November 25, 2007

Three distinctions for an epistemology of mathematics

In The fate of knowledge, Helen Longino makes a distinction between three senses of the knowledge -- that is, three different usages of the term `knowledge'. These three senses of knowledge can help us tease apart some notions that have been tangled together in Analytic epistemology of mathematics for the last 130 years or so.

Necessary and possible

Necessity and possibility are modal notions. Modal notions show up in many different domains of enquiry: we have logical modalities, physical modalities, mathematical modalities, epistemological modalities, metaphysical modalities, and so on. It seems like unqualified modalities should be either logical or metaphysical. Zalta has argued that some logical impossibilities are metaphysical possibilities, though I suspect many metaphysicians around here assume the negation of this.

When made in mathematics, this distinction seems to be related to the truth status of mathematical propositions and the theories built up out of those propositions: Under `what circumstances' are mathematical propositions true? `All of them', so mathematical truths are `necessary'. It's therefore tied to at least one view -- and more likely a family of views -- of the content of mathematical knowledge.

A priori and a posteriori

These notions distinguish two sorts of justification or warrant relations. When I know (a priori) that 2+2=4, I have warrant for this belief. This warrant is different from the (a posteriori) warrant I have for the belief that it's a little cold in my house right now or that I'm being appeared to redly. The distinction does not refer to the doxastic processes by which I came to have these beliefs. In a famous paper, `What numbers could not be', Benacerraf argues that realists about mathematics cannot give a suitable epistemology for mathematics because their views are not compatible with any `causal theory of knowledge'. But the causal story about how I came to have my belief that 2+2=4 is not the same as the warranting story about how I came to be warranted in my belief that 2+2=4. A causal theory of knowledge may be confusing doxastic and warranting relations.

More generally, when I have a priori warrant, a certain relation obtains between my belief that 2+2=4, the proposition (or sentence or whatever) that 2+2=4, and possibly some other things (perhaps the Platonic numbers 2, 4, and the addition relation; perhaps the abstract natural number structure; perhaps something else entirely). This relation is the warrant or justification relation. I am said to have knowledge (at least in part) by standing in this relation. A priori and a posteriori therefore refer to the second, relational, sense of mathematical knowledge.

Analytic and synthetic

Until the eighteenth century, analysis and synthesis were two different and complementary mathematical methods. Analysis was the method of `breaking down' ideas, while synthesis was the method of `building up' complex ideas from simpler ones. In Ancient geometry, for example, one first analysed the relations between the given geometrical objects, and then assembled the simple relations thereby discovered into a rigorous proof of the desired claim. By the late nineteenth century, the terms were adjectives -- analytic and synthetic -- and not nouns. They were identified with logical relations, which in turn were bundled together with a priori and a posteriori. This is seen most clearly in Carnap's most important books, Der logische Aufbau der Welt and Logische Syntax der Sprache. These were the basis for Ayer's Language, truth, and logic, which provided the framework within which most Anglophone philosophers have been working for the past sixty years. The transition between these two uses of the terms is identified most prominently with Kant, whose usage is, (in)famously, neither the same as a priori/a posteriori nor easy to understand. Until Kant scholars provide us with a better understand of the use of the distinction during the eighteenth century, it is probably better to stick with the pre-Enlightenment methodological understanding of analysis and synthesis.

With this sense of the distinction in place, each of these terms refers to either one of two different knowledge-producing processes or one of two complementary parts of a single knowledge-producing process. This corresponds to Longino's third sense of knowledge.

As an epistemologist, I am unusual in that I most interested in the third sense: what are the processes by which mathematical knowledge is produced? Most contemporary philosophers of mathematics are primarily interested in the first sense, the content of mathematical knowledge, and the closely related metaphysical problems. That is, they want to understand what mathematics is about. A few philosophers have approached epistemology from the second side, and attempted to give an account of how mathematical beliefs can be warranted. For realists, these generally involve either a Platonic-Goedelian direct intuition of mathematicals or a quasi-Aristotelean abstraction of mathematical knowledge from perception. Kantian apriorist, Millean empiricist, and Fregean logicist views also show up occasionally, but are nowhere near as prominent in the literature. Obviously these five are related to views about the processes by which mathematical knowledge is produced.

Generally speaking, no philosopher of mathematics is at all happy with the proposals offered by any other philosopher of mathematics, and often a given philosopher of mathematics is not all that happy with his (the subdiscipline is ridiculously male-dominated) own views either.

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