The remarkable effectiveness of math

Difficulty    

by Jonas Helsen

So this post will be a bit more, let’s say, philosophical. I’d like to share some of my thoughts on a particular subject which has always struck me when I was studying physics and also now while I’m doing it in what might be called a professional fashion. That subject is mathematics. More precisely it is mathematics as applied to physics. Now I won’t pretend to be anything close to a real mathematician, but when you need a math-person and there are no mathematicians around you can probably do worse than a theoretical physicist. In physics, and also in computer science, we use math; a lot of it. In fact I would say that, and I think most physicists would agree with me, that mathematics is the language the universe is written in. Or at least the only language capable of describing it in an efficient manner. People often marvel at the ability of mathematics to capture physical phenomena in an extremely accurate and efficient manner, often waxing philosophically about the inherent simplicity of the universe. Here I’d like to give some of my, fragmented and incomplete, thoughts on the matter. While I certainly think that the fact that nature is describable at all is a fact worth pondering over long and hard I think the prevalence of math in physics and its remarkable effectiveness is at least partly due to decidedly more down to earth cultural forces present throughout the history of mathematics.

Math is simple

One of the reasons I think math is used so expansively in physics is its simplicity. That’s right, I just called math, for many the most hated and complicated subject in the world, simple. But hear me out. When I say math is simple I don’t mean that it’s not tremendously difficult, it is. What I mean is that, and this has always struck me as somehow noteworthy, many mathematical objects have very short description lengths. The definition of a mathematical object, when clearly posed, is often only a few lines long. But these short descriptions often clearly capture some very deep and important essence. This is almost never the case in other human endeavours: ask a lawyer or a legal theorist what the definition of “property” is and you will most likely get an answer that is at least several pages long. Ask a mathematician what the definition of a manifold is, a key math concept capturing the notion of “an object with curvature” and you will get at most a few lines of text. You probably won’t understand half the words in the text though.

A rock in the tidal waters of history

This brings me to what I think is one of the key reasons that math is so effective at describing things succinctly. I think that math is good at this because collectively mathematicians are ruthless in their pursuit of simplicity, of elegance. A new mathematical object, when it is invented (or discovered, depending on how you feel about that) can often be quite complicated. But after a while other mathematicians start using the concept in their endeavours, often replacing parts of the internal machinery or labeling things in a different way. Often a descriptive method gets discarded entirely in favour of a new and shinier one. Generations of mathematicians will ebb and flow over the definition until all sharp edges are washed of and what is left is a smooth, easy to work with surface. This process is vital to mathematics because it opens up these concepts as ready to use tools to other mathematicians, who might not have the time to learn some obscure ruleset, for use in new ways.

When in Mathland, do as the mathematicians

This process often parallels itself in physics. As new mathematical concepts become available the descriptions of physical behaviours become easier to use, less complicated. A good case study of this are the famous Maxwell laws of electromagnetism. These laws, which most physicists will be able to write down in some form if prodded enough, usually form the high point of the second year of physics education. Written down in their most succinct form, which has the added advantage of making their Lorentz Invariance readily apparent these laws take the form of a single equation. However, when Maxwell wrote down his laws in his “A Dynamical Theory of the Electromagnetic Field” from 1865, he identified 20 equations. It would take until 1884 for Oliver Heaviside to condense these equations into 4 equations that an undergrad would easily recognise, and it would take another 30 years before they were unified into a single equation, fully setting up the equivalence between magnetic and electric fields and setting Einstein on the path of discovering special and general relativity. However none of these reformulations would have been possible without mathematical advances introducing new and more efficient concepts and objects for physicists to play with.

So in summary, behind every elegant equation beautifully capturing the essence of some physical property lie generations of physicists and mathematicians slowly chafing away the rough edges to produce the mathematical diamond underneath.

Authors Note [30/06/17]: Several readers have mentioned historical works that might be interesting to interested readers so I thought I would add them explicitly here. If anyone has more interesting links let me know in the comments!

“The unreasonable effectiveness of Mathematics in the natural sciences – Eugene Wigner” https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

(Also the inspiration for the title of this post!)

“Il Saggiatore – Galileo Galilei” https://en.wikipedia.org/wiki/The_Assayer

About Jonas Helsen

Jonas Helsen

Jonas Helsen is an aspiring theorist in the Wehner group where he works on verifying quantum computers. In his free time he enjoys improvisational theater and pretending to be a superhero. He likes the Netherlands but wishes they wouldn’t put peanuts in everything.


2 thoughts on “The remarkable effectiveness of math”

  1. Nice post, Jonas. And especially, nice thoughts! This is one of the topics that I’m also, often, wondering about. How can mathematics be so compact and why does it describe the physical world so effectively, even though math is just an abstract language? Did you read “the unreasonable effectiveness of mathematics in the natural sciences” by Eugene Wigner (I suppose you have, reading your blog;-)). For me that raised even more questions.

    As for now, the blogpost implies that mathematics is effective due to its simplicity. But why would simplicity always provide the ‘correct answers’? And, if we would have had different mathematics, would we perceive the world differently?

  2. “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” JVN

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