Archive for May 13th, 2011

May 13, 2011

Mathematical knowledge

by Neil Rickert

Philosophers seem to be a bit puzzled as to how there can be such a thing as mathematical knowledge.  And what puzzles them is the question of what mathematical entities (such as numbers) really are.  It’s a bit strange that they are puzzled, but then philosophy is a bit strange anyway.

The philosophical account of knowledge typically begins with ontology, the question of what exists.  And ontology is taken to be part of metaphysics.  So if mathematical entities, such as numbers, are fictions (as a fictionalist claims), then they don’t really exist.  Or if numbers exist only in a Platonic world of ideal forms (as the mathematical Platonist asserts), then their existence seems a tad dubious.  Once having established what exists, the philosopher typically says that knowledge is in the form of true beliefs (or believed facts).  However, according to philosophers, those beliefs have to be about something that exists.  If a belief is about fictional entities, then it isn’t about anything in the world, so it could not possibly be a fact.

We begin to see the difficulty.  Philosophers want statements to be propositions, descriptive statements about the world, before they can be considered to be knowledge.  But since mathematical statements purportedly describe only fictional entities or entities in an imagined Platonic world, they do not seem to fit the requirement to be descriptions.

The idea that mathematical truths are not facts seems rather strange to most mathematicians.  For those mathematical statements are among our most certain facts.  Moreover, many of those mathematical facts have proven enormously useful in the sciences.

The mistake is to treat mathematical statements as if they were descriptions.  Mathematics is not descriptive; it is methodological.  Mathematical statements are statements about important methods that are used.  And, among other things, those methods are often used to form factual statements (descriptions).

As an illustration, consider simple integer arithmetic (sometimes referred to as number theory).  The idea of arithmetic arises from counting, and counting is a method that is used to acquire factual information about the world.  The addition operation in arithmetic comes from the idea of counting in groups, and then combining the counts of the groups to yield a total count.  And multiplication comes from the idea of counting in groups in a very systematic way, so that all of the groups have the same count.

Arithmetical statements about numbers are really statements about counts.  We use numbers as place holders for the counts, so that our theoretical analysis of the consequences of counting does not depend on whether we are counting sheep or counting beans or counting people.  Arithmetic is so broadly useful, because it is applicable to any kind of counting.  The fact that numbers are abstract, are really fictional counts used as place holders to stand for real counts, is what makes arithmetic so general and so broadly useful.  And yet it is this very feature, this aspect of mathematics that makes it so general and so useful, that causes philosophers to question the epistemic status of mathematics.