The most powerful language for mathematics according to M. Gromov

[Patrik Eklund]

> As you will see, he is not satisfied with Sets as a foundation, nor 
> with Categories as a foundation.

I'm not a Fan or Greater Fan of MacLane but the following papers by 
MacLane might be interesting as well in this context:

   The Health of Mathematics 
(https://link.springer.com/article/10.1007%2FBF03026510)
   To the Greater Health of Mathematics 
(https://link.springer.com/article/10.1007%2FBF03026636)

I do think it is interesting to see how MacLane drove "algebraism" in 
Category Theory. After the group around Herrlich had published their 
second book on category theory (Abstract and Concrete Categories, the 
first one being just Category Theory), MacLane is known to have said 
"now they go from abstract to concrete nonsense". Herrlich's 
illuminating examples are drawn so that (also) topologists feel 
comfortable about reading category theory. MacLane indeed follows the 
algebraic route, moving into monoidal categories and such things, 
whereas the Herrlich side "algebraic categories" are different, but they 
do say things like "algebra and topology meet in the ultrafilter monad", 
which indeed is supported by the fact that the Eilenberg-Moore category 
of that monad is isomorphic to the category of compact Hausdorff spaces.

Cheers,

Patrik



On 2020-02-05 09:08, Louis H Kauffman wrote:
> 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.
> Best,
> Lou
> 
>> On Feb 4, 2020, at 10:34 AM, katzmik at macs.biu.ac.il 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
>> papers).
>> 
>> 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:
>> 
>> 1.
>> 
> http://www.tau.ac.il/~corry/publications/articles/Bourbaki%20-%20OHHM.html
>> 
>> 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
>> mental
>> undercurrents 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
>> the
>> higher 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.
>> 
>> References:
>> [1] Gromov, M., 2013. In a search for a structure, part 1: On
>> entropy.
>> Entropy, 17, pp.1273-1277.
>> URL =
>> 
> https://pdfs.semanticscholar.org/3137/66d8f87b29eeae9004d8c1eb5f8a8fb26cf3.pdf
>> 
>> [2] Lecture: Mikhael Gromov - 2/6 Probability, symmetry, linearity
>> URL = https://youtu.be/Vci3C6yAzRE?t=1307
>> <https://youtu.be/Vci3C6yAzRE?t=1279>
>> 
>> [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).