Jeffrey Roland has an article (a subscription to Philosophia is required to view the full text) arguing that the prospects aren’t good for extending my 2005/2008 account of arithmetical knowledge to set theory. The purpose of this post is to sketch a reply to his main argument. (I’ll also then mention a couple of quibbles, because I can’t resist.)
Roland’s main challenge is that the concept set cannot be grounded in sensory input. He says that ‘sensory input is too coarse-grained to make the distinction between the concepts of set and class’, since ‘these concepts are coextensive in the (finite) world of our observations’. This, I think, is closely related to the kinds of underdetermination worry I discuss in my book (pp. 227-30).
Roland correctly anticipates that I will not accept that the kind of ‘coextensiveness’ he gestures to makes grounding for either or both of these concepts impossible. For one thing, as he notes, I think that any concept that appears essentially in our best scientific theories is one we have reason to believe is grounded. If all of mathematics is part of best science, then the concept of set is looking pretty good by these lights. I’m pretty confident that at least enough of mathematics is part of best science to make that concept indispensable to best science.
But a more illuminating way, I think, of replying to this kind of worry is to point out that empirical concept grounding is not supposed to be construed in a simplistic way, whereby some particular identifiable batch of experiences grounds each particular concept. If that was the view, maybe it would be a problem that the same sets and classes look the same, sound the same, and so on. But what I have in mind is that the concepts of set and class may prove their worth (individually or – more likely - in tandem with each other and with other concepts) by enabling us to make best sense of our experience of the world, and hence of the world. I want to understand this process in a holistic (or at least molecularistic) way. By analogy, triangular figures and trilateral figures look the same, sound the same, etc.; but that doesn’t make trouble for the view that the two concepts trilateral and triangular are both really useful for making best sense of all of our experience.
If concepts like set and class are useful in the way envisaged here, then in my opinion the best explanation of their usefulness is that there is something about world that both of those concepts (or perhaps their ultimate constituents, if they are compounds) are mapping onto. And our responsivness to that is what makes those concepts grounded.
A related point worth making in response to underdetermination worries is that empirical concept grounding – like the empirical grounding of theories - should be construed as involving virtues besides empirical adequacy. Conceptual virtues (simplicity, fruitfulness, appearance/capacity to appear in virtuous theories …) might be drawn upon to help explain why a concept is grounded despite the apparent inadequacy of the experience which grounds it.
Contra Roland’s suggestion on my behalf, however, I certainly would not propose that we use such virtues as a basis on which to choose between the concepts set and class. Rather, I should say that it is the possession of such virtues that makes each of those concepts a useful, and hence grounded, one.
Finally, two quibbles:
(1) I don’t propose an ‘analysis of knowledge’, just a necessary and sufficient condition which I hope is illuminating.
(2) Roland asks: ‘[W]hy think that our concepts of such (transfinite) numbers are even apt for acquisition in response to sensory input, let alone actually grounded?’ I don’t actually require acquisition in response to sensory input. Even innate concepts can be empirically grounded on my view.
Recent Comments