What is the role of mathematics in science?

This question seems to puzzle some people. Personally, I never found it at all puzzling. Much of the development of mathematics was done by scientists, because they needed that mathematics in their science.

Science is not simply a matter of describing or representing nature, as some seem to believe. Describing is the job of journalists, not of scientists. Of course, science does make descriptions. But descriptions do not come for free. Facts do not just pop into our heads. Science has had to develop many methodologies, in order to make it possible to describe various aspects of reality. And that development of suitable methodologies is a large part of science, and a large part of what distinguishes science from other areas of human endeavor.

In a recent post, I suggested that mathematical knowledge was mainly knowledge of the consequences of following methodologies. I illustrated that by suggesting that arithmetic is a study of the consequences of following the methodology of counting. I could equally have pointed to traditional geometry (or ruler-compasses geometry) as a study of the consequences of using a portable measuring rod. The initiative to study these methodologies came from the importance of those methodologies. And the mathematics thus developed has proved very useful.

More generally, scientists attempt to be as systematic as possible, in their development and use of methodologies appropriate to the area that they are studying. One could almost say that “Be systematic, young man” is one of the important principles of science. Mathematics, with its interest in patterns and regularities, is very much the abstract study of systematicity. So it should be no surprise that mathematics is useful to scientists. And it should be no surprise that scientists sometimes look to systematic methods studied in abstract mathematics, so as to see if those systems can be adapted for use by science.