Categorical Foundation of Mathematics? (Gromov's point of view)

[José Manuel Rodríguez Caballero]

Harvey Friedman wrote:

> Generally speaking, philosophical coherence is required in order for a
> foundational scheme to have any traction in the wider intellectual and
> scientific community. Set theory has this, when presented using modern
> f.o.m. tools, but anything like that PNAS article certainly does not.


There are many lectures in which Mikhail Gromov expresses his preference
for the categorical language and his admiration for the work of
Grothendieck in category theory. Indeed, according to Gromov (see second
reference below), there are extremely few people who were able to make good
definitions in the history of mathematics and one of them was Grothendieck,
because of the use of category theory as a foundation of mathematics. So,
if we follow Gromov's point of view, any attack on category theory should
start by attacking the foundations of Grothendieck's work rather than other
authors who may have betrayed Grothendieck's original vision. The theme of
betrayal of his vision by his successors is constant in Grothendieck's
life, e.g., the betrayal of his program to solve Weil's conjectures, the
betrayal of his idea of topos, and many more.

Here are two quotes from Gromov:

from the manuscript

Mikhail Gromov, "In a Search for a Structure, Part 1: On Entropy - IHES",
July 5, 2012
https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/structre-serch-entropy-july5-2012.pdf

Arguably, the category language, some call it "abstract", reflects mental
> undercurrents that surface as our "intuitive reasoning"; a comprehensive
> mathematical description of this "reasoning", will be, probably, even
> farther removed from the "real world" than categories and functors


and from a recent lecture (minute 37:46 of the video)

Mikhail Gromov, "1/2 Probability by Homology", Lecture at Institut des
Hautes Études Scientifiques (IHÉS), May 17, 2022
https://youtu.be/buThBDcUYZI?t=2266


> People tried to understand Boltzmann from the point of view of set theory,
> the corresponding probability theory was axiomatized by Kolmogorov [...]
> for me it is not adequate for most purposes. Moreover, it was never made
> rigorously [...] because it depends on  Zermelo-Fraenkel theory and
> Zermelo-Fraenkel theory never was exposed 100% I guess, because you use
> sets with measure zero and the set of them is a [...] continuum. How can we
> work with this? For me, it is a big mystery, it is not a nonsensical theory
> but it is certainly very strange stuff. Probability is not about that,
> probability is about numbers, it is more combinatorial [...] Boltzmann was
> going to the next step, in mathematics, actually, two steps[...] if you
> translate it to the modern language, it was two ideas: one was
> functoriality of certain operation and secondly use of infinitesimals [...]
> if you reinterpret Boltzmann in these terms, you have a completely
> different setting for probability theory: he was saying that there are
> systems, systems that are not sets, but systems-objects and you can
> interact with them, you have morphisms, a certain functoriality [...] in
> particular there are functors from the category of his objects, which are
> not dynamical systems, but a kind of objects imitating dynamical systems in
> ensembles of particles, by the way, this was the language used, they were
> ensembles, they were not sets, they were objects of categories [...] from a
> mathematical point of view, there were a kind of natural functors in
> certain category and this category was introduced by Boltzmann. This is my
> understanding, it was never formalized what I am saying, but they are
> pretty certain the way things are.


Kind regards,
Jose M.
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