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 some 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). Very big topics for a relatively short, four chapter, book. Though Burgess in fact 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 equally accessible to mathematical readers without too much background in philosophy. But how far can we get, how deep can we go, in such a short book written for a dual audience and assuming so little background? 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, Burgess suggests, 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 contrasts. Burgess’s first chapter ‘Rigor and Rigorization’ aims to sketch in just enough history to do the job.
Now, I imagine that most readers of this blog won’t in fact need this element of scene-setting and so will be able to skim or skip through this chapter at considerable speed. There are, rather predictably, two main themes we can pick out from Burgess’s story: (a) the development of analysis, (b) the development of geometry, Euclidean and non-Euclidean.
(a) On analysis, Burgess says something about the roots of the calculus in the manipulation of infinitesimals and also about the early 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.
Now, in 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): so 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, again better analysis of modes of reasoning 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 of reasoning 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 gleaned just a very little history of mathematics.
Now, I’m not complaining in reporting that Burgess is here going over familiar ground. Those students who don’t already know the familiar stories could well find the opening fifty-page chapter useful, as it is all pretty lucidly done. Still, many other readers — surely including most readers of this blog — will be able to skip. So let’s move on …
To be continued.