Logic Matters

While I’m in Christchurch, we’re having a reading group on Charles Parsons’s Mathematical Thought and Its Objects. I’ve blogged about this challenging book before, when we had a reading group in Cambridge. But, with inspiration from new NZ colleagues and some very helpful emails from Charles Parsons himself, I’m having a bash at rewriting (some of) my earlier notes, and trying to come to a better understanding of both the book and the topics it is discussing.

So, for anyone interested, here’s my revised comments on the first chapter.

I have to start singing for my supper today, and earning my keep as an Erskine Fellow. This  means giving a dozen lectures on Gödel, plus a few talks to various seminars. The Gödel lectures kick off today, and I’ve started updating/extending the notes to accompany the lectures. The notes are now called — perhaps more honestly — Gödel Without (Too Many) Tears. The first instalment of the new version can be found here, where further episodes will be posted at the rate of two a week for the next six weeks.

The guidebooks say that the four hour train journey from Christchurch, up through the Southern Alps over Arthur’s Pass and on down to Greymouth is one of the great railway journeys of the world. The guidebooks are right. Especially when the weather is a perfect summer’s day.

Another rainy day   — so we made a first foray to the rather impressive Christchurch Art Gallery.

The building is terrific: it sits handsomely into the surroundings, without shouting its modernity too loudly. And inside it is just wonderfully light and airy.

The contrast with the modern buildings of the Los Angeles County Museum of Art couldn’t be greater. Those are just depressingly blocky, and thoroughly unwelcoming: inside the galleries are bleak and barnlike. Here, the curves of the building draw you in and the foyer is a delightful space: and galleries are divided into rooms on a human scale where you want to linger.

There couldn’t be much more contrast between the curating, either. LACMA has some wonderful pictures. But they are, so to speak, just plonked on the walls, in a take-it-leave it spirit, without description or scene-setting or comment. Christchurch of course has to try a whole lot harder, as the collection lacks the Rembrandts and the Picassos. But the result is that it thoughtfully displays what it does have with intelligent and illuminating commentaries. Or it puts together little groupings of old and new, of New Zealanders visiting Europe and Europeans visiting New Zealand, of western-influenced and Maori-influenced in ways that make you look. (Though some of the really contemporary stuff was the usual pseud-ery, worth about twelve seconds contemplation.)

So we’ll certainly be back to do more of the galleries the next wet day: and the restaurant wasn’t bad either!

The view from my office in Christchurch is quite terrific — over the wonderful trees on the campus out towards the Banks Peninsula (I think!). The best office view I’ve had since the long off days in the Hugh Owen Building in Abersytwyth where I had a panoramic view looking over Cardigan Bay.

It’s the beginning of the academic year here, and the campus is overflowing with lost-looking freshers. Somehow they look younger (and seem to be trying less hard to be cool) than their Cambridge equivalents. Almost deserted when we arrived last week,  there’s now a nice buzz about the place.

NZ coffee culture note. More or less every place we’ve drunk coffee so far has had what looks like a proper espresso machine. And suitable tamping and steaming and frothing all seems to be going on. So you take ceremonious delivery of what is supposed to be an espresso macchiato. But not once — yet — have I had anything that would pass muster as even an approximation to what you get in the humblest corner cafe bar in Italy. Very odd. But I’m reliably told that the bar opposite the art gallery does the real thing: so here’s hoping …

A surprising coincidence: Christchurch is 43° 32′ south, Siena (where we’ve spent a lot of time in recent years) is 43° 33′ north. The latitude explains why, when the sun does come out, it is burning hot. But since we’ve been here it has mostly been overcast, and the temperature then drops sharply.

We’ve been consoling ourselves for the lack of summer weather by sampling bottles of Otago Pinot Noir. The local and rather impressive supermarket has some pretty upscale bottles if we can trust these ratings: so far, though, the bottles we’ve tried have been quaffable rather than outstanding — and seemingly all put on the market too young. When I’m feeling zippier, I’ll have to find a wine shop with earlier vintages on sale: but at the moment, we’re still just a touch jet-lagged and slightly weary.

Hence concentration hasn’t been terrific for work. But the last day or so, to get into the swing again, I’ve been re-reading Smullyan’s First-Order Logic. I’ve been struck once more what a fantastic book that is (and three cheers to Dover for republishing it too): if you don’t know it, and are logically inclined, put it on your “must read” list.

There couldn’t be much greater a contrast between LA and Christchurch — and, for us, the differences all favour Christchurch, hands down. It is human-scaled, wonderfully green, relaxed, and somehow comfortingly familiar in its almost-Englishness.

Though there are more than enough small differences to savour. For example, the early settlers obviously didn’t mess about when clearing the native bush. So the great park near the centre of Christchurch could have been transported from an English city, with avenues of planes, and great oaks and limes (and there are mallard ducks on the ponds, and house sparrows hopping around as you sit outside a cafe). But, for all that, as you walk through the shade on a sunny summer’s day, the racket of the cicadas sounds entirely unEnglish.

It is nice, too, to be understood first time in shops and cafes (LA was occasionally  hard work — The Daughter reports the same from Miami, where she has to get her Italian partner to arrange complicated things as the locals find his English a lot easier to understand). And what they say about Kiwi friendliness is all true.

I think I’m really going to enjoy it here.

Even after two full days at the Getty, we were very tempted to spend a third there. But we were spending an arm and a leg on taxis, so we decided to try the Los Angeles County Museum of Art instead, which was within walking distance. Their collection is quite astonishing — from some terrific Dutch landscapes and three more Rembrandts, through twenty five Picassos, to some appalling kitsch by the egregious Jeff Koons. And turning a corner, there was that old philosophers’ favourite, Magritte’s La Trahison des images. Not, of course, a case where you get anything particular out of seeing the original, but a nice surprise all the same.

The galleries themselves are rather bleakly unwelcoming and unhelpful. It doesn’t seem to have occurred to the curators that you might welcome a bit more information than title, artist, country and date (or even that you might occasionally want to sit down). The comparison with the feel of the Getty couldn’t be more marked. But the collection is still very much worth a visit.

Style note #1. To get to the LACMA we walked through some back streets full of small houses. English half-timbered cottages, mini-castles (with battlements), miniature arts-and-crafts houses, mexican shacks, sea-side-bungalows, all jumbled together. The air of good-humoured individualism gone slightly mad was, however, rather spoilt by the little signs planted in the front gardens by the owners threatening armed response if you mess with them.

Style note #2. Once upon a time, as the galleries remind you, children were dressed like little adults. Nowadays, looking at our fellow visitors, most American men — at least when not at work — dress like big children (from the playshoes upwards). I wonder what that is all about?

Hermitage Garden, Maison Rouge 1877, by Camille Pissarro

We’ve been in LA for two days now, stopping over en route to NZ. Or more exactly, we been visiting the Getty Museum for two days, pretty much from opening time to closing time. We’ve been bowled over.

The buildings, the setting, the views over the city to the sea, are breathtaking. For a collection put together relatively recently, there are some quite wonderful paintings on display (including Rembrandts and Titians). And there’s a simply fantastic, and very illuminating, special comparative exhibition of drawings by Rembrandt and his pupils.

But if we could smuggle just one picture home from the Getty, I think it would just have to be Pissarro’s moment of domestic peace in a summer garden.

I’d hoped to have time, before setting off for a “term” (i.e. half a semester) in New Zealand, to comment in a bit of detail on Berto’s discussion of the dialetheist riff on Gödel’s theorem. But that’s not going to happen. And since I’m not going to pack Berto’s book, I’m afraid — if I ever get round to commenting at length — it won’t be for nearly three months.

The trouble is, however, — and no doubt this is the reason I’ve not got round to doing the job before — I do find it jolly difficult to take dialetheism here seriously.

This isn’t to dismiss dialetheism out of hand, across the board. Perhaps there’s a just-so story to be told about how, when we add a minimalist truth-like predicate to a language without prior explicit semantic apparatus, the smoothest thing to do — all things considered — allows some extraordinary sentences containing this new predicate to then come out both “true” and “false”. So be it. But the dialetheist line on Gödel incompleteness, when the wraps are off, is committed to saying that there’s a number (an ordinary, common-or-garden, natural number) which both does and does not satisfy some primitive recursive condition (a complicated condition, to be sure, but still primitive recursive in an entirely straightforward way). Here’s a sketch of why, in my words.

Recall: the Routley/Priest suggestion is that our overall informal mathematics — the body of assumptions and deductive processes that mathematicians take take to lead to proofs that establish mathematical truth — should be susceptible to being regimented as a recursively axiomatized theory T (recursively, because negotiable by us limited humans). But T is consistent (because a body of truths) and includes enough arithmetic for Gödel’s theorem to apply. So, fixing on a scheme of Gödel-numbering, there is a Gödel sentence G, true if and only if unprovable-in-T, which is indeed unprovable-in-T, and hence true. In principle, we could spell out that informal reasoning for the truth of G in our all-embracing theory T which, by hypothesis, includes all informal mathematics. So there’s a T-proof of G, which will have Gödel-number g. But, as is familiar G (truly) “says” that no number numbers a T-proof of G. So g is also not the number of a T-proof of G. But numbering a T-proof of G is a primitive recursive property.

That conclusion — that there’s a number  which both does and does not satisfy some primitive recursive condition — I, for one, just find incomprehensible.

“But an incredulous stare is not an argument!” Indeed. But I’m not incredulous in the sense of understanding what is being said to hold, but then treating the suggestion as beyond belief (“Another concrete world, as real as this one,  in which there are talking donkeys? Come off it, David, pull the other one!”). My trouble, to repeat, is that I just don’t understand what it would be for a perfectly ordinary number both to satisfy a primitive recursive condition and not to satisfy it. Not so much incredulous stare as incomprehending boggle.

Now I don’t pretend that that’s the end of the matter. But it is, so to speak, the beginning of the matter. And I guess my main complaint about Berto in this chapter is that — although he cheerfully takes the dialetheist to be committed in the way I’ve described — he doesn’t get far enough past the beginning and  explain what that commitment could mean. True, Berto says something briefly about strict finitism — but that’s not going to help, unless we have a proof that, whatever the system of Gödel numbering in question, the relevant number g will have to beyond a sensible finitist’s ken (so can be treated as an “inconsistent number”). And I don’t see any reason to suppose that’s true. So I’m left boggling! As will, surely, be most of Berto’s readers.