Sunday, 9 June 2013

Mochizuki on the Type of a Mathematical Object (Abstract Structure)

Over the last year or so, I posted a couple of times on the problem related to what I've called "Leibniz abstraction". That is, to find a mapping $\mathcal{A} \mapsto \hat{\mathcal{A}}$ that takes us from specific models (i.e., those with a specific domain/carrier set) to their "abstract structure", such that the following holds:
Leibniz abstraction
$\hat{\mathcal{A}} = \hat{\mathcal{B}}$ iff $\mathcal{A} \cong \mathcal{B}$
The entity $\hat{\mathcal{A}}$ would be the abstract structure of $\mathcal{A}$, in the sense that isomorphic copies of $\mathcal{A}$ get mapped to the same entity. This is, of course, the mysterious entity that all implementations of the naturals, or integers, or the reals, etc., "have in common", or "exemplify". It's the abstract graph that all labelled graphs have in common. ...

[There is a second-order version of this, with $\hat{\mathcal{B}}$ a first-order object, which is known to be inconsistent, as it leads to the Burali-Forti paradox (Hodes 1984).]

Since last year, I think I've got a nice approach to this problem, which I posted on a couple of times, the propositional diagram conception of abstract structures, and have given a couple of talks about. Given a fixed, set-sized, model $\mathcal{A}$, one defines a purely logical, categorical, second-order formula $\Phi_{\mathcal{A}}(\vec{X})$ (in a possibly infinitary language) using a method analogous the notion of a "diagram" in model theory, and this formula defines the isomorphism type of $\mathcal{A}$. Then one identifies the abstract structure of $\mathcal{A}$ with the propositional content of $\Phi_{\mathcal{A}}(\vec{X})$, which I write as $\hat{\Phi}_{\mathcal{A}}$. Because of categoricity, this satisfies:
Leibniz abstraction
$\hat{\Phi}_{\mathcal{A}} = \hat{\Phi}_{\mathcal{B}}$ iff $\mathcal{A} \cong \mathcal{B}$
But I was very interested when, a few weeks ago, someone posted on the FOM list the following quote from Mochizuki's recent work (which unfortunately no one yet seems to have gotten their head around!):
Finally, in §3, we examine — albeit from an extremely naive/non-expert point of view! — certain foundational issues underlying the theory of the present se- ries of papers. Typically in mathematical discussions [i.e., by mathematicians who are not equipped with a detailed knowledge of the theory of foundations!] — such as, for instance, the theory developed in the present series of papers! — one defines various “types of mathematical objects” [i.e., such as groups, topological spaces, or schemes], together with a notion of “morphisms” between two particular examples of a specific type of mathematical object [i.e., morphisms between groups, between topological spaces, or between schemes]. Such objects and morphisms [typically] determine a category.
On the other hand, if one restricts one’s attention to such a category, then one must keep in mind the fact that the structure of the category — i.e., which consists only of a collection of objects and morphisms satisfying certain properties! — does not include any mention of the various sets and conditions satisfied by those sets that give rise to the “type of mathematical object” under consideration.
I think I follow this. And I think that by "type of mathematcal object", Mochizuki means to include what I meant above by "abstract structure" (of a specific model). But the next part is the most interesting:
For instance, the data consisting of the underlying set of a group, the group multiplication law on the group, and the properties satisfied by this group multiplication law cannot be recovered [at least in an a priori sense!] from the structure of the “category of groups”.
It seems to me that Mochizuki means a particular group, say $(G, \circ)$. So, he means that, from the structure of the category of groups, the properties of $(G, \circ)$ itself "cannot be recovered". I'm not sure if this is correct though.
Put another way, although the notion of a “type of mathematical object” may give rise to a “category of such objects”, the notion of a “type of mathematical object” is much stronger — in the sense that it involves much more mathematical structure — than the notion of a category. Indeed, a given “type of mathematical object” may have a very complicated internal structure, but may give rise to a category equivalent to a one-morphism category [i.e., a category with precisely one morphism]; in particular, in such cases, the structure of the associated category does not retain any information of interest concerning the internal structure of the “type of mathematical object” under consideration.
Unfortunately, I don't quite get what this means (because I don't know enough category theory).

The following section, sketching the positive proposal, seems interesting, but I can't pretend to understand it:
In Definition 3.1, (iii), we formalize this intuitive notion of a “type of mathematical object” by defining the notion of a species as, roughly speaking, a collection of set-theoretic formulas that gives rise to a category in any given model of set theory [cf. Definition 3.1, (iv)], but, unlike any specific category [e.g., of groups, etc.] is not confined to any specific model of set theory. In a similar vein, by working with collections of set-theoretic formulas, one may define a species-theoretic analogue of the notion of a functor, which we refer to as a mutation [cf. Definition 3.3, (i)]. Given a diagram of mutations, one may then define the notion of a “mutation that extracts, from the diagram, a certain portion of the types of mathematical objects that appear in the diagram that is invariant with respect to the mutations in the diagram”; we refer to such a mutation as a core [cf. Definition 3.3, (v)].

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