[hist-analytic] Normativity and Intuition in Mathematics
Baynesr at comcast.net
Baynesr at comcast.net
Wed Oct 27 18:04:29 EDT 2010
There is a great deal of admirably technical discussion in the literature pertaining to normativity, and yet the idea is seldom defined with any precision, despite the flourish of references to big shots who know only one topic: meaning. Once discussion of normativity escapes the barrier of philosophy of language, it becomes mired in largely pointless drivel about ethics - not that ethics is drivel only this approach. So I'm embarking on another approach.
Normativity is more interesting in its applications than in itself. Or, at least, that's how I see it. Some might say the same thing about arithmetic. It's arguable, anyway. A good thought experiment, is to apply whatever conception you have of normativity to broadly metaphysical issues. This isn't impossible, although the "usual suspects" can emit their predictable condescending laughs; let's set them aside ("gooby") I believe there is a sense in which normativity interfaces the cognitive side and the best place to see this is in the Kantian view of mathematics. More exactly, I believe there are aspects of mathematics that are not matters of convention (a hair brained substitute for a mind substitute, done in fear of subjectivity, etc.), relying instead on a peculiar interrelation of normativity and intuition. I suspect Husserl - forget for now the usual knee-jerk blather about how he was wrong in his idealism, unless you are going to get textually specific. Let's begin with the idea of intuition in mathematics.
Let's go for an example is not '1+1 = 2'; actually, I think this is a very bad example. Consider '1/x' (or the ratio of the number one to some unspecified number coming from the domain of reals. I find this a "clear and distinct idea'. But there are intuitions that are more than clear and distinct: they are judgments (here, for now, I depart from Kant because I have a lot to say on the relation of intuition (in Kant's sense) and experience). What sort of judgments. Take an example: "As x approaches infinity, 1/x goes to 0." I know this, and I believe I know this without any authority higher than intuition. Now for the normativity part.
First, we have to be clear that 'normativity' is a slippery, largely, unexamined idea. I will begin by using it this way: a norm is a standard arrived at by decision or a property that exists only because it is intuited. Hold back on the obvious criticisms of the last remark, because I have not, as yet, discussed intuition, sufficiently. Recall I said 'as x approaches infinity'. Now if I am right this idea is not definable in purely mathematical terms. (Quine was wrong in "Truth by Convention). Take another example: '1-1/x'. I say: "As x approaches infinity the number being referred to goes to 1." It may never get to 1; nor, in the earlier case 0. This is all familiar to those among you who are mathematicians (or have some familiarity with the concept of a limit). So, we have two truths: one involving the clear and distinct idea of '1/x' (even though 'x' is a variable) and an intuitively based belief that it approaches 0 as x gets bigger. Moreover, much can be the same of '1-1/x'. However, the intuition supporting the second - on this view - is not as clear and distinct because it involves an additional operation etc. However. It IS equally intuitively justified once it is understood. As for normativity. I will have more to say on this. For those interested, the animus of some of my remarks are reflections on Russell in PoM and G. H. Hardy's definition of a limit. Interestingly, the idea of a limit is no less interesting than, say, the derivative for philosophical purposes. Russell knew this. I might discuss this a bit more in relation to Russell's take on related topics in Introduction to Mathematical Philosophy.
Regards
STeve Bayne
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