Okay I will admit that’s a bad pun, specially talk I won’t talk about maths… Maybe a bit, but not much. Now, I tend to waste my time trying to understand the physical principles of living things, but on the mean time, I like watching and read some mathematical content (youtube is a great starter, although it does take you into a rabbit hole).  Another thing I do notice as well is that a lot of people when first going into physics have that idea of not really needing a lot of maths, only to be met by juggernauts like Calculus and Linear Algebra (since I had those classes with math students, they were proof-based and many times it would feel out of place, like I just couldn’t understand why we would need some Cauchy’s definition of limit or Weierstrass’s, and why we even needed to know the two of them. BTW, I prefer Cauchy’s definition, but that’s just a personal preference). But I don’t wanna talk about how much math you can find in physics. Today I want to look at things from a different perspective.

Why is physics not a part of maths?

Now, I could argue that there is mathematical physics, which is typically considered has being an applied math field, but that’s not where I want to go. Both physicists and mathematicians may use a particular theorem to reach a certain conclusion, and yet you either have mathematicians who are not satisfied with physicist’s lack of rigor, and physicists who are thrilled by mathematical beauty when there is no pratical observation to be made (String theory!!). And what I was wondering was less about whether string theory is right or not (I couldn’t care less, I have proteins to deal with), but about this whole mathematical rigor vs predictability. It is a question that I was certain that someone else asked, and finding the answer, could help me not only understand why is physics not maths, but also it may lay the foundations for both physics and all other natural and applied sciences.

Now, onto the search. If we are to truly pinpoint the birth of physics, if we are to consider astronomy has being part of physics, it would be in the Sumerian civilization or perhaps, even earlier (Some theorize that structures like the Stonehenge where built for not only religious cerimony, but also astronomical observations). And yes even back then, people used maths to predict astronomical events like eclipses (it was not topology nor group theory, but they needed some intuitive knowledge of geometry and probability). But again, we can argue that wasn’t rigorous physics and maths. And for a very long period after that, while there was some occasional maths and physics intersection, the connection was not as deep as nowadays for instance. Even up until a couple of centuries, physics wasn’t even a science of its own, being part of natural philosophy alongside what would later become biology and physiology. This natural philosophy has with most other branches of philosophy, was the study of nature, by logical reasoning and observation. We have many examples of natural philosophy from ancient Greece, with some of its views being discarded, and others confirmed (Democritus and his atomist view of the world for instance, the guy foresaw the advent of atoms before they were cool).

By the way, fun fact, some pre-socratic philosophers were already trying to figure out the way the world and nature behaved and worked, without needing myths, and they were called physikoi (singular physikos) or as Aristotle called them, physiologoi (singular physiologos), both coming from the greek word for nature (physis), and that is why today, physiology and physics have so similar names. Both are in some way concerned with how nature works, just from different scales and perspectives.

For a very long time, physics was a branch of philosophy, and while certainly observations were made and some theory was definitely established, it was never fully grounded on mathematical logic, but mostly philosophical logic (there is a difference between these two, but I won’t talk about it here).

But natural philosophy still exists in some colleges (like Cambridge University) as a degree you can get. And at least according to Wikipedia (yeah, I’m not going to read a book on natural philosophy just to know what it studies), a lot of things studied under Natural Philosophy back then are still common for any physics undergrad: They studied the nature at the large scale (astronomy and cosmology), the intrinsic and extrinsic causes (etiology in modern philosophy), also studied chance, probability and randomness (Anyone who has had probability theory, statistics or statistical physics would recognize these subjects), elements (insert chemical joke here), matter (pretty much condensed matter and particle physics right here, to name a few), motion and change (yes, mechanics), physical quantities, relations between physical entities and the philosophy of space and time. Some of these are still taught to philosophy students at colleges and universities, particularly those in metaphysics or gnoseology.

The turning point that led to the development of “modern” physics (if you can call it modern) was the so-called development of the scientific method, where in the shortest way I can put it, you would first see something, that doesn’t have an explanation, you have an hypothesis, you test that hypothesis a bunch of times, and if to a certain error, the predicted results are obtained, your hypothesis is now a theory that explains that phenomena. And yes, a theory may not correspond to the actual reality, but it is the most accurate description you would have of said reality. That is, until you find a more accurate theory.

Now while physics incorporated more mathematical reasoning by the inclusion of the scientific method (arguably the work that quickstarted it was Newton’s Mathematical Principles of Natural Philosophy, essentially showing that instead of philosophical dialectics and reasoning, “maybe you could start doing some actual maths, good sir?”), the scientific method imposed a constraint that mathematicians didn’t have. And not only that, but also a philosophical branch still remained implicitly in physics and that is ontology. Ontology (greek for study of being)(according to the Merriam-Webster dictionary) is a branch of metaphysics concerned with the nature and relations of being, or this second one which I think is good for our purposes, “a particular theory about the nature of being or the kinds of things that have existence“. Those bold words there, those are what fundamentally distinguish between the most theoretical of physics and the least theoretical of maths. The fact that physics is fundamentally about understanding existence and being, things that are often ignored in mathematics. Mathematicians are not concerned whether or not our space as 2 or 3 or even 26 dimensions. They have been working with bigger dimensions for longer than when these questions first appeared in physics. It is irrelevant to a mathematician whether some function has an infinite where it is not supposed. “What do you, it is not supposed to reach infinity here? That’s just a property of the function.”

So after all this, what is the main difference between physics and mathematics?

While physics certainly draws a lot from mathematics, it’s main purpose is to describe nature and its laws, with mathematics being more like the tool used to describe it. A good analogy in social studies would be law and linguistics. One is more concerned with understand the laws of a society (and the philosophy behind said law), while another is concerned with the language used to describe said laws, and it may have more questions, like the nature of language, and how different languages share certain characteristics, about the sounds, the letters, the classes of words, the meaning behind words. This means that just like in law and linguistics, we have multiple fields in physics and mathematics, due to their own vastness. And if we still care about the mathematics behing physics, there is a field in applied mathematics that looks at mathematically rigorous descriptions of nature, and to mathematical properties of physical laws.

An example I can think of is how Maxwell’s Equations can be explored in Vector calculus and Analysis, by concepts like rotational and divergence, Gauss’s Divergence theorem and Stokes Theorem, or even in Tensor Calculus, by the definition of the Faraday Tensor in Special Relativity. There are many other cases of physical concepts being explored in the realm of mathematics, and mathematical theorems and conjectures being used in physical laws.

So, in the end, it may be that after all, physics and mathematics are not so different. It’s all about whether we study it for itself, or for a being.

Some of my sources:

• Applications of Mathematics (Archived link)
• Journal of Physics A: Mathematical and Theoretical (from Institute of Physics) – it has free access for the articles. For lighter reading, I suggest Physics World, also from IOP.
• Ontology (Wiki)