Peter Smith's Blog, page 93

Encore #15: Maddy, realism and arealism

Continuing the rather random selection of reposts, as the blog’s tenth birthday approaches, here’s a post from 2011 on a fascinating book by Maddy: 

Thin Realism, Arealism, and other Big Ideas (May 30, 2011)

Penelope Maddy’s recent Defending the Axioms is my sort of book. It is short (150 pages), beautifully clearly written (if there are obscurities, they are in the philosophy, not the prose), and I’m in fact rather sympathetic to her overall approach (I share her doubts about ‘First Philosophy’, I share in particular her doubts about the force of supposedly a priori arguments for nominalism and against abstracta, and I rather like her post-Quinean conception of the task for the ‘Second Philosopher’).

I’m not sure, however, that there is much in the book that will be a major surprise to the reader of Maddy’s previous two books. Still, the brevity and the tight focus ‘On the Philosophical Foundations of Set Theory’, to quote the subtitle, might help to make her ideas available to new readers daunted by the length and sweep of Second Philosophy. How well does Maddy succeed in doing that?

I’ve a lot of questions. The book is dedicated to ‘The Cabal’ of Californian set theorists. And — talking things over with Luca Incurvati — one interesting issue that came up is whether Maddy’s vision of the working methods of set theorists isn’t rather skewed by a restricted diet of examples from her local practitioners. But here I’m going muse about the central metaphysical theme in the book: does the book succeed in selling Maddy’s story about that?

Maddy discusses two positions that she suggests are prima facie open to those with no first-philosophical axes to grind. One she dubs ‘Thin Realism’. Starting from the mathematized science of the early nineteenth century, mathematical ideas came to be pursued increasingly for their own sake, leading inter alia to the great unifying endeavour which is set theory. This is hugely productive of ideas and results and is mathematically deep. If you are not already in the grip of some extra-mathematical prejudices, what’s not to like? So as good second philosophers, who don’t pretend to have special extra-mathematical reasons to criticize successful mathematical practice,  we should say that there are sets, and that set theory tells us about them — and also that there is no more to be said about them than what set theory says (other, perhaps, than negative things such as that they aren’t already-familiar bits of the causal, spatiotemporal, order). A gung-ho ‘Robust Realist’ is tempted to say more: she is gripped by an extra-mathematical picture of the sets as genuinely ‘out there’ quite independently of the natural world, forming a parallel world of entities sitting in a platonic heaven, with a great gulf fixed between the mathematical abstracta and the sublunary world. The harder she pushes that picture, the tougher it is for the Robust Realist to account for why we should think that the methods we sublunary mathematicians use should be a reliable guide to the lie of the land beyond the great gulf. It can then seems that our methods need backing up by some kind of certification that they will deliver the epistemic goods (and what could that look like?). Maddy, by contrast, thinks that the demand for such a certification is misplaced: why — as a second philosopher, working away in the thick of our best practices in science and maths — suppose that perfectly standard mathematical reasoning should stand in need of the sort of external supplementation that a Robust Realism seems to imply it requires?

Now, this might make it sound as if Maddy’s Thin Realist is going to end up with the sort of thin line about truth that we find e.g. in Crispin Wright. Then the thought would be: here is a discourse in good order, with appropriate disciplines and standards for making moves within it, so we can construct a minimalist truth-predicate for it. So we can not only say e.g. that the set of natural numbers has a powerset, but that it is true that there exists such a powerset, etc. However, this quick route to a thin realism isn’t Maddy’s line. Indeed, she explicitly contrast her Thin Realism with Wright’s minimalism:

Wright’s minimalist takes set theory to be a body of truths because it enjoys certain syntactic resources and displays well-established standards of assertion that our set-theoretic claims can be seen to meet: the idea is that a minimalist truth predicate can be defined for any such discourse in such a way that statements assertable by its standards come out true. In contrast, the Thin Realist takes set theory to be a body of truths, not because of some general syntactic and semantic features it shares with other discourses, but because of its particular relations with the defining empirical inquiry from which she begins.

It is important for Maddy’s Thin Realist, then, that our set theory — however wildly abstract it seems — has its connections to less abstract mathematics, which in turn has its connections … ultimately to the messy business of engineering-level science. Set theory, the rather Quinean thought goes, is an outlying but not entirely disconnected part of a network of enquiry with empirical anchors.

But put like that, we might wonder whether this kind of Thin Realist protests too much. To be sure, looking at the historical emergence of modern mathematics, we can trace the slow emergence from roots in mathematized science of purely abstract studies driven increasingly by a purely mathematical curiosity, and can see the (albeit very stretched) lines of connection. Starting from where we now are, however, the picture changes quite sharply: here we are with feet-on-the-ground physicists doing their thing using one bunch of methods and over there are the modern set theorists doing their improvisatory thing with a quite different bunch of rules of play (ok, let’s not worry about where string-theory fantasists fit in!). Physicists and set theorists are, it now seems, playing an entirely different game by different rules. We might ask: whatever the historical route by which we got here, is there really still a sense, however stretched, in which the physicists and the set theorists remain in the same business, so that we can sensibly talk of them both as trying to ‘uncover truths’?

The new suggestion, then, is that mathematicians have such very different fish to fry that it serves no good purpose for the second philosopher to say that the mathematicians too are talking about things that ‘exist’ (sets), or that set-theoretic claims are ‘true’. And note that it isn’t that the mathematicians should now be thought of as trying to talk about existents but failing: to repeat, the idea is that they just aren’t in the same world-tracking game. No wonder, then, that — as Maddy herself puts it on behalf of such a difference-emphasizing philosopher — our ‘well-developed methods of confirming existence and truth aren’t even in play here’. Call this second line, according to which set theory isn’t in the truth game, ‘Arealism’.

So what’s it to be for the second philosopher, Thin Realism or Arealism? What’s to choose? In the end, nothing says Maddy. Here’s modern science and its methods; here’s modern maths and its methods; here’s the developmental story; here’s a chain of connections; here are the radical differences between the far end points. Squint at it one way, and a sort of tenuous residual unity can be seen: and then we’ll incline to be Realists across the board — while, of course, eschewing over-Robust mythologies. Squint at it all another way, and the modern disunity will be foregrounded, and (so the story goes) Arealism becomes more attractive. There’s no right answer (rather, what this all goes to show is that the very notions of ‘truth’ and ‘existence’ are more malleable that we sometimes like to think).

Thus, roughly put, goes a central line in Maddy’s thought here, if I’m following aright.

But I wonder what underpins Maddy’s hankering here after a more-than-logical conception of truth? The Thin Realist, recall, thinks that Wrightian minimalism about truth isn’t enough: she wants to talk of set-theoretic truth so as to point up the (albeit distant) links from the maths to good old empirical enquiry. The Arealist doesn’t want to talk about set-theoretic truth because she wants to point up the differences between maths and good old empirical enquiry. But look at what they share: they both assume that the idea of truth needs anchoring in some way in the notion of the correct representation of an empirical world (rather, than, say in some cross-discourse formal role for the idea). Why so?

What we need here, if we are going to make progress from this point, is more reflection on the very concepts of truth and existence. And I don’t mean that we need an unwanted injection of first philosophy (so I’m not begging any question against Maddy)! Grant that our malleable inchoate ideas about truth can indeed be pressed in different directions. Still, the naturalistic second philosopher can take a view about the best way to go. She will want to look at our practices of talking and thinking and inferring, and she will want to have a theory about what is going on in various areas of discourse (empirical chat, moral chat, pure-mathematical chat, etc.). Her preferred developed notion of truth should then be the one that does the best theoretical work in her story about those discourses. And it certainly isn’t obvious at the outset how things should go. Will she end up more like a Blackburnian projectivist, privileging a class of representational discourse as the home territory for a basic notion of truth (so that other discourses are playing a different game, and have to earn their right to borrow the clothes that are cut to fit the representational case)? Or will become a more thorough-going pragmatist (holding that there is no privileged core). Or — in a different key — will she end up more like Wright’s minimalist? Certainly, the reader of Defending the Axioms isn’t given any reason to suppose that things must fall out anything like the first way, keeping room for a more substantial notion of truth.

And that gap in the end makes the current book rather unsatisfactory as a stand-alone affair (which isn’t to say it is not a fun read). Of course Maddy herself has written extensively about truth elsewhere so as to fill in something of what’s missing here. But this means that, after all, you really will have to go back to read the weighty Second Philosophy to get the whole story, and hence the full defence of the line Maddy wants to take about sets.

You can read the book review that Luca Incurvati and I wrote for Mind here.

Sophie Bevan and the Dunedin Consort

A quite terrific Bach/Handel concert in Trinity College Chapel — the Dunedin Consort with the soprano Sophie Bevan.

There’s a very warm review here in the Guardian of the same programme in Glasgow earlier in the week, which writes of Sophie Bevan’s singing: “no artifice, no fuss, a healthy wit, a refreshing kind of virtuosity that’s grounded and almost casual but still totally dazzling.” And how dazzling she was, her voice ringing round the chapel.

The highest points in an evening of high points? I should certainly mention a performance of the Brandenburg Concerto No. 4, with simply magical playing from the Dunedins’ leader Cecilia Bernardini, making the so-familiar a renewed delight. And the concert ended with Handel’s operatic Gloria with Sophie Bevan by turns exuberant, poignant, and at the end — in Handel’s long playful Amen — infectiously enjoying herself no end.

OK: you couldn’t be there. But if the Dunedins and/or Sophie Bevan come your way, snap up tickets: and meanwhile, the Dunedins have recorded the Brandenburgs, and Sophie Bevan has recorded the Gloria …

 

Structure and categoricity

‘Structure and Categoricity: Determinacy of Reference and Truth-Value in the Philosophy of Mathematics‘ by Tim Button and Sean Walsh has been posted on academia.edu. The paper nicely organizes the various options in a very clear and revealing way, and throws light on a number of discussions in the literature. Very good, very useful.

Encore #14: The Pavel Haas Quartet reviewed

We had tickets to hear the Pavel Haas Quartet three weeks ago, and again for today when they were due to play the Schubert Quintet. Very sadly, they have had to cancel their series of four London lunchtime concerts this month, because of serious family illness. Here is what I wrote about their recording of the Quintet.

Review: the Pavel Haas Quartet, Schubert ‘Death and the Maiden’ and the String Quintet (October 6, 2013)

Readers of this blog will know how greatly I have admired the Pavel Haas Quartet for a while. And it isn’t just me who finds them absolutely compelling both in live performance and on disc. Their previous four CDs have rightly been hugely praised, with the most recent Dvorak disk even winning The Gramophone Recording of the Year for 2011. And live, they are the most exciting quartet to watch and hear.

Now, at last, we have a new recording, of the Schubert D minor quartet, “Death and the Maiden”, and the String Quintet.  Michael Tanner writing in the BBC Music Magazine, another philosopher who weighs his words carefully,  called these “great performances … essential listening for anyone who loves Schubert”. The Times reviewer wrote “If CDs had grooves I would already have worn out these marvellous recordings  … the perfect fusion of virtuosity and profundity.”  Indeed. These performances are of a quite unworldly quality, deeply felt yet utterly thought-through, the most passionate you have heard but with moments of haunting delicacy,  with an overarching architectural vision always holding it all together.

The Pavel Haas launch into “Death and the Maiden” with fierce attack and astringent (almost vibrato-less) tone.  And they start as they mean to go on. The recent Takacs and the Belcea versions — good though they are — now seem slightly restrained in contrast (this is the still-young Schubert confronting death here, and the still-young Pavel Haas respond with apt intensity). The obvious comparison would be with the Lindsays’ great recording from twenty five years ago, which I would previously have said was the finest post-war version. But the Pavel Haas’s controlled passion, their even more moving account of the variations of the second movement, and their vehement drive to the end of the quartet, makes — I think — for an unparalleled performance.

As for the Quintet, this performance with Danjulo Ishizaka as the second cello is perhaps even finer. For any players, the problem — isn’t it? — is to maintain a shape to the whole piece: a bit too ethereal with the second movement and a bit too cheery with the last movements, and the Quintet is in danger of seeming unsatisfyingly unbalanced. But here, the whole hangs together better than any other interpretation I know. Although the playing is more expansive, within a few bars of the opening, the Pavel Haas have again built an extraordinary sense of tension. This is not comfortable listening — but then much of late Schubert isn’t. And the underlying tension is then maintained in a driven, uncompromising, way to the very end, with the slow movement giving only some partial relief (and there, the central section is played with a yearning fierceness, and the playing when the original theme returns is heart-stopping).  This makes for an exploration of the music at a level of intensity that again more than bears comparison with the Lindsays’ historic recording. Surely, a truly great interpretation of this great musical exploration of our humanity and mortality.

After a series of changes of second violin over the years, the Pavel Haas have never sounded better. Hopefully they will now stay happily together as they are.

Added The Gramophone reviewer writes of their “fearless risk-taking , their fervency” and “insanely memorable” phrasing; the Pavel Haas are “absolutely mesmerising” (in the close of the slow movement of the Quintet); “raw, visceral, and with an emotional immediacy that is almost unbearable” (at the ending of Death and the Maiden), and more. Yes!

Added  In the months since this review was first written, I’ve found that these performances more than live up to repeated listening.

 Added: Hopefully, their next CD will be of the Smetena Quartets: their live performances which I’ve heard have been nothing short of astonishing.

And indeed, their next CD was their stunning, award-winning, disc of the two Smetena Quartets.

Encore #14: PHQ

We had tickets to hear the Pavel Haas Quartet three weeks ago, and again for today when they were due to play the Schubert Quintet. Very sadly, they have had to cancel their series of four London lunchtime concerts this month, because of serious family illness. Here is what I wrote about their recording of the Quintet.

Review: the Pavel Haas Quartet, Schubert ‘Death and the Maiden’ and the String Quintet (October 6, 2013)

Readers of this blog will know how greatly I have admired the Pavel Haas Quartet for a while. And it isn’t just me who finds them absolutely compelling both in live performance and on disc. Their previous four CDs have rightly been hugely praised, with the most recent Dvorak disk even winning The Gramophone Recording of the Year for 2011. And live, they are the most exciting quartet to watch and hear.

Now, at last, we have a new recording, of the Schubert D minor quartet, “Death and the Maiden”, and the String Quintet.  Michael Tanner writing in the BBC Music Magazine, another philosopher who weighs his words carefully,  called these “great performances … essential listening for anyone who loves Schubert”. The Times reviewer wrote “If CDs had grooves I would already have worn out these marvellous recordings  … the perfect fusion of virtuosity and profundity.”  Indeed. These performances are of a quite unworldly quality, deeply felt yet utterly thought-through, the most passionate you have heard but with moments of haunting delicacy,  with an overarching architectural vision always holding it all together.

The Pavel Haas launch into “Death and the Maiden” with fierce attack and astringent (almost vibrato-less) tone.  And they start as they mean to go on. The recent Takacs and the Belcea versions — good though they are — now seem slightly restrained in contrast (this is the still-young Schubert confronting death here, and the still-young Pavel Haas respond with apt intensity). The obvious comparison would be with the Lindsays’ great recording from twenty five years ago, which I would previously have said was the finest post-war version. But the Pavel Haas’s controlled passion, their even more moving account of the variations of the second movement, and their vehement drive to the end of the quartet, makes — I think — for an unparalleled performance.

As for the Quintet, this performance with Danjulo Ishizaka as the second cello is perhaps even finer. For any players, the problem — isn’t it? — is to maintain a shape to the whole piece: a bit too ethereal with the second movement and a bit too cheery with the last movements, and the Quintet is in danger of seeming unsatisfyingly unbalanced. But here, the whole hangs together better than any other interpretation I know. Although the playing is more expansive, within a few bars of the opening, the Pavel Haas have again built an extraordinary sense of tension. This is not comfortable listening — but then much of late Schubert isn’t. And the underlying tension is then maintained in a driven, uncompromising, way to the very end, with the slow movement giving only some partial relief (and there, the central section is played with a yearning fierceness, and the playing when the original theme returns is heart-stopping).  This makes for an exploration of the music at a level of intensity that again more than bears comparison with the Lindsays’ historic recording. Surely, a truly great interpretation of this great musical exploration of our humanity and mortality.

After a series of changes of second violin over the years, the Pavel Haas have never sounded better. Hopefully they will now stay happily together as they are.

Added The Gramophone reviewer writes of their “fearless risk-taking , their fervency” and “insanely memorable” phrasing; the Pavel Haas are “absolutely mesmerising” (in the close of the slow movement of the Quintet); “raw, visceral, and with an emotional immediacy that is almost unbearable” (at the ending of Death and the Maiden), and more. Yes!

Added  In the months since this review was first written, I’ve found that these performances more than live up to repeated listening.

 Added: Hopefully, their next CD will be of the Smetena Quartets: their live performances which I’ve heard have been nothing short of astonishing.

And indeed, their next CD was their stunning, award-winning, disc of the two Smetena Quartets.

Encore #13: Tasks for philosophers of mathematics?

Here again is one of my very first blog-posts, musing about what philosophers of mathematics with my cast of mind might usefully get up to …

Tired of ontology? (May 13, 2006)

It requires a certain kind of philosophical temperament — which I do seem to lack — to get worked up by the question “But do numbers really exist?” and excitedly debate whether to be a fictionalist or a modal structuralist or some other -ist. As younger colleagues gambol around cheerfully chattering about these things, wondering whether to be hermeneutic or revolutionary, I find myself sitting on the side-lines, slightly grumpily muttering under my breath ‘And who cares?’.

To exaggerate a bit, I guess there’s a basic divide here between two camps. One camp is primarily interested in analytical metaphysics, or epistemology, or the philosophy of language, and sees mathematics as a test case for their preferred Quinean naturalist line or their Kantian framework (or whatever). The other camp is puzzled by some internal features of the practice of real mathematics and would like to have a satisfying story to tell about them .

Well, if you’re tired of playing the ontology game with the first camp, then there’s actually quite a bit of fun to be had in the second camp, and maybe more prospect of making some real progress. In the broadest brush terms, here are just a few of the questions that bug me (leaving aside Gödelian matters):

How should we develop/improve/augment/replace Lakatos’s model of how mathematics develops in his Proofs and Refutations?
What makes a mathematical proof illuminating/explanatory? (And what are we to make of unsurveyable computer proofs?)
Is there a single conceptual grounding for the standard axioms of set theory? (And what are we to make of the standing of various large cardinal axioms?)
What is the significance of the reverse mathematics project? (Is it just a technical “accident” that RCA_0 is used a base theory in that project? Can some kind of conceptual grounding be given for that theory? Would it be more principled to pursue Feferman’s predicative project?)
Is there any sense in which category theory provides new foundations/suggests a new philosophical understanding for mathematics?

There’s even a possibility that your local friendly mathematicians might be interested in talking about such things!

That still strikes me as quite a good list of questions that still interest me (particularly, at the moment, the last!). But what really good has been published on these in the intervening ten years? Suggestions please!

Encore #12: Advice on the book problem

During the lifetime of the blog, I have retired, which has meant that I lost the use of a university office to keep books — and indeed I lost all the shelves in an adjacent teaching room which I also used. A major book cull had to follow, and then another. I posted my sorrows a number of times. Some excerpts … leading to some wise advice about how to downsize your library!

I’m having to try to sort out my philosophy library — I can’t start shelving yet another wall at home — and that’s a painful business. It’s not just that there is a ridiculous number to sort through. It’s also a matter of encountering long-past philosophical selves, and not quite wanting to wave them goodbye. In the seventies, I was mostly interested in the philosophy of language (though there are also many ancient philosophy books dating from then, and a lot of Wittgenstein-related stuff); from the eighties there is a great number of books on the philosophy of mind; from the nineties a lot of philosophy of science and metaphysics. Digging through these archeological layers I’m reminded of past enthusiasms — not just of mine, but often quite widely shared enthusiasms which seemed philosophically rewarding at the time, but some of which now seem rather remote and even in some cases quite odd misdirections of energy. What creatures of fashion we are!

But at least in those more academically relaxed days I could follow my then interests wherever they led or didn’t lead (I never got bored). Young colleagues now don’t have the luxury: to get even their first permanent job they have to specialize, concentrate their resources, carve out a niche, build a research profile: and it takes more of the same to get promoted. Some feel constrained, lucky to have a job but trapped. The structures that we philosophers have allowed to be imposed on “the profession” (as we are now supposed to think of it) have thus come to be in real tension with the free-ranging cast of mind that gets many people into philosophy in the first place. What Marxists used to call a contradiction …

I’m still trying to sort out my (tiny) study at home. It all takes ridiculously long. Not just because I have to decide what to give to the library/students/Oxfam (though that’s difficult enough). But I find it impossible not to keep stopping over a book I haven’t opened in years and begin reading. I’m of the same mind as Churchill, who wrote

If you cannot read all your books, at any rate handle, or as it were, fondle them — peer into them, let them fall open where they will, read from the first sentence that arrests the eye, set them back on the shelves with your own hands, arrange them on your own plan so that if you do not know what is in them, you at least know where they are. Let them be your friends; let them at any rate be your acquaintances. If they cannot enter the circle of your life, do not deny them at least a nod of recognition.

And those nods of recognition as I move the books from one shelf to another all take so much time!

Chez Logic Matters, approximately

So you start buying books — I mean academic, work-related, books of one kind or another — in your late teens. As retirement age looms you’ve been doing it for the better part of fifty  years. Suppose you average a couple of books a month. Not very difficult to do! You buy a few current books on topics that you are working on; books for reference; books you feel you should read anyway, given the ripples they are producing; books for seminars or reading groups you belong to; books it is useful to have to hand for teaching (the textbooks the kids are reading, or just useful collections of articles, before the days when everything was online). It very soon mounts up. Add in a few review copies, freebies, books given by friends, serendipitous finds rescued from the back of obscure second-hand bookshops (I got a set of Principia Mathematica that way). Then without any effort at all your modest library is steadily growing at over thirty books a year or more. But go figure: that’s already around 1500 books as you get to the end of your career. I’ve been a bit more incontinent than some, but actually not a lot (especially as my interests have rather jumped about). Say I’ve acquired 1750 over the years. I’ve got rid of a few books from time to time, of course, though I’ve been absurdly reluctant to let them go: but overall, I’ve still probably got not far short of 1600. Which, I agree, is a quite stupid number to end up with — but (as we’ve seen!) it’s easy enough to end up there without a ridiculously self-indulgent rate of book-buying as you go along.

Soon enough, I’m going to finally lose an office; and we’re trying to declutter at home anyway. So over the coming weeks and months I need to cut that number down. A lot. Halving is the order of the day. What to do?

Most of the Great Dead Philosophers and the commentaries can go — I can’t see myself ever being overwhelmed by a desire to re-read Locke’s Essay, for example (and anyway I can always get the text online). But that doesn’t make much of a dent, as I was never much into the history of philosophy anyway. I can get rid of some of the books-for-teaching, and old collections of articles whose contents are now instantly available on Jstor. But that doesn’t help particularly either. So now it gets difficult.

It could just be neurotic attachment of course! But I like to think that there is a bit more to it than that. I’m sure I’m never going to seriously work on chaos again, so — though it was great fun at the time — I guess I will let the chaotic dynamics books go fairly easily. I’m also pretty sure that I’m never going to seriously work on the philosophy of mind again, and I’ve never done anything in epistemology: but just axing the phil. mind and theory of knowledge books seems to go clean against how I think of philosophy, as the business of trying to understand “how things in the broadest possible sense of the term hang together in the broadest sense of the term” as Sellars puts it. And anyway, some of the issues I’d like to understand better in the philosophy of mathematics seem to hang together with broader issues about representation and about knowledge. So perhaps I need to hang on to more of  the mind and knowledge books after all ….?

No, no, that way madness lies (or at any rate, swamping by unnecessary books). After all, Cambridge is not exactly short of libraries, even if I do dump something I later find myself wanting to read again! So, I’m just going to have to be brutal. A few old friends apart, if I haven’t opened it in twenty years, it can certainly go. If it is just too remote from broadly logicky/phil mathsy stuff, it really better go too. Sigh.

… But now it is getting harder. I’m slowing down, and it is all getting more discombobulating. In some cases it is a matter of regretfully having to acknowledge that — being realistic — I am never going to have a year or so to really get my head again round X or Y. I’d love to really get to the point where I was back on top of the state of play in the philosophy of quantum mechanics (say); but it is never going to happen — or at least, it’s never going to happen if I am to have half a chance of finishing some logicky projects. So that whole area will have to remain a closed book, or rather a small pile of closed books. A cheering reminder of faded hopes, eh?

Then there are the books to which I still feel an odd attachment and find difficult to let go for no reason I can easily articulate. Irrational, as I’ve not read them for decades, and I’m surrounded by Cambridge libraries. For instance, I’ve just found myself rereading some of  Cornford’s Unwritten Philosophy, which I must have bought in 1967, and not had occasion to read much since. I’m sure it is all very creaky: ancient philosophy has come such a very long way since when Cornford was writing (the essays date from the thirties and forties). I’ve long since lost touch, and my Greek has quite disappeared. And yet, and yet … The charm of his writing still weaves its magic. No; this I think I will keep, just for a bit longer.

Back to the pile for sorting …

Hello. My name is Peter and I am a bookaholic …

Well, perhaps it isn’t quite as bad as that. But I’ve certainly bought far too many books over the years. I’ve now given a great number away, but retiring and losing office space means there is still a serious Book Problem at home. We want to do some re-organization, which will mean losing book shelving there too. So lots more must go. Dammit, the house is for us, not the books. One hears tell of retiring academics who have built an extension at home for their library or converted a garage into a book store. But that way madness lies (not to mention considerable expense).

“A little library, growing larger every year, is an honourable part of a man’s history. It is a man’s duty to have books. A library is not a luxury, but one of the necessaries of life.” Yes. But let “little” be the operative word!

Or so I now tell myself. Still it was — at the beginning — not exactly painless to let old friends go, or relinquish books that I’d never got that friendly with but always meant to, or give away those reproachful books that I ought to have read, and all the rest. After all, there goes my philosophical past, or at any rate the past I would have wanted to have (and similar rather depressing thoughts).

But I think I’ve now got a grip (so at last, here’s my advice to anyone else in the same position, needing to downsize). It’s a question of stopping looking backwards and instead thinking, realistically, about what I might want to think about seriously over the coming few years, and then aiming to cut right down to (a still generous) working library around and about that. So instead of daunting shelves of books reminding me about what I’m not going to do, there’ll be a much smaller and more cheering collection of books to encourage me in what I might really want to do. The power of positive thinking, eh?

At least, that’s the plan. I’ll let you know how it goes.

And in the event, I surprised myself. A few years on, I can’t remember one occasion when I’ve been kicking myself, really regretting some book I got rid of. Though I confess I do seem to have acquired more books. Can’t imagine how that happened. Time, I suppose, for another sort-out … 

“As we approached Florence …”

“As we approached Florence, the country became cultivated to a very high degree, the plain was filled with the most beautiful villas, and, as far as the eye could reach, the mountains were covered with them; for the plains are bounded on all sides by blue and misty mountains. The vines are here trailed on low trellises of reeds interwoven into crosses to support them, and the grapes, now almost ripe, are exceedingly abundant. You everywhere meet those teams of beautiful white oxen, which are now labouring the little vine-divided fields with their Virgilian ploughs and carts. Florence itself, that is the Lung’ Arno, (for I have seen no more) I think is the most beautiful city I have yet seen. It is surrounded with cultivated hills, and from the bridge which crosses the broad channel of the Arno, the view is the most animated and elegant I ever saw. You see three or four bridges, one apparently supported by Corinthian pillars, and the white sails of the boats, relieved by the deep green of the forest, which comes to the water’s edge, and the sloping hills covered with bright villas on every side. Domes and steeples rise on all sides, and the cleanliness is remarkably great. On the other side there are the foldings of the Vale of Arno above; first the hills of olive and vine, then the chesnut woods, and then the blue and misty pine forests, which invest the aerial Apennines, that fade in the distance. I have seldom seen a city so lovely at first sight as Florence.”

Percy Bysshe Shelley to Mary Shelley, 20th August, 1818.

The approach to Florence

“As we approached Florence, the country became cultivated to a very high degree, the plain was filled with the most beautiful villas, and, as far as the eye could reach, the mountains were covered with them; for the plains are bounded on all sides by blue and misty mountains. The vines are here trailed on low trellises of reeds interwoven into crosses to support them, and the grapes, now almost ripe, are exceedingly abundant. You everywhere meet those teams of beautiful white oxen, which are now labouring the little vine-divided fields with their Virgilian ploughs and carts. Florence itself, that is the Lung’ Arno, (for I have seen no more) I think is the most beautiful city I have yet seen. It is surrounded with cultivated hills, and from the bridge which crosses the broad channel of the Arno, the view is the most animated and elegant I ever saw. You see three or four bridges, one apparently supported by Corinthian pillars, and the white sails of the boats, relieved by the deep green of the forest, which comes to the water’s edge, and the sloping hills covered with bright villas on every side. Domes and steeples rise on all sides, and the cleanliness is remarkably great. On the other side there are the foldings of the Vale of Arno above; first the hills of olive and vine, then the chesnut woods, and then the blue and misty pine forests, which invest the aerial Apennines, that fade in the distance. I have seldom seen a city so lovely at first sight as Florence.”

Percy Bysshe Shelley to Mary Shelley, 20th August, 1818.

Another categorial week

I’m at rather a troublesome point in putting together the Gentle Introduction. That’s because I really can’t put off any more writing a couple of introductory chapters, one on ‘Structures’ and one on ‘Sets’, to say something first about the very idea of abstract structures (in some sense, the topic of a lot of modern mathematics), and then to say something about the minimal notion of set that non-set-theorists need. Neither is exactly a non-contentious topic; and in due course category theory is supposed by many to throw light on both. So it will be a balancing act, trying to do enough scene-setting but not foreclosing options. Surprise, surprise, it’s going slowly. In the meantime, here are three things I’ve learnt about category theory this week:

Jean-Pierre Marquis’s From A Geometrical Point of View: A Study in the History and Philosophy of Category Theory is a disappointment. I’d tried to read this a few years ago when first getting interested in category theory, and found it tough going, so set it aside to return to when I knew more. Well, now I do know quite a bit more category theory, and more about some of the background too. And it is still tough going; too often, it is much less than perfectly clear, even impenetrable when you try to work out exactly what is being said in plain terms. I like my exposition transparent and simple-minded. So this is an opportunity lost, I think, as Marquis plainly knows a lot about a lot and has read very widely through the early days of category theory. But he’s just not getting his story across, even to this sympathetic reader coming better primed than many philosophers, I would guess. Of course, there are many potentially useful episodes, either pointing to bits of the history or suggestive of conceptual points to try to think through. However, not a successfully achieved book by my lights.
Our small reading group working through Leinster’s book (which is successful) has been continuing to be both fun and illuminating, with Part III students giving presentations ranging from the very helpful to the terrific — in some cases not in their first language too. I’m impressed. This week, we were looking at the proof of the General Adjoint Functor Theorem, and Patrick Stevens gave a lovely talk through a route to making it all seem entirely natural: here’s his written-up version. I’ll steal from him in due course! Next week, on beyond Leinster, to monads.
I was quite surprised to find, in an hour’s browsing though various UK university maths departments websites, that there are almost no courses on category theory out there (indeed standard books are missing from the university library collections of the University of X and Y). Drat. Turning my notes (if I ever do) into a book isn’t going to make my fortune then! Odd though it isn’t taught more, given how combinatorially easy and how conceptually neat elementary category theory is, and how very widely basic categorial ideas are assumed elsewhere. But I suppose the first textbooks are only about fifty years old, and maths departments are notoriously conservative …

But enough procrastination. Back to thinking about what mathematicians mean by talk of structures (as opposed to what some philosophers think they ought to mean).