Notes on Category Theory, (partial) version #2

After a bit of a gap, I’ve been able to get back to writing up my notes. The current instalment of the notes (61 pp.) corrects some typos in the first six chapters — and it is those needed corrections that prompt me quickly to post another version even though I’ve only added two new chapters this time. So far, then, I cover

  1. Categories defined
  2. Duality, kinds of arrows (epics, monics, isomorphisms …)
  3. Functors
  4. More about functors and categories (and the category of categories!)
  5. Natural transformations (with rather more than usual on the motivation)
  6. Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
  7. The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
  8. The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).

It took me a while to see how best(?) to split the proof of the Yoneda Lemma into obviously well-motivated chunks: maybe some others new(ish) to category theory will find the treatment in Chs 7 and 8 helpful.

Download the new version of the notes here

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Logic books of the year?

It is the time of year when the more serious newspapers invite panels of authors, reviews editors, and others to pick out their books of the year, leaving the rest of us to feel hopelessly out of touch and wondering how to find the time to read more … (Only a few months late, I did greatly enjoy and admire one of last year’s oft-chosen books, Donna Tartt’s The Goldfinch. I try to alternative reading novels old(ish) and new(ish), and the returned-to-modern-classic that I got lost in, and wished hadn’t come to an end, even though it is one of the longest single novels in the language, was Vikram Seth’s A Suitable Boy.)

But what about the logic books of 2014 (mathematical or philosophical)?

My patience with philosophy seems frankly to be getting less and less. I was disappointed by Stewart Shapiro’s Varieties of Logic, and haven’t yet read Penelope Maddy’s new The Logical Must. I’m sure Roy Cook’s The Yablo Paradox is a good thing, but again I haven’t mustered the enthusiasm to tackle that. But what else broadly in the area of philosophy-of-logic/philosophy-of-maths has newly appeared this year? I’m probably being forgetful, but as I look along my shelves I can’t recall anything that got me excited!

As for more technical stuff, however, I can be much more positive. The stand-out book for me is

Tom Leinster, Basic Category Theory (CUP, viii + 183 pp.).

To be sure, this is not for everyone who visits Logic Matters, for it is a mathematics text (published in the Cambridge Studies in Advanced Mathematics series), and also it won’t tell you about the more specifically logic-related topics in category theory. But the book’s treatment of the basic topics that it does cover strikes me as a particularly fine expository achievement, balancing economy of scale with accessibility. So that‘s my logic book of the year for 2014.

What are your logic/phil maths book highlights of the year?

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Notes on Category Theory, (partial) version #1

As I said in my last post, I’ve been following some lectures on category theory since the beginning of term. The only way of really nailing this stuff down is to write yourself some notes, work through the proofs, etc. Which I’ve been doing. And then I’ve done some polishing to make the notes shareable with others following the course:

So here are my current notes (50 pp.) on the topics of the first quarter of the course.

Warning: the course I’m following is for the Part III Maths Tripos (i.e. a pretty unrelenting graduate level course for mathematicians with a very strong background). My notes are easier going because I proceed quite slowly and pause to fill in all the proofs where the blackboard notes might well simply read “Exercise!”. But still, this is maths which requires some background to follow (even if perhaps less than you might think).

To be sure, I want to be thinking more in due course about some of the philosophical/foundational issues that category theory suggests: but for the moment my aim is to really get my head round the basic maths more than I’ve done in the past. Hence the notes, which maybe some others might find useful. So far, I cover

  1. Categories defined
  2. Duality, kinds of arrows (epics, monics, isomorphisms …)
  3. Functors
  4. More about functors and categories (and the category of categories!)
  5. Natural transformations (with more than usual on the motivation)
  6. Equivalence of categories (again with a section on the motivation)

Enjoy! (And even better, let me know where I’ve gone wrong and what I can improve.)

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Back to school …

Once upon a lifetime ago, I took Part III of the Maths Tripos.

In fact, rather alarmingly, I started exactly fifty years ago this term. And it was tough. You had to aim to do over the year (the equivalent of) six courses of 24 lectures, which were lectured at a helter-skelter, take-no-prisoners, pace. The blackboard notes gave you just the barest skeleton, and you had to spend a great deal of time working on them between classes in order to keep up, and then a lot more time in the vacations to really get on top of the material. I remember it as the time in my life I had to work by far the hardest, though it all worked out well.

Things, it seems, have changed astonishingly little. I’ve been turning out — Mondays, Wednesdays and Fridays at 9! — to go to this year’s Part III Category Theory lectures (given by Rory Lucyshyn-Wright. The course is still lectured at a cracking pace, with blackboard notes giving you a bare skeleton, and leaving a great deal of work required if you are to put enough flesh onto the bones to get the real shape of what’s going on. No pre-digested handouts here!

I’m just about hanging on in there. I’m trying to write up quite detailed notes to fix ideas, and I’m already falling behind with those — and this despite the fact that I’ve read around a bit the subject in the past. But, as we all know, in maths in particular there is all the difference between a casual read and really working your way into a topic. And that’s what I want to try to do, at least for the beginnings of category theory. (Well, why not?)

OK, I’m no doubt slower on the uptake than I was back in the day, and the kids around me are among the world’s best mathematicians of their age, have a lot more energy and function more hours in the day. But they are having to keep up with three times as much this term, and will do it all again next term. We can only be impressed.

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Saffron Hall/Brodsky Quartet

12438262893_fb2726ff9a_zWe’ve just made our first visit to a concert at Saffron Hall, less than forty minutes from Cambridge. This is a multi-purpose hall newly built as part of Saffron Walden County High School and opened at the very end of last year. Once upon a long time ago, Mrs Logic Matters was at school there, and as we ordered interval drinks at the bar, she found herself standing again outside the head’s office, remembering being torn off a strip inside (mascara, too much; skirt, too short …).

But I digress. The Hall itself is large and the stage huge, as you can see, so the four music stands for the string quartet (and raised seat for the cellist)  looked very lost and lonely on the bare expanse. I worried that this space wasn’t going to work for such a small ensemble. Quite wrongly. The hall dimmed to leave the performers in a central pool of light. The acoustics were simply wonderful (apparently, there are state-of-the-art adjustable  acoustics). The sight lines were perfect. The general ambience was very engaging, with particularly friendly front-of-house volunteers.   There’s even a lot of parking. We were very impressed indeed with the Hall.

And the Brodskys? They began with Stravinsky’s short Three Pieces for String Quartet, new to me, and then gave a magical account of the fifth Shostakovich Quartet before the interval. Haunting and sensitively done. Bowled over. After the interval, however, the quartet played Death and the Maiden, and — by contrast — neither of us particularly warmed to their performance. (I had my doubts about the suitability of first violin’s playing style, and there wasn’t enough youthful fierceness either in the opening movement or the closing dance of death.) Still, the Shostakovich alone was more than worth the journey.

The main reason for posting this, however, is very warmly to encourage anyone within striking distance of Saffron Walden to check out the Hall’s programme of concerts over the next few months: Maria Joao Pires, The Sixteen, Ian Bostridge, Paul Lewis …  in a number of cases repeating a programme from the Wigmore Hall a few days before or after. A rather astonishing line up. The efforts of the new Director in her first season are more than worth supporting.

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Quick book note: Pollard’s Mathematical Prelude to the Philosophy of Maths

Unknown “This book is based on two premises: one cannot understand philosophy of mathematics without understanding mathematics and one cannot understand mathematics without doing mathematics.” Thus the blurb of Stephen Pollard’s recent book A Mathematical Prelude to the Philosophy of Mathematics (Springer, 2014: xi + 202pp).

I certainly agree that if you want to study the philosophy of X, then it is good idea to know something about X! And that applies in particular when X = science or X = mathematics. It is hard work teaching philosophy of science to students whose ignorance of science is profound. Philosophers of maths can have things a bit easier. For in fact many serious philosophy departments do actually teach some relevant maths in-house, in the guise of mathematical logic courses. Students can encounter core first-order logic up to a smidgin of model theory, variations/extensions such as intuitionistic logic and second-order logic, something about theories of arithmetic, ideas about computable functions, and bit of set theory. True, these topics won’t necessarily be taught in the style of a hard-core maths course: the emphasis might be more on the Big Ideas and on conceptual foundations rather than on rather tricky problem-solving. But still, philosophy students who do tangle with the traditional menu of mathematical logic topics should acquire enough first-hand knowledge of enough serious mathematics for their philosophy of maths course to have something to work on.

Of course, if elementary mathematical logic is all the maths you ever get to know, you’ll end up with a rather skewed view of the mathematical enterprise. But at least you’ve made a start. It is then a nice question what other maths it would be good for a budding philosopher of maths to acquire some small acquaintance with. Now, Pollard’s title, and  then his talk in the Preface of the book’s “motley” character, might perhaps suggest for a moment that we are going to get an interestingly varied menu of topics, including some out of the usual run. But this isn’t really how things go. We in fact get three chapters on set theory, preceded by two chapters on arithmetic, and succeeded by another chapter on arithmetic and a chapter on intuitionistic logic. So in fact it is business pretty much as usual — mostly covering, though briefly, the sort of topics mathematical logicians typically cover for their philosophy students — albeit with some twists in the treatment of arithmetic which we’ll come to in a moment.

How then does this compare with other accounts of the familiar topics? Starting with the set theory chapters, Pollard fusses a lot at the outset, stressing that we shouldn’t be misled by unhappy metaphors of the “sets are like boxes …” variety, and recommending that we think of them as more like unordered list-types. And then he runs and runs with this idea, talking about “Zermelian lists” and more before reverting to standard set talk. But I didn’t find this particularly well done. And I frankly don’t think this, as an expository ploy, would be likely to work any better as an introduction to set theory than the standard approach of e.g. those wonderfully lucid entry-level books on set theory that I recommended in the TYL Guide.

The preceding chapter on first-order Peano Arithmetic is more conventional, but also rather compressed, and again there are significantly better options out there.

So that leaves three chapters to say just a little more about. The first chapter of the book is indeed unusual, for it starts by discussing Hilbert’s stroke arithmetic, so we get a discussion of tokens and types of tally marks, and operations on them, and then aims to develop primitive recursive arithmetic on this basis. The metaphysics of types here seems to get rather murky (types can be uninstantiated on p. 6, so they seem to be platonic universals at that point, but on p.7 it seems they are worldly enough for their existence to vary between worlds, so maybe not so platonic after all; and things aren’t really sorted out in Sec. 1.9 “Some Philosophy” ). Maybe we can work at extracting a clear position, but is this what we want to be doing at this early point in what is supposed to be a maths book? And indeed the ensuing development of PRA really could be clearer too. So I’m not sure I’d want to recommend this chapter either.

But Pollard in his Preface does invite readers to pick and choose. And I so choose the last two chapters! The first of these takes us back to arithmetic after the excursus on set theory, but now to Frege Arithmetic. Students encountering the (neo)logicist programme in their philosophy of maths course could well find this presentation quite useful, as a companion piece to set alongside the Stanford Encyclopedia article. (I should note that, as in other chapters in the book, there are lots of comprehension-testing exercises as you go along, which some will find helpful.)

Then the final chapter goes off somewhat at a tangent, presenting a certain approach to intuitionist logic. This approach has its roots in the work of Gentzen, Prawitz, Dummett and Tennant, aiming to show that there is a particular naturalness to intuitionist logic if we think of the meaning of logical operators as given by their introduction rules, with elimination rules required to be harmonious (with this idea developed against the background of a certain way of thinking about negation and absurdity). But the particular version of these ideas we get here is due to Jaroslav Peregrin, in his ‘What is the Logic of Inference?’ Studia Logica 88 (2008) 263-294. This review isn’t the place to argue whether or not Peregrin’s is the best version of that general line of argument for the special naturalness of intuitionist logic. But Pollard’s exposition is done with verve, and a student ought to find it intriguing and thought-provoking. So this is the best and most novel chapter in a rather patchy book, I think.

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More logic, phil. maths, foundations blogs?

Brian Weatherson has an interesting new blogroll of “active philosophy blogs” with “substantive” content [update: where Logic Matters now gets a mention!]. Inspired by that, I thought it was more than time to update the blogroll here. So alongside are now two short blogrolls to be going on with — scroll down the sidebar. One lists a few blogs with (occasional) logic/phil. maths/foundations of maths/or just maths content, the other links to a few other random favourites. (Hover your cursor over the blog title for a mini-description.)

The first list of logicky blogs is surprisingly short [Added: still short after an update, though I am being a bit selective]. Which probably simply reflects that I haven’t looked hard enough. So, folks, what am I missing? Any recommendations for (still active) blogs with good logic-related content??

Though actually, now I think about it, even Weatherson’s wide-ranging philosophy list is quite short, given the number of enthusiastic, energetic, philosophers out there. Perhaps the cool kids have moved on and blogs are no longer the done thing. A pity if so. They can be fun and illuminating for readers, and writers do get some ideas out there into the wider world (even a modest effort like Logic Matters counts its visitors per day in the many hundreds on the least generous of the stats counters).

[Added: with many thanks to @logicians on twitter, I’ve added a few more links, some I’d forgotten about, and one new to me.]

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Quick book note: Stewart Shapiro’s Varieties of Logic

9780199696529_450Stewart Shapiro’s very readable short book Varieties of Logic (OUP, 2014) exhibits the author’s characteristic virtues of great clarity and a lot of learning carried lightly. I found it, though, to be uncharacteristically disappointing.

Perhaps that’s because for me, in some key respects, he was preaching to the converted. For a start, I learnt long ago from Timothy Smiley that the notion of consequence  embraces a cluster of ideas. As Smiley puts it, the notion “comes with a history attached to it, and those who blithely appeal to an ‘intuitive’ or ‘pre-theoretic’ idea of consequence are likely to have got hold of just one strand in a string of diverse theories.” Debates, then, about which is the One True Notion of consequence are likely to be quite misplaced: for different purposes, in different contexts, we’ll want to emphasize and develop different strands, leading to different research programmes. As Shapiro puts it, the notion(s) of consequence can be sharpened in different ways — and taking that point seriously, he suggests, is already potentially enough to deflate some of the grand debates in the literature (e.g. about whether second-order logic is really logic).

And I’m still Quinean enough to find another of Shapiro’s themes congenial. Do we say, for example, that ‘or’ or ‘not’ mean the same for the intuitionist and the classical mathematician? Or is there a meaning-shift between the two? Shapiro argues that for certain purposes, in certain contexts, with certain interests in play, yes, we can say (if we like) that there is meaning shift; given other purposes/contexts/interests we won’t  say that. The notion of meaning is maybe too useful to do without in all kinds of situations; but it is also itself too shifting, too contextually pliable, to ground any grand debate here.

Put it this way, then. I’m pretty sympathetic with Shapiro’s claims that some large-scale grand debates are actually not very interesting because not well-posed. What that means, I take it, is that we’ll in fact find the interesting stuff going on a level or two down, below the topmost heights of cloudy generality, in areas where enough pre-processing has gone on to sharpen up ideas so that questions can be well-posed.

Here’s the sort of thing I mean. Take the very interesting debate between those like Prawitz, Dummett and Tennant who see a certain conception of inference and the logical enterprise as grounding only intuitionistic logic (leaving excluded middle as a non-logical extra, whose application to a domain is to be justified, if at all, on metaphysical grounds), and those like Smiley and Rumfitt who argue that that line of thought depends on failing to treat assertion and rejection on a par as we ought to do. This debate is prosecuted between parties who have agreed (at least for present purposes) on how to sharpen up certain ideas about logic, consequence, the role of connectives, etc.,  but still have an argument about how the research programme should proceed.

Shapiro doesn’t mention that particular debate. Absolutely fair enough (I just plucked out something that happens to interest me!). The complaint, though, is that he doesn’t supply us with much by way of other illustrations of investigations of varieties of logic at a level or two below the most arm-waving grand debates — i.e. at the levels where, by his own account, the real action must be taking place. Hence, I suppose, my general disappointment.

Shapiro does however mention a number of times one interesting example to provide grist to our mills, namely smooth infinitesimal analysis. This, if you don’t know it, is a deviant form of infinitesimal analysis — deviant, at any rate, from the mathematical mainstream. (If you look at Nader Vakil’s recent heavy volume Real Analysis Through Modern Infinitesimals  in the CUP series Encyclopedia of Mathematics and Its Applications, then you’ll find smooth analysis gets the most cursory of mentions in one footnote.) The key idea is that there are nil-potent infinitesimals — at a rough, motivational level, quantities so small their square is indeed zero, even though they are not assumed to be zero. More carefully, we have quantities \delta such that \delta^2 = 0 and \neg\neg(\delta = 0), but — because the logic is intuitionistic — we can’t assert \delta = 0. And then, the key assumption, it is required that for any function f, and number x, there is a unique number f'(x) such for any nil-potent \delta, f(x + \delta) = f(x) + f'(x)\delta. So looked at down at the infinitesimal level, f is linear, and f'(x) gives its slope at x — so is the derivative of f. Now it turns out that, with enough assumptions in place, this theory allows us to define integration in a correspondingly natural way, and then we can readily prove the usual basic theorems of analysis.

Now that is indeed interesting. But — and here’s the rub — the internal intuitionistic logic is absolutely crucial. The usual complaint by the intuitionist is that adding the law of excluded middle unjustifiably collapses important distinctions (in particular the distinction between \neg\neg P and P).  But in the case of smooth analysis, add the law of excluded middle and the theory doesn’t just collapse (by making all the nil-potent infinitesimals identically zero) but becomes inconsistent. What are we to make of this? In particular, what can the defender of classical logic make of this?

I guess there is quite a lot to be said here. It is a nice question, for example, how much sense we can make of all this outside the topos-theoretic context where the Kock-Lawvere theory of smooth analysis had its original home. To be sure, as in John Bell’s A Primer of Infinitesimal Analysis, we can write down various axioms and principles and grind through deductions: but how much understanding ‘from the inside’ does that engender? Shapiro says just enough to pique a reader’s interest (for someone who hasn’t already come across smooth analysis), but not enough to leave them feeling they have much grip on what is going on, or to help out those who are already puzzling about the theory. And that’s a real disappointment.

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Philosophers being offensive

If philosophers want to be really offensive, at least do it with style.

I’m reminded of a story about my favourite Cambridge philosopher, C.D. Broad. Not entirely a nice man.

A long time colleague of his at Trinity was the organic chemist Frederick Mann, whom Broad evidently thought an uncultured dullard. One evening at High Table, Broad finds himself unfortunately sitting opposite Mann. Broad beams cherubically. Pauses. And, to no-one in particular, sighs “Ah, where every prospect pleases …”.

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The Very Short Teach Yourself Logic Guide

I seem to have gone full circle! The very first instalment of the TYL Study Guide was a short blogpost here. Then things grew. And grew. Until we get to the current 100 page PDF monster — and that’s only as short(!) as it is because some material has been exported to the Appendix and a supplementary webpage on Category Theory. Of course, the long version is full of good things, explains why the chosen texts are recommended, explains why others aren’t, suggests alternatives and supplementary reading, and more. But still, some might be interested in just getting the headline news.

So now, rather in the spirit of the original post, there’s an encouraging short and snappy page, The Very Short Teach Yourself Logic Guide, which just gives you the winners, the top recommendations for entry-level reading on the various areas of the core math. logic curriculum. (If you want to know why they are the winners, then you will have to look in the corresponding section of the full version of the Guide.)

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