Peter Smith's Blog, page 104
January 4, 2015
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 …)
January 1, 2015
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 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.
December 31, 2014
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 Rhianoff 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!
December 28, 2014
Philosophical remnants/Notes on Category Theory v.3
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
Categories defined
Duality, kinds of arrows (epics, monics, isomorphisms …)
Functors
More about functors and categories (and the category of categories!)
Natural transformations (with rather more than usual on the motivation)
Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
An aside on Cayley’s Theorem
The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
Representables (definitions, examples, universal elements, the category of elements).
December 23, 2014
A Christmas card
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
December 8, 2014
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
Categories defined
Duality, kinds of arrows (epics, monics, isomorphisms …)
Functors
More about functors and categories (and the category of categories!)
Natural transformations (with rather more than usual on the motivation)
Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
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.
Notes on Category Theory, instalment #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
Categories defined
Duality, kinds of arrows (epics, monics, isomorphisms …)
Functors
More about functors and categories (and the category of categories!)
Natural transformations (with rather more than usual on the motivation)
Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
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.
December 1, 2014
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 and new(ish), and the returned-to-classic that I got lost in, and wished hadn’t finished (even if 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’s Basic Category Theory (CUP). This is not for everyone who visits Logic Matters, perhaps, for it is a mathematics text, and it won’t tell you about the more 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. So that‘s my logic book of the year for 2014.
What are your logic/phil maths book highlights of the year?
November 18, 2014
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
Categories defined
Duality, kinds of arrows (epics, monics, isomorphisms …)
Functors
More about functors and categories (and the category of categories!)
Natural transformations (with more than usual on the motivation)
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.)
Notes on Category Theory, instalment #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
Categories defined
Duality, kinds of arrows (epics, monics, isomorphisms …)
Functors
More about functors and categories (and the category of categories!)
Natural transformations (with more than usual on the motivation)
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.)
