The most powerful language for mathematics according to M. Gromov

Patrik Eklund peklund at cs.umu.se
Wed Feb 5 05:19:12 EST 2020

Hi José Manuel,

You may have seen my example on the term monad in other e-mail. I repeat
it here. If we start with a signature of 'sets' of sorts and operators,
the signature "resides" in set theory. We have shown how both can be
objects in a more general monoidal category, but it is practical only in
the case of operators. When we formalize the term construction
inductively, we can mix set theory and category theory by using
transfinite induction with the set functors, i.e. when staying over SET,
the category of sets. When we go more general, having induction purely
as a coproduct as a limit within the underlying category, generally a
type of monoidal category, induction is not 'transfinite', but as
"infinite as it can be" for limits in category theory.

---

Topos is another story. Truth values become objects, and propositional
style operators are morphisms. But in this case, there are no
"predicative toposes", i.e., a "topos logic" is not capable of managing
an underlying universal algebra.

---

In Ref [3], exocytosis and endocytosis is mentioned on p. 69. Examples
mentioned on pp. 73-74 are also very general. There is no biomolecular
detail on cell biology, so that one could see how mathematics connects
with it. On p. 75, it says: "A neuron, for example, would be represented
in this theory by an organismic set of the first order, in spite of its
higher complexity for information processing than other less specialized
cells of the same organism, such as the stem, or neuroglial cells." This
is very shallow, and it does not embrace at all how a neuron is built
and how it develops, renews, repairs, etc. And there is not just one
type of neuron. Some are simple, some more complicated like the Auerbach
ones.

What I don't like about these papers is that they fix there math, and
then they take whatever they need to illuminate their math. They don't
do it the other way around, for instance taking epithelial cells in the
inner lining of the small intestines, see how they communicate e.g. the
help of immunonutrients, and then observe some disease oriented problems
(like Crohn or Celiac), and from there move on to phenomena to be
modelled by math. Doing so would mean the problem driving the math, not
the other way around. 99.9% of mathematicians do it the wrong way, I am
blunt to say.

---

Best,

Patrik

On 2020-02-05 10:56, José Manuel Rodriguez Caballero wrote:
> 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.
>
> URL = https://youtu.be/aJAQVletzdY
>
> [2] nlab “natural number object”
>
> URL = https://ncatlab.org/nlab/show/natural+numbers+object
>
> [3] Baianu, I.C., Brown, R., Georgescu, G. et al. Axiomathes (2006)
> 16: 65. https://doi.org/10.1007/s10516-005-3973-8
>
>
> Sent from my iPhone
>
>> On Feb 5, 2020, at 09:08, Louis H Kauffman <kauffman at uic.edu> 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