Bernays on sets and classes

Prompted by his two pieces on Bernays as philosopher, I’ve found myself pausing before reading on in Parsons’s book to step sideways and remind myself about Bernays the set-theorist.

I confess I’d not before dipped into Bernays’s little 1958 book Axiomatic Set Theory (long available as a Dover reprint). This aims to give a first introduction to the kind of set theory Bernays had developed in that long series of papers ‘A System of Axiomatic Set Theory’ published in seven instalments in the JSL between 1937 and 1954 (a promised second volume elaborating on further issues from those papers never appeared).

What we get here is, of course, NBG set theory, with classes as well as sets – which is certainly familiar to those of us who learnt our logic from the first edition of Mendelson’s textbook. Now, Mendelson’s presentation of the motivation for this style of theory, as I recall it anyway, isn’t compelling. And certainly, careless classroom talk can make this style of theory seem puzzling (“if we can have a class of all sets, why not a superclass of all classes — isn’t it just ad hoc to allow one but not the other?”). So I was struck by Bernays’s lucid explanation here — albeit in slightly fractured English — of what is going on. In summary,

Th[e] distinction between sets and classes is not a mere artifice but has its interpretation by the distinction between a set as a collection, which is a mathematical thing, and a class as an extension of a predicate, which in comparison with the mathematical things has the character of an ideal object. This point of view suggests also to regard the realm of classes not as a fixed domain of individuals but as an open universe, and the rules we shall give for class formation need not to be regarded as limiting the possible formations but as fixing a minimum of admitted processes for class formation.

In our system we bring to appear this conception of an open universe of classes, in distinction from the fixed domain of sets, by shaping the formalism of classes in a constructive way, even to the extent of avoiding at all bound class variables, whereas with regard to sets we apply the usual predicate calculus. So in our system the existential axiomatic method is joined with a constructive formalism.

By avoiding bound class variables we have also the effect that the class formation \{x \mid \mathfrak{A}(x)\} is automatically predicative, i.e. not including a reference by a quantifier to the realm of classes … Further the conception of classes as ideal objects in distinction from the sets as proper individuals comes to appear in our system by the failing of a primitive equality relation between classes. (pp. 56-57)

And so on, in the same vein. There is, of course, helpful modern elucidation about NBG and other theories with classes as well as sets out there in the literature, in e.g. the last two appendices of Michael Potter’s Set Theory and its Philosophy. But I do wish I’d known Bernays’s own presentation a long time ago!

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Pavel Haas Qt play Smetena 2nd Qt


The Pavel Haas, playing Smetena’s 2nd Quartet live on the BBC, starting about 36 minutes into this programme. [Radio programme now longer available.]

They were en route to Aldeburgh: here’s a review of their performance. I foolishly left trying to get tickets until too late, so we weren’t there — but we’ve heard the Pavel Haas play the Janacek and Smetena Quartets at various concerts and the review chimes very much with our experiences of them. If you get a chance to hear them live, grab it.

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Veronese again


We went to the Veronese exhibition at the National Gallery just after it opened, and were quite bowled over. So we revisited it yesterday, and are more than glad we did so. It was a particular delight to see again the portraits of Iseppo da Porto and his wife Livia, he with a protective arm round their oldest son, she with one of their daughters who is still looking at us quizzically across four and a half centuries (click to enlarge). Now Livia lives on in Baltimore, and Iseppo is in the Contini Bonacossi Collection you can visit one day a week in Florence. But the works belong together — they are not just wonderfully executed but also make a quite extraordinarily humane and very touching pair.


By the datings in the catalogue, Veronese was just 24 when he painted the da Porto portraits in 1552. Which is surely remarkable. And below, painted near the end of his life some 30 years later, is another of our favourites, Venus, Mars and Cupid (from the Scottish National Gallery). Bolder, so richly coloured, but yet for gods at play still rather touchingly human.

Some huge altarpieces, teeming with life and movement, had been allowed to travel from Italy, and the National displayed them very dramatically. By using some of large galleries upstairs (rather than the smaller rooms in the basement of the Sainsbury Wing usually used for special exhibitions), we were allowed vistas through one room to another to see pictures framed by great doorways. The whole exhibition was a triumph (and so it was a surprise that we could just walk in without booking or even queuing, two days before it closed).

So yes, sorry, if you haven’t seen it, and can’t be in London tomorrow, you have missed your chance (the exhibition closes on June 15th). But I can warmly recommend the catalogue as a typically fine piece of book-production by the Gallery (the text is more historical than art-critical, but none the worse for that).  The paperback seems to be sold-out at the moment, but the hardback is still something of a bargain. And is full of delights such as this …


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Parsons #3: Bernays (continued)

We noted a couple of the most familiar early papers by Bernays, and picked out a prominent theme –  a kind of anti-foundationalism (as Parsons labels it). Perhaps we can give finitary arithmetic some distinctive kind of justification (in intuition? in ‘formal abstraction’?), but classical infinitary mathematics can’t receive and doesn’t need more justification than the fact of its success in applications — assuming it is consistent, of course, but (at least pre-Gödel) we might hope that that consistency can be finitarily checked. And for a second theme, we remarked that while Bernays endorses classical modes of reasoning — if not across the board, then in core analysis and set theory — he is not a crude platonist as far as ontology goes: if anything there is a hint of some kind of structuralism.

The anti-foundationalism and hint of structuralism  are exactly the themes which are picked up in Parsons’s paper ‘Paul Bernays’ later philosophy of mathematics’.

In discussing the a priori in a number of places, the later Bernays certainly “distance[s] himself from the idea of an a priori evident foundation of mathematics” (to quote Parsons);  rather in “the abstract fields of mathematics and logic … thought formations are not determined purely a priori but grow out of a kind of intellectual experimentation” (to quote Bernays himself). But what does this mean, exactly? Bernays proves elusive, says Parsons, when we try to discern more of how he views the “intellectual experience” which is involved in the growth of mathematics. Still,

There’s no doubt that [Bernays] continued to accept what has been called default realism … which amounts to taking the language of classical mathematics at fact value and accepting what was been proved by standard methods as true. … In fact a broadly realist attitude was part of his general approach to knowledge in the post-war years. In one description of the common position of the group around Gonseth [including Bernays], he emphasizes that the position is one of trust in our cognitive faculties. He also introduces the French term connaisance de fait; the idea is that one should in epistemology take as one’s point of departure the fact of knowledge in established branches of science. The stance is similar to the naturalism of later philosophers, though closer to that of Penelope Maddy … than to that of Quine.

True, at least from the evidence Parsons provides, Bernays’s proto-naturalism remains rather schematic (in the end, negatively defined, perhaps, by the varieties of foundationalism he is against). But the theme will strike many as a promising one.

As for the structuralist theme, Parsons says that Bernays “in later writings … took important steps towards working it out.” On the evidence presented here, this is rather generous. Certainly Parsons finds no hint of the key thought, characteristic of later structuralisms, that mathematical objects have no more of a nature than is fixed by their basic relationships in a structure to which they belong. What we do get is a discussion of mathematical existence questions, proposing that these typically concern existence-relative-to-a-structure (an idea that might seem much more familiar now than it would have done when Bernays was writing). Still that can’t be the end for the story since, as Bernays recognises, there remain questions about the existence of structures themselves. According to him, with those latter questions

We finally reach the point at which we make reference to a theoretical framework. It is a thought-system that involves a kind of methodological attitude: in the final analysis, the mathematical existence posits relate to this thought-system.

But then, Parsons says, “Bernays is not as explicit as one would wish as to what this framework might be”, though Bernays seems to think that there are different framework options. There are hints elsewhere too of views that might come close to Carnap’s. But then Parsons also doubts that Bernays would endorse any suggestion that choice of framework is merely pragmatic (and indeed, that surely wouldn’t chime too well with the naturalism). So where are we left? I’m not sure!

“I have claimed,” says Parsons, “some genuine philosophical contributions, but their extent might be disappointing, given the amount Bernays wrote.” Still, Bernays comes across as an honest philosophical enquirer striking off down what were at the time less-travelled paths, paths that many more would now say lead in at least promising directions. So I remain duly impressed.

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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 …

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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.

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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” …

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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.

Continue reading

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This wireless world


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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!

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