What’s so great about sets?

Here’s something I wrote a while back to answer a question on math.stackexchange about why sets and set theory should (or shouldn’t) be thought to have a special place in maths. Following a link on a related matter I found myself directed back to this piece of my own: I think I still quite like it. So here it is again …

There is a long, fascinating, and often-told story about the nineteenth century project for the rigorization of analysis, and about the re-construction of classical mathematics in terms of natural numbers and sets of natural numbers and sets-of-sets of natural numbers, etc. etc. And if we are feeling particularly austere we can even re-construct the naturals in a pure set theory which lacks urelements, so everything gets implemented in pure set theory. There are lots of good recountings of the story — here’s a short one with lots of pointers to more: http://plato.stanford.edu/entries/settheory-early/

I mention the history because it explains why set theory has long been thought to have a special “foundational” place in the architecture of mathematics. But does it really? Can category theory (for example) provide an alternative foundation? And anyway, now we’ve got over our wobbles from about a hundred-and-twenty years ago, when some thought classical mathematics was threatened by paradoxes of the infinite, does mathematics in any sense need universal “foundations”?

Big questions indeed, and the general question about some supposed need for “foundations” is not wanted I wanted to comment on here. But here’s one line of thought that I’ve encountered from mathematicians, not so often mentioned by philosophers, which perhaps underlies some of the continuing nods to the special place of set theory.

Suppose working on Banach spaces, or algebraic topology, or whatever, I conjecture all widgets are wombats. And then the bright young grad students try to prove or disprove Smith’s Conjecture.

Next week, Jane turns up to class claiming to have refuted the conjecture by finding a structure in which there is a widget which isn’t a wombat.

Well, what are the rules of the game here? What kit is Jane allowed to use in her structure building? To give her a best shot at refuting the conjecture, she perhaps ideally wants some kind of all-purpose kit that only minimally constrains what she can build. She wants the mathematical equivalent of a Lego kit where you can pretty much attach anything onto anything, rather than the equivalent of a building kit you can only make toy houses from, or one you can only make toy cars from. (Perhaps Smith’s Conjecture still works fine for, so to speak, houses and cars.)

What the standard sets of the iterative hierarchy seems to provide is just such an all-purpose mathematical Lego kit. We start with some things (or if you like, with nothing at all), and then we are allowed to put them together however you like into new things, and then we are allowed to put what we’ve got together however we like ad libitum, and to keep on going as long as we like. Precisely because the rules for building new sets allow maximising at every step (the idea is at each level we are allowed every possible new combo, and there is no limit to the levels), we really do get an all-purpose structure-building kit. And having such a mathematical Lego kit is just what Jane ideally needs if she is to have untrammelled free rein in coming up with her widget which isn’t a wombat.

Or so the story goes, in outline …

Posted in Logic, Phil. of maths | 5 Comments

Reasons to be cheerful

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New CDs are announced from some musicians whom I admire immensely. For a start, later this month, David Fray is releasing a disk of Schubert’s great G major piano sonata, the four-hand Fantasia D940 and “Lebenssturme” D947 (with Jacques Rouvier). As I’ve said here before, I think Fray’s previous disk of the Moments Musicaux and Impromptus Op.90 is simply mesmerising, so I can’t wait to hear more Schubert from him.

imageNext, Alina Ibragimova recorded the Bach Violin Concertos with Jonathan Cohen and his Arcangelo ensemble back in August, and the disk will come out this year. Ibragimova’s earlier disks of the Bach sonatas and partitas are just wonderful (as one review put it, “Her playing proclaims authority… In slow movements, [her] understated poise asserts an intensity all her own. The famous D minor Chaconne here becomes an absorbing saga unto itself.”). This new disk too should surely be equally compelling.

rachel7Rachel Podger has been playing Vivaldi’s L’Estro Armonico with her Brecon Baroque ensemble in concerts over the last year or so. Hearing them perform a selection of these concerti in King’s College Chapel was one of our concert highlights of 2014. As I said here, “they played with verve and enjoyment, playfulness and charm, and a lot of light and shade. Technically brilliant too. The performances made the case wonderfully well for Rachel Podger’s description of these works, in her lovely talk to the audience after the first concerto, as intriguingly complex and rule-bending.” A recording is due out this year.

pavel haasAnd, then there’s the Pavel Haas Quartet — who were from the beginning a really fine quartet, but who have undergone that alchemical transformation into a truly great quartet. They have now recorded the Smetena Quartets. Anyone who has heard the PHQ’s live performances of these quartets will know that the disk is destined to be an instant classic.

(My review of their Schubert recording from the previous year was still the most-viewed blog-post here during 2014. It would be really rather good to think that I’ve encouraged some readers to listen to the PHQ.)

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Category Theory 2015

My main logical resolution for 2015 is to get to know quite a bit more category theory. Well, it’s fun, I find it aesthetically very appealing, there are some super-smart category theory people here in Cambridge — and there seem to be enough lurking conceptual issues to engage the philosophical bits of my brain, though I want to know (a lot) more category theory before sounding off about them.

I’ve therefore now started a new section of Logic Matters on categories  where I’ll be posting various stuff — starting with my slowly-expanding Notes on Category Theory, and eventually some book notes, links to on-line resources on Category Theory, and so on. Enjoy!

(Other New Year’s resolutions? One is to stop wasting time and endangering my blood pressure reading the comments on various Well Known Philosophy Blogs, comments which seem too often to be getting increasingly bonkers, unpleasant and — let’s hope — unrepresentative. Feeling better already …)

Posted in Category theory | 4 Comments

Teach Yourself Logic 2015

Back to logic after the festivities. Easing myself in very gently, here’s

A new version of the Teach Yourself Logic Guide.

This is largely a ‘maintenance’ release of the study guide to the literature for upper undergrads/grads wanting to teach themselves some mathematical logic (or to supplement their courses). But it is slightly re-organized, has a couple of added recommendations from 2014 publications, and is actually a few pages shorter.

Last year, the Very Short Teach Yourself Logic Guide single webpage was visited 150K times (with a big spike in visitors due to an honourable mention on Reddit — I suspect most of those visitors, however, were looking for something much more elementary). But even versions of the full TYL Guide  were downloaded over 5K times. Since you have to click a link in an explanatory blog-post (like this one) or go to the TYL webpage to get the full version,  I guess that most of these downloads are purposeful, indicating that there is indeed a real need for something like the Guide. So although it isn’t my top priority, I’ll keep updating it when the spirit moves me, or when I get some good suggestions/helpful comments. Many thanks to everyone who has provided input over the last few years.

Posted in TYL | 1 Comment

The last dance

It isn’t all High Culture chez Logic Matters. Oh no. Perish the thought. For a start, from September to December we are devotees of Strictly. 

That’s Strictly Come Dancing (the original, BBC, version of Dancing with the Stars, Ballando con Le Stelle, and over forty other versions). So it is all glamour and glitter, sequins and sexy outfits, fake tans and gallons of hair products, tears and tantrums. And that’s just the guys.

Each season we start watching with the same amused detachment. We tell  ourselves that this year we won’t get hooked, this year the quarter-celebrities (most of whom we’ve never heard of) are an uninteresting/unattractive bunch, this year the silliness of the whole palaver is just too much …

And yet …

Like millions others, we find ourselves tuning in every Saturday. And as the no-hopers and the joke participants are voted off the show, we get more enthused. We start watching It Takes Two  — the admirably warm and amusing weekday programmes interviewing participants, explaining the finer points of choreography, and generally having fun. Which can reveal that an impossibly glamorous pro dancer has a very sharp self-deprecating wit and is endearingly happy to send herself up, while a seemingly equally glamorous member of a boy band is self-doubting and charming. And that this soap actor is as two-dimensional as his character, but that that footballer’s wife is plainly a sweet girl who is delightfully surprised to find that she can dance. You get engaged with the contestants (and indeed with some of the pro dancers), with the ‘journeys’, with the increasingly terrific dances. Fellow devotees will know how it goes …

… And if you don’t, well, here for your holiday delight is the very final dance of the series. Down to the last three couples, the final contestants can choose their favourite dance to perform again. Here Simon Webbe and Kristina Rihanoff reprise their Argentine tango. They didn’t win Strictly 2014. But this was the dance of the series, from a man who three months before seemingly had two left feet and zero confidence. Who has come so far. And the result is really rather moving … Enjoy!

And Happy New Year!

Posted in This and that | 2 Comments

Philosophical remnants/Notes on Category Theory v.3

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So, over the last months, quite a few more large boxes of books have gone to Oxfam. I have kept almost all my logic books. But in three years I must have given away some three quarters of my philosophy books. Very largely unmissed, if I am honest. Those works of the Great Dead Philosophers are no longer reproachfully waiting to be properly read. The more ephemeral books of the last forty years (witnessing passing fashions and fads) are largely disposed of. I’m never going to get excited again e.g. about general epistemology (too arid) or about foundations of physics (too hard), so all those texts can go too. I’m left with Frege, Russell, Wittgenstein, Quine; an amount of philosophical logic and philosophy of maths and related things; and an eclectic mix of unneeded books that somehow I just couldn’t quite bring myself to get rid of (yet). I’m not sure why among the philosophical remnants, Feyerabend for example stays and Fodor goes when I’ll never read either seriously again: but such are the vagaries of sentimental attachment.

But if I’m still rather attached to some authors and topics and themes and approaches, I’m not quite so sure about ‘philosophy’, the institution. Still, that’s another story. And anyway, those lucky enough to have philosophy jobs in these hard times certainly don’t need ancients grouching from the comfort of retirement: they have problems enough. True, judging from what’s been churning around on various Well Known Blogs over the last year, some might perhaps do well to recall Philip Roth’s wise words about  that “treacherous … pleasure: the ecstasy of sanctimony”. But being the season of goodwill, I’ll say no more!

Instead, for your end-of-year delight, here’s an updated version of the Notes on Category Theory (still very partial though now 74 pp.). Newly added: a section on comma categories to Ch.4, a short chapter between the old Chs 7 and 8, and a chapter on representable functors. 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. An aside on Cayley’s Theorem
  9. The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
  10. Representables (definitions, examples, universal elements, the category of elements).

Download the new version of the notes here

Posted in Academic life, Books, Category theory | Leave a comment

A Christmas card

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Giorgione (c. 1477/8–1510) Adoration of the Shepherds National Gallery of Art, Washington, D.C.

All good wishes for a happy and peaceful Christmas

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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

Posted in Category theory, Logic | 1 Comment

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?

Posted in Books, Logic | 13 Comments

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.)

Posted in Category theory | 1 Comment