Tuesday, October 20, 2009

Mathematical Concepts and Learning From Experience

I've been reading Susan Carey's new book on the development of concepts, which features a lot of interesting stuff about the development of children's reasoning about number. The last two chapters are philosophical though, and bring up an important point, which it had not occurred to me needed to be stressed:

Learning from experience need not take the form of someone explicitly forming a hypothesis, and then letting experience falsify it/doing induction to conclude the hypothesis is true.

If this were all that experience could do, it would be hopeless to appeal to it to help explain how we could get mathematical knowledge. For, plausibly, you only count as having the concept of number, once you are willing to make certain kinds of applications of facts about the numbers, reason about the numbers largely correctly etc. So, by the time that experience could falsify hypotheses containing the mature concept of number, you would already have to have lots of mathematical knowledge.

Instead, experience helps us correct and hone our mathematical reasoning all through the process of "developing a concept". How can this be?

Well, firstly, think about the way students are normally introduced to the concept of set. No one makes a hypothesis that there are sets, nor do math profs attempt to define sets in other terms. Rather the professor just demonstrates various ways of reasoning about sets, ways of using these claims to solve other mathematical problems etc. and gets the students to practice. Given this, the student's usage and intuitions conform more and more to standard claims about the sets, and eventually they count as having the concept of set.

I propose (and I think Carey would agree) that the original development of many concepts in mathematics works similarly, only with trial and experience playing the role of the teacher.

You start out not having the concept, and try various usages. Here, however, rather than having a professor to imitate, you just have your general creativity/trial and error/analogical reasoning to suggest ways of reasoning about "the X"s and then an ability to check whatever kinds of consequences and applications you expect at a given time. Often this kind of creative trying and analogical reasoning will turn out to fail in some way, such as leading to contrdiction, or underspecifying something important. But then you can correct it. Inconsistent reasoning about limits in the 19th century and sets in the early 20th would be examples of the former. And the kind of process of refinment of the notion of polygon in Lakotosh's Proofs and Refutations would be an example of the latter.

We try out various patterns of reasoning about the world (e.g. calling certain things Xs, trying to apply the analogue of good reasoning about one domain to another) -with perhaps a nudge from brain structures subject to evolution effecting which patterns we are likely to try- and experience corrects these inference patterns until they cohere enough that we count as genuinely having some new concept. And note that no conscious scientific reasoning must be assumed to start this process, all we need some disposition to go from seeing things to making noises to doing things, together with a playful/random/creative inclination to try extending those dispositions in various ways!

p.s. I haven't emphasized this point the past, because I think questions like 'when exactly does someone start having the concept of X?', don't generally cut psychology or metaphysics at their joints. I mean: when exactly did people start having the modern conception of atom? The interesting facts are surely facts about when people started accepting this or that idea atoms "atoms", or reasoning about "atoms" in this or that way. Coming up with a decision about exactly what amount of agreement with us is necessary for people to count as having the same concept is a matter of arbitrary boundary setting.

But I realize now that ignoring the whole issue of concepts can be confusing. So let me just say:

When I say mathematical knowledge is a joint product of mathematically shapted problems in nature, correction by experience, the wideness of the realm of mathematical facts and the relationship between use and meaning, "Correction by experience" doesn't just mean what happens when hypotheses consciously proposed by people who already count as having all the right mathematical concepts get refuted. Rather, "correction by experience" includes what happens when you are inclined to reason some way, you get to an unexpected conclusion, and then subsequently become disposed to draw slightly different inferences/feel less confident when engaging in some of the processes that lead you there. You might or might not count as revising some hypothesis, phrased in terms of fully coherent concepts, when you do this.

p.p.s. The idea that experience helps us form coherent mathematical concepts, (while not figuring in the justification of our beliefs) is also a central theme in Carrie Jenkins' 2009 Grounding Concepts: an empirical basis for arithmetical knowledge.

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