Mathematics and science

by Neil Rickert

In his blog post today:

thonyc gives some interesting history on the use of mathematics in science. I found this quite interesting, and it corrects some of my own misunderstandings. Popular books which touch on the history of mathematics tend to gloss over much of the detail.

We tend to see the use of mathematics in science as relatively sudden. But thonyc’s account shows that it was actually more gradual. In a way, that makes a lot of sense and is perhaps what we should have expected.

The experimental method

Discussions of the scientific method usually emphasize the idea of experimental testing. That’s how I was introduced to science in elementary school. Many internet discussions of science emphasize the experimental method. This can be a way of distinguishing between science and religious creationism, because the so-called scientific creationists do not use the kind of experimental testing that we see in science.

In reality, though, experimental testing is not limited to science. A good cook tests her concoctions. A tennis player tests his strokes. Experimental testing is ubiquitous in life, and is better thought of as part of pragmatism. Even a religious creationist tests his ideas by seeing how his intended audience responds to his stories.


Another important aspect of science, is that it is systematic. Nobody denies this, but it usually gets very little emphasis. And that’s where the mathematics comes in. We can think of mathematics as a theory of systematicity. Science use mathematics because it provides them with good ways of being systematic in their explorations.

As an example, think of an ordinary ruler. The actual length, whether 1 foot or 1 meter is pretty much arbitrary as a standard. That we use the same standard everywhere is an aspect of being systematic. And science has long since moved to the metric standard. Ordinary life, with the USA still using the foot, is not nearly as systematic as science.

But we don’t stop with that one standard. We divide the ruler up into parts, with calibration marks. This can be done using the geometric construction for dividing a line into equal parts. So mathematics is already involved in the calibration of the ruler. We then number these parts, so we have brought in arithmetic for naming the different lengths. And because we did this in such a systematic way, we can now add the lengths of two things to get a combined length.

Mathematics is sometimes described as a science of pattern. And it is often said that scientists find patterns in the world. However, it is far from clear what “pattern in the world” would actually mean. Scientists don’t go around looking for patterns in the world. However would they do that. Rather, they may find patterns in the data. But we don’t get data directly. We only get data because of our systematic ways of conceptualizing and measuring the world.

The periodic table illustrates this. The scientists, particularly the chemists, have been able to divide the material world into elements. That dividing is a systematic way of categorizing materials. By purifying the elemental materials, then measuring them, we come up with interesting data such as the conductivity of copper, the conductivity of aluminum, etc. If we had just studied pieces of mineral that we came across, we would find far fewer patterns. By systematically purifying them, then studying each in isolation, we learn so much more.

Philosophy of science

Systematicity is central to the practice of science. And mathematics is a theory of systematicity. When we put those together, we begin to realize that mathematics is really a central philosophy of science.

Unfortunately, academic philosophers already use the name “philosophy of science” for an exercise in epistemology. But most scientists do not find that particularly useful in their work. It is mathematics, the real philosophy of science, that they rely on.

One Comment to “Mathematics and science”

  1. Many scientists do not delve into the mathematical underpinnings or implications of their concepts beyond simple statistical notions. Biologists in particular, I am noting lately. If they did,they would be far less sanguine about the feasibility of Neo-Darwinian theories concerning how all present (and past) organisms arose via exclusively random mutation events. As David Gelerntner and David Berlinski have pointed out… to kneejerk scorn.


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