I want to flesh out the idea, which I expressed yesterday, that a priori knowledge is knowledge of rules which exist solely by virtue of knowledge of them, and which therefore is not justifiable in principle. The idea is a little obscure, but I think I can make it plainer. First, we have to recognize the distinction between propositional knowledge (representational knowledge, often called "knowledge that") and non-propositional knowledge (competence, often called "knowledge how"). A priori knowledge is not a matter of representational verisimilitude, but a matter of competence.

Proceduralism is the view that mathematical truths express cognitive procedures. The equation "2+2=4" does not represent a fact which could be corroborated by empirical observation, but a procedure which exists solely by virtue of the minds which carry it out. In that sense, it is a rule which is known, but which exists solely by virtue of the fact that it is known--that there are minds which can carry out the procedure.

A difficulty arises when we consider that "2+2=4" is a mathematical expression: it represents a procedure, and so we seem to have knowledge that "2+2=4" is a rule. So even if we adopt a proceduralist position, we want to recognize that we have propositional knowledge of mathematical rules. Still, there is a difference between knowing the procedure and knowing that it is a procedure. We need mathematical formulae (or other linguistic expressions) to express our knowledge that something is a rule; but what that knowledge is about--what the expression represents--is a variety of knowledge which itself is non-representational.

We cannot believe that 2+2=4 in the sense that we can believe Obama is POTUS. Knowledge of the procedure which "2+2=4" represents is not belief, but competence. Yet, we can have beliefs about the representation. We can have beliefs about what procedure "2+2=4" represents, for example, or about whether or not "2+2=4" is a well-constructed formula. Those beliefs are contingent and they are empirically justifiable. Such knowledge about mathematical rules is not a priori. But mathematical knowledge itself is a priori, because it is knowledge which constitutes those very rules; it exists only as procedures (or, rather, as dispositions to carry out procedures) in minds. This is not just the case for mathematical procedures, but all instances of a priori knowledge.

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## Saturday, April 16, 2011

### Procedures and the A Priori

Posted by Jason Streitfeld at 9:03 AM

Labels: epistemology, Philosophy of Mathematics

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