Peter Smith's Blog, page 54

Interest in Gödel? Just a bit!

Yesterday, just after I posted here about the availability now of my Gödel book as a free PDF download, I tweeted the same news, and that tweet was retweeted dozens of times.

After 24 hours, the book has been downloaded … over 47 thousand times.

Which is a bit crazy.

The Daughter immediately had the explanation. Someone had posted a link on Hacker News, giving just the title with a direct link (but not explaining that An Introduction to Gödel’s Theorems is a long book!). And that link stayed on the front page of Hacker News for quite a while. No doubt most of those who clicked through, not reading the comments that pointed out that it is a book, and perhaps thinking they were going to download a quick read, were surprised by what they found!

But still, I guess that it is really rather good to find that so many were tempted to follow up a link to something on Gödel’s theorems. The topic (rightly!) remains of considerable interest, even fascination. Which certainly encourages me to crack on and finish reworking the cut-down version of the book, the lecture notes Gödel Without Tears. And who knows, a few people might even read the book!

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Gödel news!

As I said in my previous post, I’ve been working on a revised version of Gödel Without Tears. Predictably, I’ve not been able to resist tinkering more than I had planned to do. So it will take a few weeks more to finish the job. But I’m not complaining: it’s an enjoyable enough project!

GWT is a sort of cut-down introduction to some of the themes of my  Gödel book. Quite unexpectedly, I have now acquired the rights to that book. I think the friendly thing to do is to immediately make the second edition of  An Introduction to Gödel’s Theorems available as a free PDF downloadable from Logic Matters, taking the opportunity to correct three dozen minor misprints. This way, students and other readers anywhere can get free access. So here it is! Spread the word …

(I may in due course also make this corrected version of the book available as an inexpensive print-on-demand book via Amazon, for those who want a physical copy. But I doubt that there would be a big demand for that, so one step at a time.)

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Not so structural … so back to Gödel instead!

A week with many domestic distractions; so until today I’ve not been able to settle for any length of time to such logical matters as reading on in the Pre-History of Mathematical Structuralism.

And now returning to that book, I do confess I find that my interest is waning. Having read (increasingly speedily) the next four essays, I’m not particularly enthralled. When commenting on the piece by Ferreirós and Reck on Dedekind, I worried that (at least  by my lights) it was done with too broad a brush to paint in enough mathematically satisfying detail. I’ve had rather similar reactions to some of the other essays.  I may well yet return to jot down some notes on more of the essays in this book. But for the moment, I’m finding myself much more enthusiastic about a quite different project — a revised edition of Gödel Without (Too Many) Tears, which I started working on a little while back.

I’m making some minor revisions/clarifications of the content, but also changing the format as I go along. And I’ve been encouraged to make it available as a print-on-demand book. Which I will probably do;  so watch this space for an announcement, perhaps in a month’s time, of a 150 page book. If I get the hang of this self-publishing malarkey. What fun!

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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.

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From a small corner of Cambridge, 9

What news from the Rialto? Who knows? We are still avoiding the Rialto. As indeed are most of our friends. The hopelessly wavering and confused messaging from the  government, its perceived incompetence, has engendered among our generation very little confidence in the safety of re-embracing anything like the old ‘normal’. So, for example, where we would once have wandered into the city centre across the common perhaps every other day, we now go out to various places in the country to walk. And indeed we wonder why we (rather unthinkingly) had stuck to our old more urban habits for so long; we are unlikely to ever re-adopt them. Anecdotally, lockdown has prompted similar ‘resets’ for many: small changes for individuals that are, however, going to sum to very significant economic effects for the city (and not just the coffee shops and restaurants).

The local National Trust properties continue to be a delight. No doubt because there are fewer gardening volunteers, the borders seem a little less well-ordered, the meadows more unkempt, but all the lovelier for that. We have never seen so very many butterflies there.

The latest issue of the London Review of Books arrives. Of course, I could already have read it on the iPad. But there still is something very pleasing about sitting over coffee with the elegantly produced paper edition (even if this is at home rather than out at a café). Yes, the LRB can be idiosyncratic and provoking (though mostly in a good way). I will usually dive into most of it. Sometimes — though, trying to be realistic, not as often as I’d like — I’m even spurred on to buy one of the books reviewed. A piece in the current issue entices me to send off for a copy of the poet A. E. Stalling’s recent collection Like. I confess I had not heard of her before.

In the same post as the LRB, a copy of the newly paperbacked Fleishman is in Trouble by Taffy Brodesser-Akner. Praise was heaped when first published. But Mrs Logic Matters, fifty-something pages in, is not at all impressed; she rarely gives up on a book, but has already passed it on to my pile.

I’ve just been looking for our copy of Tony Judt’s wonderful The Memory Chalet. Where can it possibly have gone? It surely can’t accidentally have been given away in one of those many piles of books that have gone to Oxfam in the last few years. I find losing or misplacing a book that I’ve been emotionally affected by to be strangely dismaying, even if the copy can be readily replaced. We’ve both searched high and low, and still no sign.

Of recently online concerts, we found the Pavel Haas Quartet’s performance from Litomyšl particularly moving. There is of course something special about being present, there in the audience for a live concert. But compare having to traipse up to London and back, perhaps in bleak mid-winter, to get perhaps a distant view with slightly distracting neighbours, perhaps in any case on a day you are not particularly feeling like going to a concert, compare all that with viewing a well-produced streamed video of the live concert at home, watching and re-watching when the mood takes. Put it this way: if (say) Wigmore Hall starts offering paid access to streamed videos — or free to members, perhaps! — then the prospect of going to far fewer live concerts is not overall such a bad one.

But who knows when live concerts will resume to be streamed! Looking at the blank future diaries of (say) the Pavel Haas, and the Doric, Belcea and Chiaroscuro quartets, and of Cuarteto Casals, is depressing indeed. Heaven knows how such great musicians are faring in these times.

The third instalment of Hilary Mantel’s Tudor trilogy remains on the coffee table. But it’s  time to start it, as we have now both read Wolf Hall and Bring Up The Bodies again. And we have independently both been very struck by how much we’ve got out rereading them — appreciating them, if anything, even more the second time around. The writing really is stunningly good. So that’s this week’s (very unoriginal) semi-lockdown reading recommendation: if you have been wondering about rereading those first two books of Hilary Mantel’s trilogy, don’t hesitate. You’ll really enjoy it!

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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.

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Exercises!

Thirty-six of the chapters in IFL2 have end-of-chapter exercises. Thirty-one of these  sets of exercises now have on-line answers, often with quite detailed discussion, available here. That includes all the chapters up to and including the chapters on QL proofs.

The question sets are also available online, in a form that means that most of them can be used independently of the book. Indeed from the download stats pre-publication, they already seemed to being so used, which is good to see.

Something else I recently discovered from those stats is that in fact the sets of exercises and worked answers I started for the Gödel book (which I stopped developing because I thought there was no interest in them) are in fact being downloaded quite often. So once all the IFL2 answers are all in place, I might yet turn back to putting together more Gödel-related exercises, which could be quite a fun project at least for the later parts of the book.

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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!

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The Prehistory 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!

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Update to the TYL Guide

I have uploaded a half-year “maintenance upgrade” to the Teach Yourself Logic Guide. There are just a few additional entries, a few changes in existing entries, minor re-writing here and there, and then some re-arrangement of material to make it easier for the two different readerships — philosophers and mathematicians — to navigate the beginning of the Guide.

I really must give this project some more love and attention over the coming months (it’s been rather neglected as I have been concentrating on IFL2). In the first half of this year the Guide was downloaded over 15K times from this site, and looked at another 5K times on my academia page (I do still find those stats rather startling). So obviously — patchy and half-baked though it is in many places — there remains some real need for such a Guide. So I guess I should do my best to make it as good as I can. And it is fun enough to work on when in the right frame of mind.

Any suggestions for improvement are of course always welcome!  I surely must have missed some recent texts which might be worth looking at. Though I do suspect that the current culture of “research assessments” — where writing such books (as opposed to papers read by eleven people) can count for so very little — puts many people off devoting their energies to writing introductory or mid-level texts.

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