Logic Matters

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 a rather 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 do 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:

1. Categories defined
2. Duality, kinds of arrows (epics, monics, isomorphisms …)
3. Functors
4. More about functors and 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. Categories of categories: issues of size
8. The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
9. An aside on Cayley’s Theorem
10. The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
11. Representables (definitions, examples, universal elements, the category of elements).
12. First examples of limits (terminal objects, products, equalizers and their duals)
13. Limits and colimits defined (cones, limit cones: pullbacks etc.)
14. The existence of limits (in particular, having finite products and equalizers implies having all finite limits).
15. Functors and limits (preserving, reflecting, creating limits: hom-functors preserve limits, etc.)
16. [NEW] Galois connections (warming up for the general discussion of adjoint functors by looking at a special case, functions that form a Galois connection)
17. [NEW] An aside for logicians, concerning a well-known example, the Galois connection between syntax and semantics.
18. [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. in the last main segment of the Part III Tripos category theory course this year, the last chapter of Awodey’s book). Watch this space.

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.

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!

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 ideas and constructions (which may in fact have their natural home just in one area of mathematics, in particular in analysis) — e.g. allowing functions which associate arguments with values in ways that go beyond what is finitely specifiable or allowing the repeated taking of limits or constructions that involve infinite sequences of choices. These may seem rather compellingly natural extensions of classical ideas yet their 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 claimed with plausibility 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.

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:

1. Categories defined
2. Duality, kinds of arrows (epics, monics, isomorphisms …)
3. Functors
4. More about functors and 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. [New] Categories of categories: issues of size
8. The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
9. An aside on Cayley’s Theorem
10. The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
11. Representables (definitions, examples, universal elements, the category of elements).
12. First examples of limits (terminal objects, products, equalizers and their duals)
13. Limits and colimits defined (cones, limit cones: pullbacks etc.)
14. The existence of limits (in particular, having finite products and equalizers implies having all finite limits).
15. [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!

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?

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.

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.

In his Preface, Burgess says that in putting together his book it became clear that he

would need to explain not only what rigour is, but also how set theory came to occupy the position … of being in some sense the accepted foundation or starting point for rigorously building up the rest of mathematics. Such … explanations are given in Chapter 2.

The chapter begins with more observations on the development of modern mathematics which could have as easily been in the first chapter. Burgess touches on the way that a familiar space or number system may get expanded by the addition of ideal elements (e.g. points at infinity, complex numbers). Since we can’t carry over habits of thought acquired when working with the old familiar systems to the augmented systems, we’ll have to pay careful attention to what does and doesn’t follow from our assumptions about the new ideal elements — another driver towards increased rigour.

But other developments involve learning more about the familiar not by augmenting our original system with ideal elements but rather by making connections between seemingly very different bits of mathematics  — e.g. as in Galois’s theory connecting solutions of polynomials (an old topic) which a theory of permutations (part of the theory of groups).

And it is here, in the fact of the interconnectness of different branches of mathematics, that Burgess locates “crucial implications for the project of rigorization”. Making the connections requires us to get more precise about the systems we are connecting. But local precision isn’t enough. More  tellingly (for present purposes)

To guarantee that rigor is not compromised in the presence of transferring material from one branch of mathematics to another, it is essential that the starting points of the branches being connected should at least be compatible.

Burgess gives a nice example of what he means: there’s evidently a problem about using analytic methods in Euclidean geometry if your geometry assumes the Archimidean axiom (given any two lengths, there is an integer n such that n times the shorter length exceeds the greater length) yet your analysis essentially depends on infinitesimals (which don’t obey the  Archimidean principle). But what does that show? Burgess writes:

The only obvious way to ensure compatibility of the starting points of different branches is ultimately to derive all branches from a common, unified starting point. The material unity of mathematics, constituted by the interaction of its various branches at their higher levels, virtually imposes a requirement of formal unity, of development within the framework of a common list of primitives and postulates, if the rigorization project is to be carried to completion.

Well, suppose we do adopt a standard set theoretic framework. Within that framework, we can develop various forms of geometry, Archimidean and non-Archimidean, and various forms of analysis with and without infinitesimals. All can be “derived from a common, unified starting point”: but that’s plainly not enough to warrant transporting results from a version of analysis to a version of geometry, to stick for a moment to Burgess’s prime example. We need to make the right compatible pairings. But how is compatibility established? In this case, perhaps, by interpreting the geometrical theory into analysis. Now, interpreting both in a background set theory may be an aid to this: but to repeat, it can’t be enough just to interpret both ‘vertically downwards’ into set theory — the interpretations must make the right interpretative links ‘horizontally’ between a particular axiomatized geometry and the axiomatized analysis. And it is the horizontal links that matter if our theories are to be compatible: yet surely such links can in this case, and in some other cases too, be made without dog-legging through a full-blown set theory in which all branches of mathematics can be derived.

What about the example of elementary Galois theory — another of Burgess’s main examples — where we attack a problem about polynomial equations with some elementary group theory? Here it isn’t like a situation where we have two theories — as was the case with a synthetic and an analytic geometry — purporting to be about the ‘same’ things, lines, conic sections and so on, and hence a situation where straightforward issue of consistency can arise. Rather we have some abstract algebraic apparatus about permutations (any permutations of any finitely many objects) which makes merely conditional claims — if anything satisfies such-and-such, it is so-and-so. And then we find we can apply this conditional apparatus, initially surprisingly, to a case where it wasn’t at all immediately obvious that that there are significant permutations to get a grip on, investigation of which will deliver key results about the polynomial equations. If that, at an arm-waving level, is what is going on, then there is a question of the useful applicability of group theory in a certain domain. But it isn’t immediately clear what kind of issue of compatibility arises here in any sense which would press us to look for an all-encompassing unified starting point for all mathematics.

So Burgess surely has, to say the least, more work to do to make out the claim that the very project of rigorization requires a formal unification of mathematics.

But we have a lot more of Ch. 2 to come: to be continued …

Modern pure mathematics is characterized by the rigour of its methods, and by its special subject matter, i.e. abstract structures. Or so the story goes. But exactly what is meant by rigour here? What, exactly, is meant by saying that modern mathematics is about structures? And what is the relationship between the drive to rigour and the drive to a sort of abstract structuralism (if that’s what it is)?

These are the topics of John Burgess’s new book Rigor and Structure (OUP, 2105). Big topics for a relatively short, four chapter, book. Though Burgess says that he originally intended to write even more briskly, planning just a long paper to make and defend his key novel claim, about which more in due course. But in the writing, he found that for some audiences he needed to say something about how we got to where we are. So the defence of his main claim doesn’t come until Ch. 3, with a corollary then drawn  out in Ch. 4. First, though, we get Ch. 1 about how the ideal of rigour developed and how it is realised in present-day mathematics, and then Ch. 2 explores how set theory came to occupy the special position of being “in some sense the accepted foundation or starting point for rigorously building up the rest of mathematics”.

Burgess’s topics couldn’t be more central to the philosophy of mathematics, and he writes with an engaging clarity and directness. The book is intended to be, and mostly should be, accessible to philosophical readers without too much detailed background in mathematics and indeed accessible to mathematical readers without too much background in philosophy.  But how far can we get in a short compass from almost a standing start? Well,  let’s dive in and see! I plan to comment here as I read through.

If we want to understand what characterizes rigorous mathematics, it would be good to have a foil, significant examples of not-so-rigorous-mathematics: but  we will not find many of those looking round the contemporary scene in pure mathematics. So if we want throw modern standards of rigour into relief, it is natural to look to the history of mathematics to provide some constrasts. Burgess’s first chapter ‘Rigor and Rigorization’ aims to sketch in just enough history to do the job.

I imagine that quite a few readers of this blog won’t need the scene-setting and will be able to skim or skip through this chapter at speed. But here’s the headline news. There are, rather predictably, two main themes we can pick out from Burgess’s story: (a) the development of analysis, (b) geometry, Euclidean and non-Euclidean.

(a) On analysis, Burgess says something about the roots of the calculus in the manipulation of infinitesimals and about the reliance  on “geometric intuition” (in something like a Kantian sense — not mere hunches, but e.g. the “perception” that if this curve tracing a function goes from a negative value to a positive value then it must cross the axis x = 0 and so for some value the function takes a zero value). He also describes what he calls the use of “generic reasoning” (extending the application of reasoning patterns known to have exceptions, without any clear story about what makes for favourable ‘safe’ cases) — though some more examples would have been good to have.

Using infinitesimals, naively construed, we have to delicately alternate between treating them as strictly non-zero and allowing them to vanish: how are to make sense of this? We need better conceptual analysis of what is going on in taking differentials and integrating here (and the account that won out in fact eliminated the need to take infinitesimals seriously). Again, geometric intuition gets it wrong about e.g. the relation between continuity and differentiability (once we see that we can define an everywhere continuous nowhere differentiable curve): how can we avoid relying on fallible intuition? By better analysis of concepts again and by closer analysis of reasoning. Failures of generic reasoning also force rigorization in the guise of exploring just what underpins various proof methods so we can understand their proper domain of application etc.

(b) As for Euclidean/non-Euclidean geometry, there’s a familiar story to be told about what happens in Euclid, its successes and failures in rigour. And then the development of non-Euclidean geometry (especially in the process of exploring what can and can’t be proved from Euclidean geometry minus the parallels axiom) forces what Burgess calls a “division of labor” between mathematical geometries and physical theorizing about which geometry is best suited to be recruited to describe the world. But then, if geometric/physical intuition can no longer play its role inside pure mathematical geometries, how else can we  conceive their exploration other than (in modern vein) the deductive elaboration, ideally with gap-free proofs, of various suites of axioms (where being an axiom now means only being a starting point, not being a basic truth about the world).

So familiar themes emerge. The growing importance of sharp conceptual definitions, explicitness about what axioms and principles we are working with (precision being essential, truth-to-the-world not), the emerging neo-Euclidean ideal of gap-free proofs from those explicitly acknowledged starting points (with no smuggled-in extras or reliance on “intuition”)  — these will be very familiar tropes to those philosophy or maths students who have  looked at just a little history of mathematics. But I’m not at all complaining in saying that Burgess is here going over familiar ground. This is all worth repeating, and is here clearly enough explained, and those many students who don’t already know the territory should find the opening fifty-page chapter pretty useful.

Still, life being short,  many other readers can happily start reading the book at Ch. 2. So let’s move on to that.

To be continued.