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.