Year: 2009

Pro-vice-chancellorian bollocks

Once upon a very long time ago, I spent a year or more of my life doing nothing else much but mathematics problems, preparing for the then entrance scholarship to get into Trinity to do maths. The standard was fairly stratospheric, and it was quite difficult to find enough practice questions on e.g. projective geometry. But one source was selected questions from various university examinations. Yes, degree-level questions from elsewhere being used as Cambridge entrance paper practice … which rather undermined any belief I might have had in the myth of close equality of level of enterprise across different universities.

Well, things have no doubt changed in all sorts of ways since those distant days. But one thing remains the same. As almost everyone knows perfectly well, there are wide differences between what is intellectually required to get (say) an upper second-class degree in a particular subject at different UK universities.

But there are always those concerned to deny the obvious. Thus, in the correspondence column of the Guardian today, the Pro-vice-chancellor of London South Bank University writes

All universities work to a common understanding of degree classification (supported by an external examiner system which is common to all).

And another correspondent writes

There is, in fact, close equivalence of degree-level standards across institutions. It is effectively maintained through the Quality Assurance Agency, which asserts that the standard “should be at a similar level across the UK”. The much-admired external examiner system is a major feature of this.

Which is all, of course, complete bollocks.

For a start, external examiners are very largely swapped between universities at similar levels in the pecking order. I bet, for example, that London South Bank University rarely uses examiners from Cambridge, or vice versa. And moreover, even when you do act as an external examiner for another university, your main job is to certify that the examining board there is behaving properly in following its own rules, and is behaving in a transparent and principled way. You might hope, where appropriate, for some rough comparabilities — but only pretty rough. And rough comparability is patently not transitive. Standards at A might be roughly comparable with standards at B, and standards at B might be roughly comparable with standards at C, and so on down the chain. But standards at A may of course be in a different ballpark to standards at Z.

And so they certainly ought to be. For example, here in Cambridge we spend a great deal of effort in trying to recruit the brightest and best; we then give them maybe seventy hours or more one-to-one ‘supervisions’ (i.e. tutorials) in philosophy over three years, not to mention all sorts of other formal and informal small group teaching on top of lectures — a quantity and quality of provision that most universities can only dream about. It would simply be a scandal if our students by the end weren’t in general performing at a level a good few notches above what it is reasonable to expect in many other places. And what we demand of them in Tripos quite rightly reflects that. That “close equivalence of degree-level standards” remains a myth.

Logic matters website updated

I’ve not been at all attentive to the Logic Matters website recently — it’s one of those things that sometimes I’m quite in the mood to do, and at other times it seems to drop right off the bottom of the “to do” list.

Anyway, there is now a new page “Other logic” linking to some recent papers and expository handouts. There’s more to be added, which I’ll want to tidy very slightly before irrecoverably letting them lose onto the net. Putting this page together was a rather cheering experience — I seem to have done more recently than I remembered!

I hope to get round to finishing the port of the LaTeX for Logicians pages from the old form to the new one sooner rather than later — at the moment some internal navigation is a bit flakey, though the requisite info is mostly in there somewhere.

Parsons, the whole story, at last

I have been blogging on and off for quite a while about Charles Parsons’s Mathematical Thought and Its Objects, latterly as we worked through the book in a reading group here. I’ve now had a chance to put together a revised (sometimes considerably revised) version of all those posts into a single document — over 50 pages, I’m afraid. You can download it here.

I’ve learnt quite a bit from the exercise. I’ll be very interested in any comments or reactions.

Parsons’s Mathematical Thought: Sec 55, Set theory

The final(!) section of Parsons’s book is one of the briefest, and its official topic is about the biggest — the question of the justification of set-theoretic axioms. But, reasonably enough, Parsons just offers here some remarks on how the case of justifying set theory fits with his remarks in the preceding sections.

First, on “rational intuition” again. We can work ourselves into sufficient familiarity with ZFC for its axioms to come to seem intrinsically plausible — but such rational intuitions (given the questions than have been raised, by mathematicians and philosophers) “fall short of intrinsic evidence“. Which isn’t very helpful.

And what about Parsons’s modified holism? In the case of set theory, is there “a dialectical relation of axioms and their consequences such as our general discussion of Reason would suggest”? We might suppose not, given that (equivalents of) the standard axioms were already “essentially in place in Skolem’s address of 1922”. Nonetheless, Parsons suggests, we do find such a dialectical relation, historically in the reception of the axiom of choice, and perhaps now in continuing debates about large cardinal axioms, etc., where “the role of intrinsic plausibility” is much diminished, and having the right (or at least desirable) consequences are an essential part of their justification. But, Parsons concludes — the final sentence of his book — “apart from the purely mathematical difficulties, many problems of methodology and interpretation remain in this area”. Which is, to say the least, a rather disappointing note of anti-climax!

Afterword Later this week, I’ll post (a link to) a single document wrapping up all the blog-posts here into a just slightly more polished whole, and then I must cut down 30K words to a short critical notice for Analysis Reviews. I feel I’ve learnt a lot from working through (occasionally, battling with) Parsons — but in the end I suppose my verdict has to be a bit lukewarm. I’m unconvinced about his key claims on structuralism, on intuition, on the impredicativity of the notion of number, in each case in part because, after 340 pages, I’m still not really clear enough what the claims amount to.

MacBook Air, six months on

Anyone out there who is still wavering about getting a MacBook Air might be interested in some comments from a delighted owner who has now had one for six months (this updates my “after one month” post from last August). Everyone else can, of course, just cheerfully ignore this again!

As background, I make heavy academic use of a computer (particularly using LaTeX, and reading a lot of papers, even books, onscreen, as well of course as the usual surfing, emailing etc.) but don’t really use one as a media centre except for light iPhoto use, and occasionally ripping CDs into iTunes for transfer to an iPod.

  1. The portability is fantastic. No question. Just to compare: I’ve had in the past a 15″ Titanium PowerBook, a 15″ G4 PowerBook, and a 17″ MacBook Pro before; and they’ve of course been portable in the sense I could heave them from home to my office and back. But all of them were honestly just too heavy/bulky to make that particularly convenient. I very rarely bothered to take them elsewhere, e.g. to a coffee shop, or even a library. (You might well ask why on earth, in that case, I had portables at all! Answer: Partly because our Cambridge house is very small, my “study” is the size of a large cupboard, and I very much like to be able to work on the kitchen table for a change of scene, or answer emails with a computer on my knees in the living room in the evening. So I certainly want “local” portability. And partly I need to be able to drive data projectors when lecturing.) Anyway, by contrast with the earlier portables, I can and do cheerfully tote the MBA (in its snug protective sleeve) anywhere, without really thinking about it, whether or not I’m definitely planning to use it. It just is so light and convenient. Much lighter than the new aluminium 13″ regular MacBook which I’ve tried out in the Apple Store too. Yet the MBA always feels remarkably sturdy. There’s not a sign really of six months of constant use.
  2. Some early reviews complained about the MBA’s footprint, saying that it isn’t a genuine ultraportable. People still complain about that. Well, true, I can imagine e.g. it wouldn’t be that easy to use in the cramped conditions of an airline seat. But that sort of issue just doesn’t arise for me. It’s not the footprint but the lightness and thinness which means that you can carry it so very comfortably in one hand, and of course the larger-than-ultra footprint goes with the stunningly good, uncramped, screen and the generous keyboard. In my kind of usage now, I’ve never found the footprint an issue.
  3. I rarely use the MBA to do anything very processor-intensive for a prolonged period of time, and it normally remains cool — though the fans can sometimes kick in a bit enthusiastically e.g. when backing up. And the battery life seems just fine: well over three hours for writing, text-browsing, reading. Recharging though is pretty slow: but if you need to take it with you, then the MBA’s charger is very small and portable (though I’ve bought a second one for my office in the faculty, and so don’t find in practice I need to carry it around).
  4. One main reason I traded up a couple of years ago from the 15″ G4 machine was that LaTeX ran pretty slowly: nearly 30 seconds to typeset my Gödel book on the G4, about 4 seconds on the new intel MacBook Pro. The MBA, despite its slower chip, seems almost as fast running LaTeX , and indeed in most other ways: occasionally, e.g. when opening an application, the MBA is noticeably slower — but it has never been a particularly irritating issue. So this is plenty fast enough.
  5. And the reason, when I traded up the previous time, I chose the 17″ MBP model was to have enough “real estate” to have a TeXShop editing window and the PDF output window side-by-side and comfortably readable. Obviously, I’m now looking at 1280 x 800 pixels, rather than 1680 x 1050 (so that’s just 58% as much). But I’ve surprised myself by getting very used to working with overlapping windows again, and the screen quality is really terrific. The best I’ve ever had by far. Of course it is nicer e.g. for extended on-screen reading to plug in an external monitor as well. But not any sort of necessity — and indeed I seem these days pretty often not to bother even if I’m sitting next to the external monitor.
  6. What about the paucity of ports, mentioned critically by all the reviewers, or the absence of an onboard CD drive? Really not an issue. I’ve a couple of times wished there were two USB ports, I bought a little one-to-two-port splitter, for very occasional home use, but in fact even when I don’t have it with me, I’ve never been seriously annoyed. Of course, if I had one of the new version MBA’s I’d be tempted with one of the new displays that also acts as a USB hub: but that would be an indulgence. I’ve latterly bought an external CD/DVD drive built for the MBA, for when I occasionally need one. (The one caveat concerns day one, long before I got the external drive. Since there is no firewire port, you can’t migrate files from your old computer to your new MBA using the usual firewire connection. And using a wireless connection to migrate is both painfully slow and seems flaky. Is that a problem? I didn’t find really it so. I installed new versions of necessary additional software, like the LaTeX installation, over the web, and then copied my documents folder and other bits and pieces from a SuperDuper! clone of the old hard disk on an external drive. Quick to do, and resulting in a clean and tidy MBA.)
  7. So that’s all very, very positive. Are there any negatives? The flat keyboard is surprisingly nice to use (much better than I imagined it would be). But, unlike the almost silent similarly flat new iMac keyboards, I find the MBA version to does seem a bit noisier (and a bit more so than the MBP keyboard). But that’s a very marginal disappointment.
  8. I thought, when I bought the MBA a month ago, I’d be using it very much as a second machine, carrying on using the 17″ MPB (with external monitor) as a main, quasi-desktop, set-up. In fact I find I now almost never use the MPB.
  9. So, assuming a three year life cycle (and it seems very well built so should last longer with a battery refresh after a while), the MBA after education discount costs much less than half a pint of beer a day. Put like that, how can you resist?
  10. And then, of course, there is the “Wow!”-factor …

Parsons’s Mathematical Objects: Sec. 54, Arithmetic

How does arithmetic fit into the sort of picture of the role of reason and so-called “rational intuition” drawn in Secs. 52 and 53?

The bald claim that some basic principles of arithmetic are “self-evident” is, Parsons thinks, decidedly unhelpful. Rather, “in mathematical thought and practice, the axioms of arithmetic are embedded in a rather dense network … [which] serves to buttress [their] evident character … so that in that respect their evident character does not just come from their intrinsic plausibility.” Moreover, there is a subtle interplay between general principles and elementary arithmetical claims — a dialectic “between attitudes towards mathematical axioms and rules and methodological or philosophical attitudes having to do with constructivity, predicativity, feasibility, and the like”. Which, as Parsons notes, is all beginning to sound rather Quinean. How is his position distinctive?

Not by making any more play with talk of “rational intuition”, which made its temporary appearance in Sec. 53 just as a way of talking about what is intrinsically plausible: indeed, the idea that the axioms of arithmetic derive a special status from being grounded in rational intuition is said to be “in an important way misleading”. Where Parsons does depart from Quine — and it is no surprise to be told, at this stage in the book! — is in holding that some elementary arithmetic principles can be intuitively known in the Hilbertian sense he discussed in earlier chapters. And the main point he seems to want to make in this chapter is that, as we move to more sophisticated areas of arithmetic which cannot directly be so grounded, so “the conceptual or rational element in arithmetical knowledge becomes much more prominent”, the web of arithmetic isn’t thereby totally severed from intuitive knowledge grounded in intuitions of stroke strings and the like. It is still the case that “an intuitive domain witnesses the possibility of the structure of numbers”.

Of course, how impressed we are by that claim will depend on how well we think Parsons defended his conception of intuitive knowledge in earlier chapters (and I’m not going to go over that ground again now, and nor indeed does Parsons). And what grounds the parts of arithmetic that don’t get rooted in Hilbertian intuition? To be sure, those more advanced parts can get tied to other bits of mathematics, notably set theory, so there is that much rational constraint. But that just shifts the question: what grounds those theories? (There are some remarks in the next chapter, but as we’ll see they are not very unhelpful.)

So where have we got to? Parsons’s picture of arithmetic retains a role for Hilbertian intuition. And unlike an “all-in” holism, he wants to emphasize the epistemic stratification of mathematics (though his remarks on that stratification really do little more than point to the phenomenon). But still, “our view does not differ toto caelo from holism”. And I’m left really pretty unclear what, in the end, the status of the whole web of arithmetical belief is supposed to be.

Parsons’s Mathematical Objects: Secs 52-53, Reason, "rational intuition" and perception

Back to Parsons, to look at the final chapter of his book, called simply ‘Reason’. And after the particularly bumpy ride in the previous chapter, this one starts in a very gentle low-key way.

In Sec. 52, ‘Reason and “rational intuition”‘, Parsons rehearses some features of our practice of supporting our claims by giving reasons (occasionally, he talks of ‘features of Reason’ with a capital ‘R’: but this seems just to be Kantian verbal tic without particular significance). He mentions five. (a) Reasoning involves logical inference (and “because of their high degree of obviousness and apparently maximal generality, we do not seem to be able to give a justification of the most elementary logical principles that is not in some degree circular, in that inferences codified by logic will be used in the justification”). (b) In a given local argumentative context, “some statements … play the role of principles which are regarded as plausible (and possibly even evident) without themselves being the conclusion of arguments (or at least, their plausibility or evidence does not rest on the availability of such arguments).” (c) There is there is a drive towards systematization in our reason-giving — “manifested in a very particular way [in the case of mathematics], though the axiomatic method”. (d) Further, within a systematization, there is a to-and-fro dialectical process of reaching a reflective equilibrium, as we play off seemingly plausible local principles against more over-arching generalizing claims. (e) Relatedly, “In the end we have to decide, on the basis of the whole of our knowledge and the mutual connections of its parts whether to credit a given instance of apparent self-evidence or a given case of what appears to be perception”.

Now, that final Quinean anti-foundationalism is little more than baldly asserted. And how does Parsons want us to divide up principles of logical inference from other parts of a systematized body of knowledge? His remarks about treating the law of excluded middle “simply as an assumption of classical mathematics” suggest that he might want to restrict logic proper to some undisputed core — though he doesn’t tell us what that is. Still, quibbles apart, the drift of Parsons’s remarks here will strike most readers nowadays as unexceptionable.

Sec. 52, ‘Rational inuition and perception’, says a bit more to compare and contrast intuitions in the sense of statements found in a given context of reasoning to be intrinsically plausible — call these “rational intuitions” — and intuitions in the more Kantian sense that has occupied Parsons in earlier chapters. As he says, “intrinsic plausibility is not strongly analogous to perception [of objects]”, in the way that Kantian intuition is supposed to be. But perhaps analogies with perception remain. For one thing, there is the Gödelian view that intrinsic plausibility for some mathematical propositions involves something like perception of concepts. And there is perhaps is another analogy too, suggested by George Bealer: reason is subject to illusions that, like perceptual illusions, persist even after they have been exposed. But Parsons only briefly floats those ideas here, and the section concludes with a different thought, namely there is a kind of epistemic stratification to mathematics, with propositions at the lowest level seeming indisputably self-evident, and as we get more general and more abstract, self-evidence decreases. Which is anodyne indeed.

Nerdy stuff

Just for fellow Macaholics …

  1. I’ve just noticed that a new version of TeXShop has been released in the last couple of weeks, with a couple of useful little tweaks.
  2. I had a pre-release trial copy of Things for a while, and now the first proper release is out. Very neat and very simple: so it is, for once, “task management” software — ok, a fancy way of keeping To Do lists — that I actually do use.
  3. Oh, and I’ve just got an Iomega eGo Helium external drive. Very small and no power block, so easy to tote, and no fan so very quiet. It’s a bit sad to get even mildly pleased by a hard drive. But still, it is rather pretty …
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