This article is part of a series of essays I’ve written for an Introduction to the Philosophy of Science course at KTH on the fall of 2017.
Henri PoincarĂ© (a French mathematician known for the PoincarĂ© conjecture and for pulling off a fantastic goatee) once said: “Mathematics is the art of giving the same name to different things”.
This quote summarizes quite clearly the immense impact mathematics can have on studying the complex and intertwined laws of physics and thus making it a prime candidate for being able to join what, by definition, can’t be joined. Its powerful abstractions allow scientists from different fields to use the same language in an attempt to explain and predict the laws of nature.
As Wigner said in his paper on “The Unreasonable Effectiveness of Mathematics”[1], it is true that physics does not use all mathematical concepts to describe its inner workings, but the mysterious part lies in the fact that all of its laws use mathematics in one way or another.
Examples of this phenomenon can be quite suprising sometimes to say the least: quantum mechanics which is a physics theory that manages to describe with high fidelity, interactions between subatomical particles uses a complex Hilbert space.
As complex numbers were invented with no particular links to the natural world (where we might think that the circle was based on actual circles found in nature…), the fact that we can use them to describe centuries after, the complex interaction between particles is truly astounding.
After explaining how mathematics can be so effective in describing the universe, we may be tempted in trying to explain the reason why it holds such a prime position amongst all human endeavours.
One good answer comes to us from Yanofsky’s paper “Why Mathematics Works So Well”[2] and is related to the concept of symmetry.
Simply put, mathematics are effective because they deal with the same kind of regularity as physics do.
When physicians notice symmetries and similarities in nature, where for example, one can swipe a quantity by another and still get the same result (the ideal gas law is given as an example), they express them in mathematics because it is another tool that deals with regularities. In fact, mathematics are constructed from the start in this way, with symmetries in mathematical statements being the foundations of the discipline.
A further question would be asking why there is regularities in the universe for us to even notice.
An answer lies according to Yanofsky from the anthropic principle. That is, if humans are alive, it must be because there is a certain amount of symmetries in the universe to even allow for life to happen.
So while there can be a universe with no symmetry, maybe we would’t have existed to notice it in the first place.
If mathematics have clearly proven to be so powerful at describing the laws of nature, the answer to why this would be the case haven’t been as clear. While certain mysteries remain, we can still be sure that our physical universe can naturally be expressed with the power of mathematics.
[1] Wigner E (1960) The unreasonable effectiveness of mathematics in the natural sciences.
[2] Yanofsky NS (2015) Why mathematics works so well?