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 …