After much discussion, Mario Carneiro wrote:

So I avoid any notion of “standard foundations of mathematcs”, and prefer a multi-foundational approach, where the “true” theorems are those that make sense in any (and hence all) foundational systems.

I replied:

One nice thing your comment clarifies is that different people have very different attitudes toward foundations, which need to be discussed before true communication about the details can occur.

Harvey Friedman seems to believe that foundations should be “simple” (a concept that begs further explication) and sufficiently powerful to formalize all, or perhaps most, mathematics. For example, he wrote:

When I go to the IAS and ask in person where the most elementary place is that there is a real problem in formalizing mathematics using the standard f.o.m….

as part of an explanation of his “sieve”.

Most people interested in categorical foundations have a completely different attitude. This is why nothing they do will ever pass through Friedman’s sieve.

First, they — I might even say “we” — want an approach to mathematics that formalizes how modern mathematicians actually think, with a minimum of arbitrary “encoding”. For us, it is unsatisfactory to be told that − 3 \sqrt{-3} is a set. In ZFC it is encoded as a set, and it’s very good that this is possible, but mathematicians don’t usually think of complex numbers as sets, and if you repeatedly raised your hand and asked what are the members of various complex numbers, you’d be laughed out of a seminar.

Second, when I speak of “how modern mathematicians think”, this is not a monolithic thing: there are different historical strata that need to be considered. There seems to be a huge divide between people like Lawvere, who were content with topoi, and people who are deeply involved in the homotopification of mathematics, for whom anything short of an ( ∞ , 1 ) (\infty,1) -topos seems insanely restrictive. Homotopy type theory is mainly appealing to the latter people.

Anyone who has not been keeping up with modern mathematics will not understand what I mean by “homotopification”. Yuri Manin explained it very nicely in an interview:

But fundamental psychological changes also occur. Nowadays these changes take the form of complicated theories and theorems, through which it turns out that the place of old forms and structures, for example, the natural numbers, is taken by some geometric, right-brain objects. Instead of sets, clouds of discrete elements, we envisage some sorts of vague spaces, which can be very severely deformed, mapped one to another, and all the while the specific space is not important, but only the space up to deformation. If we really want to return to discrete objects, we see continuous components, the pieces whose form or even dimension does not matter. Earlier, all these spaces were thought of as Cantor sets with topology, their maps were Cantor maps, some of them were homotopies that should have been factored out, and so on. I am pretty strongly convinced that there is an ongoing reversal in the collective consciousness of mathematicians: the right hemispherical and homotopical picture of the world becomes the basic intuition, and if you want to get a discrete set, then you pass to the set of connected components of a space defined only up to homotopy. That is, the Cantor points become continuous components, or attractors, and so on — almost from the start. Cantor’s problems of the infinite recede to the background: from the very start, our images are so infinite that if you want to make something finite out of them, you must divide them by another infinity.

This will probably sound vague and mysterious to people who haven’t learned homotopy theory and haven’t seen how it’s transforming algebra (e.g. the so-called “brave new algebra”), geometry (“higher gauge theory”) and other subjects. Until the textbooks are written, to truly understand this ongoing revolution one must participate in it.

Again, all this stuff can be formalized in ZFC, but only at the cost of arbitrary encodings that do violence to the essence of the ideas. A homotopy type can be encoded as a topological space, or as a simplicial set, or as a cubical set, or as a globular ∞ \infty -groupoid… but all of these do some violence to the basic idea, which is — as Manin points out — something much more primitive and visual in nature. A homotopy type is a very flexible blob, not made of points.

Third, because people interested in categorical foundations are interested in formalizing mathematics in a way that fits how mathematicians actually think, and different mathematicians think in different ways at different times, we tend to prefer what you call a “multi-foundational” approach. Personally I don’t think the metaphor of “foundations” is even appropriate for this approach. I prefer a word like “entrance”. A building has one foundation, which holds up everything else. But mathematics doesn’t need anything to hold it up: there is no “gravity” that pulls mathematics down and makes it collapse. What mathematics needs is “entrances”: ways to get in. And it would be very inconvenient to have just one entrance.