Does mathematics need a philosophy? — 2

A few more thoughts after the TMS meeting (mainly for non-philosophers) … 

‘Does mathematics need a philosophy?’ The question isn’t exactly transparent.  So, to ask one of those really, really annoying questions which philosophers like to ask, what exactly does it mean?

Well, here’s one more focused question it could mean (and it was in part taken to mean in the TMS discussion): should mathematicians take note of, care about, the philosophy of mathematics as currently typically done by paid-up philosophers of mathematics? Both Imre Leader and Thomas Forster had something to say about this. And they agreed. The answer to this more focused question, they said, is basically “no”. Thomas went as far as saying,

The entirety of “Philosophy of Mathematics” as practised in philosophy departments is — to a first approximation — a waste of time, at least from the point of view of the working mathematician.

Fighting talk, eh?! But is that a reasonable assessment?

Well, I suppose it could have been that much of the philosophy is a waste of time  because philosophers just don’t know what the heck they are talking about when it comes to mathematics. But that’s rather unlikely given how many professional philosophers have maths degrees (when I was in the Philosophy Faculty, a third of us had maths degrees, including one with a PhD and another with Part III under their belts). So it probably isn’t going to be just a matter of brute ignorance. What’s going on among the philosophers, then, that enables Imre and Thomas to be quite so sniffy about the philosophy of mathematics as practised?

Here’s my best shot at making a case for their shared view. There’s a lovely quote from the great philosopher Wilfrid Sellars that many modern philosophers in the Anglo-American tradition [apologies to those Down Under and in Scandinavia …] would also take as their motto:

The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term.

Concerning mathematics, then, we might wonder: how do the abstract entities that maths seems to talk about fit into our predominantly naturalistic world view (in which empirical science, in the end, gets to call the shots about what is real and what is not)? How do we get to know about these supposed abstract entities (gathering knowledge seems normally to involve some sort of causal interactions with the things we are trying to find out about, but we can’t get a causal grip on the abstract entities of mathematics)? Hmmmm: what maths is about and how we get to know about it — or if you prefer than in Greek, the ontology and epistemology of maths — seems very puzzlingly disconnected from the world, and from our cognitive capacities in getting a grip on the world, as revealed by our best going science. And yet, … And yet maths is intrinsically bound up with, seems to be positively indispensable to, our best going science. That’s odd! How is it that enquiry into the abstract realms of mathematics gets to be so empirically damned useful? A puzzle that prompted the physicist Eugene Wigner to write a famous paper called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”.

Well, perhaps it’s the very idea of mathematics describing an abstract realm sharply marked off from the rest of the universe — roughly, Platonism — that gets us into trouble. But in that case, what else is mathematics about? Structures in some sense (where structures can be exemplified in the non-mathematical world too, which is how maths gets applied)? — so, ahah!, maybe we should go for some kind of Structuralism about maths? But then, on second thoughts, what are structures if not very abstract entities? Hmmmm. Maybe mathematics is  really best thought of as not being about anything “out there” at all, and we should go for some kind of sophisticated version of Formalism after all?

And so we get swept away into esoteric philosophical fights, as the big Isms slug it out (there are more guys than I’ve mentioned waiting on the sidelines to join in too: I’ll come back to them in the next post).

Now: the sorts of questions that ignite the Battle of the Isms do look like perfectly good questions … for philosophers. But they are questions which seem to get a lot of their bite, as I say, from worries about how maths hangs together with other things we tend to believe about the world and our knowledge of it. And the working mathematician is likely to think that, fine questions though they may be, s/he has quite enough nitty-gritty problems to think about within mathematics, thank you very much, and is far too busy to pause to worry about how what s/he’s up to relates to other areas of enquiry. So it’s division of labour time: let the philosophers get on with their own thing, building broad-brush ontological and epistemological stories about Life, the Universe, and Everything (including the place of maths); and let the mathematicians get on doing their more particular things. The philosophers had better know a smidgin about maths so their stories about how it fits into the Big Picture aren’t too unrealistic. But the mathematicians needn’t return the compliment, ’cos Big Picture stuff  frankly isn’t their concern.

Right ….

Doesn’t that actually look a pretty sensible view, which would sustain the line that Imre and Thomas took (and indeed, between them, they made a few remarks suggesting this sort of picture)?

However, I’ll suggest in the next instalment of these comments that there is after all some reason to think that mathematicians are inevitably, like it or not, entangled with some Big Picture questions. But first, in the rest of this instalment, let’s note a rather different reason why we (qua mathematicians) should hesitate to be quite so quick to ignore the philosophers.

The simple truth is that philosophers in fact talk about much more than the Big Picture stuff. To be sure, the beginning undergraduate curriculum tends to concentrate in that region: e.g. for an excellent textbook see Stewart Shapiro’s very readable Thinking about Mathematics (OUP, 2000). [And by the way, Shapiro has interesting things to say in the very first chapter of the book relevant to the general question whether mathematical practice is underpinned by philosophical assumptions.]  But the philosophers also worry about more specific questions like this: Have we any reason to suppose that the Continuum Hypothesis has a determinate truth-value? How do we decide on new axioms for set theory as we beef up ZFC trying to decide the likes of the Continuum Hypothesis? Anyway, what’s so great about ZFC as against other set theories (does it have a privileged motivation)? In  what sense if any does set theory serve as a foundation for mathematics? Is there some sense in which topos theory, say, is a rival foundation? What kind of explanations/insights do very abstract theories like category theory give us? What makes for an explanatory proof in mathematics anyway? Is the phenomenon of mathematical depth just in the eye of the beholder, or is there something objective there? What are we to make of the reverse mathematics project (which shows that applicable mathematics can be founded in a very weak system of so-called predicative second-order arithmetic)? Must every genuine proof be formalisable (in the sort of way I talked about in the last post), and if so, using what grade of logical apparatus? Are there irreducibly diagrammatic proofs? …

I could go on. And on. But the point is already made. These questions, standing-back-a-bit and reflecting on our mathematical practice, can still reasonably enough be called philosophical questions (even if they don’t quite fit Sellars’s motto). They are more local than what I was calling the Big Picture questions — they don’t arise from looking over our shoulders and comparing mathematics with some other form of enquiry and wondering how they fit together, but rather the questions are internal to the mathematical enterprise. Yet certainly they are discussed by mathematically-minded people who call themselves philosophers as well as by philosophically-minded people who call themselves mathematicians (sometimes it is difficult to remember who is which, and some people such as Solomon Feferman and Steve Awodey are in both camps — it is worth having a look at their lists of papers to see what they get up to).  And the sort of questions  we’ve listed surely are worth some mathematicians thinking about some of the time. Which, thankfully, they do.

So let’s not, after all, be as dismissive of the philosophers as Thomas was!

To be continued …


Does mathematics need a philosophy? — 1

At last week’s meeting of the Trinity Mathematical Society, Imre Leader and Thomas Forster gave introductory talks on “Does Mathematics need a Philosophy?” to a startlingly large audience, before a question-and-answer session. The topic is quite a big one, and the talks were very short.  But here are just a few after-thoughts, in three instalments (primarily for mathmos such as the members of TMS, but others might be interested …).

Imre did very briskly sketch a couple of philosophical views about mathematics,  which he called platonism and  formalism. And he suggested that  mathematicians tend to be platonist in their assumptions about what they are up to (in so far as they presume that  they are exploring a determinate abstract mathematical universe, where there are objective truths to be discovered) but they turn formalist when writing up their proofs for public consumption.

Now, Imre characterized formalism as an account of the nature of mathematics  along the lines of “it’s all juggling with meaningless symbols, a game of seeing what symbol strings you can ‘deduce’ from other strings according to given rules”. It is worth remarking that (at least as far as serious players in the history of philosophical reflection about mathematics go) this is something of a straw position: for example the great David Hilbert is usually taken to be the paradigm formalist; but his position was a lot more nuanced than that.  But ok, I’ve heard other mathematicians too describe the same naive kind of it’s-all-symbol-juggling line: so, for present purposes let’s understand ‘formalism’ as Imre did, though nothing really hangs on this. And the point I want to make is that it is a mistake to conflate endorsing formalism of any kind, naive or sophisticated, with something quite different, namely pursuing the project of formalization. Since I’ve also more than once heard others just make the same conflation, it’s worth pausing to pick it apart. (If some of the following sounds familiar to regular readers of this blog, it is because I’m shamelessly plagiarizing my earlier self.)

Start from the observation that, in presenting complex mathematical arguments, it helps to regiment our propositions into mathematical-English-plus-notation in ways which are expressly designed to be precise, free from obscurities, and where the logical structure of our claims is clear [think of the way we use the quantifier/variable notation — as in $latex \forall\epsilon\exists\delta$ — to make the structure of statements of generality crystal clear]. Then we try to assemble our propositions into something approximating to a chain of formal deductions. Why? Because this enforces honesty: we have to keep a tally of the premisses we invoke, and of exactly what inferential moves we are using. And honesty is the best policy. Suppose we get from the given premisses to some target conclusion by  inference steps each one of which is obviously valid (no suppressed premisses are smuggled in, and there are no suspect inferential moves). Then our honest toil then buys us the right to confidence that our premisses really do entail the desired conclusion. Hooray!

True, even the most tough-minded mathematics texts are written in an informal mix of ordinary language and mathematical symbolism. Proofs are very rarely spelt out in every formal detail, and so their presentation still falls short of the logicians’ ideal of full formalization. But we will hope that nothing stands in the way of our more informally presented mathematical proofs being sharpened up into fully formalized ones. Indeed, we might hope and pray that they could ideally be set out in a strictly regimented formal language of the kind that logicians describe (and which computer proofs implement), with absolutely every tiny inferential move made totally explicit, so that everything could be mechanically checked as being in accord with some overtly acknowledged rules of inference, with the proofs ultimately starting from our stated axioms.

True again, the extra effort of laying out everything in complete detail will almost never be worth the cost in time and ink. In mathematical practice we use enough formalization to convince ourselves that our results don’t depend on illicit smuggled premisses or on dubious inference moves, and leave it at that — our motto is “sufficient unto the day is the rigour thereof”. Here are local heroes Whitehead and Russell making the point in Principia:

Most mathematical investigation is concerned not with the analysis of the complete process of reasoning, but with the presentation of such an abstract of the proof as is sufficient to convince a properly instructed mind.

(A properly instructed mind being, like them, a Trinity mathmo.)

Let’s all agree, then:  formalization (at least up to a point) is a Good Thing, because a proof sufficiently close to the formalized ideal is just the thing you need in order to check that your bright ideas really do fly and then to convince the properly instructed minds of your readers. [Well, being a sort-of-philosophical remark, you’ll be able to find some philosophers who seem to disagree, as is the way with that cantankerous bunch. But the dissenters are usually just making the point that producing formalizable proofs isn’t the be-all and end-all of mathematics — and we can happily agree with that. For a start, we often hanker after proofs that not only work but are in some way explanatory, whatever exactly that means.]

So Imre would have been dead right if he had said that mathematicians are typically (demi-semi)-formalizing when they check and write up their proofs. But in fact, having described formalism as the game-with-meaningless-symbols idea, he said that mathematicians turn formalist in their proofs. Yet that’s a quite different claim.

Anyone who is tempted to run them together should take a moment to recall that one of the earliest clear advocates of the virtues of formalization was Frege, the original arch anti-formalist. But we don’t need to wheel out the historical heavy guns. The key point to make here is a very simple one. Writing things in a regimented, partially or completely symbolic, language (so that you can better check what follows from what) doesn’t mean that you’ve stopped expressing propositions and started manipulating meaningless symbols. Hand-crafted, purpose-designed languages are still languages. The move from ‘two numbers have the same same sum whichever way round you add them’ to e.g. ‘$latex \forall x\forall y (x + y = y + x)$’ changes the medium  but not the message. And the fact that you can and should temporally ignore the meaning of non-logical predicates and functions while checking that a formally set-out proof obeys the logical rules [because the logical rules are formalized in syntactic terms!], doesn’t mean that non-logical predicates and functions don’t any longer have a meaning!

In sum then, the fact that (on their best public behaviour) mathematicians take at least some steps towards making their proofs formally kosher doesn’t mean that they are being (even temporary) formalists.

Which is another Good Thing, because out-right naive formalism of the “it’s all meaningless symbols” variety is a pretty wildly implausible philosophy of mathematics. But that’s another story ….

To be continued

Stefan Collini writes again about the attack on universities

In the latest London Review of Books, Stefan Collini writes again from the heart and with critical incisiveness about the privatisation disasters befalling British universities. Here’s his peroration:

Future historians, pondering changes in British society from the 1980s onwards, will struggle to account for the following curious fact. Although British business enterprises have an extremely mixed record (frequently posting gigantic losses, mostly failing to match overseas competitors, scarcely benefiting the weaker groups in society), and although such arm’s length public institutions as museums and galleries, the BBC and the universities have by and large a very good record (universally acknowledged creativity, streets ahead of most of their international peers, positive forces for human development and social cohesion), nonetheless over the past three decades politicians have repeatedly attempted to force the second set of institutions to change so that they more closely resemble the first. Some of those historians may even wonder why at the time there was so little concerted protest at this deeply implausible programme. But they will at least record that, alongside its many other achievements, the coalition government took the decisive steps in helping to turn some first-rate universities into third-rate companies. If you still think the time for criticism is over, perhaps you’d better think again.

Read the article, weep, … and then if you are still in a UK academic job get a grip and do something!

Today wonderful reading: good fun!

If you follow @PeterSmith on Twitter — and why not? — these are apparently the delights that await you …

Logic and Schubert, eh? Can’t be bad.

Added: HT to Rowsety Moid for spotting the tagline I should have used to begin with!

Review: the Pavel Haas Quartet, Schubert ‘Death and the Maiden’ and the String Quintet

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 (as Michael Tanner also remarks). 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.

GWT and TYL on hold …

A few weeks ago, I said I’d this month be starting to post weekly instalments of a new version of Gödel Without (too many) Tears, updating and expanding the previous version to match the new edition of the Gödel book.

Well, things have conspired to prevent this. A temporarily hospitalized very aged mother, lots of visits at some distance, arranging a nursing home, etc., have taken up/will for a while take up a great deal of time and energy, so I just haven’t been able to do the work I’d planned on GWT2. And I don’t want to do a rushed or second-rate job. So I’ll have to delay starting the new sequence of GWT posts until next term/next semester.

For the same reason, it will be a while before I can update the Teach Yourself Logic study guide. I’m planning next a long entry on Peter Hinman’s well-regarded blockbuster Fundamentals of Mathematical Logic, but that too is on hold.

What Frege and Kripke didn’t tell you

Why is Venus star multinominous and called both Phosphorus and Vesper?

Venus is multinominous, to give example to her prostitute disciples who so often, either to renew or refresh themselves towards lovers, or to disguise themselves from magistrates, are to take new names. It may be she takes many names, after her many functions. For as she is supreme monarch of all love at large (which is lust) so is she joined in commission by all mythologists with Juno, Diana, and all others, for marriage. It may be, because of the diverse names of her affections, she assumes diverse names to her self. For her affections have more names then any vice, to wit Pollution, Fornication, Adultery, … Incest, Rape, Sodomy, Masturbation, and a thousand others. Perchance her diverse names shew her appliableness to diverse men. For Neptune distilled and wept her into Love, the Sun warmed and melted her, Mercury persuaded and swore her; Jupiter’s authority secured, and Vulcan hammered her. As Phosphorus she presents you with her bonum utile, because it is wholesomest in the morning; as Vesper, with her bonum delectabile because it is pleasantest in the evening. And because industrious men rise and endure, with the Sun, their civil business, this star calls them up a little before, and remembers them again a little after for her business.

John Donne, Problem XI, Paradoxes and Problems (c. 1590, published 1633) [Oddly, he in fact has “Hesperus” when he should have written, as here, “Phosphorus”!]

A query about countability

Suppose we are working in an elementary  context where e.g. we don’t want to rush to invoke infinitary choice principles, and want to keep background assumptions modest. What should our attitude be to the idea of countability? Countability is defined by a quantification – X are countable if there is a function f : N → X which enumerates them. But quantification over which functions?

I’m not raising Skolemite concerns here. Even if you fully buy into a rich set-theoretic background, taken at face value, different set theories will supply different enumerating functions. Thus the so-called constructible reals are uncountable according to the theory ‘ZFC + V = L’ but countable according to the theory ‘ZFC + there exists a Ramsey cardinal’. More needs to be said even by the orthodox who identify functions with sets, if it isn’t to be left somewhat indeterminate what objects are countable. But suppose we fall short of  endorsing the orthodoxy because we don’t (in the context, anyway) want definitely to buy into the wildly infinitary assumptions of set theory: we might initially seem to be in danger of making the notion of countability too indeterminate to be comfortable with. For if we are leaving it open just which functions we are prepared to countenance, we leave it correspondingly open which enumerating functions we are aiming to quantify over when we say that some objects are countable.

Yet mathematicians — at least when writing in fairly elementary contexts — cheerfully talk about the countable as if that’s unproblematic. How come? Is that just carelessness?

Well, no. Elementary talk about the countable tends (doesn’t it?) to feature  in three sorts of context:

  1. There are claims that certain objects are indeed countable, defended by showing that the objects in question are unproblematically counted by producing a nice enumerating function. (Consider, for a familiar simple example, how we show that the positive rationals m/n are countable by actually constructing the ‘zig-zag’ enumerating function for ordered pairs m, n, and so counting them.)
  2. There are claims that certain objects are uncountable, defended by reducing the assumption that they can be counted to absurdity. (Consider, for another familiar simple example, the usual diagonal argument that the infinite binary sequences are uncountable.)
  3. There are conditional claims of the kind if X are countable, then …, supported by general arguments that are insensitive to how exactly we delimit (or fail to delimit) the countable.

(The second is a special case of the third, of course, but perhaps worth highlighting.) In none of these kinds of case, at any rate, does such indeterminacy as we might be leaving in the extent of the countable become problematic. So if we proceed with due caution – restrict ourselves to these cases — we can continue to talk about the (un)countable safely enough. And in elementary contexts we do exercise such caution.

Or at least, so it seems. Query: is that a fair description of ‘ordinary’ mathematical practice in elementary, non-set-theoretic, areas? If not, what is going on? While if I’m right, can you think of some texts which overtly ’fess up to the need for this element of caution?

Aldous Huxley being prescient (almost)

Here’s Aldous Huxley writing in 1936:

To a considerable extent browsing has become,  for almost all of us, an addiction, like cigarette-smoking. We browse, most of the time, not because we wish to instruct ourselves, not because we long to have our feelings touched and our imaginations fired, but because browsing is one of our bad habits, because we suffer when we have time to spare and no websites with which to plug the void.

OK, Huxley has “reading/printed matter” rather than “browsing/websites”. But the thought is surely even more true now.

TYL, #17: The Teach Yourself Logic Guide updated

After taking a bit of a rest from it, I’ve been getting back to work on the Teach Yourself Logic Guide. Here then is Version 9.2 of the Guide, newly updated (pp. iii +  62).  Do spread the word to anyone you think might have use for it.

The main new additions are one-page reviews of Dirk van Dalen’s Logic and Structure and Shawn Hedman’s A First Couse in Logic, but there has also been some tinkering throughout.

The previous version from 1 June has been downloaded over 2250 times in three months.  That encourages me to continue putting some time and effort into the project.

[Added  One small error: Manzano’s Model Theory is not out of print — just gone to the limbo where absurdly expensive now-print-on-demand books eke out a ghostly after-life.]

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