Tuesday, July 29, 2008

Sounding off (x 50)

I've just noticed that I've now written 50 responses as a panelist for the admirable askphilosophers.org. You can read my efforts here. It's been fun. And contributing is probably a rather more productive way of procrastinating on the web than most.

A blast from the past

After an electrical storm, my new MacBook Air suddenly stopped recognizing the wireless network at home. It could pick up a signal from about six neighbouring houses, but not the Airport base station a few feet away, while the old laptop had no problems. I tore my hair trying everything I could think for a few frustrating hours. No luck.

Eventually I phoned Apple Care. And after re-trying one or two other things, the guy at the end said "Restart, holding down Command-Option-P-R". Wow. Zapping the PRAM!

Which took me right back to 1990 and my first Mac (a IIsi, since you ask). In those days, zapping the Parameter RAM was a fairly frequent dodge, used whenever the poor thing got a bit confused. But I can't have done it for a decade or more. I'd long forgotten it was even an option. Did the trick though, so here I am again ...

Logic's Lost Genius again

The last appendix to Eckart Menzler-Trott's book on Gödel is a thirty-six page essay "From Hilbert's Programme to Gentzen's Programme" by Jan von Plato. As you'd expect from this author, this is accurate and useful, as far as it goes. But actually, although it would have made a long book even longer, it would have been rather good to have fifteen rather than five pages on the key results about the consistency of arithmetic. For von Plato's discussion is a bit frustratingly short of details about the proof-strategies of Gentzen's four proofs. Students who aren't planning to do a heavy-duty course on proof-theory will find it difficult to pick up quite enough about the basics of what is going on in the proofs to satisfy their intellectual curiosity. (All the same, like Smorynski's piece, this essay will be well worth putting on reading lists -- and so libraries should certainly get a copy of this book.)

I should add -- to counterbalance remarks in an earlier posting -- that Menzler-Trott's Introduction does very clearly say "This is the story of the outer life of the mathematician and logician Dr.habil. Gerhard Gentzen; it is not a book about proof theory or its development." (p. xviii). So perhaps we shouldn't judge a story of the "outer life" alongside Dawson's account of Gödel or the Fefermans on Tarksi which do try to tell something of the inner stories. But Menzler-Trott's Preface also says "I hope that this book will be judged as a contribution to the philosophy and the history of logic and proof theory and not merely to the art of biography of mathematicians." (p. xiv). So in the end I'm still unsure quite what the book is aiming for. But it is obviously a labour of love by its avowedly amateur author, with a great (sometimes excessive) amount of detail and -- despite its oddities -- it is certainly worth dipping into.

Thursday, July 24, 2008

MacBook Air!

I'm a pretty minimal user of the mobile phone, and don't use my iPod that much either ... so it really would be an expensive self-indulgence to buy an iPhone and contract. OK, ok, let's be honest, it's still very tempting! But I've manfully managed to resist, and I've put some of the consequent savings together with some computing money from the Faculty to buy a MacBook Air instead. (Is there a financial fallacy buried in there somewhere? Well, let's not examine that too closely, ...) It will be genuinely useful to have a much more portable laptop.

It would be really boring to bang on here about what a great machine this seems to be, even after just a few hours close acquaintance. I'm perhaps not entirely convinced by the keyboard (which could be even quieter for library use???). But the screen quality is stunning. And it remains a surprise to the hand every time you lift it that something that -- when opened up in use -- looks so sizeable and well-built is so light. It's pretty speedy too -- indeed, it seems to compile the LaTeX file for my Gödel book about as fast as a MacBook Pro whose clock-speed is 50% more.

I think this might be the beginning of a beautiful friendship. So I'll try not to throw it away.

Wednesday, July 23, 2008

Smorynski on Hilbert's Programme

As I mentioned before, Menzler-Trott's biography of Gentzen has a number of appendices, including a fifty page essay "Hilbert's Programme" by Craig Smorynski. (A better title might have been "The slow emergence of Hilbert's Programme from Hilbert's intermittent work on foundational questions up to 1930/31, and in particular from his disputes with Brouwer and Weyl." But I can see why Smorynski stuck to his snappy title!)

I found this essay a terrific read, very helpful and illuminating, at least for someone who makes no pretence of knowing much about the history here. This should now go on any reading list for philosophy of maths students touching on Hilbert's Programme. And so spread the word: it would be a great pity if Smorynski's efforts went largely unread because buried at the back of a rather oddly written biography. (The rationale for having this piece appended to the biography is that it sets the scene for Gentzen's work -- but actually, as I noted, Menzler-Trott doesn't engage very closely with that work, so he doesn't really join up the dots. An opportunity missed.)

Monday, July 21, 2008

Parsons's Mathematical Thought: Sec 13, Nominalism and second-order logic

A general comment before proceeding. Parsons himself says that this book has been a very long time in the writing. And I suspect that what we are reading is in fact a multi-layered text with different passages added at different times, without the whole being finally reorganized and rewritten from beginning to end. This does make for a bumpy read, with the to-and-fro of argument not always ideally well signalled.

Anyway, Sec. 13 falls into two parts, both related to nominalist takes on second-order logic. First, Parsons offers some remarks on the Fieldian project of using mereology to do the work of second-order logic. The key thought is this. For mereology to do all the work Field wants, it needs an (impredicative) comprehension principle: "Given a predicate of individuals that is true of at least one individual, there is a sum of just the individuals of which the predicate is true, and moreover, the admissible predicates will be closed under quantification over all individuals, including those very sums." (Cf. the principle "Cs" in Field's "On Conservativeness and Incompleteness".) But what entitles Field to such a strong comprehension principle? Well, Parsons notes that it's not clear that Field can offer any direct a priori argument (but then, I wonder, would he want to?). The justification will be that "the comprehension principle is a hypothesis justified by its consequences in systematizing the geometrical basis of physics". But then "Field's view, on this reading, puts him in a position in which we have found other formulations of nominalism: making the justification of mathematics turn on some hypothesis about the physical world, which is more vulnerable to refutation than the mathematics."

But how troubled will a Fieldian be by that complaint? Suppose we decide that our physical theory of the world doesn't require such a strong comprehension principle (we can get away with recognizing a less wide-ranging plurality of regions). That's not at all implausible, actually, given that (nearly) all the mathematics required for physics can be reconstructed in a weak second-order arithmetic like ACA_0 with only predicative comprehension. Then the Fieldian response will (surely?) be just to demote the full mathematical apparatus of the classical reals from its status in Science without Numbers as a supposedly justified tool for getting more nominalistically acceptable consequences out of our best physics. It is no longer so justified. In that sense, for the Fieldian, the "justification" of a bit of mathematics is wrapped up with our hypotheses about the physical world, and Parsons's complaint will seem question-begging. [Or am I missing something here?]

The second part of Sec. 13 considers Boolos's attempt to make second-order logic ontologically tame by giving a plural reading to the second-order quantifiers. The thought under scrutiny is that plural quantification is ontologically innocent because, in plurally quantifying over Fs, we are just committing ourselves to Fs (not to sets or to Fregean concepts). Parsons's discussion [or again, am I missing something here?] initially advances familiar sorts of worries about this claim of innocence. But Parsons does make one point towards the end of the section that I find very congenial (i.e. I've argued similarly myself!).

Consider (say) the range of second-order arithmetics that Simpson discusses in SOSOA. As we advance through theories with stronger and stronger comprehension principles, then -- on a standard platonist construal -- we are countenancing more and more sets of numbers. If we reconstrue the second-order quantifiers plural-wise, then, as we go from theory to theory, we are countenancing more and more .... well, more what? It is tempting to say "pluralities". And indeed it is convenient to give an informal gloss of the plural reading using talk of pluralities. But -- if this isn't to smuggle back reference to pluralities-as-single-entities, i.e. sets -- this convenient way of talking needs to be eliminable (cf. Linnebo's nice article on plural logic). So how do we eliminate it here? We might, I suppose, trade in talk of countenancing more and more pluralities for talk of allowing more and more different ways we can take numbers together: but this seems tantamount to re-instating Fregean concepts as the values of the second-order variables -- which is fine by me, but then the supposed ontological gain of interpreting the second-order quantifiers via plurals is lost.

The question then is this: if we accept the pluralist's contention that we can treat second-order numerical quantifiers as ontologically committing just us to numbers, period, then how are we to think of the surely varying commitments we take on with varying strengths of comprehension principle. As Parsons puts it, "If there is no enlargement of ontological commitment [my emphasis] as one passes to less restricted versions of the comprehension schema, then perhaps that speaks against the importance of the notion."

Saturday, July 19, 2008

Once upon a time ...

Recently, we cleared out the loft, preparatory to having some building work done. And since a couple of big boxes of vinyl records had been sitting up there untouched for a decade, we gave the lot to Oxfam (there were probably a few collectors items there, but the local shop assured me they had someone expert to sort through them).

But I did keep just one disk -- not to play, but for its iconic sleeve.

I watched Jules et Jim again a year or two back on one of its occasional television outings, the first time for many years. And I found it a strange experience, feeling at such a distance from my much earlier self who once upon a time thought it so wonderful. The film perhaps wears less well than some of its era.

But one moment did magically draw me back in again, when Jeanne Moreau sings Le Tourbillon. So here she is again, especially for readers of a certain age ... (And those of a sentimental disposition might enjoy this too.)

Friday, July 18, 2008

Logic's Lost Genius

It has to be said that Eckart Menzler-Trott's Logic’s Lost Genius: The Life of Gerhard Gentzen is a strange work.

For those who haven't seen it, a word about the structure. The main part (pp. 1-283) of this large-format, small-print, book is notionally a biography of Gentzen, but as I'll note in a moment there are very long digressions. Then there are four appendices. The first (pp. 285-292) is a note by Craig Smorynski (who is also the main translator from the original German version of the book) on an elementary but neat proof in geometry re-discovered by the school-boy Gentzen. Next (pp. 293-343) there is a long essay by Smorynski on Hilbert's Programme. The final appendix (pp. 369-405) is another essay, almost as long, by Jan von Plato, called "From Hilbert’s Programme to Gentzen’s Programme". Lastly, rather oddly sandwiched in between those last two essays, Menzler-Trott includes three lectures by Gentzen himself. These are

  1. The Concept of Infinity in Mathematics (item #6 in Szabo's Collected Papers),
  2. The Concept of Infinity and the Consistency of Mathematics,
  3. The Current Situation in Research in the Foundations of Mathematics (item #7 in Szabo).
1 and 3 are newly retranslated. 2 is just over two sides long (and is little more than a summary of 1).

I'll comment in later posts on the essays by Smorynski and von Plato. But what about the main biography?

Well, as I said, this really is rather strange. For a start, one very long chapter (pp. 141-232, 'The Fight over “German Logic” from 1940 to 1945: A Battle between Amateurs') concerns Nazi attitudes about the "decadence" of mathematics supposedly due to Hilbert. This tells us just a bit about how Gentzen's work was viewed in some quarters. But most of the discussion is only distantly relevant (pages and pages go by without Gentzen being even mentioned). In fact, however interesting this all will be for those researching on the politicization of academic life under the Nazi regime, it tells us almost nothing about Gentzen's intellectual development.

And the other chapters, which really are on Gentzen, are oddly written (and I'm not talking about the translation into an English replete with far too many sentences no native speaker would use). Rather the text too often reads like unprocessed working notes, stringing together remarks on intellectual events, or on unrelated family affairs, with excerpts from Gentzen's letter and reviews. For a particularly staccato example, on p. 94 we read [and yes, these are consecutive mini-paragraphs]:
In December 1937 Gentzen informed at least Paul Bernays that he had carried out his consistency proof in a simpler and more thorough form.

Since December 1937, Gentzen’s sister, Waltraut, and her husband lived in Liegnitz/Niederschlesien (today: Lignice, Poland).

On 3 January 1938 Bernays wrote from Besenrain Str. 30 in Zürich that he had finished §11 of the foundations book for two weeks, but it was not yet typeset: “As soon as the copy is made, I will send it to you.”

In Zentralblatt für Mathematik 17 (1938), p. 242, there appeared: [and Menzler-Trott then reproduces Gentzen's review of Barkley Rosser's 1937 JSL paper 'Gödel theorems for non-constructive logics'.]
So no one can call this a gripping, well-structured, story!

But stylistic complaints (and the very long aside on Nazi attitudes) apart, is the biography at least illuminating? The answer, I'm afraid, is "not very", at least not if you are looking for an account of Gentzen's intellectual trajectory. And this is because -- unlike Dawson on Gödel or the Fefermans on Tarski -- Menzler-Trott doesn't just engage enough with Gentzen's logic. For example, if you don't already know about his work on natural deduction and sequent calculi, you'll hardly get any sense of what Gentzen was up to here and why it matters. (Ok, Gentzen's work is going to be addressed more directly in the two long appendices by Smorinski and von Plato: but it does feel as if Menzler-Trott narrative has a large hole at the centre.)

Still, there is a lot of detail about Gentzen's milieu, about whom he met when, about who influenced him, and whom he was corresponding with. And this scene-setting is interesting enough. Moreover, quite a few letters are quoted, which have some interest -- though note that there is little by way of e.g. novel informal exposition here. Hence you won't get a new understanding of Gentzen's results from them. But you'll at least come away with a better sense of the external shape of the context in which his papers were written. You'll have to wait until the appendices, however, to learn more about the internal dynamic of the ideas there.

I should add, though, that the last main chapter, on Gentzen's death in prison, contains moving accounts from fellow prisoners: do read this chapter even if you don't read the rest of the book.

Thursday, July 17, 2008

What language is this?

I'm settling down to a serious read of Eckart Menzler-Trott's biography of Gentzen. Supposedly the English version. But what language is this?

  • The examination of the mathematics using means and methods of other sciences or humanities is still disgusting for many.
  • One could still learn mathematics today in substance in the writings of Euclid ...
  • A bright and conceivable history of modern logic isn’t understandable without one’s biography using conceptual and contextual ideas.
  • ... without clarification of historical facts, the different forms of evolving mathematical treatments, methodical and resulting knowledge, and epistemic configurations or its reflection are not once meaningfully describable.
Those are just a selection from just two pages of the Introduction. I'm really grateful to have a rendition of Menzler-Trott's book into something I can understand. But what on earth were the AMS series editors doing letting this misbegotten prose through?

Wednesday, July 16, 2008

Parsons's Mathematical Thought: Sec. 12, Nominalism

This is a short and rather insubstantial section, which I'm just taking separately to get out of the way, because the next section is weighty (and one of the longest in the book).

Parsons understands 'nominalism' Harvard-style -- no surprise there, then! -- to mean the rejection of abstract entities and the eschewing of (ineliminable) modality. What hope, then, for giving a response to the potential-vacuity problem for eliminative structuralism about arithmetic (say) which meets nominalist constraints? We can't, by hypothesis, go modal: so what to do?

Well, as the physical world actually is (or so we might well now believe), there are in fact enough physical things -- e.g. space time points -- and suitable physical orderings on them to give us physically realized 'simply infinite' structures. But Parsons is unhappy with this way of meeting the vacuity worry, and for familiar reasons: "[S]hould it be taken as a presupposition of elementary mathematics that the real world instantiates a mathematical conception of the infinite? This would have the consequence that mathematics is hostage to the future possible development of physics."

But (although I have no particular nominalist sympathies myself), I'm not sure how worried the nominalist eliminative structuralist should be about giving such hostages to fortune. As things are, given how we believe the world actually to be, he can reasonably continue to speak with the vulgar and treat arithmetical claims as true or false. Even if the worst happens, so we come to believe the world is ultimately grainy and finite in all respects, it's not that 'school-room' arithmetic is going to get undermined. At most, it is the idealizing rounding out of school-room arithmetic which insists on an infinitude of numbers. And if it should emerge that the rounding out, construed the eliminative-structuralist way, collapses in vacuity -- well, formal arithmetic can still be played as an intriguingly entertaining game. It's just that then, after all, the nominalist eliminative structuralist who is relying on physical realizations for structures can no longer readily construe idealized arithmetic's claims as true or false, and so the nominalist has to sound a bit more revisionary. But, he'll say, so what? (Parsons says "a great deal of the historically given mathematics would have to be jettisoned in this case" -- but that's too quick. Talk of 'jettisoning' covers over a slide. For no longer thinking of arithmetic as construable as literally true by the eliminative structuralist manoeuvre is not the same as throwing arithmetic into the trash-can, as any fictionalist will insist.)

What about the other line that offered to the nominalist at the end of Sec. 11? -- i.e. sidestep the vacuity problem by going modal in an anodyne way ("interpret the theories in an if-thenist way, but deal with the problem of possibility by appealing to consistency, nominalistically interpreted"). Well, again Parsons sees trouble, this time arising from the fact that there might be physical limitations in how big a proof-token could be, and so a theory could count as (nominalistically) consistent -- because no proof of an inconsistency could be tokened -- even if we can show that there is a process which, given world enough and time, would produce an inconsistency. But again, I'm not sure that the obstreperous nominalist couldn't swallow that too.

At the end of this section, Parsons revisits the question of how to frame an eliminative structuralism for arithmetic. He looked at a move from a set-theoretic formulation to a more 'logical', second-order formulation. But could we go first-order, in a way more congenial no doubt to those of nominalist inclinations? The trouble is, of course, that we won't get categoricity (whatever we build into the axioms), so the eliminative structuralist who goes first-order runs up against the intuition that the natural numbers have a unique structure. But how secure, in fact, is that intuition? Parsons raises that excellent question (too often passed over in silence), but only to shelve it until Ch. 8. So we'll have to return to that later.

Monday, July 14, 2008

Telling your epis from your monos.

Ok, so how do you remember which are the epimorphisms, which are the monomorphisms, and which way around the funny arrows get used?

Since the textbooks don't seem eager to offer helpful mnemonics, I offer a forgetful world the following.

It's the LM/PR rule. L-for-left goes with M-for-mono, and P-almost-for-epi goes almost next to R-for-right. OK?

But what does that mean? Simple. A mono is of course a left-cancellable morphism, and you signal one using an arrow with an extra decoration (a tail) on the left. Dually, an epi is a right-cancellable morphism, and you signal one of those using an arrow with an extra decoration (another head) on the right.

Easy, huh? Well, it works for me -- and these days, I'm grateful for all the props I can get ... [As always, click on the image to get a full sized version.]

Declutter your Mac!

This will only be of interest to (a few) other Mac users. But, for what it is worth ...

For years, I've taken the easy option and just installed one version of Mac OS on top of another, and migrated files from one computer to another. And, all credit to Apple, the easy option has worked just fine. Well, almost. Still, there was a lot of legacy software cluttering up my laptop, loads of ancient files buried in the Library, even bits and pieces of OS 9 stuff, and it wasn't always clear what could and couldn't be trashed. And there was a growing number of small glitches (at the level of e.g. some DevonThink scripts not working, Skype always forgetting my account details, a newsreader never quitting gracefully, and so on -- you know, the sort of thing you decide you can live with after you've spent the first hour failing to sort it). But the newest glitch was the new MobileMe sync service just not recognizing the laptop. And unlike the others, this bug was more seriously annoying. So yesterday I thought the time had perhaps come to clean things up and get back to basics.

I took the nuclear option. With some trepidation. So I archived calendars and address book, made a backup of the whole drive (a second proper clone, not a TimeMachine archive), did an erase-and-install for Leopard, and ran the system updates. Moved back Safari bookmarks, address book, calendars (the mail lives on me.com anyway). Installed iLife. Copied back the main documents folder -- which was in any case in a reasonably tidy state -- and the iPhoto library. Installed the latest MacTeX LaTeX distribution. Downloaded the latest versions of NoteBook, DevonThink Pro and SuperDuper (the three bits of non-Apple software I've bought and still make serious use of), and then the free TextWrangler, Camino and Skype.

And that's about it, apart from syncing with my iPod. (If I find I actually need anything else, I'll reinstall it from the backup, as and when. Since you can QuickLook at Word documents, I think I can probably even manage without Open Office.)

It took about five hours in all. Everything is working again now just fine. I have oodles more hard disk space. The little glitches I knew about have disappeared. MobileMe seems very happy. And a lot of other things are just a bit snappier (or is that imagination?). So, it all seems to have been very well worth doing. And the process was painless.

So if like me, you have a cluttered Mac, with annoying little bugs here and there, it really is worth drawing a deep breath, hitting the erase button, finding a good book to read as you watch the progress bars, and putting back together what you actually need. And -- being kinda useful, even if not what you most ought to be doing -- it makes for another great bit of structured procrastination.

Sunday, July 13, 2008

You can fool most of the people most of the time.

I've mentioned before the estimable Ben Goldacre's Bad Science column from the Guardian. In fact, his blog is in the list of links on the left; it is well worth following regularly. But this week's column touches of something of more direct interest to philosophers than usual. Here's an excerpt

In 1973 a group of academics noticed that student ratings of teachers often seemed to depend more on personality than educational content. They wanted to find out how far this effect could be stretched: what if you had an impressive, charismatic and witty lecturer, who knew nothing at all about the subject on which they were lecturing? Could plausibility alone make an audience feel satisfied that they had learned something, even if the information delivered was deliberately inconsistent, irrelevant, and even meaningless?

They hired a large, affable gentleman who “looked distinguished and sounded authoritative”. They called him “Dr Myron L Fox” and he was given a long, impressive, and fictitious CV. Dr Fox was an authority on the application of mathematics to human behaviour.

They slipped Dr Fox on to the programme at an academic conference on medical education. His audience was made up of doctors, healthcare workers, and academics. The title of his lecture was Mathematical Game Theory as Applied to Physician Education. Dr Fox filled his lecture and his question and answer session with double talk, jargon, dubious neologisms, non sequiturs, and mutually contradictory statements. This was interspersed with elaborate diversions into parenthetical humour and “meaningless references to unrelated topics”. It’s the kind of education you pay good money for in the UK.

The lecture went down well. At the end, a questionnaire was distributed and every person in the audience gave significantly more favourable than unfavourable feedback. The comments were gushing, and yet thoughtful: “excellent presentation, enjoyed listening”, “good flow, seems enthusiastic”, and “too intellectual a presentation, my orientation is more pragmatic”.

The researchers repeated the performance. Time and again they got the same result: the third group consisted of 33 people on a graduate-level university educational philosophy course. Twenty-one had postgraduate qualifications. They loved it: “extremely articulate”, “good analysis of subject that has been personally studied before”, “articulate”, and “knowledgable”, they said.

Nobody can check everything, we’re all interdependent for information, and sometimes you might find yourself in a soulful, detached state, wondering whether everything you think you know is grounded in nothing more than a string of half-remembered assertions from people like Dr Fox.
If you want to read the research report, here it is. It is notable, as the original researchers say, that their sophisticated audiences (including those educational philosophers) failed badly as "competent crap detectors".

Which makes you wonder. Philosophers of an analytic stripe like to think that they are rather good at detecting intellectual rubbish. But how competent are we really? Still, perhaps one moral to be drawn is that the extended, vigorous, no-respecter-of-persons, test-to-destruction, highly sceptical, all-in-intellectual-wrestling, with which visiting speakers get mauled at least at some UK philosophy departments (Moral Sciences Club, anyone?) does serve an essential intellectual function. It makes it less easy to get away with the crap.

Friday, July 11, 2008

A Tuscan wine list ...

Before it all becomes too distant, a few -- ignorant and purely subjective! -- wine memories from our Tuscany trip, mostly local wines from around Castelnuovo Beradenga. Quite a few of these wines are available from good merchants in the UK and USA, so these notes aren't just of idle interest. Do go and indulge! The stars -- as in (*) -- represent the number of bicchiere in the Gambero Rosso wine guide. One star is pretty good, and three is a classic.

  • Fèlsina, Beradenga Chianti Classico '05 (*). Still a bit closed(?) but opens up nicely after a few hours. I can get this in Cambridge and maybe I'll put a few bottles under the stairs for a while. (Felsina's recent top wines are by all accounts amazing, but we didn't splash out this trip. I was going to say that this is their entry level wine. But actually, you go round the back of their winery, and can get last year's unbottled at 1.80 euro a litre into your plastic box, and that's pretty good too!)
  • Fèlsina, Beradenga I Sistri '05 (*). Their chardonnay: very different from New World chardonnays and indeed from French ones. But I thought the '04 we had last year was better. This is just a bit too heavy perhaps with surprisingly little nose. (But I've bought another bottle here, just to check, you understand ...)
  • Poggio Bonelli, Chianti Classico '01 (later years get * or **). This was recommended by our local restaurant, and comes from just down the road. Inexpensive but perhaps the best Chianti we drank all month. The bottle age made it very rounded, almost unusually smooth for sangiovese, without losing character. Excellent!
  • San Felice, Chianti Classico '05 (*). Rather undistinguished, I thought, though others thought it better of it. Maybe I was just getting picky.
  • San Felice, Il Grigio, Chianti Classico Riserva '04 (*). Rather better but again I wasn't particularly impressed.
  • San Felice, Pugnitello ['04 I think]. Now this was something else. "Rediscovered" old Tuscan grape-variety. Quite excellent. Purple, complex, very full in the mouth, but not overwhelming. Very drinkable!
  • Ricasoli, Castello di Brolio, Chianti Classico '04 (***). Very good indeed. A quintessential "modern" Chianti. (I suppose you might say it was a bit "middle of the road", but it has enough character and texture -- and I bet will be terrific in a few years).
  • Dievole, La Vendemmia Chianti Classico '05 (*). Gambero Rosso says "easy drinking", and yes, it was. Good for a light meal.
  • Dievole, Broccato ['04 I think] (*). This is a sangiovese blend, much fuller bodied. I think the Gambero Rosso underestimated this. Excellent for a heavier Tuscan meal! (An honourable mention too, by the way, to Dievole's Rosato, which is terrific hot-weather quaffing wine -- which we'd have drunk more of if the weather had been better.)
  • Villa Arceno, Chianti Classico '05 (*). This is the really local wine, which our restaurant gives you as their wine-by-the-glass. Nothing outstanding, but as-it-were essence of good-ordinary-Chianti.
  • Lornano, Commendator Enrico '04 (**). Sangiovese/merlot which we usually drink at the Bottega di Lornano. Seriously good for accompanying Tuscan-style food.
  • Castello del Terriccio, Lupicaia '04 (***). No. Philosophers aren't paid that much. This was by courtesy of a very generous son-in-law! Even so young was sumptuous. Classic. Words fail. And in a few more years must be unbelievable. (Drank this at Bottega del 30, surely one of the best restaurants in the world, just a couple of miles away. Sigh.)

Parsons's Mathematical Thought: Secs 8 - 11

Back, after rather a gap, to Charles Parsons's book and on to the first half of his second chapter, "Structuralism and nominalism".

(Sec. 8) Parsons says that he himself thinks that "something close to the structuralist view is true". But structuralist in what sense? It is often said, perhaps in a Bourbachiste spirit, that mathematics is the study of structures. But -- as Parsons stresses -- that leaves it wide open what picture we should adopt of the ontology of mathematical objects. He is more concerned with structuralism(s) with more ontological bite -- something along the lines suggested by "the objects of mathematics are positions in structures, [and] have no identity or features outside of a structure" (to quote from Michael Resnik's well-known 1981 Nous paper).

(Sec. 9) But what are structures? The usual modern mathematical story sees these as sets (or classes) with distinguished elements, equipped with relations and/or functions. So it looks as though an account of mathematical objects as positions in structures already presupposes familiar kinds of objects (sets, classes) to build structures out of, and explaining their nature in structuralist terms threatens circularity. But Parsons puts this worry on hold for the moment.

(Sec. 10) So go with the set-theoretic conception of structure, just pro tem, and consider as an exemplar Dedekind's treatment of the natural numbers. Dedekind defines what it is for a set N, with distinguished element 0, and a mapping S: N -> N - {0} to be "simply infinite". Abbreviate those (categorical) conditions Ω(N, 0, S). With some effort, an ordinary statement of arithmetic can be correlated with a version A(N, 0, S) whose primitives are again N, 0, S. And on one reading of Dedekind -- the eliminative reading -- the suggestion is that the ordinary statement can be treated as elliptical for

For any N, 0, S, if Ω(N, 0, S) then A(N, 0, S).

This is 'eliminative' in that a statement apparently about one kind of thing, numbers, is treated as in fact a disguised generalization about other kinds of things. The suggestion neatly sidesteps "multiple reduction" problems for more straightforward attemps to reduce arithmetic to set theory. But (on the face of it) it faces the worry that if there are no simply infinite systems then any ordinary arithmetical statement comes out as vacuously true and arithmetic is inconsistent. True, that first worry won't be pressing if we already buy into a background universe with enough sets, but it will become more urgent when we try to repeat the trick and give an eliminative structuralist account of them. And there's a related second worry. Ω(N, 0, S) will involve quantification over sets, as indeed will a typical A(N, 0, S) as we give explicit definitions of e.g. recursive arithmetical functions. Do we want really want a structuralist account of a particular familiar kind of mathematical object, numbers, to tells us that we've been generalizing about some other rather less familiar kind of object all along? (Parsons wonders: Maybe we need to generalize over structures to state structuralism as a general thesis: but does a structuralist account of a particular kind of object have to similarly generalize over structures?)

(Sec. 11) Well, we can sidestep the second of those worries, and the worries of Sec. 9, perhaps, by trading in an explicitly set-theoretic presentation of Dedekind's eliminative structuralism for a version couched in second-order logical terms. We get a new second-order definition of being simply infinite, Ω'(N, 0, S), a new correlate of an ordinary arithmetical claim, A'(N, 0, S), and correspondingly a new suggestion that the ordinary statement can be treated as elliptical for

For any N, 0, S, if Ω'(N, 0, S) then A'(N, 0, S).

where now 'any N' and 'any S' are treated as second-order. If we are relaxed enough about second-order quantification, we might find this easier to swallow that the previous version (though that's quite a big "if"). However, this kind of 'if-thenism' is still threatened by the possibility of vacuity. What to do?

One option is to read the conditional as stronger-than-material, e.g. by discerning a governing modal operator. But that opens up another set of problems. What kind of modality is involved here? Can we e.g. give a modest possibility-as-consistency reading? Perhaps "we interpret the theories in an if-thenist way, but deal with the problem of possibility by appealing to consistency, nominalistically interpreted." The suggestion is to be pursued critically in Sec. 12.

OK, so much by way of brisk summary of these sections (I didn't find them entirely easy to follow, but I hope I've fairly represented the way the discussion develops). I don't think I have much to add by way of commentary: in fact, the dialectic so far is a pretty familiar one.

Saturday, July 05, 2008

Geektastic: Finite Simple Group (of Order Two)

Luca Incurvati gave me a link to this (you see, it's non-stop serious work in the philosophy grad. centre). Yep, ok, it seems to be have been around for a good while: but I've only just seen it, and guess that it might be new to someone else too. Enjoy!