Logic Matters

Smorynski on Hilbert's Programme

2008-07-23T08:44:00.006Z

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.)

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

2008-07-21T08:23:00.005Z

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."

Once upon a time ...

2008-07-19T14:07:00.004Z

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.)

Logic's Lost Genius

2008-07-18T18:47:00.006Z

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

The Concept of Inﬁnity in Mathematics (item #6 in Szabo's Collected Papers),
The Concept of Inﬁnity and the Consistency of Mathematics,
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.

What language is this?

2008-07-17T09:02:00.004Z

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?

Parsons's Mathematical Thought: Sec. 12, Nominalism

2008-07-16T14:41:00.005Z

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.

Telling your epis from your monos.

2008-07-14T18:11:00.007Z

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!

2008-07-14T08:33:00.006Z

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.

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

2008-07-13T09:11:00.004Z

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.

A Tuscan wine list ...

2008-07-11T15:46:00.007Z

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

2008-07-11T10:56:00.004Z

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.

Geektastic: Finite Simple Group (of Order Two)

2008-07-05T14:54:00.004Z

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!

An Introduction to Gödel's Theorems revamped!

2008-06-30T12:54:00.004Z

I've just, at last, got hold of a printed copy of the revised version of An Introduction to Gödel's Theorems which came into stock with CUP about six weeks ago. Of course, as soon as it went to press, three or four people told me of errors that still need correcting! -- but this version has significantly fewer typos, and also a small number of passages have been improved. The corrections page at the book's website will tell you what's been changed, and what still needs changing.

(So, tell your libraries that they just must have the shiny new improved version! And to answer Richard Zach's question -- the quick way of telling the versions apart is to glance at the imprints page, i.e. the verso of the title page. The later version notes, halfway down the page, "Reprinted with corrections 2008''.)

Galois connections again

2008-06-28T21:38:00.007Z

For some reason, I found myself minded to spend a bit of time tidying up the pages on posets and Galois connections that I was working on some weeks ago. They have the form of the first two chapters of something longer, something that may or may not get written -- but these pages should have some stand-alone interest. Here's the new version. As always, comments gratefully received. (Thanks so far to Tim Button and Luca Incurvati -- and in particular to Nathan Bowler who gave a talk in which he explained the basic idea of syntax/semantics as a Galois connection, which inspired me to write these pages.)

[Later: Monday 30th June] And that link is now to a better version, with some paragraphs about quantifiers as adjunctions, and some superfluous material removed for later use.

The Review of Symbolic Logic

2008-06-25T20:01:00.002Z

The first issue of The Review of Symbolic Logic is out and available online (if your library has the right subscription). And two Cambridge friends have papers in it, both on set theory. Luca Incurvati has a piece on Kripke semantics in set theory, and Thomas Forster discusses the iterative conception. Excellent stuff.

Conference: Computation and Cognitive Science

2008-06-25T09:32:00.002Z

There's an upcoming conference in Cambridge on Computation and Cognitive Science with an impressive line-up. The plan is for the papers to be made available in advance, for there to be no formal presentations, but for the sessions to be devoted to discussion. The papers are beginning to appear online here.

Back in Cambridge. Sigh.

2008-06-21T16:40:00.009Z

So this is the borgo we've left behind for a while (the little white patch in the centre distance in the centre of the photo is the distant dome of the duomo in Siena, or rather its covers during restoration works). Ah well. Back in September, we hope. It has taken us a few days to re-adjust to Cambridge (which was particularly grey and damp when we returned). But the sun is out today, and the place is almost at its best, so that is cheering. Logic postings will resume here in the next day or two, now I'm back in the mood again!

Tripos results came out, both for philosophy and for maths, just as I got back, posted on the Senate House boards. It was good to see that all the names I looked for of students I thought ought to get a first were there in the right class. Justice, of course, is always done ...

Postcard from Siena - 8

2008-06-16T11:08:00.000Z

The meteo predicts that really good weather will start on Thursday. Since we are leaving on Wednesday, this is just a bit galling. This morning it was so cold we put the heating on again. And jazz last night in the little village piazza under our window was good, but not the balmy June night under the stars we might have expected, and the well-wrapped-up audience was understandably a bit thin.

Siena itself is like Cambridge at least in this respect: the tourists tend to stick to a small part of the city. So it can be very busy round the Campo and the Duomo. But other sights, even those the guide books warmly praise, can be more or less deserted. We did make one nice discovery a couple of days ago when it was dry in the afternoon. We found ourselves at the botanical gardens which we'd never visited before (and, predictably, they were more or less empty of people). They are very fine, cool under the trees, tumble down a steep slope, and so the views out of the city are beautiful. Recommended.

Awodey's Category Theory: Ch. 2

2008-06-12T14:03:00.003Z

The second chapter of Awodey's book is called 'Abstract Structures'. It gives the usual abstract category-theoretic definitions of epis and monos, of sections and retractions, of initial and terminal objects, of products, and so on. This would certainly be tough going if it was the first time you'd ever encountered these notions. Even as revision/consolidation it's a bit of a bumpy ride. But for all that, I did get a fair bit out the chapter (Awodey's clusters of illustrative examples can be very illuminating).

One query. In the sections on products, Awodey starts carefully, talking of a product as an object together with a pair of arrows, and rightly referring to the object A x B as part of a product. And mostly what he says about products reflects this understanding of what products are. But on p. 42 he says that any object A is the unary product of A with itself one time. Is that right? The unary product is surely not just the object but the object with its self-identity arrow.

And one suggestion. The first stage of the two-stage proof at the top of p.27 is surely unnecessarily. Just start in f(-n) = f(-n) * u = f(-n) * g(0) = f(-n) * g(n + -n) etc. [Actually the first stage too illustrates one of Awodey's quirks, a tendency to occasionally slightly abuse notation without explanation.]

Postcard from Siena - 7

2008-06-12T08:04:00.004Z

Here, everyone has to park outside the walls of the old part of the borgo. But that's no hardship. There's stone and gravel put down between the olive trees just under the house, and you park the car among them, leaving it to quietly admire the views for miles over the hills.

The trees have been brutalized since last year, obviously scaring the living daylight out of them, and as a result they are beginning to fruit like mad. I can report that the local olive oils vary, but from merely very good indeed to the amazing. (And judging from the ages on the gravestones in the village cemetery, they must have magically life-extending properties.)

Parsons's Mathematical Thought: Sec. 7

2008-06-11T15:06:00.000Z

Back in Sec. 1, Parsons says "Roughly speaking, an object is abstract if it is not located in space and time and does not stand in causal relations." In the last section of the first chapter, he returns to question of characterizing abstract objects, and suggests a distinction among them between pure abstract objects (e.g. pure sets) and those which "have an intrinsic relation to the concrete" -- Parsons calls the latter quasi-concrete.

As a paradigm example of the quasi-concrete, Parsons takes the example of sentence types: "what a sentence [type] is is a matter of what physical inscriptions are or would be its tokens". (Actually, just as an aside, I suppose we might wonder whether sentence types might be a counter-example to the claim that abstract objects lack temporal location. We might ask: did the sentence type "the cat is on the mat" really exist in 2000 BC before anyone spoke English?)

But how should we generalize from this case? Parsons writes "What makes an object quasi-concrete is that it is of a kind which goes with an intrinsic, concrete 'representation'". The scare quotes are there in Parsons -- and you can see why. Should we really say, for example, that a sentence token is a representation of its type? Your first response might be: the token isn't about the type, so isn't a representation of it. But, reading on, it becomes clear that Parsons doesn't mean representation but representative. And then, yes, we might say that the token is a representative of the type. Parsons also writes "Although sets in general are not quasi-concrete, it does seem that sets of concrete objects should count as such; here the relation of representation would be just membership." (no scare quotes!). Again, we might say the spoon in my coffee cup is a representative of the set of cutlery (though not a representation).

How clear is the idea of "having a concrete representative"? You might have supposed that the Earth's equator is a candidate for belonging with sentence types as tangled with the concrete. But does the equator have a concrete representative? Could it? What about that old Fregean example, the direction of a line. Of course there can be physical lines with that direction; but it doesn't seem quite natural to me to say a particular line is a representative of the direction. (We might say the equator or a direction could have a representation, painted on the ground!)

Parsons's discussion here thus seems to me to be rather undercooked. To be sure, it is plausible to say that some abstract objects are more purely abstract than others, but I don't think he has given a sharp characterization of the phenomenon.

But let's go, for the moment, with his notion of the quasi-concrete. Then he raises the question, are numbers quasi-concrete? We might be tempted to say yes, suggesting that the number five, for example, has the concrete representatives like: ||||| . Parsons makes two Fregean points against this. First, to take that block as representative, we have already to take it as a set or sequence of strokes (rather than as a single grid, for example). So the representative here is not strictly concrete but itself quasi-concrete. Perhaps then we can say that numbers are quasi-quasi-concrete (meaning they have quasi-concrete representatives). But second, that can't be the whole story, as numbers can number anything, including the purely abstract. (Parsons says he is going to return to talk about this in Chapter 6, so I'll say no more for the moment.)

Postcard from Siena - 6

2008-06-10T10:06:00.000Z

It's sunnier and warmer (for a while). This year the excellent restaurant just a few steps across the piazza has put a few tables outside, and will bring you a coffee and cornetto from when they open up in the morning, or an aperitivo in the afternoon. A great idea, but so far the weather has been such that we've only made use of it a few times. But this morning, sitting in the sun at half-past nine, it was already pretty hot. At last.

Parsons's Mathematical Thought: Sec. 6, 'Being and existence'

2008-06-10T06:43:00.002Z

At the outset of this section, Parsons writes that one point at which "reservations about standard first-order logic as the universal measure of ontology can affect the notion of mathematical object is the ancient question whether reference to objects is necessarily reference to objects that exist."

A comment before proceeding. Note that Parsons had earlier (Secs 1 to 4) proposed that (1) "speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification" to make serious, and indeed true, statements. And defending that view about, so to speak, the measure of what objects we are committed to falls short of saying that (2) standard first-order logic is the universal measure of ontology in general. Resisting the more sweeping claim is quite consistent with accepting Parsons's initial Fregean claim about objects. Not that I'm suggesting that Parsons thinks otherwise. I'm just emphasizing that if (e.g. as a Fregean) you are not persuaded by Parsons Sec. 5 suggestions, and hold that we are committed to entities that are not objects, then you can accept formulation (1) without accepting (2).

Anyway, what of reference to objects that is not reference to objects that exist? Parsons discusses Meinongian views in some detail (this is one of the longest sections in the book). Here's part of his final summary of the discussion.

We are left with the question whether the "true" meaning of the existential quantifier is [i] the permissive Meinongian one [allowing quantification over objects that do not exist], [ii] existence that allows freely for abstract objects but that rules out impossibilia, or [iii] something like actuality. The logic based concept of object does not decide between these alternatives, although, once it has been set forth, the case for [iii] is weakened. But in order to understand the notions of object and existence in mathematics we have to put more flesh on the bare form given by formal logic. We need to fill out the logic-based conception by looking at cases. ... [C]onsiderations proper to mathematics will not lead us to favour [i] over [ii]. General as the notion of object in mathematics is, there is still a constraint of possibility, coherence, or consistency that objects postulated in Meinongian theories are allowed to violate.

The talk here of having to "fill out the logic-based conception" might initially seems surprising given what has gone before. But, though he is not entirely clear, I assume that what Parsons means is simply this: the Fregean thesis is that objects are just whatever are we have to construe terms that behave in the right sorts of way in true sentences as referring to. So, to fill out that general template view about objects, we have to say what kinds of sentences we do in fact accept as being true. If we e.g. take statements like "Sherlock Holmes is more famous than any living detective" and "There's a fictional detective who is more famous than any living detective" at face value as true claims then (the suggestion goes) we have to accept (i) the Meinongian line that there are objects that do not exist. If we paraphrase away apparent talk of fictional objects and the like, but accept that there are true mathematical statements talking of numbers, sets, etc., then (ii) we are not committed to non-existent objects, but have to accept that there are abstract objects which aren't "actual". If we insist on also paraphrasing away apparent straight talk of numbers (e.g. construing it as governed by an operator "in the arithmetical fiction ..."), then perhaps (iii) we may only be committed to actual objects.

Parsons is sceptical about whether we have any need "to admit into the range of our quantifiers such objects as the golden mountain, the round square, Pegasus and Sherlock Holmes", though it is not his concern to argue for this here. But he does argue that "considerations proper to mathematics" don't give any impetus for preferring the Meinongian views (i) over (ii). Mathematics doesn't countenance impossibilia like the round square, or present itself as fictional discourse. As to (iii), I assume Parsons thought is that a critic of our common-or-garden standards of mathematical truth on the basis of a metaphysical repudiation of abstract objects is (in danger of) getting things upside down, at least by the lights of the truth-first, "logic-based conception" of objects, according to which we don't have a handle on the notion of an object except via a prior grip on the notion of truth for the relevant object-referring statements.

If this reading of Parsons is right, then I agree with him.

Postcard from Siena - 5

2008-06-08T14:49:00.003Z

We normally never watch breakfast TV, but here we have the excuse of trying to pick up more Italian: and actually it isn't at all bad. The weekend show we watch has a nice slot visiting different places around Italy and talking at length about their local produce, and demonstrating a characteristic recipe. That -- followed by walking through the woods onto the estate of Villa Arceno and alongside their vineyards -- worked up appetites for Sunday lunch at a favourite restaurant, La Bottega di Lornano. But by then the weather was getting too threatening again to eat outside (even under their big awning). Still, a terrific meal as always, in Tuscan quantities, and we drank a favourite wine, Dievole's Broccato. Prices in Italy are going up, and the pound is going down against the euro, so this is not quite the stunning bargain it would have seemed three years ago. But we still ate much better for less than the cost of a second-rate chain restaurant meal in England. Which is why we very rarely bother to eat out at home.

Awodey's Category Theory: Ch. 1

2008-06-08T14:47:00.001Z

I bought Steve Awodey's book Category Theory (Oxford Logic Guides, Clarendon Press, 2006) when it first came out. Awodey says that his book is aimed, inter alia, at "researchers and students" in philosophy; I'd been impressed and intrigued by a couple of his lucid contributions to Philosophia Mathematica, and had hoped for an equally approachable book. But, whatever Category Theory's virtues, easy approachability isn't one of them, and after reading a fair bit of it, I had to put the book aside for when I had enough time to work through it again more slowly. At last, I've got back to it, and I'll give some reactions here.

I have to say immediately (as in fact I said here before) that I can't imagine that there are many philosophers who would be equipped to dive straight in and cope with this book. Meeting Cayley's Theorem (about representing groups as permutations on sets) at p. 11 or free monoids at p. 16 is going to be quite a challenge to those without a background in mathematics. It isn't that those ideas are intrinsically very difficult; but you surely won't grasp their point or feel comfortable with the ideas just from their brisk presentations here. Likewise, I bet no one will understand Remark 1.7 (p. 12) on concrete categories who hasn't already met the idea of "test objects" from elsewhere. By the time the reader gets to the first example of a "universal mapping property" at pp. 17-18, most philosophers surely will be floundering: Awodey's explanations of what is going on are too terse to help the not-so-mathematical. And things seem only to get worse as the book progresses. I'm pretty sure, then, that this book wouldn't work as a first introduction to category theory e.g. for philosophy graduate students interested in logic and the philosophy of maths (unless they have an unusually strong background in pure maths already). Although Awodey says in the preface that, if Mac Lane's book is for mathematicians, his is for 'everyone else', in fact Category Theory is actually orientated to students who are, as they say, 'mathematically mature'.

So, from now on, I'll be taking the book as in fact operating at (so to speak) a level up from the one Awodey says that it is designed for, i.e. as a follow-up text for mathematically ept readers, to read after mastering e.g Lawvere and Rosebrugh's Sets for Mathematicians -- a follow-up which starts again from scratch to consolidate some basic ideas and then pushes things on deeper and further.

How does the introductory first chapter work on this level? Well, to be frank, still not entirely brilliantly. For example, the whys and wherefores of the first example of a universal mapping property are not really explained that well (nor why we should be particularly interested in free categories). However, on the other side, I like the way that the idea of a functor between categories is introduced early; and some of the illustrative examples of categories and functors between categories in the chapter are illuminating. And the idea of "forgetful functors" comes across nicely.