Year: 2009

There’s Something about Gödel, Chs 10 & 11

After a couple of rather unsatisfactory philosophical chapters, Berto does quite a bit better in the next two. (Which is good, because these comments were getting unseasonably grumpy!)

Ch. 10 is called “Mathematical Faith”, and discusses what we should learn from the unprovability of consistency of a theory (a consistent theory containing enough arithmetic) within that theory. Does it show that the mathematician has to rely on blind faith in some worrying sense? To which the right answer is “no!”. Here Berto pretty closely follows a good discussion by Franzen. There’s nothing that adds much to Franzen’s similarly introductory discussion, but equally Berto doesn’t go astray.

Ch. 11 is on the Lucas/Penrose argument. Again Berto’s discussion is sane and sensible. The ur-Lucas argument is sabotaged by the familiar Putnam riposte. Souped up versions are sabotaged by souped up versions of that riposte. But it remains that something can be learnt from Gödel incompleteness about the nature of the mind — namely the disjunctive conclusion of Gödel’s Gibbs lecture (prefigured also in Benacerraf’s old discussion). This is rapidly done, though: for example, what on earth will the beginner with the thin background Berto is officially presupposing make of the invocation of transfinite ordinals on p. 183? Still, this chapter could make for helpful introductory reading for some students working towards on an essay on this topic.

There’s Something about Gödel, Ch. 9

Chapter 9 of There’s Something about Gödel is about Platonism. There are two tricky issues here — first getting clear about what Gödel’s own views were and how they changed over the decades between 1931 and his late philosophical papers, and then second assessing those views. Berto’s chapter is only sixteen pages long. And five of those are an explanation of Tarski’s theorem on the indefinability of truth. Unsurprisingly then, the rest is too rapid and superficial to get very far. But does it at least start off in the right direction?

Berto writes: “Gödel appears to have believed [that] the Incompleteness Theorem … refutes the idea of mathematics as pure syntax, and validates the metaphysical claim that numbers are real, objective entities in the timeless Platonic sky”. The first half of that is right. Gödel did believe that the Theorem refutes the idea of mathematics as pure syntax: but work is needed if we are to characterize a general notion of “mathematics as syntax” that is wide enough to cover potentially attractive programmatic views but sharp enough to be vulnerable to a crisp refutation (which is probably why there are six drafts of his paper on Carnap left in the Nachlass). But the second half of Berto’s claim about the “timeless Platonic sky” is just crass. It takes real work too to tease out the non-metaphorical content of Gödel’s Platonism — and just ramping up the level of metaphor and talking of timeless Platonic realms (which as far as I can recall, Gödel never does) is no help at all in doing that work, which is left entirely undone here. If to talk about objects in a Platonic sky is to “treat the analogy between the existence of physical objects and the existence of mathematical ones seriously as a literal account of the way things are” (as Michael Potter puts it), then it is highly arguable — and has been argued — that this quite badly misrepresents Gödel.

Berto doesn’t mention that at all, but instead seems rather keen on Rebecca Goldstein’s crude account in her Incompleteness: The Proof and Paradox of Kurt Gödel, and quotes her as giving “a summary interpretation” of a supposed Platonic interpretation of the First Theorem (p. 158). That’s a pretty extraordinary choice, given that Goldstein’s book is frankly awful. It’s not just my view that “the book as a whole is marred by a number of disturbing conceptual and historical errors” — those words are from Feferman’s damning review.

Striking out for ourselves just for a moment, here’s what we establish in proving the First Theorem applied to PA (making the usual assumption about omega-consistency). There’s a primitive recursive relation two-place relation Prf, and a number g, such that for all numbers x, it isn’t the case that Prf(x,g), i.e. ∀x¬Prf(x,g): but PA can’t prove or refute ∀x¬Prf(x,g), where Prf(x,y) formally represents Prf and g is the formal numeral for g. There’s no metaphysically loaded notion of truth involved in stating that theorem, because there is no notion of truth involved, full stop. Of course, since ∀x¬Prf(x,g) expresses that, for all numbers x, it isn’t the case that Prf(x,g), and indeed for all numbers x, it isn’t the case that Prf(x,g), we can say ∀x¬Prf(x,g) is true, and so say that the Theorem shows that there is a truth that can’t be proved (nor, thankfully, disproved) in PA. But this use of the notion of truth is anodyne and basically disquotational, still without metaphysical ooomph. If we start generalizing, and talking about not just PA but suitable axiomatized theories more generally, then we can again say that more generally that for each such theory there will be truths that can proved in that theory. But still, the notion of truth involved remains metaphysically anodyne, and not distinctively platonistic: an anti-realist can be content so far. Which suggests that — without more philosophical input as side premisses — the First Theorem doesn’t have specifically Platonistic implications.

Did Gödel think otherwise? At a second pass, with some further premisses, can we after all draw Platonist morals from the incompleteness phenomenon? Well, a useful theme to pursue would be this. Dummett famously argues that that Gödelian incompleteness is tied up with indefinite extensibility, and taking indefinite extendability seriously should leads us to be anti-realists, specifically intuitionists, about mathematics. The later Gödel, however — particularly in work first published after Dummett’s famous paper — seems to take the extendability of the notion of set, for example, to be a count in favour of his conceptual realism. Where exactly is it, then, that Dummett and Gödel disagree? However, this kind of investigation — which is the sort of thing we need to throw some more light on what constitutes Gödel’s Platonism — takes us a long way from anything that Berto touches on (or even mentions in notes or bibliography).

There’s Something about Gödel, Ch. 8

The first seven chapters of Berto’s book are exposition of the formalities: I’ve said my piece on those. The last five chapters are philosophical essays, which can be read independently of the particular preceding exposition, as long as you know something about Gödel’s theorems. And indeed the philosophical chapters can be read independently of each other. (As it happens, I read the very last chapter of the book before I read any of the rest of the book, as someone asked me what I thought of the kind of paraconsistent line explored there.)

I’ll take the philosophical chapters in turn, starting with Chapter 8, on “The Postmodern Interpretations”. The first half of this chapter touches on some of the wilder things pomos have said, and some exaggerated claims about Gödelian incompleteness showing that there can’t be a physical theory of everything. Berto briefly says the right kind of things, but he’s largely drawing on Torkel Franzen’s excellent and deservedly familiar demolitions of such daftness, and Franzen does it better.

The second half of the chapter isn’t so much about pomo interpretations in particular as about what we are to make of the phenomenon of non-standard models. There are indeed serious issues here (though of course not all specifically to do with Gödel’s theorems): but Berto’s discussion is too quick and shallow to be useful. And by the end of the chapter he seems to have forgotten the issues that he was supposed to be discussing, namely whether arithmetical statements can sensibly be said to be true only in a relative sense (as in “true-in-the-standard-model). For Berto ends up talking not about deviant models of what we thought was the canonical theory (PA) but about the possibility of deviant theories. So nothing much is achieved here.

There’s Something about Gödel, Chs 5–7

Berto’s book is cheerfully sub-titled “The complete guide to the incompleteness theorem”. That, I take it, is a joke. But he is perhaps trying to give a sense of how it can be proved in roughly Gödel’s way, which is complete enough to provide a decent platform for ensuing philosophical discussion. And on the whole, I think his basic choice of path here is a good one (see my book, or the cut-down version Gödel Without Tears, which takes a similar route in significantly more detail). But how well does he execute the expository task of filling in sufficient detail?

Well, to continue, Chapter 5 of Berto’s book is about the idea of representing properties, relations and functions in arithmetic, and he states (but only states) the result that all primitive recursive functions are representable in PA. Two quibbles about this short chapter. First, a minor techie point: given Berto’s definition of what it is to represent a function, it isn’t exactly “easy to show” that a property is representalble if and only if its characteristic function is. Second, the box diagram on p. 85 is misleading, for it implies that the representability of general recursive functions in PA is essentially involved in the proof of its incompleteness: not so, of course — it’s the representability of primitive recursive functions that does the job, as Berto says elsewhere.

The next chapter, titled “I am not provable”, does the construction of a Gödel sentence for PA, and gives the syntactic proof of incompleteness. The chapter is disappointing. The basic construction strikes me as being presented in an unnecessarily laboured way: my bet is that the reader who has struggled this far will find this pretty tough. There are smoother presentations on the market! Given this chapter is pivotal, more work could have gone into make this stuff maximally accessible.

Berto has an odd conception of what makes a chapter. I’ve already mentioned his presenting PA and the idea of Gödel numbering in the same chapter. Chapter 7 is another case in point. The first two sections are about how you prove the Second Theorem. Then there’s a crash of gears, and we move to general discussion about the “immediate”, i.e. relatively non-controversial, consequences of the first and second theorems. The obvious presentational glitch here is that Berto officially uses the labels “The First Incompleteness Theorem” and “The Second Incompleteness Theorem” for results specifically about PA. That’s misleading, if only because that’s not how the labels are usually used. But worse, when Berto starts talking a bit about the incompleteness theorems implications (at p. 107), he already has in mind the theorems in the usual sense, i.e. the generalizations to any appropriately strong theories. For instance he writes there “The First Theorem only shows that one cannot exhibit a single sufficiently strong formal system within which all the mathematical problems expressible in the underlying formal language can be decided.” That’s not true of the First Theorem as he has stated it, and it is only five pages later that we meet the generalization that warrants the claim.

Overall, my sense, to be honest, is that the expository half of this book has been put together in a bit of a rush, to get to the interesting philosophical discussions in the second half, and the chapters have not been tried out on enough hyper-critical students and colleagues. Take for example this: “One has (or believes oneself to have) some grasp of the structure N, which one tries to capture by means of the numerals, the syntactic characterizations, etc. [of PA]. However Gödel’s First Theorem entails that this is an illusion.” Grammatically, the “this” ought to refer to the “some grasp” of the structure of N. Well, Gödel’s First Theorem of course doesn’t entail that it is an illusion that you have some grasp. And indeed, it doesn’t entail either that it is an illusion that you  in fact have a grasp sufficient to to fix on a determinate structure. The most that can be said is that the idea that you can determinately fully capture the structure of N by a first-order theory like PA is shown to be an illusion: which is probably what Berto meant, but isn’t what his sentences actually say. This sort of casual carelessness is too frequent.

The verdict on the first part of the book, then? Berto’s expository chapters are not too bad, and could (as I said before) initially be helpful for beginners as orientation. But some will struggle as explanations fly past: and those who can cope with the tougher bits should move on to reading a proper presentation.   To be continued

There’s Something about Gödel, Chs 1–4

A number of people have now asked me what I think of the fairly new There’s Something About Gödel by Francesco Berto. Well, I’ve been meaning to look at the book anyway: I’m teaching a course on incompleteness in NZ starting in six weeks, and I thought I should know whether I can recommend it as introductory reading for students. So I’ve started the book, and though I plan to crack through at a fast pace, I might as well make a few comments here as I go along.

The book divides into two parts. The first seven chapters are pretty informal exposition (though aiming to give some real sense of how the incompleteness theorems are proved). The last five chapters are more philosophical. I’ll talk about just the first four chapters in this serving of comments.

Chapter One is a whistle-stop tour through some background (introducing e.g. the Liar Paradox and Russell’s Paradox, explaining what an axiomatic theory, what an algorithm is, and so on). This is done with a light touch, but not always with ideal accuracy. The most misleading claim is probably that “the amazing developments of mathematics in the nineteenth century” allowed “the reduction of higher parts of mathematics to elementary arithmetic” (p. 14). Given the informal definitions that precede, it is also misleading to say that every decidable set is enumerable (p. 27). And it won’t really do to say that properties can be considered as sets (p. 23) and then say that Russell’s Paradox shows that not every property “delivers the corresponding set” (p. 32). The concluding pages on set theories (pp. 37-38) are far too fast to be useful — but equally they are not necessary either.

The next chapter starts by outlining Hilbert’s programme. It would unfortunately take a more-than-usually-careful reader to pick up that (1), when doing Hilbertian metamathematics, we are to concentrate on the syntactic features of the relevant formal axiomatized theory we are theorizing about and ignore its semantic properties — rather than it’s being the case that (2) the formal axiomatized theories that Hilbert wants us to study simply  lack such properties (are “merely” formal). Berto writes e.g. “In the formalist’s account of these notions, axioms and formal systems are not considered descriptive of anything” (p. 41), which unfortunately sounds like (2). And then the reader will then be puzzled about why, later in the chapter, we are back to talking about semantic features of formal systems.

This chapter also gives first informal presentations of the semantic version of Gödel’s first theorem, and of the second theorem.

Chapter Three both introduces standard first order Peano Arithmetic (I do rather deprecate following Hofstadter and calling it “TNT” for Typographical Number Theory). Most students would I’m sure have appreciated more explanation of the induction schema TNT7 (p. 58) — if the substitution notation has been explained before, then I’ve forgotten that, so I guess that students will too! Also Berto’s half-hearted convention of boldfacing some symbols inside the language of PA but not others is not a happy one: if you are going to do this sort of thing, than use the same face/font choice throughout for all symbols of the formal language. This quite short chapter also introduces Gödel numbering. You’d have to try this out on various student readers, but I’d suspect that this is going a bit too fast for comfort.

Chapter Four is called “Bits of Recursive Arithmetic”; and I think this is rather fumbled. One trouble is that it again will certainly  go too fast for most of the intended readers. Berto jumps immediately to a general specification of a definition of one function by primitive recursion in terms of two others, rather than works up to it via a few simple examples: not a good move, in my experience. More seriously, Berto officially sets out to define the full class of recursive functions, mentions minimization, gives no sense of why this keeps us inside the class of computable functions, and then says “I shall skip further development” because “for a proof of Gödel’s Theorem one can take into account only the primitive recursive functions” (p. 75). But then, in the next section, Berto moves on Church’s Thesis which requires us to understand the general notion of a recursive function. Given this isn’t necessary for the incompleteness proof, and that Berto hasn’t actually defined such functions anyway, this would seem to be a recipe for confusion.

Verdict so far: Berto’s exposition in these opening chapters isn’t outstandingly good, but it could initially be helpful for some students as orientation. To be continued

A chill wind coming

It’s hardly the most important world news. But the UK university cuts announced a couple of days ago will mean a pretty bleak time for universities. Sure, more money has gone into universities here in recent years, but then so have more students: so what is being talked about is a sharp squeeze on per capita expenditure, hence a yet further worsening of staff-student ratios, with all that entails. No doubt it will be a lot worse further down the pecking order: but I imagine a pretty chill wind will be blowing in Cambridge too. It will be a fight (at poor odds) to get my post filled when I have to leave at the end of next academic year. Cambridge hasn’t exactly an encouraging track record of supporting small humanities departments — despite the growth in student numbers we have no more established lecturing staff in philosophy than we had twenty years ago.

And I suspect that — given the government’s emphasis on business-useful skills etc. — it could be a particularly chill wind for many other philosophy departments too. Overall, philosophy departments have done rather spectacularly well over the last fifteen or more years, partly driven by a big increase in the numbers of students opting for the subject. But I’ve never heard that increase plausibly explained, nor heard any reason given why the numbers shouldn’t evaporate again as quickly (think sociology). After all, what students actually get in most philosophy courses, once they are enrolled a philosophy degree, is only tangentially related to what many of them think they have signed up for (in some cases, even those who have done some philosophy at school).

I get sent a publication called Discourse produced by the “Subject Centre for Philosophical and Religious Studies” [why that marriage?] of the “Higher Education Academy”. Issues of Discourse are indeed rather disheartening documents, as they seem to a significant extent about how to teach an approximation to philosophy to students who are neither particularly committed nor particularly bright. Are those students really going to stick around? OK, philosophy can tick that “transferrable skills in critical thinking” box. But we are hardly unique in that.

So I’m not optimistic. It would be ironic if — despite all our grumbling about  RAEs and QAAs and mounds of administrivia — the last decade  soon looks like something of a silver age for UK philosophy.

The Autonomy of Mathematics, §3.4

Hilbert’s consistency proof in the Foundations of Mathematics is model-theoretic. But of course the later Hilbert seeks consistency proofs that don’t depend on model-construction: what then are the resources can be brought to bear in syntactic proof-theory? Well, §3.4 on “Hilbert’s proto-proof theory”, discusses in particular how far induction can be used if we want a non-circular consistency proof of arithmetic. (Note, by the way, following up a comment on the previous section, in this current section when Franks talks of “arithmetic”, he does indeed seem to mean it in the narrow sense of the arithmetic of the natural numbers.)

Poincaré famously challenges the very possibility of a non-circular justification of induction. In response Bernays, in 1922, writes

two types of complete induction are to be distinguished: the narrow form of induction, which relates only to something completely and concretely given, and the wider form of induction, which uses either the general concept of whole number or the operating with variables in an essential manner. [Quoted by Franks, p. 76.]

Now, I make no pretence at all to be any kind of historian. So maybe I’m just reading back into this remark something that only became really clear to the Hilbert school much later. But it seems to suggest the following thought. It is one thing to accept induction over predicates that are sufficiently “concrete” (whose application to a particular case can be determined in a direct way), and something else to accept induction over predicates that e.g. themselves embed variables. It is the latter than we need to get full-strength induction in arithmetic. But perhaps the pre-Gödel Hilbertian hope is that it is the former — which e.g. we might regiment in PRA — that could suffice for proof-theory.

Franks however says this:

[T]he true threat of circularity — and the one Poincaré apparently advances — concerns the strength of the principle used and not the subject matter to which one applies it. Bernays’ defense is therefore inadequate to meet this threat, for he distinguishes only the ranges of application in the narrow and wide uses of induction and not their deductive strengths.

So Franks reads Bernays as just drawing a distinction between domains of application of the same inductive principle (“concrete-intuitive”, as Bernays puts it, vs. arithmetical?) rather than a distinction between the inferential strength of two different principles, i.e. induction for quantifier free predicates and induction for arbitrarily complex predicates. But why is this the right reading? For all I know, it might be. But since Franks doesn’t here even mention what seems — with hindsight? — to be the alternative natural reading, I just don’t know why he prefers a reading which has Bernays making a rather feebly question-begging response to Poincaré rather than read him as pointing to a potentially more promising response (a response that of course must fail, but it will take Gödel to show that). Franks quotes a passage from Hilbert on p. 78 which seems again to invite what I’m calling the natural reading, for it seems to contrast “contentual” induction with induction in cases which essentially involve (non-schematic) variables. But Franks says that Hilbert too is drawing a distinction which “seems to have more to do again with the proper subject matter of the two principles than with their comparative inferential strength”. Again I don’t see why.

There’s another passage I don’t understand near the end of the section. Franks writes that

[E]ven if one follows Hilbert in treating formulas and proofs mechanically, without attributing any meaning to them or the symbols from which they are built up, a statement about proofs and formulas — Bernays’ formulation of consistency for example — is itself not such a formula but rather an ordinary statement. To confirm or refute a statement like this, either one’s proof theory must be sufficiently informal to treat such statements directly, or to a purely mathematical Beweistheorie one must append some extra-mathematical reasoning to determine when informal metatheoretical questions have been settled by mathematical results. The lack of purity in his program thus stemmed from the informality of the consistency statement he aimed to verify. (p. 82)

But what’s the complaint here? We formalize a given mathematical theory T, clarifying its axoims, deductive apparatus etc. Then its syntactical proof-properties, in particular, themselves become possible objects for mathematical enquiry (which doesn’t mean that T lacks content, of course: it is just that investigating T‘s syntactic proof-properties is done without reference to that content). Now the mathematical enquiry about T‘s syntactic properties gives us another theory — call it ‘M‘ for metatheory. And, if we want, M of course can itself be regimented more or less into the ideal of a formalized theory: compare, for example, the theories of first-order syntax given in different levels of logic book. And whether relaxedly informal or souped-up to the most rigorous standards, either way M still treats T‘s syntax “directly”. It isn’t the case that only an informal theory has semantics, or that a mathematics of syntactic structures needs to be supplemented by something else before it can be about those structures. Franks seems here to be sliding between “informal” and “contentful”, “formalized” and “contentless”. But that would just be a mistake — and not one he’s given us reason to suppose, either, that Hilbert was guilty of. Assuming Franks isn’t making that mistake, I don’t see what his point is.

Imogen Cooper’s Schubert continues …

Six months ago — oh, where has the time between fled to? — I very warmly recommended the first volume of Imogen CooperVol2.jpg Cooper’s new cycle of live recordings of the later Schubert piano music. The second volume has been out for a while, and is equally terrific.

The second CD ends with the four Impromptus D935, and her performances — it seems to me — more than stand comparison with Schnabel, Fischer, and Brendel’s earlier versions among the classic greats, not to mention Schiff, Uchida, Maria João Pires and the later Brendel among more recent recordings. There are so many heart-stopping moments. I loved Imogen Cooper’s earlier cycle, but now her playing has that perfect authority — while you are listening, you are entirely swept into her vision of the music, and you feel ‘this is how it must be played’. Wonderful.

Why Miss Jones …

It’s all been a bit serious here lately, what with Hilbert and Gödel, visits to the Louvre, and rereading Emma. But it isn’t all high culture in the Smith household. Oh no. The last couple of days we’ve been laughing out loud at the wit and wisdom of Miss Jones, reading out choice entries from her brilliant blog, and exchanging emailed snippets with Miss RumAndReason who is also an instant fan. Great fun, but with a squeeze of lemon juice.

Scroll to Top