Peter Smith's Blog, page 101

Notes on Category Theory v.6a

A number of people have very kindly sent corrections to the last version of Notes on Category Theory. There were some possibly confusing typos and also a downright wrong proof. Embarrassing. So here’s a “maintenance upgrade”, making some needed repairs.

The Open Logic Text

As you will very probably have already seen, The Open Logic Project (a team of serious and good people) has now made available an early public version open-source collaborative logic text, somewhat ploddingly called the Open Logic Text. 

There are two things to comment on here (eventually!), namely the Text itself — or at any rate, the current snapshot of an evolving text — and the open-source nature of the enterprise.

At a first quick glance, the Text does look rather uneven: there are 77 pages on first-order logic and beyond (some at quite an elementary level), 100 pages on computability, incompleteness, etc. (this looks like a solid graduate course), and then just 21 pages on sets (at a very much lower level of sophistication). Still, this is obviously exactly the sort of thing that should be covered in the Teach Yourself Logic Study Guide. So when I’ve had a chance to take a serious look, I’ll report back with my two pennies’ worth, maybe in a mid-year update to the Guide.

As the Project site says of the Text,

… you can download the LaTeX code. It is open: you’re free to change it whichever way you like, and share your changes. It is collaborative: a team of people is working on it, using the GitHub platform, and we welcome contributions and feedback.

I will be really interested to see how this pans out in practice. Using GitHub is a notch or three above my current nerdiness grade. But I simply don’t know if this is me just not keeping up with everyone — or whether it is pretty typical for logicians to know a smidgeon of very basic LaTeX, with that being about their geek limit. Maybe, at least as a bit of exercise to keep the brain from entirely rusting up, I should take a look at this GitHub malarky about which I’ve heard tell before (any useful pointers to an idiot’s guide?). Then I could also report back about how the collaborative aspect looks to a complete beginner. Again, watch this space.

Notes on Category Theory v.6

Here is an updated version of my on-going Notes on Category Theory, now over 150 pp. long. I have added three new chapters, at last getting round to the high point of any introduction to category theory, i.e. the discussion of adjunctions. Most people seem to just dive in, things can get a bit hairy rather quickly, and only later do they mention, more or less in passing, the simpler special case of Galois connections between posets (which transmute into adjunctions between poset categories). If there’s some novelty in the Notes at this point, it’s in doing things the other way around. We first have a couple of chapters on Galois connections — one defining and illustrating this simple idea, the other discussing a special case of interest to the logic-minded. Only then do we get round to generalizing in other natural way. We then find e.g. that two equivalent standard definitions of Galois connections generalize to two standard definitions of adjunctions (presented without that background, it isn’t at all so predicatable that the definitions of adjunctions should come to the same).  I think this way in to the material is pretty helpful: I’ll be interested, eventually, in knowing how readers find it.

So we this is what we now cover:

Categories defined
Duality, kinds of arrows (epics, monics, isomorphisms …)
Functors
More about functors and categories
Natural transformations (with rather more than usual on the motivation)
Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
Categories of categories: issues of size
The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
An aside on Cayley’s Theorem
The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
Representables (definitions, examples, universal elements, the category of elements).
First examples of limits (terminal objects, products, equalizers and their duals)
Limits and colimits defined (cones, limit cones: pullbacks etc.)
The existence of limits (in particular, having finite products and equalizers implies having all finite limits).
Functors and limits (preserving, reflecting, creating limits: hom-functors preserve limits, etc.)
[NEW] Galois connections (warming up for the general discussion of adjoint functors but looking at a special case, functions that form a Galois connection)
[NEW] An aside for logicians, concerning a well-known example, the Galois connection between syntax and semantics.
[NEW] Adjoints introduced. [Two different definitions of adjoint functors, generalizing two different definitions of Galois connections; some examples of adjunctions; a proof that the two definitions are equivalent.]

There will certainly be a few more chapters on adjoints. But don’t hold your breath, with a family holiday coming up and some other commitments. I haven’t decided yet whether eventually to add a chapter or two on monads (for monads seem a standard next topic to cover — e.g. the last main segment of the Part III Tripos category theory course this year, the last chapter of Awodey’s book). Watch this space.

Barnaby Sheppard’s The Logic of Infinity — website

I haven’t yet reviewed Barnaby Sheppard’s The Logic of Infinity (CUP 2104) here — and I don’t know if I will, for even if time may be infinite, that allotted to me certainly isn’t! But when I dipped into the book, it looked a Really Good Thing which could of be real use to its intended audience of near beginners in mathematics whose imaginations might be captured by foundational questions.

Now I know only too well what it is like to publish a book with technical aspects and then find the inevitable typos and thinkos and sheer mistakes. At least the internet makes it possible to ease the pain a bit by giving you a second chance to explain what, really, you meant to say. But readers need to know where to look. So as a friendly gesture to a fellow author, let help me spread the word that Barnaby Sheppard has now set up a small website for errata for his book.

Brilliant indeed

I was going to post about the delights of Amsterdam as a place to visit for a week — the cityscapes, the cafes, the restaurants, the museums large and small, the whole urban experience, all even better than we had hoped. But more or less as soon as we got back, I was felled by a nasty attack of a recurrent problem, about which all I will say is thank heavens for penicillin. Though industrial quantities of antibiotics do leave you feeling still pretty flattened, so it has been a few days of staggering from bed to sofa and back. But as I begin to feel more human I’ve had plenty of time to finish the book I’d just started before going away. Like Amsterdam, this too has been lauded to the skies by those who know it. It has been a delight, in both cases, to find that other people’s really warm recommendations are more than deserved (it doesn’t always happen!). And since you certainly don’t need me to tell you more about Amsterdam, but you might not have heard of Elena Ferrante’s My Brilliant Friend — I hadn’t until a couple of months ago, from the much better read Mrs Logic Matters — maybe I’ll just sing its praises instead.

It really is absolutely wonderful. But I’m not going even to try, in my limping way, to say why. Rather let me point you to this New York Review of Books review of Ferrante’s oeuvre by Rachel Donadio, and/or this review from the New Yorker by James Wood. If these don’t get you reading, nothing will!

Burgess, Rigor and Structure — 3

As we saw, Burgess holds that the very project of rigorization calls for the development of a single unifying foundational system with “a common list of primitives and postulates”; but I suggested that the initial reasons he gives for this, at least, don’t seem particularly compelling. But there’s more. Burgess next notes that a number of constructions — e.g. taking ordered pairs, forming products, taking quotients, etc. — are used and re-used in various cases of manufacturing new  spaces or number systems or whatever out of old ones. So there is here, he seems to think, another reason for providing a unified general framework in which all these constructions can be uniformly carried out. But at first blush it isn’t clear why this would take us in the direction of a single foundational system: you might instead think that what this suggests is that, inspired by seeing similarities in procedures in different areas, we should aim to develop a more general structural framework which makes it easier to spot more such similarities and port techniques developed in one area to another area (category theory, anyone?).

So if there is a drive to a unified foundational system in the vicinity, I don’t think it can be just in the observation that such (surely anodyne) constructions as forming ordered pairs or taking quotients can be found across different areas of maths — these constructions, at least, seem pretty unproblematic. If there is a drive to seek foundations hereabouts it comes, surely, from much more specific concerns about certain distinctively infinitary constructions (which may in fact have their natural home just in one area of mathematics, in particular in analysis) — e.g. the repeated taking of limits or constructions that involve infinite sequences of choices, which seem to involve compellingly natural extensions of classical ideas yet whose legitimacy is open to challenge. We might now reasonably worry about consistency. (Burgess’s initial reasons for supposing that rigorization might lead us to seek a unified foundational theory had to do with consistency too — but those worries could be met by relative consistency proofs, it would seem, without seeking foundations. Now, however, we are in novel territory where we are tangling with the infinite in new ways that are simultaneously enticing and worrying, and so we feel a more pressing need to discipline them by reflecting on the principles underlying the new constructions.)

Which takes us to Cantor’s own route in to his set theory, and which Burgess (eventually) gets round to discussing. And now Burgess’s story becomes pretty conventional. First, there’s a lightning tour through some Cantorian themes, eventually noting Cantor’s own worrying falling short from rigour — his unacknowledged invocation of choice principles, his recognition (as we would put it) that not all predicates can have sets as extensions while lacking any sharp way of demarcating the “inconsistent multiplicities” from the kosher ones. Then there’s something about Russell’s vs Zermelo’s way of dealing with the paradoxes that threaten Cantor’s set theory, with Zermelo’s approach becoming the canonical one, to the point where it can be said that “From the 1950s onward, classical mathematics had just one deductive system, namely, first-order Zermelo-Fraenkel set Theory  with Choice” [that’s Wilfrid Hodges, quoted by Burgess]. This is all done, however, very rapidly — most readers of this blog won’t need the reminders, while students for whom this is actually news might well find it all too quick to be very useful.

So far, then, I don’t think Burgess’s Ch. 2 is particularly satisfactory: but there is still more to come. In the second half of the chapter, Burgess turns to discuss some opponents of the project of rigorization when conceived as the project of regimenting mathematics into classical ZFC set theory. So that’s our next topic.

Notes on Category Theory v.5

Here is an updated version of my on-going Notes on Category Theory, now 130 pp. long. I have done an amount of revision/clarification of earlier chapters, and added two new chapters — inserting a new Ch. 7 on categories of categories and issues of size (which much expands and improves some briefer remarks in earlier versions), and adding at the end Ch. 15 saying something about how functors can interact with limits. There’s quite a bit more that could be said in this last chapter, and I’ll have to decide in due course whether to expand the chapter, or return to the additional topics later, or indeed to only mention some of those topics in the end (I’m trying to keep things at a modestly introductory level). But for the moment I’ll leave things like this and move on to a block of chapters on adjoints and adjunctions. So here’s where we’ve got to:

Categories defined
Duality, kinds of arrows (epics, monics, isomorphisms …)
Functors
More about functors and categories
Natural transformations (with rather more than usual on the motivation)
Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
[New] Categories of categories: issues of size
The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
An aside on Cayley’s Theorem
The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
Representables (definitions, examples, universal elements, the category of elements).
First examples of limits (terminal objects, products, equalizers and their duals)
Limits and colimits defined (cones, limit cones: pullbacks etc.)
The existence of limits (in particular, having finite products and equalizers implies having all finite limits).
[New] Functors and limits (preserving, reflecting, creating limits: hom-functors preserve limits, etc.)

Don’t hold your breath for the chapters on adjoints, though. After a very busy time for various reasons, I’ve a couple of family holidays coming up!

Books published in the first quarter of 2015?

What books on logic (mathematical or philosophical logic) and/or philosophy of maths have been published in the first quarter of 2015? There’s John Burgess’s Rigor and Structure which I have started blogging about here. But what else has appeared so far this year?

Not that I’m lacking things to read! But I’d like to know what I’m missing, and I’m probably not alone. So maybe you would like to share any recommendations of recent titles which might be of interest to readers of Logic Matters?

Mathematical depth

In our Mind review of Penelope Maddy’s Defending the Axioms, Luca Incurvati and I were rather skeptical about whether she could really rely on the notion of mathematical depth to do as much work as she wants it to do in that book. But we did add “We agree that there is depth to the phenomenon of mathematical depth: all credit to Maddy for inviting philosophers of mathematics to think hard about its role in mathematical practice.”

Since then, there has been a workshop on mathematical depth at UC Irvine co-organized by Maddy, and now versions of the papers there have been made available as a virtual issue of Philosophia Mathematica which will remain freely available until November this year. Looks interesting.

The Chiaroscuro Quartet: Mozart and Mendelssohn

This is currently my favourite late-evening listening among recent releases — it’s the third CD by the Chiaroscuro Quartet. Each CD couples one of the Mozart Haydn quartets with another work: this time it is Mozart’s Qt 15, K. 421 with Mendelssohn’s Qt 2, op. 13. The performances are extraordinarily fine.

The Chiaroscuro are friends with other musical careers, who come together for the pleasure of playing together — and oh, how it shows! There’s a sense of listening in to private music making of exploratory intensity. The leader is Alina Ibragimova whose solo work is stella beyond words, but here Ibragimova in no way overshadows Pablo Hernan Benedi, Emilie Hörnlund and Claire Thirion: the togetherness, the shared style and understanding, is astonishing indeed.

If you haven’t heard their previous CDs then initially their sound is a shock: they are playing on gut strings, almost without vibrato. So the timbre is spare, the period sound unadorned: it can take a couple of hearings to get used to it. And if — like me — you already know the Mozart well and the Mendelssohn hardly at all, then another surprise is how the Chiaroscuro bring the works much closer in their worlds than you have previously heard them. The Mozart is more troubled, the 18-year-old Mendelssohn more austere: but this makes for a revelatory and satisfying programme.

You can listen to excerpts on the Quartet’s website here. Very warmly recommended.