What to read?

What recent, new, and forthcoming logic/phil. maths  books have caught your attention?

This first little book, forthcoming in January from CUP in the very mixed quality Cambridge Elements series, looks promising. Maddy, at least, reliably writes interestingly and well (hopefully, she keeps her co-author from getting too mired in technicalities). And the topic is a great one. “This Element addresses the viability of categoricity arguments in philosophy by focusing with some care on the specific conclusions that a sampling of prominent figures have attempted to draw … It begins with Dedekind, Zermelo, and Kreisel, casting doubt on received readings of the latter two and highlighting the success of all three in achieving what are argued to be their actual goals. These earlier uses of categoricity arguments are then compared and contrasted with more recent work of Parsons and the co-authors Button and Walsh. …  the Element concludes that categoricity arguments have been more effective in historical cases that reflect philosophically on internal mathematical matters than in recent questions of pre-theoretic metaphysics.” So this is certainly on my list.

I recently looked at Justin Khoo’s  The Meaning of If  published last year by OUP. Anyone writing an intro logic book (and I still hanker after a third edition of IFL) wants — or ought to want — to have something sensible to say about the relation of  ‘if’ and ‘⊃’, though some do duck the task. So I’m always interested to see what people are writing these days on the topic. But I can’t say I got much out of this. One of the phenomena here is that, however ‘if’s work in the wider world, in mathematics regimenting them by a connective  ‘⊃’ governed by the usual rules (acceptable to classical logic and constructive logic alike) seems to work a treat, at least once we distinguish plain ‘if’s from the ‘imply’s we regiment using turnstiles. But there isn’t a word about this in Khoo’s book (you look in vain for anything about mathematics, or indeed about “conditional proof”, or “supposition”, and so on). So whatever the virtues of this book — which I confess didn’t impress me — it will be of no real interest to logicians.

Erik Stei’s Logical Pluralism and Logical Consequence was published early this year (at a disgraceful price) by CUP. From the blurb: “The logical pluralist challenges the philosophical orthodoxy that an argument is either deductively valid or invalid by claiming that there is more than one way for an argument to be valid. In this book, Erik Stei defends logical monism, provides a detailed analysis of different possible formulations of logical pluralism, and offers an original account of the plurality of correct logics that incorporates the benefits of both pluralist and monist approaches to logical consequence.” OK, that looks as if it should be just up my street, as the topic is basic and important and I’m all for calming down debates by trying to draw out what each side has got right. My first impressions, though, on reading early pages have so far not been that encouraging. But I’ll certainly try again, and let you know.

On my desk right now, though, is Introducing String Diagrams: The Art of Category Theory by Ralf Hinze and Dan Marsden, recently published by CUP. This is a comp. sci. book in origin, and it is taking me a while to get the measure of it. But the book comes much praised, so I shall press on in the hope of pennies starting to drop with satisfying clunks …

Meanwhile, I’ve revised a couple chapters of my own entry-level (and hyper-conservative?) Category Theory II. Both these chapters, one on categories of categories and one on functor categories, have been much revised and in places simplified, so I hope work much better. There remain two groups of chapters to revise, one group on the Yoneda lemma and related stuff, one group on adjunctions. Fun topics. I remain quite undecided, though, about how things will go after this initial round of revisions of old material.

6 thoughts on “What to read?”

  1. A possibility: Andrew Bacon, A Philosophical Introduction to Higher-order Logics.

    I’m not far into Introducing String Diagrams: The Art of Category Theory but have two questions.

    1. What is the “abundance of type information” in commutative diagrams (p ix)? At first, I thought it might be the different arrow symbols. Instead (p 1), it seems to be about domains and co-domains:

    Two arrows can be composed if their types match: if f ∶ A → B and g ∶ B → C , then g ⋅ f ∶ A → C (pronounced “g after f ”)

    I don’t think I’ve seen that before, and there doesn’t seem to be an explanation in the book. Why are those ‘types’?

    2. If “the aesthetics of string diagrams are important” (p x), why did they pick those colours? Other colour combinations are available, and one of the authors (Dan Marsden) uses some here: Category Theory Using String Diagrams.

    1. Bacon’s book could be interesting (so I hope his talk of “Zermelo-Frankle” set theory doesn’t betoken other kinds of carelessness …!).

      The more I get into Introducing String Diagrams the less I like it. The quality of exposition seems increasingly dire.

    2. For your first question, it’s the confluence of compositions that you get from matching target and source, which can tell you when certain functions are and aren’t possible to write, when two types are isomorphic, when natural transformations exist between functor types, etc. I think it’s justifiable that they’re using ‘type’ language given who their audience is and what their goals are, among them chiefly being the introduction and use of the category of types and functions. Without a doubt, using that language loses the category theoretic foundations perspective of taking objects and arrows to be undefined primitives, but no part of me thinks that’s what the goal of this book is: “This is not a book on the mathematical foundations of string diagrams,” (p x). The book is an introduction to string diagrams through category theory for computer scientists, and in that light I don’t see how what they said is confounding. Is saying “f: A -> B can’t compose with h: A -> C, that’s a type mismatch” to a room full of computer scientists really that obtuse? It might be wrong at a technical level, but I don’t see how it’s confusing.

      It covers much less ground, but Robin Piedeleu and Fabio Zanasi’s An Introduction to String Diagrams for Computer Scientists is excellent and might be more amenable: arxiv.org/abs/2305.08768

      “This document is an elementary introduction to string diagrams. It takes a computer science perspective: rather than using category theory as a starting point, we build on intuitions from formal language theory, treating string diagrams as a syntax with its semantics. After the basic theory, pointers are provided to contemporary applications of string diagrams in various fields of science.”

      But I will say that theirs as well is not a category theory foundations perspective, they assume you know what a category is and don’t spend time talking about it.

      1. Thanks for the pointer to Piedeleu and Zanasi’s Introduction. It looks like it quickly moves to an algebraic perspective, which is not ideal for me. It does look interesting and useful, though!

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