Thursday, November 02, 2006


As a wise man once said, "Behold! I am full of post!"

So, while grading a midterm this afternoon/evening, one of the other GSIs and I had a brief cconversation on this notion of mathematical platonism. It turns out I am something of platonist--that is, I tend to believe that mathematical objects are real in some sense--but the reason I feel this is almost worth reporting to you, my esteemed public, is that I think the "mathematical objects are real" characterization is not the right way to think about this. Obviously, the natural numbers, \NN, (the counting numbers or nonnegative whole numbers) don't exist in the same way that my shirt exists--I can't, for example, spill food on the natural numbers. But if they don't exist in some obvious way, in what way can they be said to exist at all? Well, I think it may not be correct to think of the structure \NN as existing. Rather, I think that \NN is somehow the canonical or inevitable construction that beings need for reasoning about counting; that is to say, counting exists, and if we want to reason about this notion, then the construction of \NN is imposed on us. As far as I've thought it through (not very far), this is not identical to supposing that \NN has some kind of obscure or spiritual existence--in other words, I've made a quite mundane notion of transcendence to allow for the existence of certain transcendental objects as "natural ways of thinking about things." In a similar vein, then, I can allow for the real numbers \RR to exist as the canonical structure for thinking about a continuum (as for example, in the motion of an object through space).

Further back, I went to a logic colloquium talk on the structure "knowing an implication." Now, I'm not expert enough to really comment on the quality of the presenter's notions. Naturally, "belief" figured into this construction, but what really bothered me is that I'm not sure that I believe that people ever actually believe in implication in the mathematical/logical sense. I'm almost sure that people actually believe only in causation, and implication is formal way to talk about causation without making a mess. For example, I realize that theorems are generally stated as implications, but we generally follow the statement of a theorem with a proof, wherein we show how the hypotheses cause the conclusion to be true. Indeed, we often aren't particularly satisfied with a proof that has no intuitive (read: causative) content. A better toy model maybe is that of street lights at intersections. Certainly, one could be said to believe the implication "If my light is green, then the opposing light is red"; and certainly, it sounds silly to say, "my light being green causes the opposing light to be red." However, I'm prepared to claim that what a person really "believes" is that there is an agent causing the situation to be as it is. I think that worrying about belief in or knowledge of an implication is beside the point because implication is not the sort of thing that people can believe in--people know and believe in causative relationships and only subsequently hedge their bets or admit to ignorance of the nature of the cause by stating these beliefs as implications. Hence, as I'm not necessarily that interested in the properties of an ideal rational agent, I'm not sure why we are discussing knowledge of implication as such.