> We already have a perfectly good foundation of mathematics. It's called ZFC [1]
This is completely missing a core point of category theory and type theory (which is about categories equipped with certain extra structure), which is to work is settings which can be flexibly tightened or loosened in order to do constructions which are as general as possible. A theorem in HoTT is far more widely applicable than a theorem in ZFC; HoTT can be interpreted in any higher Gröthendieck topos, ZFC can be only intepreted in a certain class of 1-toposes. In HoTT, we can simply assume choice and LEM to internally reproduce ZFC. In ZFC, univalence is false and cannot be assumed.
> It's not at all clear HoTT, or type theory in general, is the best solution for formalizing mathematics
Xena project isn't too hot on HoTT indeed, but they will tell you that type theory in general is absolutely the best solution for formalizing mathematics.
> in particular denying the law of excluded middle
MLTT does not deny LEM, nor HoTT. The point is not to deny LEM, but to work in a more general system which does not necessarily assume it.
This is completely missing a core point of category theory and type theory (which is about categories equipped with certain extra structure), which is to work is settings which can be flexibly tightened or loosened in order to do constructions which are as general as possible. A theorem in HoTT is far more widely applicable than a theorem in ZFC; HoTT can be interpreted in any higher Gröthendieck topos, ZFC can be only intepreted in a certain class of 1-toposes. In HoTT, we can simply assume choice and LEM to internally reproduce ZFC. In ZFC, univalence is false and cannot be assumed.
> It's not at all clear HoTT, or type theory in general, is the best solution for formalizing mathematics
Xena project isn't too hot on HoTT indeed, but they will tell you that type theory in general is absolutely the best solution for formalizing mathematics.
> in particular denying the law of excluded middle
MLTT does not deny LEM, nor HoTT. The point is not to deny LEM, but to work in a more general system which does not necessarily assume it.