Parsons’s Mathematical Objects: Secs 52-53, Reason, "rational intuition" and perception

Back to Parsons, to look at the final chapter of his book, called simply ‘Reason’. And after the particularly bumpy ride in the previous chapter, this one starts in a very gentle low-key way.

In Sec. 52, ‘Reason and “rational intuition”‘, Parsons rehearses some features of our practice of supporting our claims by giving reasons (occasionally, he talks of ‘features of Reason’ with a capital ‘R’: but this seems just to be Kantian verbal tic without particular significance). He mentions five. (a) Reasoning involves logical inference (and “because of their high degree of obviousness and apparently maximal generality, we do not seem to be able to give a justification of the most elementary logical principles that is not in some degree circular, in that inferences codified by logic will be used in the justification”). (b) In a given local argumentative context, “some statements … play the role of principles which are regarded as plausible (and possibly even evident) without themselves being the conclusion of arguments (or at least, their plausibility or evidence does not rest on the availability of such arguments).” (c) There is there is a drive towards systematization in our reason-giving — “manifested in a very particular way [in the case of mathematics], though the axiomatic method”. (d) Further, within a systematization, there is a to-and-fro dialectical process of reaching a reflective equilibrium, as we play off seemingly plausible local principles against more over-arching generalizing claims. (e) Relatedly, “In the end we have to decide, on the basis of the whole of our knowledge and the mutual connections of its parts whether to credit a given instance of apparent self-evidence or a given case of what appears to be perception”.

Now, that final Quinean anti-foundationalism is little more than baldly asserted. And how does Parsons want us to divide up principles of logical inference from other parts of a systematized body of knowledge? His remarks about treating the law of excluded middle “simply as an assumption of classical mathematics” suggest that he might want to restrict logic proper to some undisputed core — though he doesn’t tell us what that is. Still, quibbles apart, the drift of Parsons’s remarks here will strike most readers nowadays as unexceptionable.

Sec. 52, ‘Rational inuition and perception’, says a bit more to compare and contrast intuitions in the sense of statements found in a given context of reasoning to be intrinsically plausible — call these “rational intuitions” — and intuitions in the more Kantian sense that has occupied Parsons in earlier chapters. As he says, “intrinsic plausibility is not strongly analogous to perception [of objects]”, in the way that Kantian intuition is supposed to be. But perhaps analogies with perception remain. For one thing, there is the Gödelian view that intrinsic plausibility for some mathematical propositions involves something like perception of concepts. And there is perhaps is another analogy too, suggested by George Bealer: reason is subject to illusions that, like perceptual illusions, persist even after they have been exposed. But Parsons only briefly floats those ideas here, and the section concludes with a different thought, namely there is a kind of epistemic stratification to mathematics, with propositions at the lowest level seeming indisputably self-evident, and as we get more general and more abstract, self-evidence decreases. Which is anodyne indeed.

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