Peter Smith's Blog, page 130

Four lectures on the First Incompleteness Theorem

Better late than never — what I said, more or less, in four lectures earlier this term on the First Incompleteness Theorem. The lectures were to maths students, but only the last of the four requires a bit of background in computability theory.

IGT, 2nd edn again — and an unofficial bribe!

As I said in a post here a couple of weeks ago, I'm negotiating with CUP about a second edition for An Introduction to Gödel's Theorems. (And if you have any suggestions/comments about what improvements you'd like to see, do please either email me, or comment here.)

Now, to kick start the process, I need to give CUP a few names of people who might advise on the (currently pretty unspecific) proposal for a new edition. So if you use (and like!) the book for teaching, and would be prepared for CUP to contact you to comment briefly, could you please email me? I'd be most grateful. [The Press do tend to pay for such help not ungenerously, in nice shiny new CUP books of your choice, not of course that you'd be swayed by that thought ...]

Thin Realism, Arealism, and other Big Ideas

Penelope Maddy's recent Defending the Axioms is my sort of book. It is short (150 pages), beautifully clearly written (if there are obscurities, they are in the philosophy, not the prose), and I'm in fact rather sympathetic to her overall approach (I share her doubts about 'First Philosophy', I share in particular her doubts about the force of supposedly a priori arguments for nominalism and against abstracta, and I rather like her post-Quinean conception of the task for the 'Second Philosopher').

I'm not sure, however, that there is much in the book that will be a major surprise to the reader of Maddy's previous two books. Still, the brevity and the tight focus 'On the Philosophical Foundations of Set Theory', to quote the subtitle, might help to make her ideas available to new readers daunted by the length and sweep of Second Philosophy. How well does Maddy succeed in doing that?

I've a lot of questions. The book is dedicated to 'The Cabal' of Californian set theorists. And — talking things over with Luca Incurvati — one interesting issue that came up is whether Maddy's vision of the working methods of set theorists isn't rather skewed by a restricted diet of examples from her local practitioners. But I'll leave that issue for Luca to pursue, even though it is more interesting than my topic here: for he knows a heck of a lot more about the ins and outs of recent set theoretic practice than I do. Here, rather, I'm going muse about the central metaphysical theme in the book: does the book succeed in selling Maddy's story about that?

Maddy discusses two positions that she suggests are prima facie open to those with no first-philosophical axes to grind. One she dubs 'Thin Realism'. Starting from the mathematized science of the early nineteenth century, mathematical ideas came to be pursued more for their own sake, leading inter alia to the great unifying endeavour which is set theory. This is hugely productive of ideas and results and is mathematically deep. If you are not already in the grip of some extra-mathematical prejudices, what's not to like? So as good second philosophers we should say that there are sets, and set theory tells us about them — and also that there is no more to be said about them than what set theory says (other, perhaps, than negative things such as that they aren't already-familiar bits of the causal, spatiotemporal, order). A gung-ho 'Robust Realist' is tempted to say more: she is gripped by an extra-mathematical picture of the sets as genuinely 'out there' quite independently of the natural world, forming a parallel world of entities sitting in a platonic heaven, with a great gulf fixed between the mathematical abstracta and the sublunary world. The harder she pushes that picture, the tougher it is for the Robust Realist to account for why we should think that the methods we sublunary mathematicians use should be a reliable guide to the lie of the land beyond the great gulf. It can then seems that our methods need backing up by some kind of certification that they will deliver the epistemic goods (and what could that look like?). Maddy, by contrast, thinks that the demand for such a certification is misplaced: why — as a second philosopher, working away in the thick of our best practices in science and maths — suppose that perfectly standard mathematical reasoning should stand in need of the sort of external supplementation that a Robust Realism seems to require?

Now, this might make it sound as if Maddy's Thin Realist is going to end up with the sort of thin line about truth that we find e.g. in Wright. Then the thought would be: here is a discourse in good order, with appropriate disciplines and standards for making moves within it, so we can construct a minimalist truth-predicate for it. So we can not only say e.g. that the set of natural numbers has a powerset, but that it is true that there exists such a powerset, etc. However, this quick route to a thin realism isn't Maddy's line. Indeed, she explicitly contrast her Thin Realism with Wright's minimalism:

Wright's minimalist takes set theory to be a body of truths because it enjoys certain syntactic resources and displays well-established standards of assertion that our set-theoretic claims can be seen to meet: the idea is that a minimalist truth predicate can be defined for any such discourse in such a way that statements assertable by its standards come out true. In contrast, the Thin Realist takes set theory to be a body of truths, not because of some general syntactic and semantic features it shares with other discourses, but because of its particular relations with the defining empirical inquiry from which she begins.

It is important for Maddy's Thin Realist, then, that our set theory — however wildly abstract it seems — has its connections to less abstract mathematics, which in turn has its connections … ultimately to the messy business of engineering-level science. Set theory, the rather Quinean thought goes, is an outlying but not entirely disconnected part of a network of enquiry with empirical anchors.

But put like that, we might wonder whether the Thin Realist protests too much. To be sure, looking at the historical emergence of modern mathematics, we can trace the slow emergence from roots in mathematized science of purely abstract studies driven increasingly by a purely mathematical curiosity, and can see the (albeit very stretched) lines of connection. Starting from where we now are, however, the picture changes quite sharply: here we are with feet-on-the-ground physicists doing their thing using one bunch of methods (ok, let's not worry about string-theory fantasists!) and over there are the modern set theorists doing their improvisatory thing with a quite different bunch of rules of play. They are, it now seems, playing an entirely different game. We might ask: whatever the historical route by which we got here, is there really still a sense, however stretched, in which the physicists and the set theorists remain in the same business, so that we can sensibly talk of them both as trying to 'uncover truths'?

The new suggestion, then, is that mathematicians have such very different fish to fry that it serves no good purpose for the second philosopher to say that the mathematicians too are talking about things that 'exist' (sets), or that set-theoretic claims are 'true'. And note that it isn't that the mathematicians should now be thought of as trying to talk about existents but failing: to repeat, the idea is that they just aren't in the same world-tracking game. No wonder, then, that — as Maddy puts it on behalf of such a difference-emphasizing philosopher — our 'well-developed methods of confirming existence and truth aren't even in play here'. Call this second line according to which set theory isn't in the truth game 'Arealism'.

So what's it to be for the second philosopher, Thin Realism or Arealism? What's to choose? In the end, nothing says Maddy. Here's modern science and its methods; here's modern maths and its methods; here's the developmental story; here's a chain of connections; here are the radical differences between the far end points. Squint at it one way, and a sort of unity can be seen: and then we'll incline to be Realists across the board — while, of course, eschewing over-Robust mythologies. Squint at it all another way, and the modern disunity will be foregrounded, and (so the story goes) Arealism becomes more attractive. There's no right answer (rather, what this all goes to show is that the very notions of 'truth' and 'existence' are more malleable that we sometimes like to think).

Thus, roughly put, goes a central line in Maddy's thought, if I'm following aright. But I wonder what underpins Maddy's hankering here after a more-than-logical conception of truth? The Thin Realist, recall, thinks that Wrightian minimalism about truth isn't enough: she wants to talk of set-theoretic truth so as to point up the (albeit distant) links from the maths to good old empirical enquiry. The Arealist doesn't want to talk about set-theoretic truth because she wants to point up the differences between maths and good old empirical enquiry. But why this shared basic anchoring of the idea of truth not to the cross-discourse formal role of the notion but (so it seems) to the idea of correct representation of the empirical world?

What we need here, if we are going to make progress, is more reflection on the very concepts of truth and existence. And I don't mean that we need an unwanted injection of first philosophy! Grant that our malleable inchoate ideas can indeed be pressed in different directions. Still, the naturalistic second philosopher can take a view about the best way to go. She will want to look at our practices of talking and thinking and inferring, and she will want to have a theory about what is going on in various areas of discourse (empirical chat, moral chat, pure-mathematical chat, etc.). Her preferred developed notion of truth should then be the one that does the best theoretical work in her story about those discourses. And it certainly isn't obvious at the outset how things should go. Will she end up more like a Blackburnian projectivist, privileging a class of representational discourse as the home territory for a basic notion of truth (so that other discourses are playing a different game, and have to earn their right to borrow the clothes that are cut to fit the representational case)? Or will she end up like a more thorough-going pragmatist (there is no privileged core), or — in a different key — like Wright's minimalist? Certainly, the reader of Defending the Axioms isn't given any reason to suppose that things must fall out anything like the first way, keeping room for a substantial notion of truth.

And that gap in the end makes the current book rather unsatisfactory as a stand-alone affair (which isn't to say it is not a fun read).

Of course Maddy herself has written extensively about truth elsewhere so as to fill in something of what's missing here. But this means that, after all, you really will have to go back to read the weighty Second Philosophy to get the whole story, and hence the full defence of the line Maddy wants to take about sets.

Last chance to see

The exhibition 'Jan Gossaert's Renaissance' is on at the National Gallery for another ten days. Warmly recommended. Especially to those, like us, who are really pretty ignorant of the Northern Renaissance. Quite a revelation. We learnt a lot, and some of the pictures are just stunning. Beautifully presented (as usual), but also (not so usual for major exhibitions downstairs in the Sainsbury Wing) very quiet — which was important as a lot of the pictures and all the drawings are small scale and so you need to get up close. So, as I said, much recommended.

A First Eleven, 1975

Retirement looms/beckons (depending on whether it's a day that I feel it is to be regretted or welcomed). So I've started really clearing out my office (as opposed to the occasional half-hearted efforts in the past). Most of a filing cabinet of assorted admin papers and dog-eared xeroxes has gone. Now it's the turn of folders of old lecture notes.

Among some handouts from long back, I found a two-page end-of-term squib from 1975 headed "A First Eleven". As I say in the preamble

Philosophers often indulge in the pastime of picking their world-beating first eleven of all-time greats or a team of contemporary philosophers … Here is my selection of contemporary 'greats'. The team I've picked is of philosophers in the strict sense, still alive and influencing contemporary philosophy in a dominant way …

Then came the list with a few comments and suggestions for key reading against each name. The headline list read

Quine (Captain)
Davidson
Dummett
Feyerabend
Grice
Kripke
Popper
Putnam
Rawls
Sellars
Strawson

Twelfth man: Bernard Williams.

If it is surprising to see Feyerabend's name there, then remember that  the pieces that were later collected in his two 1981 volumes of collected papers were then much read — and are good serious stuff. It isn't all Against Method. Perhaps the name that surprises me the most is Strawson's: I can't recall ever being a particular fan, or even reading him much (I really struggled with the Bounds of Sense, while I loved Bennett's Kant's Analytic).

Still, I'm not too ashamed of my earlier self's enthusiasms revealed here!

Intro to Gödel's Theorems, Mk 2

I've just heard that CUP will probably look kindly on a proposal for a second edition of my Gödel book, to come out two or three years hence. There have already been a couple of (quite heavily) corrected reprints, but this time I'd do an end-to-end rewrite, and the whole thing could perhaps be up to 10% longer. So if you have bright ideas about how to make the thing better/more useful, or views about stuff I'd left out that ought to be in the book, I'd be delighted to hear them. It will be good to have a limited retirement project to fall back on when other work isn't going so well.

Gödel lectures

Notes for the first of the four lectures I'm giving this term on Gödel's theorems (for mathematicians, not philosophers) are now available here. I hope the rest will follow very promptly!

I'm not sure I'm judging these lectures well. So even more than usual, comments, corrections and suggestions for improvement are most welcome.

AV vs FPTP

Apart from sounding off occasionally about Higher Education Bollocks (there really is a lot of it about), I tend to steer off politics here. My half-baked and (these days) ill-informed prejudices would be of little interest. Still, I've always been pissed off by our current crazy voting system. Which isn't to say that I think that AV is a best buy. But it sure beats the (so-called) "first past the post" system, and that's the only choice on offer at the moment. So I'm voting "yes" in the referendum. For a lot of reasons, see this quite terrific piece by the estimable Tim Gowers.

TTP, 9. §2.IV A map of the terrain

Weir, to summarize once more, wants to develop a position that allows him to say sincerely, speaking with the vulgar mathematicians (and not having to cross his fingers behind his back, or do that little dance with the fingers that signals scare-quotes, or do some radical reconstrual of what they "really" mean), "there are infinitely many prime numbers", even though while amongst the learned, or at least amongst the metaphysicians, he consistently asserts "THERE ARE no numbers". He hopes to have softened us up for the idea that there is room for such a have-your-cake-and-eat-it position by considering (i) how projectivism (supposedly) allows us to agree sincerely with the vulgar that "X is G" (for certain G) while also agreeing with metaphysicians who say "THERE IS no such property as being G", and considering (ii) how a certain line on fiction (supposedly) allows us to agree with the vulgar reader of the stories that "Sherlock lived in London" while agreeing with metaphysicians who say "Sherlock never EXISTED". Not that Weir want to be a projectivist or a fictionalist about maths: but the idea is that the prima facie tenability of those accounts elsewhere indicates that there is perhaps room for a similar ontological anti-realism about mathematics, one which rests on the key idea that in making mathematical assertions (as when making fictional assertions or "projective" assertions) we are playing a different game from when we are in the business of representing the world.

But Weir, as he now emphasizes again, wants more. He wants to combine ontological anti-realism about mathematical entities with "metaphysical realism" in the Putnamian sense of allowing for the possibility of evidence-transcendent truth in maths. Of course, this isn't exactly a novel combination. The modal structuralist is similarly concerned to eliminate commitment to a distinctive ontology of mathematical abstracta, which he does by translating away mathematical claims into modal quantified truths, and he can allow that it is evidence-transcendent what the modal truths are. However, unlike the modal structuralist, Weir wants to take mathematical talk at face value (he doesn't want to go in for telling mathematicians what they "really" mean by translating away their ostensible commitments). So he wants a brand of ontological anti-realism for mathematics akin to projectivism or his sort of account of fictional discourse — we again aren't in the representational business — while allowing evidence-transcendent truth.

But it can't be said that we've been softened up for that combination. Certainly, it is difficult to see how there could be e.g. evidence-transcendent truths about what is tasty! Maybe a projectivism about probability could be developed in such a way as to allow for evidence-transcendent truths in this case: but Weir doesn't say anything about such a case — and, in sum, I think we get no illumination on the ontological-anti-realism/metaphysical-realism combo from anything he says about projectivism (have I missed something?). However, Weir does think his account of fiction gives us something to go on:

There is no incoherence in holding to this anti-realism [about fiction] while viewing truth in general as evidence-transcendent — perhaps even fictional truth, if the fact that S follows in the right way from the text, and thus is true, can be evidence-transcendent.

But what does Weir have in mind here?

Earlier, he talked about S following in the right way — "flowing from" the text — if "experienced readers would, on reflective consideration, judge [that S] must form part of the story if it is to make overall sense." But that notion of flowing from the text, where what flows depends on our best judgements, would hardly make room for evidence transcendence! But perhaps the idea is that things may follow logically or indeed mathematically from the text, but in an evidence-transcending way. Thus suppose "2 is the least number such that P" is an evidence-transcendent mathematical truth. Then I guess we have "The number of Dmitri Karamazov's half-brothers is at least as large as the least number such that P" as a truth about the fiction which would be evidence transcendent. But then the evidence transcendence of the fictional truth would be dependent on the evidence transcendence of the mathematical truth (and so we couldn't use the possibility of former fact as illustrating how the latter could be possible). Well, maybe there are other cases we could think about here: but that's enough to suggest that Weir's one-sentence jab at persuading us that his story about fiction gives us a useful illustration of the desired ontological-anti-realism/metaphysical-realism combo is just too quick.

But let that pass. We now have some sense of where Weir wants to end up about mathematics: ontological anti-realism without radically reconstruing maths (we continue to take it "at face value"), to be achieved by seeing assertion in maths as playing a different role to representational assertion, BUT also "metaphysical realism", in the sense of allowing for evidence-transcendent truth. The work of spelling out his attempted "neo-formalist" articulation of such a position starts in the next chapter.

TTP, 8. §2.III Reduction

The projectivist about e.g. judgements of tastiness explains how "X is tasty" (as an ordinary judgement made in the restaurant, not the philosophy class) is an assertion that can be correct or incorrect even though there is no such property-out-there as tastiness, so the assertion isn't representationally-true (or correspondence-true, if your prefer). Or so the story goes.

The theory about fiction that Weir sketches explains how "Sherlock Holmes lived in Baker Street" (as an ordinary judgement made in discussing the stories, not the in history class) is an assertion that can be correct or incorrect even though there is no such person-out-there as Sherlock (or Meinongian substitute), so again the assertion isn't representationally-true. Or so the story goes.

The projectivist line about tastiness or goodness or beauty, the theory about fiction, allow us to speak with the vulgar but think with the learned (assuming the learned have a naturalistic bent). We can legitimately talk as if there is a kosher property of tastiness or as if there are fictional beings such as Sherlock, while not being really ontologically committed to such things. If we use small caps to signal when we are making assertions in full-on, stick-by-it-even-in-the-metaphysics-classroom, genuine-representation mode, then we can say (ordinary conversation) "Marmite is tasty" even though (when in Sunday metaphysical mode) we can agree "Marmite IS NOT TASTY"; and likewise we can say (conversationally) "Holmes lived in London" while (on Sundays) agreeing that "Holmes NEVER EXISTED". Hence the projectivist story and the story about fiction allow us to eliminate some of our ostensible ontological commitments in talking with the vulgar (Weir calls this "ontological reductionism", but I've grumbled before about that label). So the story goes.

What does Weir add to the story in this section, to further set the scene before trying to paint a comparable picture of mathematics as non-representational? (1) Some remarks about what makes for the difference between a representational mode of assertion and a non-representational one. (2) Some remarks about why the difference between "Holmes existed" and "Holmes EXISTED" shouldn't be confused with a lexical or structural ambiguity. (3) Some remarks about what a projectivist should say about the likes of "If sentient beings had never existed, there would still have been beautiful sunsets".

Concerning (1), I'd have thought the way to go is to illustrate the kind of basic semantic story that applies to canonical examples of "representational" discourse, and then say that non-representational discourse is whatever needs some different kind of semantic story (I'm not saying that's easy to do! — but Weir's p. 59 seems to go off in a slightly skew direction.)

Concerning (2), I agree. I'm not sure it is helpful then to go on to talk about 'metaphysical ambiguity' (but maybe that's just complaining about Weir's taste in labels again).

Concerning (3), Weir discerns a wrinkle, but also thinks that is doesn't carry over his promised non-projectivist but analogously anti-realistic account of mathematics. So we needn't pause over this.