each other in interesting ways. Will anything important ever come of String Theory? Maybe, but from a mathematical point of view no theory needs to correctly model what is real or have demonstrated application to be worthy of exploration and study.
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. There is no particular reason to choose one or the other, but set theory is the most common foundational substrate. Depending on how this is done it may be more simple, easy, or direct to express or make proofs regarding some particular idea or domain. But Wolfram seems to be seeing things in terms of there being a big space with defined paths through it. Instead of leaning on set theory or logic theory to get to a particular expression or result there should be a global basis for expressions and results. That different models can be used for the same phenomena doesn't seem to be considered valuable or even a possibility.