Phil. of maths

More Cambridge Elements: Mark Wilson on maths, David Liggins on abstract objects

I overlooked Mark Wilson’s Innovation and Certainty when it was published in 2020. But I didn’t miss much. The topic is an excellent one — just what is going on when we make innovations like adding points and lines at infinity in geometry (to take a reasonably comfortable but still instructive example), and just how are these extensions justified? How can we be sure they don’t lead us astray? But heavens, Wilson’s discussion is arm-wavingly pretentious and tediously obscurantist. It is just a dreadful piece of writing, and it baffles me that the series editor let it pass. Don’t waste time on this.

By contrast, David Liggins’s brand new mini-book on Abstract Objects  is the very model for how Elements should surely be.  It is admirably lucid and plain-speaking, approachable by an undergraduate student, yet the way Liggins organizes the material (evidently reflecting a good deal of thought) ought to be useful too for readers with rather more background who, for example, want to revisit the area and perhaps return to thinking about it.

I do have quibbles. Well, more than quibbles. I suspect that we have pretty different views on Hale/Wright abstractionism (which is touched on), and on the value of the ideas in e.g. Charles Parsons’s Mathematical Thought and Its Objects or Øystein Linnebo’s Thin Objects, (neither of which is mentioned). And if I weren’t telling myself that I must concentrate on other projects, I’d certainly be moved to engage properly here. But disagreement is only to be expected with a fifty-page essay. And I’d still happily put this into the hands of a student. Wilson’s effort, not so much.

(Abstract Objects is still free to download from CUP for another 48 hours or so.)

Two new Cambridge Elements on Phil. Maths

Just briefly to note that there are two new short contributions in the Cambridge Elements series in the Philosophy of Mathematics, both free to download for another week. The Euclidean Programme by Alex Paseau and Wesley Wrigley critically examines the traditional idea that mathematical knowledge is obtained by deduction from self-evident axioms or first principles. How much of that idea can be rescued?

And Number Concepts by Richard Samuels and Eric Snyder takes an interdisciplinary approach to reviewing and critically assessing work on number concepts in developmental psychology and cognitive science. (And after all, shouldn’t philosophers of arithmetic be interested in the concepts deployed by folk arithmeticians?)

So far, contributions in this series have been, it seems to me, a rather mixed bunch. So naive induction is little guide, in this case, as to how worthwhile these new efforts will prove to be. But let’s live in hope. When I’ve had a chance to take a proper look, I’ll let you know what I think. But I thought I would post a quick note straight away, while these two Elements are still freely downloadable, and you can judge them for yourself.

Maddy and Väänänen on categoricity arguments

There’s a new short book in the Cambridge Elements series — Penelope Maddy and Jouko Väänänen have written a very interesting contribution on Philosophical Uses of Categoricity Arguments. Here’s their Introduction:

Mathematicians and philosophers have appealed to categoricity arguments in a surprisingly varied range of contexts. One familiar example calls on second-order categoricity in an attempt to show that the Continuum Hypothesis, despite its formal independence, has a determinate truth value, but this doesn’t exhaust the uses of categoricity even in set theory, not to mention its appearance in various roles in discussions of arithmetic. Here, we compare and contrast a sampling of these deployments to get a sense of when these arguments tend to succeed and when they tend to fail. Our story begins with two historical landmarks, Dedekind and Zermelo, on arithmetic and set theory, respectively, and ends with leading contemporary writers, Charles Parsons and the coauthors Tim Button and Sean Walsh, again on arithmetic and set theory, respectively. In between, we pause over the well-known contribution of Georg Kreisel. In each case, we ask: What does the author set out to accomplish, philosophically? What do they actually do (or what can be done), mathematically? And does what’s done (or can be done) accomplish what they set out to do? We find this focus on context illuminating: these authors have qualitatively different philosophical goals, and what works for one might not work for another.

Their scorecard? “Dedekind has successfully achieved his goal” (p. 6), and “In the end … Zermelo accomplished more than he set out to do -– and ultimately more than he could have realized at the time – so this application of categoricity arguments must be counted as a resounding success” (p. 15). As for Kreisel, properly read “determinateness of CH wasn’t his target in the first place. At his actual goal – elucidating the independence phenomenon – he succeeds” (p. 21). Next, “In the end, there seems room for doubt that our shared concept [of number], Parsons’s own Hilbertian intuition of the endless sequence of strokes, is as clear and determinate as we think it is. And if there is this room for doubt, formal categoricity theorems don’t seem to be the kind of thing that might conceivably help. Given these open questions, both mathematical and philosophical, Parsons’s appeal to categoricity arguments to establish “the uniqueness of the natural numbers” can’t yet be judged a success.” (p. 38, after a particularly useful discussion.) Finally, “We conclude that Button and Walsh have not succeeded in establishing that internalist … concerns over the status of CH are “difficult to sustain” (p. 49).

Along the way, we get pointers to some significant first-order results due to Väänänen, and the book concludes

Perhaps unsurprisingly, we think the first-order theorems do make an important philosophical point: an outcome that was thought to require secondorder resources – namely, categoricity theorems – can actually be achieved by suitable first-order means. … This is a useful discovery, which supports our general moral: a bit of mathematics that fails at one task might succeed (and even be aimed) at another.

I hope that’s enough to pique your interest in what does seem to be one of the best so far of the logic/philosophy of mathematics Elements; I enjoyed a quick first reading — it is only 50 small pages — and will want to return to think more carefully about some of the interpretations and arguments.

(A minor but welcome point: unlike some earlier Elements, this looks to have been properly LaTeXed so the symbols aren’t garbled.)

This little book should be readily available if your library has a suitable Cambridge Core subscription. And until the end of today the CUP version is freely available for download here. But there is also (as pointed out in a comment below) a version which looks to be more or less identical on the arXiv here.

Annals of Mathematics and Philosophy

This may by now be old news, but there is a recently started journal Annals of Mathematics and Philosophy. It aims to encourage dialogue between contemporary mathematics and philosophy. Issues are (it seems) to be freely available online. The first issue is out at the journal’s website here.

I confess I didn’t find this issue an encouraging start, being mostly low-level stuff that wouldn’t have got near publication in e.g. Philosophia Mathematica. Except that there is a substantial, characteristically lucid, and very interesting piece by Timothy Gowers, “What makes mathematicians believe unproved mathematical statements?” which I much enjoyed reading.

Does Mathematics need a Philosophy?

At a meeting some years ago of the Trinity Mathematical Society, Imre Leader and Thomas Forster gave introductory talks on “Does Mathematics need a Philosophy?” to a startlingly large audience, before a question-and-answer session. The topic is a very big one, and the talks were very short.  After the event, I wrote up a few after-thoughts (primarily for maths students such as the members of TMS, though others might be interested …). I had occasion to revisit my remarks just recently. Rough and ready though they were, I’m happy enough to stand by their broad message, so here they are again, just slightly tidied up for new readers! …

Philosophy of mathematics — a reading list

A few people recently have quite independently asked me to recommend some introductory reading on the philosophy of mathematics. I have in fact previously posted here a short list in the ‘Five Books’ style. But here’s a more expansive draft list of suggestions.

Let’s begin with an entry-level book first published twenty years ago but not yet superseded or really improved on:

  1. Stewart Shapiro, Thinking About Mathematics (OUP, 2000). After introductory chapters setting out some key problems and sketching some history, there is a group of chapters on what Shapiro calls ‘The  Big Three’, meaning the three programmatic ideas that shaped so much philosophical thinking about mathematics for the first half of the twentieth century — i.e. varieties of logicism, formalism, and intuitionism. Then there follows a group of chapters on ‘The Contemporary Scene’, on varieties of realism, fictionalism, and structuralism. This might be said to be a rather conservative menu — but then I think this is just what is needed for a very first introduction to the area, and Shapiro writes with very admirable clarity.

By comparison, Mark Colyvan’s An Introduction to the Philosophy of Mathematics (CUP, 2012) is far too rushed to be useful. And I would say much the same of Øystein Linnebo’s Philosophy of Mathematics (Princeton UP, 2017). David Bostock’s Philosophy of Mathematics: An Introduction (Wiley-Blackwell, 2009) is more accessible, but — apart from a chapter on predicativism — covers similar ground to the earlier parts of Shapiro’s book, but has little about more recent debates.

A second entry-level book, narrower in focus, that can also be warmly recommended is

  1. Marcus Giaquinto, The Search for Certainty (OUP, 2002). Modern philosophy of mathematics is still in part shaped by debates starting well over a century ago, springing from the work of Frege and Russell, from Hilbert’s alternative response to the  “crisis in foundations”, and from the impact of Gödel’s work on the logicist and Hibertian programmes. Giaquinto explores this with enviable clarity: this is really exemplary exposition and critical assessment. A terrific book.

Then, before moving on, I have to mention that most accessible of modern classics:

  1. Imre Lakatos, Proofs and Refutations (originally published in 1963-64, and then in expanded book form by CUP, 1976). Textbooks tend to present developed chunks of mathematics in a take-it-or-leave-it spirit, the current polished surface hiding away the earlier rough versions, the conceptual developments, the false starts. Proofs and Refutations makes for a wonderful counterbalance. A classic exploration in dialogue form of the way that mathematical concepts are refined, and mathematical knowledge grows. We may wonder how far the morals that Lakatos draws can be generalized; but this remains a fascinating read (I’ve not known a good student who didn’t enjoy it).

Mathematical Structuralism, Essay 3

The third essay in The Pre-history of Mathematical Structuralism is by José Ferreirós and Erich H. Reck, on ‘Dedekind’s Mathematical Structuralism: From Galois Theory to Numbers, Sets, and Functions’. The title promises something rather more exciting than we get. Why? Let’s work backwards by quoting from (some of) their concluding summary. They write:

From … Dirichlet and Riemann, Dedekind inherited a conceptual way of doing mathematics. This involves replacing complicated calculations by more transparent deductions from basic concepts.

“Replacing”? That seems rather misleading to me: isn’t it more a matter of a change of focus to new, more abstract, general questions, rather than a replacement of methods for tackling the old questions that required “complicated calculations”?

Both Dedekind’s mainstream work in mathematics, such as his celebrated ideal theory, and his more foundational writings reflect that influence. Thus, he distilled out as central the concepts of group, field, continuity, infinity, and simple infinity. A related and constant aspect in his work is the attempt to characterize whole systems of objects through global properties.

From early on, Dedekind also pursued the program of the arithmetization of analysis … . A decisive triumph came in 1858, with Dedekind’s reductive treatment of the real numbers. From the 1870s on, he added a reduction of the natural numbers to a general theory of sets and mappings. This led to an early form of logicism, since he conceived of set theory as a central part of logic … [H]is attempt to execute a logicist program
had a decisive effect on the rise of axiomatic set theory in the 20th century.

Its conceptualist and set-theoretic aspects are central ingredients in Dedekind’s mathematical structuralism. But we emphasized another characteristic aspect that goes beyond both. This is the method of studying systems or structures with respect to their interrelations with other kinds of structures, and in particular, corresponding morphisms.

But again this seems slightly misleading to me. It isn’t that Dedekind and contemporaries “distilled out” the concepts of a group and of a field, for example, for the fun of it and then — as an optional further move — considered interrelations between structures. Weren’t the abstracting moves and their interrelations and their applications by relating them back to more concrete structures all tied together as a package from the start, as for example in the very case that Ferreirós and Reck consider:

A historically significant example, particularly for Dedekind, was Galois theory. As reconceived by him, in Galois theory we associate equations with certain field extensions, and we then study how to obtain those extensions in terms of the associated Galois group (introduced as a group of morphisms from the field to itself, i.e., automorphisms). …

Just a word more about this in a moment. Finally,

As we saw, Dedekind connected his mathematical or methodological structuralism with a structuralist conception of mathematical objects, i.e., a form of philosophical structuralism.

Now, in bald outline, all this is familiar enough (though of course in part because of the earlier work of writers like Reck and Ferreirós in helping to highlight for philosophers Dedekind’s importance in the development of mathematics). So does this essay add much to the familiar picture?

Not really, it seems to me. For it proceeds at too armwavingly general a level of description. There’s too much of the mathematical equivalent of name-dropping: ideas and results are mentioned, but with not enough content given to be usefully instructive.

Take the nice case of Galois theory again. If you are familiar with modern basic Galois theory from one of the standard textbooks like Ian Stewart’s or D.J.H. Garling’s, you won’t pick up much idea of just how Dedekind’s work related to the modern conception (well, I didn’t anyway). And if you aren’t already familiar with Galois theory, then you won’t really understand anything more about it from Reck and Ferreirós few paragraphs, other than it is something to do tackling questions about the roots of equations by using more abstract results about groups and fields — which isn’t exactly very helpful. By my lights, it would have been much more illuminating if our authors had devoted the whole essay to Dedekind’s work on Galois theory as a case study of what can be achieved by the “structuralist” turn.

Mathematical Structuralism, Essays 1 & 2

Here are some quick comments on the first two of the essays in The Pre-history of Mathematical Structuralism. Let me start, though, with a remark about the angle I’m coming from.

I have been wondering about getting back to work on my stalled project, Category Theory: A Gentle Introduction. And what I’d like to do is write some short preliminary chapters around and about the familiar pre-categorial idea that (lots of) mathematics is about abstract structures and their inter-relationships. What does that idea really come to? I sense that there is some disconnect between, on the one hand, what this amounts to in the nitty-gritty of ordinary mathematical practice and, on the other hand, some of the arm-wavingly generalities of philosophers with axes to grind. I’m interested then in seeing discussions of varieties of structuralism which are, perhaps, more grounded in the varieties of mathematical practice. Hence I hope that looking at some of the historical developments in maths that have led to where we are might prove to be illuminating. Let’s read on …


The book opens with an introductory essay by the editors, Erich Reck and Georg Schiemer. The first part of this essay is in effect a summary version of the same authors’ useful entry on ‘Structuralism in the Philosophy of Mathematics’ in the Stanford Encyclopedia of Philosophy. In this book, then, they touch again more briskly on the familiar varieties of structuralism as a philosophical position; perhaps too briskly? — a reader relatively new to the debates will find their longer SEP version significantly more helpful. But they also press a distinction between structuralism as a philosophical story (particularly a story about the ontology of mathematics) and what they call methodological or mathematical structuralism — a term which “is meant to capture a distinctive way of doing mathematics”. “Roughly,” we are told, “it consists of doing mathematics by ‘studying abstract structures’”, something a mathematician can do without explicitly considering metaphysical questions about the nature of stuctures. And then “One main goal of the present collection of essays is to clarify the origins, and with it the nature, of methodological/mathematical structuralism up to the rise of category theory”.

So what, in a little more detail, do Reck and Schiemer count as being involved in methodological structuralism? (1) The use of concepts like ‘group,’ ‘field,’ ‘3-dimensional Euclidean space’, which (2) are characterized axiomatically and “typically specify global or ‘structural’ properties”, and where (3) we importantly study systems falling under these concepts by relating them to each other by iso/homomorphisms, and (4) by considering ‘invariants’, and where (5) “there is the novel practice of ‘identifying’ isomorphic systems”.

[T]his can all be seen as culminating in the view that what really matters in mathematics is the ‘structure’ captured axiomatically, on the one hand, and preserved under relevant morphisms, on the other hand.

But having said this, Reck and Schiemer allow that not all of features (1) to (5) are required for what they call methodological structuralism: they say that we are dealing with “family resemblances” between cases.

Plainly, themes (1) to (4) did indeed emerge in nineteenth century mathematics. I rather discount (5), though, as the supposed novel practice of identifying isomorphic systems seems typically to amount to no more than ignoring specific differences between isomorphic Xs for certain purposes when doing the theory of Xs. For example, while focusing on the pure group-theoretic properties of some structures, we do ignore the non-group-theoretic differences between isomorphic groups. But of course, when we start to apply the group theory, e.g. in Galois theory, such differences become salient again; it is crucial, for example, that that certain groups are permutations of roots of equations, a property not shared with isomorphic groups.

There are various stripes of mathematical problem. To stick to nineteenth-century ones, at one end of the spectrum, there are very specific problems. For instance, we might be interested in finding a closed form solution for some integral: we tackle our problem using a rather specific bag of tricks we’ve developed (like substituting variables, integrating by parts, etc.). Again, we might be interested in finding an approximate solution to a specific application of the Navier-Stokes equation, and there’s another bag of tricks to develop and use. At the other end of the spectrum, there are considerably more abstract problems — is there a global solution in radicals for quintic equations? is the parallel postulate independent of the other Euclidean axioms? if a and k are co-prime, must there be an infinite number of primes in the arithmetic progression a, a + k, a + 2k, a + 3k, …? Not surprisingly, such abstract general questions often need abstract general ideas to answer them — and these ideas, being sufficiently abstract, will typically have other applications too, so may indeed well not need to be tied to a particular subject matter (i.e. they will be, in a sense, structural ideas).

So, at least those mathematicians interested in sufficiently general questions (far from all of all mathematicians, then) will naturally find themselves deploying abstract concepts as arm-wavingly gestured at by Reck and Schiemer’s (1) to (4). And the successes in doing this were impressive. But were (1) to (4) pursued with enough unity and clarity of purpose to constitute a methodological “ism”? We shall see.


The second part of Reck and Schiemer’s editorial introduction gives thumbnail sketches of the remaining fifteen essays in the book. So I won’t delay over that here, but will move on to the first contribution, Paola Cantù’s ‘Grassmann’s Concept Structuralism’.

I have to report, however, that I really got very little from this. Not knowing Grassmann’s work, I found Cantù’s sketched exposition of some his ideas quite opaque. So I imagine this essay will only be of interest to those who already know something of her topic; it certainly didn’t engage this reader. So I’ll skip over it.

The Pre-history of Mathematical Structuralism

There is a new collection of essays edited by Erich H. Reck and Georg Schiemer being published by OUP, officially next month. Here’s the book description: “Since the 1960s, there has been a vigorous and ongoing debate about structuralism in English-speaking philosophy of mathematics. But structuralist ideas and methods go back further in time; that is, there is a rich prehistory to this debate, also in the German- and French-speaking literature. In the present collection of essays, this prehistory is explored in a twofold way: by reconsidering various mathematicians in the 19th and early 20th centuries (Grassmann, Dedekind, Pasch, Klein, Hilbert, Noether, Bourbaki, and Mac Lane) who contributed to structuralism in a methodological sense; and by re-examining a range of philosophical reflections on such contributions during the same period (also by Peirce, Poincaré, Russell, Cassirer, Bernays, Carnap, and Quine), which led to suggestions about logical, epistemological, and metaphysical aspects that remain relevant today. Overall, the collection makes evident that structuralism has deep roots in the history of modern mathematics, that mathematical and philosophical views about it have often been closely intertwined, and that the range of philosophical options available in this context is significantly richer than a mere focus on current debates may make one believe.”

I’m going to be really interested to read probably most of the essays in the book. So I plan to start blogging about them here. (I certainly don’t promise to have anything especially illuminating or insightful to say: but writing blog-posts makes me read a bit more carefully and helps me fix ideas.) And now the good news: you don’t have to fork out £64 in order to follow along! For this is being published as an open access title. The collection is already free to read at Oxford Scholarship Online and is offered as a free PDF download from OUP. Which is excellent!

Hellman and Shapiro on Mathematical Structuralism

CUP are publishing a series of short books (about 100 pages) under the title Cambridge Elements in the Philosophy of Mathematics. The blurb says that the series “provides an extensive overview of the philosophy of mathematics in its many and varied forms. Distinguished authors will provide an up-to-date summary of the results of current research in their fields and give their own take on what they believe are the most significant debates influencing research, drawing original conclusions.” Which sounds ambitious. So far, just two Elements have been published, Mathematical Stucturalism (2018) by Geoffrey Hellman and Stewart Shapiro, and A Concise History of Mathematics for Philosophers (2019) by John Stillwell.

The hyper-active Stillwell has already written a well-known and accessible Mathematics and Its History (3rd edn, 2010) as well as a number of other non-specialist books (alongside his singificantly better hard-core maths texts). It seems a bit of a failure of imagination for the series editors to ask him to write another history; and to me, the result looks pretty unexciting.

Again, getting Hellman and Shapiro to write on structuralism is hardly adventurous! But I have now read their book. It’s not clear, though, who the intended readership really is. The series blurb — “up-to-date summary”, “current research”, “original conclusions” — might suggest a book aimed at e.g. graduate students. But little of the book approaches that sort of level. (One odd feature: we aren’t told who wrote what, even though some of the passages are in the first person and are characteristic of just one of the authors.)

There is a short Introduction giving initial characterizations of some forms of mathematical structuralism, and setting out some questions that we’d want any structuralism to address. Chapter 2 gives Historical Background. This is over twice the length of the next longest chapter and it is nicely done, with a good selection of quotations;   it will provide very helpful reading for undergraduates. Chapter 3 is then on Set-Theoretic Structuralism, the view that “structures are isomorphism types (or representatives thereof) within the set-theoretic hierarchy”.

Of course, this view won’t in itself give us structuralism for mathematics across the board: the status of set theory itself is left up for grabs. Indeed, on the obvious story “the foundational theory [set theory] is an exception to the theme of structuralism. But, the argument continues, every other branch of mathematics is to be understood in eliminative structuralist terms.” The authors don’t do much, however, to explain why set theory should get this foundational role, or illuminate how this reductionist story squares with the familiar fact that most working mathematicians can get by in cheerful ignorance of set theory, etc.  A student could be better pointed to e.g. some of Maddy’s work for a more nuanced account of the role of set theory in mathematics.

Chapter 4 is on Category Theory as a Framework for Mathematical Structuralism. But who is this short chapter addressed to? The philosophy of maths student (at any level) should have some initial grasp on what set theory is about. But most won’t have much clue about what Category Theory might be, and these brisk arm-waving pages are unlikely to help at all. On the other hand, the few who are in the game will be familiar with the usual suggestions from Awodey and others which are gestured to here: for them, there will be no news, certainly no “original conclusions”.

Chapter 5 and 6 discusses Structures as Sui Generis Structuralism (Shapiro-style) and The Modal-Structural Perspective (Hellman-style). Both authors have been presenting their respective lines for well over twenty years, and so we are not going to expect any exciting new insights, criticisms or developments — and in under a dozen printed pages for each chapter, we don’t get them.

The final Chapter 7 is on Modal Set-Theoretic Structuralism, in particular as developed by Øystein Linnebo. This topic is at least relatively novel and is interesting; but since the authors are nowhere near as good at explaining Linnebo’s approach as that particularly lucid author is, the sufficiently equipped student reader would do a lot better to go to the original paper “The Potential Hierarchy of Sets,” Review of Symbolic Logic (2013).

Which all sounds rather carping. But overall I found this an extremely disappointing book.

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