Mathematics is a tool used by scientists to help them construct models of how the universe works and make precise predictions that can be tested against observation. That is really all there is to it, but I had better add some more or this will be a really short essay.
For an activity to be science, it is neither necessary, nor sufficient, for it to involve math. Astrology uses very precise mathematics to calculate the planetary positions, but that does not make it science any more than using a hammer makes one a carpenter (Ouch, my finger!). Similarly, not using math does not necessarily mean one is not doing science any more than not using a hammer means one is not a carpenter. Carl Linnaeus’s (1707 – 1778) classification of living things and Charles Darwin’s (1809 – 1882) work on evolution are prime examples of science being done with minimal mathematics (and yes, they are science). The ancient Greek philosophers, either Plato or Aristotle, would have considered the use of math in describing observations as strange and perhaps even pathological. Following their lead, Galileo was criticized for using math to describe motion. Yet since his time, the development of physics, in particular, has been joined at the hip to mathematics.
The foundation of mathematics itself is a whole different can of worms. Is it simply a tautology, with symbols manipulated according to well defined rules? Or is it synthetic a priori information? Is 2+2=4 a profound statement about the universe or simply the definition of 4? Bertrand Russell (1872 – 1970) argued the latter and then showed 3+1=4. Are the mathematical theorems invented or discovered? There are ongoing arguments on the topic, but who knows? I certainly don’t. Fortunately, it does not matter for our purposes. All we need to know about mathematics, from the point of view of science, is that it helps us make more precise predictions. It works, so we use it. That’s all.
I could end this essay here, but it is still quite short. Luckily, there is more. Mathematics is so entwined with parts of science that is has become its de facto language. That is certainly true of physics where the mathematics is an integral part of our thinking. When two physicists discuss, the equations fly. This is still using mathematics as a tool, but a tool that is fully integrated in to the process of science. This has a serious downside. People who do not have a strong background in mathematics are to some extent alienated from science. They can have, at best, a superficial understanding of it from studying the translation of the mathematics into common language. Something is always lost in a translation. In translating topics like quantum mechanics—or indeed most of modern particle physics—that loss is large; hence nonsense like the “God Particle”. There is no “God Particle” in the mathematics, only some elegant equations and, really, considering their importance, quite simple equations. One hears question like: How do you really understand quantum mechanics? The answer is clear, study the mathematics. That is where the real meat of the topic and where the understanding is—not in some dreamed up metaphysics-like the many worlds interpretation.
Closely related to mathematics are logical and rational arguments. Logic may or may not give rise to mathematics, but for science, all we require from logic is that it be useful. Rational arguments are a different story. Like mathematics, they are useful only to the extent they help us make better predictions. But that is where the resemblance stops. Rational arguments masquerade as logic, but often become rationalizations: seductive, but specious. Unlike mathematics, rational arguments are not sufficiently constrained by their rules to be 100% reliable. Indeed, one can say that the prime problem with much of philosophy is the unreliability of seemingly rational arguments. Philosophers using supposedly rational arguments come to wildly different conclusions: compare Plato, Descartes, Hume, and Kant. This is perhaps the main difference between science and philosophy: philosophers trust rational arguments, while scientists insist they be very tightly constrained by observation; hence the success of science.
In science, we start with an idea and develop it using rational arguments and mathematics. We check it with our colleagues and convince ourselves using entirely rational arguments that it must be correct, absolutely, 100%. Then the experiment is performed. Damn—another beautiful theory slain by an ugly fact. Philosophy is like science, but without the experiment. Perhaps the real definition of a rational argument, as compared to a rationalization, is one that produces results that agree with observations. Mathematics, logic, and rational arguments are just a means to an end, producing models that allow us to make precise predictions. And in the end, it is only the success of the predictions that count.
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 I believe this observation comes from one of the Huxelys but I cannot find the reference.