Year: 2014

Parsons #2: Bernays

I guess for many — most? — of us (the anglophone, non-German-reading us), our initial acquaintance with Paul Bernays as a philosopher of mathematics was via his 1934 lecture ‘On platonism in mathematics’ reprinted in the Benacerraf and Putnam collection Philosophy of Mathematics. In retrospect, the important point (insight?) in that lecture is that what is characteristic of what Bernays calls “platonistically inspired mathematical conceptions” is not primarily an ontological view — not the idea of “postulating the existence of a world of ideal objects” (whatever that comes to) — but rather a preparedness to use “certain ways of reasoning”. He has in mind a willingness to apply the law of excluded middle, to apply classical logic in quantifying over infinite domains, to adopt choice principles (appealing to the existence of functions we can’t give a rule for specifying), and the like. And different domains may call for different principles of reasoning: platonism isn’t an all-or-nothing doctrine, but rather we should aim to bring about “an adaption of method” suitable to whatever it is we are investigating.

If we have met Bernays the philosopher again later, then — apart from a rather fleeting appearance in the van Heijenoort volume — it may well be through the translations in Mancosu’s extremely useful 1998 volume From Brouwer to Hilbert which includes four pieces by Bernays. The longest of these is a somewhat earlier essay from 1930 on “The philosophy of mathematics and Hilbert’s proof theory”. Here, he starts by reviewing changing views of what mathematics is about, concluding (at least as at a first pass) that

We have established formal abstraction as the defining characteristic of the mathematical mode of knowledge, that is, focusing on the structural side of objects …

So note that the junior but more philosophically sensitive member of Team Hilbert seems here to be, if anything, hinting at a proto-structuralism as far as ontology is concerned. Bernays then reflects, however, on the fact that — whatever is the case as far as grounding basic arithmetic is concerned — formal abstraction from the experienced world doesn’t get us to the (actual) infinite which classical analysis presupposes. What to do? Accept the intuitionist critique of the actual infinite? The price is too high. Turn to the logicists and try to secure the infinite by translating analysis into a purely logical framework? This doesn’t do the desired work, given the problematic role of the axiom of infinity and the even more problematic axiom of reducibility.

We “postulat[e] assumptions for the construction of analysis and set theory” based on supposed analogies between finite and infinite cases. These analogies however  don’t by themselves show that “the mathematical idea-formation on which the edifice of [analysis] rests” is in good order. But no matter. The resulting edifice turns out to prove its worth in a spectacular way by its systematic success:

As a comprehensive conceptual apparatus for theory-formation in the natural sciences, [analysis] turns out to be not only suitable for the formulation and development of laws, but it is also invoked with great success, to a degree earlier undreamt of, in the search for laws.

Is that enough? We’d like in addition a consistency proof to confirm that there are no so-far-hidden problems lurking. But note, it is not being said that a proof of consistency “suffices as a justification for this idea formation”: the main justification comes from mathematical and scientific success — so the epistemology for infinitary mathematics is, if anything, “naturalist” (as we might now put it). Still, Bernays concludes his 1930 paper as you’d now expect, explaining how the process of rigorous axiomatization makes mathematical theories themselves available as finite objects which can can be studied by finitary modes of reasoning and (we hope, pre-Gödel) shown to be not just spectacularly successful but comfortingly consistent.

Re-reading these two papers, I am struck by how congenial the take-home messages are: and note, by the way, that there is no hint here of the naive, strawman, formalism that Hilbertians used to get accused of. Ok, the ideas are not worked through as thoroughly (or always as clearly) as we would now want: but Bernays’s philosophical inclinations seem to be going very much in the right direction, by my lights anyway. So it could be very interesting to have more of his philosophical work made available by being collected together and translated into English. And yes, just such a project is under way: a volume Paul Bernays: Essays on the philosophy of mathematics edited by Wilfried Sief and others has been announced as in preparation and to be published by Open Court. 

Charles Parsons has a long-standing interest in Bernays: indeed he was the translator of that lecture on Platonism for the Benaceraff and Putnam volume, and he has had a role in preparing the forthcoming Essays volume. And Bernays now makes two extended appearances in Parsons’s own essays in Philosophy of Mathematics in the Twentieth Century. The first is in the longish, previously unpublished, opening piece in the book, ‘The Kantian legacy in twentieth-century foundations of mathematics’. This paper in fact discusses Brouwer, at some length, and Hilbert more briskly, as well as Bernays. But despite Parsons’s efforts, Brouwer’s philosophy remains as murky as ever; and the residual Kantian themes in Hilbert’s “articulation of the finitary method” have been explored before. So the interesting news from this opening piece is perhaps going to be found, if anywhere, in the treatment of Bernays.

Now, as with Hilbert, if there is a Kantian residue in Bernays’s philosophical work, we’d expect to find it in his account of the nature of finitary reasoning  and its respects in which it is “intuitive”. And as Parsons remarks, Bernays is actually not very explicit about this: indeed, “I have not found any place where Bernays gives what might count as an ‘official’ explanation of the concept of intuition”. And Parsons later remarks that Bernays e.g. “seems quite unworried” about how we get our knowledge that however far we go along the number series, we can always continue one more step.

There is a hint of admonition here, that Bernays ought to be worrying about these things. But perhaps the dialectical situation is such that these aren’t really his problem. After all, the Hilbertian hopes to work with  (a rather restricted portion of) whatever the intuitionist or predicativist or other critic  of infinitistic mathematics will allow, and show that those agreed materials are enough in fact to show the consistency of classical analysis. So, it is the critic, then, who needs e.g. a defence-in-depth of claims about the special “intuitive” status of the agreed materials which he wants to contrast with the infinitary excesses of classical analysis: it is the critic who might have a problem about whether his stringent epistemological principles allow him even to know that we can always continue one more step along the number series. At the end of the day, the Hilbertian need only say, e.g., “the kind of finitary reasoning that I am using in my proof theory counts as intuitive by your standards (in so far as I understand them), so you at least can’t complain about that“.

Be that as it may, the discussion of Bernays in the ‘Kantian legacy’ paper is perhaps not very exciting. So let’s now turn to the third piece in Parsons’s book, a substantial paper on ‘Paul Bernays’s later philosophy of mathematics’.

To be continued …

Aberystwyth sunset

Old College, Aberystwyth, at sunset
Old College, Aberystwyth, at sunset

The Guardian has published its latest rankings of UK universities. These things mustn’t be taken too seriously, of course. But I see that Aberystwyth, where I taught for the first half my career, has now plummeted from 50th overall three years ago to 106th (out of 116). And it isn’t just the Guardian which perceives a rapid falling off: the Complete University Guide has Aber dropping over the same period from 47th to  87th. Those are, surely, huge drops, signalling something pretty dire going on. And apparently, applications are dropping sharply too, with the number of new undergraduate students at the university dwindling from 3,283 in 2011 to just 2,510 in 2013, and entry requirements for some courses hitting rock bottom. (There’s now an online petition with some excoriating comments about the current vice-chancellor who, incidentally, got nearly a 10% “performance-related” pay rise for presiding over the first two years of this plunge down the ratings, and now earns 50% more than the Prime Minister …)

When I was in Aber, there were some very distinguished people scattered round the college. My colleagues in the philosophy department included D.O. Thomas, who wrote the definitive book on Richard Price, the Berkeley scholar Ian Tipton, and O. R. Jones — all extraordinarily nice men, and thoughtful serious philosophers (and very hard-working and productive too, by the standards of the day). We had some very good students too: for example,  Sue Mendus started as a student the year I arrived there.  A department very ripe for closure, then, as happened in the “Thatcher cuts” of the later 1980s.  At that time, the institution faced hard choices but seemed to jump the wrong way repeatedly, and the slow diminution as a serious place started that has now accelerated alarmingly.

Aber was the founding college of what became the University of Wales (a national institution which also no longer exists in its original form). It is sad to see the place, which used to be held in quite unusual but very understandable affection by its students, in such precipitous decline.

M.M. McCabe: the crisis of the universities

Talking together, talking to ourselves: Socrates and the crisis of the universities.

Here Prof. McCabe says with passion and eloquence and learning what many of us think and sometimes, so much more stumblingly, try to express. Three cheers!

I hope that she makes available a written version of this valedictory lecture for those — normally including me — who, if only for time reasons, don’t get round to watching lectures online.  But yes, as Richard Baron says in his comment, the live performance in this case carries an impact that indeed makes it more than worth watching.

Ah, “impact” …

Parsons #1: Predicativity

As I’ve said before, I’m planning to post over the coming weeks some thoughts on the essays (re)published in Charles Parsons’s Philosophy of Mathematics in the Twentieth Century (Harvard UP). I’m going to be reviewing the collection of Mind, and promising to comment here is a good way of making myself read through the book reasonably carefully. Whether this is actually going to be a rewarding exercise — for me as writer and/or you as reader — is as yet an open question: here’s hoping!

This kind of book must be a dying form. More and more, we put pre-prints, or at least late versions, of published papers on our websites (and with the drive to make research open-access, this will surely quickly become the almost universal norm). So in the future, what will be added by printing the readily-available papers together in book form? Perhaps, as in the present collection, the author can add a few afterthoughts in the form of postscripts, and write a preface drawing together some themes. But publishers will probably, and very reasonably, think that that’s very rarely going to be enough to make it worth printing the selected papers. Still, I am glad to have this collection. For Parsons doesn’t have pre-prints on his web page, the original essays were published in very widely scattered places, and a number of them are unfamiliar to me so it is good to have the spur to (re)read them. And I should add the volume is rather beautifully produced. So let’s make a start. For reasons I’ll explain in the next set of comments, I’ll begin with the second essay in the book, on predicativity.

Charles Parsons’s new book — let’s discuss it (again)!

OK, I’m back from Cornwall (and very nice it was too, thanks for asking), and am trying to get the philosophical corner of my mind back into gear.  Now, as I mentioned a couple of posts ago, Charles Parsons has published a new collection of some of his essays, Philosophy of Mathematics in the Twentieth Century, which (sensibly or otherwise) I’ve said I’ll review for Mind. So I better make a start on the reading. On the principle that telling people you are going to do something is a good way of keeping yourself up to the mark, I said that as I read through I’ll start posting some comments here. Please do chime with and thoughts and comments of your own.

If you want to be reading along, now I’ve had a first skim through the beginning of the book, here’s the plan for the first few instalments. For my opening effort — which I’ll post at the end of the week — I’ll look at Parsons’s second essay ‘Realism and the Debate on Impredicativity, 1917-1944’ (originally published in the Feferman festschrift edited by Sieg et al.).

Parsons’s first essay, new to the present volume, is on ‘The Kantian Legacy in Twentieth-Century Foundations of Mathematics’, and perhaps the most interesting bit of this not-very-exciting essay is on Bernays, so it seems a good notion to discuss that alongside the third essay ‘Paul Bernays’ Later Philosophy of Mathematics’ (published in Dimitracopoulos et al., eds, Logic Colloquium 2005).

Then for my third instalment I’ll look at Parsons’s next two pieces, a short piece on Gödel from the Dictionary of Modern American Philosophers, and then the substantial piece on Gödel’s essay ‘Russell’s Mathematical Logic’ written as an Introductory Note for Gödel’s Collected Works, Vol. II.

So watch this space!

Postcard from Cornwall

We have been away from Cambridge in Cornwall, where the weather has been extraordinarily kind. So there’s been a lot of sitting around in the sun by the sea, walking the coastal paths, or visiting gardens like Trelissick (which you can reach, delightfully, by ferry). Wonderful.

We did make one trip across to St Ives, which did not strike us as being now as nice a place as past reputation would have it. But it was more than worth the journey, just to see the Barbara Hepworth Museum and Sculpture Garden.

Here is Hepworth, writing in The Studio, 1946:

“I have always been interested in oval or ovoid shapes. The first carvings were simple realistic oval forms of the human head or of a bird. Gradually my interest grew in more abstract values – the weight, poise, and curvature of the ovoid as a basic form. The carving and piercing of such a form seems to open up an infinite variety of continuous curves in the third dimension, changing in accordance with the contours of the original ovoid and with the degree of penetration of the material. Here is sufficient field for exploration to last a lifetime.”

“Before I can start carving the idea must be almost complete. I say ‘almost’ because the really important thing seems to be the sculptor’s ability to let his intuition guide him over the gap between conception and realization without compromising the integrity of the original idea; the point being that the material has vitality – it resists and makes demands.”

“I have gained very great inspiration from Cornish land- and sea-scape, the horizontal line of the sea and the quality of light and colour which reminds me of the Mediterranean light and colour which so excites one’s sense of form; and first and last there is the human figure which in the country becomes a free and moving part of a greater whole. This relationship between figure and landscape is vitally important to me. I cannot feel it in a city.”

Thoughts of a lucidity that contrasts, as Mrs Logic Matters tartly remarked, with the sort of nonsense you can get from modern artists writing about their work …

Charles Parsons’s new book — let’s discuss it!

Charles Parsons has a new book out, Philosophy of Mathematics in the Twentieth Century
a collection of his essays (all but one, I think,  previously published but with some new postscripts). Here’s the table of contents. So there is a broad unity of theme, but there should be enough variation of topic between the various essays to maintain interest.

Parsons is always worth reading and thinking about, and can write extremely lucidly. So I’m going to be reviewing this for Mind, since I’ll enjoy the chance of re-reading some of these essays, and catching up with those I haven’t read before. My plan is to post about some thoughts on the essays in order, starting about three weeks hence.

I’m announcing this in advance to give you a chance to get a copy of the book (or persuade your librarian to do a rush acquisition). I do hope some will be inspired to join in discussion in the comments.

Angela Hewitt plays the Art of Fugue

Last night here in Cambridge, a truly remarkable concert. Remarkable for a start that such an austere programme — a performance of the Art Of Fugue on the piano — was sold out, and to an audience seemingly of much wider age spread than is too often the case with concerts here. Remarkable in the event for the utter concentration of the audience, silent and hardly moving in their seats for an unbroken hour and a half. But most remarkable, of course, for the performance by Angela Hewitt, which made an absolutely compelling case that this music is not just for the study, or for solitary listening with score in hand to follow the twists and turns, but can work in the concert hall. Judging from the reception at the end, the audience were more than won over. There was a shared sense that we had witnessed musicianship of a quite astonishing kind in action.

You’ll have to take my word for it for now: but Angela Hewitt has recorded a CD of the Art of Fugue for Hyperion (to add to her other stunning Bach discs) which will be out in the autumn, and you can judge for yourself then — though there was something magical about being there in the atmosphere at the live concert. I predict you will be bowled over.

The picture isn’t from last night’s concert but I think characteristic. But here, as tweeted by Angela Hewitt, visual proof that she was indeed in Cambridge!

Recursive pleasures

I’m much enjoying at the moment re-reading Hartley Rodgers’s Theory of Recursive Functions and Effective Computability. What prompts me to take the book off the shelf again is the treatment of constructive ordinals some two hundred pages in; but (one of the upsides of retirement) I’ve got the time to start reading from the beginning, and it is well worth spending the time doing so. The book is as good and illuminating as I remembered it as being. In fact more so, as I’m sure I didn’t really appreciate it, back in the day.

I bought my copy at the end of 1970, and paid seven pounds and nine shillings for it (the bookseller’s pencilled markings are still on the flyleaf). That was a lot more than we could afford, and I expect I didn’t fess up to my extravagance, for it would then have been about 8% of my monthly take-home pay. Such was my devotion to logic. Or my obsessive book-buying habit.

The book-buying has had to be much reduced, as we are pretty much constrained to a one-in, one-out policy (not of course, that it quite works like that). But I did get in the post today a copy of Rózsa Péter’s great Recursive Functions — the copy was relatively inexpensive and though it once belonged to the library at the National Physical Laboratory was seemingly hardly touched.  I’m not sure quite why, but I take real pleasure in having a copy at last.

Question (since the gender gap is vexing the philosophical interwebs these days): is Rózsa Péter the only woman so far who is the sole author of an indisputably significant mathematical logic book? Or am I having a senior moment and forgetting someone? (Even more run of the mill math. logic textbooks solely by women seem very few and far between: there’s Judith Roitman’s nice set theory text, and then ….?)

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