Philosophers, including philosophers of science, talk about representations. For example, the statement “the cat is on the mat” might be a representation of one aspect of the world. When mathematicians talk of representations, they think of a representation as a mapping of one space into another. For the way that philosophers talk of representations, a mathematician might think of that as a mapping from reality to the space of linguistic expressions.

One way of having representations of the world is to come up with ad hoc methods of representing little bit of the world, and then tossing them together, willy nilly, to provide a more comprehensive representation of the world as a whole. For want of a better term, I’ll call that way of representing the “willy nilly method” and I will refer to a representation result from such a method as a willy nilly representation. The naming of houses by the first builder in my post “The parable of the three builders” could be considered a willy nilly representation.

Scientists have usually preferred to be more systematic in the way that they form representations. The second and third builders in the parable were more systematic than the first. It will be convenient to use the expression “systematic representation” for a representation that is done following a systematic methodology.

When we think of representation as a mapping between reality and a space of representations, then there is the possibility that we can use some structure from the representation space to analyze the representations and, by implication, to analyze what is represented. This will only work when the representation space is structured, and when the structure of the representation space is a good fit (in some hard to define sense) for the world that is being represented.

We see this use of representational structure in the ordinary use of logic within philosophy. The world is represented in the space of linguistic expressions. The the logical structure of linguistic expressions can be used to analyze the representations and, by implication, to analyze the world itself. When the structuring is a poor fit, this analysis can fail, as we see happens with the sorites paradox.

Scientists, where possible, try to be systematic in the way that the represent the world. Where possible, they look for a representation space with a mathematical structure that will fit reasonably well. They do not expect it to be a perfect fit. The choice of how to construct representations is a pragmatic one. Scientists experiment to see what works. And if they can find a sufficiently structure mathematical representation space, then they can use that mathematical structure to analyze representations, and to make plausible inferences about the world that is represented.

**A simpler account**

Here’s a simpler way of saying what I said above.

Science is systematic. It prefers systematic methods of represention over willy nilly methods. Mathematics is the theoretical study systematic methodology. And that is why mathematics is so useful to the sciences.

According to some accounts, epistemology is supposed to be an abstract theoretical study of knowledge. But if we look at knowledge, as used within the sciences, then it is mathematics that is the appropriate abstract theoretical study. This is why scientists find mathematics to be very useful, but they find epistemology to be largely useless. Perhaps the correct spelling of epistemology ought to be MATHEMATICS.