The most powerful language for mathematics according to M. Gromov

Dr. Cyrus F Nourani akdmkrd at
Thu Mar 5 05:54:02 EST 2020

Louise, in our volumes that Patrik Eklund stradels on further foudnations with categories and sets
are examined on term functors, ultrafiter categories, and topos. 
Language is not the same when speaking about objects that are sets or categories, functors, functor categries,
or what have we. We can only rely on basic notations: signatures, ordinals, or perhaps cardinals that on topos
are not so measurable since we do not know what the under lying ordinals sets might be,
containments, or more generic notions.

 Product models and Ultrafilter categories: A PreviewCyrus F Nourani and Patrik Eklund, AAA
March 2018, Vienaa, Austria.
Cyrus F Nourani and Patrik Eklund (2017), Term Functors and Product Models: A Brief,
Joint MM AMS-MAM, Atlanta, Georgia meeting, abstract 1125-VJ-2308.
CYRUS F NOURANI, Filters, language topologies, and product models.
European Summer Logic Colloquium, Udine, Italy, July 2018.
Cyrus Nourani and Patrik Eklund: Sweden entitled Term Functors, Filter Computability, and
Realizability Morphisms, Lambert Publishers. Düsseldorf-Karlsruhe.

There are also two volumes on Functorial Models Theory an Enumeration models on categories that I published 2015 t0 2017 on Apple Academic Press, CRC Press Taylor and Francis that are on the topics.

cyrusfn at

Sent: Wednesday, February 05, 2020 at 8:08 AM
From: "Louis H Kauffman" <kauffman at>
To: katzmik at
Cc: "José Manuel Rodríguez Caballero" <josephcmac at>, "Foundations of Mathematics" <fom at>
Subject: Re: The most powerful language for mathematics according to M. Gromov

Dear Mikhail,
Here is an excerpt from MacLane’s book “Mathematics Form and Function”.
As you will see, he is not satisfied with Sets as a foundation, nor with Categories as a foundation.


Sets and categories together form the foundational tools of many working mathematicians.
We would not know how to think about topological quantum field theory without categories, and 
much of algebraic topology would be unintelligible. For topological quantum field theory it is crucial to understand that a topological space 
can become a morphism in a category of cobordisms. These morphisms are structural, not maps of the underlying sets. And yet the underlying sets are there
as topological spaces as well. A mixture of this sort is the common material of ongoing mathematical work.
On Feb 4, 2020, at 10:34 AM, katzmik at[mailto:katzmik at] wrote:
Thanks for this thought-provoking posting, Jose.

Type theory was of course the preferred "language" that Abraham Robinson chose
to express his framework for analysis with infinitesimals.  Moreover, Robinson
specifically and explicitly stated that he does not accept set theory as
having any special foundational claim (I have quoted him on this in some of my

One of the FoM participants asked me privately to provide some details
concerning the work of Corry and others on Bourbaki failure to adopt a
category-theoretic approach, and I thought perhaps other participants may be
interested as well.  Here are some of the references:


2. Krömer, Ralf.  Tool and object.  A history and philosophy of category
theory. Science Networks. Historical Studies, 32. Birkhäuser Verlag,
Basel, 2007.

3. Krömer, Ralf.  La "machine de Grothendieck'' se fonde-t-elle seulement
sur des vocables métamathématiques? Bourbaki et les catégories au
cours des années cinquante.  [Is the "Grothendieck machine'' based only on
metamathematical vocabulary?  Bourbaki and categories during the 1950s] Rev.
Histoire Math.  12 (2006), no. 1, 119-162 (2007).

I my original posting, I meant to mention (but forgot) that there are of
course bi-interpretability results relating set theory and category theory,
but they don't affect the practical issue concerning which foundations are
found to be more convenient by the actual practitioners in the field.

I am glad that the famed topologist Louis Kauffman joined the discussion.  It
was one of the hoped-for outcomes of my posting that experts in specific
fields might comment on this issue.  If I understood Louis' comment correctly,
he seemed to conclude that both set theory and category theory are
indispensable.  I wonder though whether the kind of set theory he has in mind
is actually the metalanguage (rather than object language).  If MacLane ever
made comments about set theory being indispensable, I would assume that he was
referring to metalanguage as well.  At any rate I would be interested in
substantiation of the claim that MacLane made such indispensability "perfectly
clear", and where.

Mikhail Katz

On Mon, February 3, 2020 03:13, José Manuel Rodríguez Caballero wrote:Dear FOM members,
 I would like to share the following quotations from M. Gromov concerning
the most powerful language for mathematics according to him. In his paper
on entropy, M. Gromov wrote (page 2 of [1]) :

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

In his second lecture about his paper [1], M. Gromov (min 21 of [2])
continues his defense of the language of categories:

If you cannot say something in categorical language, possibly go to thehigher level, but more likely you are just stupid enough that you do not
know how to say it. And people hate categories, because they do not know
how to say [...] Categories is the most primitive language available to us
and therefore the most powerful.

It would be interesting to know whether or not there are mathematicians in
this list who disagree with this point of view and which language they
would like to propose as the most powerful for mathematics?

In my opinion, the language of simple type theory is enough for the sort of
mathematics that I encounter in my personal research (quantum cryptography,
elementary number theory, context-free grammar) and it is the most powerful
for me because of automation in proof assistants. Nevertheless,
mathematicians working in other areas of mathematics may have their own
preferred languages, e.g., for Voevodsky [4], following Grothendieck [3],
it was homotopy type theory rather than category theory his preferred
language (homotopy type theory can be developed independently of category
theory [5]).

Kind regards,
Jose M.

[1] Gromov, M., 2013. In a search for a structure, part 1: On entropy.
Entropy, 17, pp.1273-1277.
URL =[]

[2] Lecture: Mikhael Gromov - 2/6 Probability, symmetry, linearity

[3] Grothendieck, A., 1997. Esquisse d'un programme. London Mathematical
Society Lecture Note Series, pp.5-48.

[4] Voevodsky, V., 2011, May. Univalent foundations of mathematics. In
International Workshop on Logic, Language, Information, and Computation
(pp. 4-4). Springer, Berlin, Heidelberg.

[5] Bauer, A., Gross, J., Lumsdaine, P.L., Shulman, M., Sozeau, M. and
Spitters, B., 2017, January. The HoTT library: a formalization of homotopy
type theory in Coq. In Proceedings of the 6th ACM SIGPLAN Conference on
Certified Programs and Proofs (pp. 164-172).

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