Note that fmap writes: "Equality of rational numbers is decidable, which means that classical reasoning is provable. And yes, even if it wasn't, it would still work."
What is meant by "even if it wasn't, it would still work" goes back to something he said earlier: type theory embeds an infinite hierarchy of axioms of choice and laws of excluded middles. If you want to do propositional-like reasoning in homotopy type theory, you can assume AC or LEM for homotopy (-1)-types, corresponding to propositional logic.
In type theory you are encouraged to drop the law of the excluded middle and the axiom of choice, because of the fact that doing so gives you potentially more expressive ways of doing things as we have said, but you have gotten the impression that you have to, which you don't.
Also, the claim in this thread was that the results from classical mathematics are provable using homotopy type theory, not that they are provable in the same way (though that holds as well, as I've said above; it's just that the mathematics might not look as clean as if you did it in a more idiomatic way). This kind of a value proposition is not exactly new: category theory loses certain axioms over set theory and mathematicians adapted to the point that category theory is now the language of modern algebra.
I want to point out that I suggested that you read the introduction to the book because it provides these same answers to the questions you are wondering about. I still suggest you do so, as it goes into more detail on what we have said here in a way that I am not able to quite as well.
I already read the introduction a couple of months ago, pretty much on the day when they officially announced that the book is available.
From what I understand: If you are a type theorist / constructivist, then certain constructions will not be provably equivalent, although their classical counterparts are. Some of those classically trivial equivalences might be recovered though via the univalence axiom. Is that correct or totally wrong? Furthermore it seems to me from the discussions with fmap, that if I am a classical mathematician, I cannot use HoTT (unless I am willing to compromise and give up some of my power).
Your understanding of univalence seems essentially correct to me.
At this point we are mostly debating what "can use" means -- it's probably enough to say that unless you reframe your thinking, perhaps radically, probably it will be hard for you to be able to use HoTT to get work done. It is possible to do classical mathematics within HoTT, in the normal way, but it would not be very fun.
Yes :) My interest in homotopy type theory is only auxiliary to my research. Designing dependent type systems in a way that balances tractability with expressiveness is a pretty hard thing to do. SMT solvers are nice because you can treat them as oracles and "see what happens". I'm not an expert on decision procedures, and I'd characterize myself more as a user of solvers than a developer of them, though of course once you get deep enough in, that line blurs.
What is meant by "even if it wasn't, it would still work" goes back to something he said earlier: type theory embeds an infinite hierarchy of axioms of choice and laws of excluded middles. If you want to do propositional-like reasoning in homotopy type theory, you can assume AC or LEM for homotopy (-1)-types, corresponding to propositional logic.
In type theory you are encouraged to drop the law of the excluded middle and the axiom of choice, because of the fact that doing so gives you potentially more expressive ways of doing things as we have said, but you have gotten the impression that you have to, which you don't.
Also, the claim in this thread was that the results from classical mathematics are provable using homotopy type theory, not that they are provable in the same way (though that holds as well, as I've said above; it's just that the mathematics might not look as clean as if you did it in a more idiomatic way). This kind of a value proposition is not exactly new: category theory loses certain axioms over set theory and mathematicians adapted to the point that category theory is now the language of modern algebra.
I want to point out that I suggested that you read the introduction to the book because it provides these same answers to the questions you are wondering about. I still suggest you do so, as it goes into more detail on what we have said here in a way that I am not able to quite as well.