The most powerful language for mathematics according to M. Gromov

José Manuel Rodriguez Caballero josephcmac at
Wed Feb 5 03:56:15 EST 2020

L. Kauffman wrote (about the languages of sets and categories)
> A mixture of this sort is the common material of ongoing mathematical work.

Why is this a mixture? Could sets be treated just as a particular kind of category (Boolean topos)?

Indeed, returning to Gromov, in his mathematical world view, ZFC is not the central framework, because he says [1] that he does not know what mathematical induction is (rhetorical expression). An example of a framework where mathematical induction does not hold in general is the theory of elementary topos (this is related to the existence of natural number objects [2]).

Among the natural sciences, the success of ZFC is mainly in physics and chemistry. In biology, this Boolean topos may not be the most natural framework. Indeed, here is an example of the applications of Likasiewicz-Moisil topos in biology [3].

Kind regards,
Jose M.

[1] M. Gromov, “Probability, symmetry, linearity (1/6)”, time 1:07:00.


[2] nlab “natural number object”


[3] Baianu, I.C., Brown, R., Georgescu, G. et al. Axiomathes (2006) 16: 65.


Sent from my iPhone

>> On Feb 5, 2020, at 09:08, Louis H Kauffman <kauffman at> 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.
> <MacLaneCatSet.pdf>
> 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
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