[szany]

It's a foundation of mathematics whose native objects are structures fundamental to higher mathematics (∞-groupoids, which are more or less equivalent to topological spaces up to homotopy), rather than awkward encodings (ZFC "sets", which are tree-like things), and where everything is automatically invariant with respect to abstract notions of equivalence (think isomorphism vs. equality).

The point is that it makes it much more realistic to check complicated math using computers, which will become more important as math inevitably gets more complicated.

Oh, and it happens to be a functional programming language. Which says something about functional programming.

[nardi]

Well, yes, it's a functional programming language. But not one like most people here are familiar with, so that's a little misleading. It's also an object-oriented language with polymorphism and inheritance.

Specifically, it's (almost exactly) the Calculus of Constructions (http://en.wikipedia.org/wiki/Calculus_of_constructions), the language at the top of the lambda cube (http://en.wikipedia.org/wiki/Lambda_cube).

[mafribe]

No, HoTT is not based on the Calculus of Constructions, but instead on intensional Martin-Loef type theory with identity typed. The novel part is the univalence axiom, which MLTT doesn't have.

If you erase types, the underlying programming language is the lambda-calculus, familiar to every programmer.

[nardi]

Yes, Vladimir Voevodsky says HoTT is based on Martin-Loef type theory, but he discovered HoTT by learning the CoC used in Coq, and is implementing his Univalent Foundations library of proofs in Coq. That implies that HoTT is implementable in Coq (with a couple of minor tweaks), which means that CoC is at least as powerful as HoTT. And anyway, according to Voevodsky, they are the same thing.

Really, Martin-Loef type theory is the mathematics foundation, and the Calculus of Constructions is the computer science foundation. HoTT ties them together.

[mafribe]

It depends on what you mean by "embedding". If one does a deep embedding then powerful calculi can be embedded in much weaker calculi. E.g. Isabelle uses a variant to LF as meta-language to embed all manner of more powerful logics. That doesn't mean LF has a lot of expressive power.

Coq's logic isn't quite the CoC anymore: impredictivity is switched off by default, Coq has universes etc.

With this in mind, I'm not sure what Voevodsky means when he says HoTT and CoC are the same thing.

[tel]

I would like to see more clarification of why you'd call HoTT OO. I have no doubt you can encode those things, but I find it strange to try to argue that it doesn't have more in common with the FP tribe than the OO tribe. Given that neither term is terrifically well-defined, I think that's the best you could argue.

[j2kun]

Inevitably? Strong words.

[sillysaurus2]

Sort of. As t approaches infinity, more and more of the simple parts of math will have been discovered. Therefore the only parts left to discover will be those which are more complicated; hence the overall idea is necessarily correct.

[siddboots]

It could instead be the case that all of maths is simple, but that complexity tends to appear whenever we have an incomplete understanding of a domain.

[j2kun]

You are misled to believe that mathematics has simple parts and not simple parts, and that simple parts are easier to "discover" than the complex parts.

In fact, the simple parts are usually the most difficult to discover. Why? Precisely because it takes extreme genius to give simple answers to mathematical questions. Over time complicated proofs and definitions get simplified and clarified (such it was with calculus, set theory, logic, group theory, and countless other topics).

So there's no precedent to believe that current mathematics won't be simplified in the future (unless, of course, you believe that all mathematics is complicated; but then you seem to be trying to make an objective point about complexity, so we'll ignore that possibility).

[PurplePanda]

This would seem to be true as long as mathematics is discovered rather than invented. Otherwise I beleive that one may be able to invent generalisations that result in a simplification of mathematics.

[dkural]

This assumes that the purpose of math is to prove things, whereas another worthy goal for me at least is enabling humans to truly understand why things are true, and not just what things are true.

[pavpanchekha]

No disrespect, but your purpose for math is entirely irrelevant unless you speak for the majority of research mathematicians.

[gjm11]

No disrespect, but dkural's idea of what mathematics is about is no more irrelevant than sillysaurus2's or yours, and dkural's view is quite well represented among research mathematicians, first-rate ones included. For a famous example, see William Thurston's "On proof and progress in mathematics" at http://arxiv.org/pdf/math/9404236v1.pdf .

[dkural]

I have a mathematics degree from Harvard, though I haven't taken a poll of research mathematicians. I do think algorithmic theorem proving is a great field of inquiry!