In Vol. 12 of the Handbook of Philosophical Logic (Springer 2005), John L. Bell contributed a long piece on ‘The development of categorical logical’ (some 62 pages, plus a 10 page bibilography, with another 8 pages of background from category theory). Here’s an online version. That piece starts with a historical sketch, and then gets down to a rather hard-core, compressed, outline of some key ideas and results. I can’t imagine that anyone has found it an easy read: in fact, I’ve thought of it as not so much a source essay to start from but as a target summary, something to aim to get to eventually understand from other readings.
It is disappointing, then, to find that the main contribution on categorical logic in Landry’s collection is a 23 page piece ‘Categorical logic and model theory’ by Bell which is mostly just a series of excerpts from that old longer paper. It doesn’t range as widely, but I’m not sure that is much more readily accessible. Will a logic-minded philosopher who knows some category theory — has tackled, say, Emily Riehl’s Category Theory in Context — get much out of this as it stands? I doubt it. To my mind, then, this is an opportunity missed.
The next paper, the eighth, is again by Jean-Pierre Marquis, ‘Unfolding FOLDS: A foundational framework for abstract mathematical concepts’. The first dozen pages are mostly, as a section title has it, ‘some generalities about abstract mathematical concepts’, with arguments for the claim that traditional set theoretic foundations ‘do not encode abstract mathematical concepts properly’. Category theory, you won’t be surprised to hear, does better; and a theory embracing a hierarchy of 1-categories, 2-categories, 3-categories, …, does better still. And then, Marquis claims, Makkai’s system FOLDS — that’s first-order logic with dependent sorts — is what we need as a formal framework ‘to describe this hierarchy directly and properly’. So in the remaining dozen pages of this paper, Marquis outlines FOLDS.
The opening pages of generalities include a sample of the sort of claims made by some categorial enthusiasts which I hesitate to sign up to. For example, what are we to make of this? —
“Let G be an abstract group.” This is a common way of talking in contemporary mathematics, say in group theory. … Our mathematician certainly thinks that the abstract group G has an underlying abstract set. An abstract set is basically a set whose elements have no structure.
Really? Who, I wonder, goes in for this ‘common’ way of talking? You won’t find the phrase ‘abstract group’ anywhere in e.g. Lang’s Algebra. Nor in Aluffi’s Algebra: Chapter 0 (an interesting case, given the categorial flavouring that Aluffi likes to give to his exposition in that lovely book). You will find a few occurrences at the very beginning of Dummit and Foote’s Abstract Algebra, and they do talk on p. 13 of ‘the notion of an abstract group’. But that can be read equally as ‘the abstract notion of a group’ — the notion we get by abstracting from various cases of permutation groups, symmetry groups and the like. I don’t think they are committed to the idea that there is, as well as those more concrete groups there is another sort of thing, an abstract group made of items which (unlike permutations or symmetries) have no nature of their own. Here’s a quote from another book, a standard undergraduate text, Rotman’s First Course in Abstract Algebra: he has just moved on from talking about permutations to introducing the general notion of a group”
We are now at the precise point when algebra becomes abstract algebra. In contrast to the concrete group Sn consisting of all the permutations of the set X = {1, 2, . . ., n} under composition, we will be proving general results about groups without specifying either their elements or their operation. … It will be seen that this approach is quite fruitful, for theorems now apply to many different groups, and it is more efficient to prove theorems once for all instead of proving them anew for each group encountered. For example, the next proposition and three lemmas give properties that hold in every group G. In addition to this obvious economy, it is often simpler to work with the “abstract” viewpoint even when dealing with a particular concrete group. For example, we will see that certain properties of Sn are simpler to treat without recognizing that the elements in question are permutations.
So yes, our mathematician can often proceed abstracting from a group’s details: but that doesn’t mean that “our mathematician certainly thinks” that they are dealing with an abstract group, something built from elements which actually lack structure.
But maybe, on a more careful reading, that isn’t quite the view that Marquis really wants to attribute to the contemporary mathematician. He has a footnote on p. 141, quoting the French mathematician Maurice Fréchet
In modern times it has been recognized that it is possible to elaborate full mathematical theories dealing with elements of which the nature is not specified, that is with abstract elements. A collection of these elements will be called an abstract set. … It is necessary to keep in mind that these notions are not of a metaphysical nature; that when we speak of an abstract element we mean that the nature of this element is indifferent …
So it isn’t after all that the elements of an abstract group have no structure, but rather that for certain generalizing purposes we are ignoring it. As Marquis comments, approvingly I think, here we have ‘a quote by a mathematician that specifies that the property of being abstract is epistemological rather than ontological’.
But if that’s the line, is it so obvious that good old-fashioned set-theoretic reductionism does ‘not encode abstract mathematical concepts properly’? If talk ‘the’ Klein four group is not to be construed ontologically, as talk about some Dedekind abstraction, an entity over and above all the concretely realized Klein four groups, why can’t we say: a claim of the form ‘the (abstract!) Klein four group is F’ is to be cashed out along the lines of ‘any (set-theoretic?) instantion of a Klein four group is F*’ some some suitable derived F*?
Ok, I can’t pursue this further here. But I’m flagging up that Marquis’s opening discussion about ‘abstract mathematical concepts’, while hinting at interesting issues, to my mind goes too fast to be very satisfactory.
What about the more technical part of the paper, on FOLDS? I can only say that I didn’t find the exposition inviting or leaving me wanting to find out more …
Marquis’s paper may or may not be a success, but at least I can see why the editor thought it belonged in this volume. Not so for the next two papers. A sixty page heavy technical paper by Kohei Koshida on ‘Categories and modalities’ belongs in one of the specialist journals. Similarly for forty pages on ‘Proof theory and the cut rule’ by J. Cockett and R. Seely. Unless I am much mistaken, I imagine the number of readers who will be interested and comprehending will be the same as for most such technical articles. Tiny.