The unreasonable effectiveness of mathematics

Previously I mentioned a famous paper by physicist Eugene Wigner titled The unreasonable effectiveness of mathematics, where, citing Wikipedia

Wigner observed that the mathematical structure of a physics theory often points the way to further advances in that theory and even to empirical predictions, and argued that this is not just a coincidence and therefore must reflect some larger and deeper truth about both mathematics and physics.

So how is it that, a product of human thought that is abstract and apparently unrelated to experience is so surprisingly successful at describing the physical world? It seems remarkable to the point of unreasonable.

Armed with our understanding of what we mean by coincidences and how they are explained, let’s turn to this problem. Recall that many times explanations of coincidences are nothing other than the establishment of causal links between their two elements, such that one follows from the other in an unsurprising way.

So automatically we have a recipe to try to tackle the above. There are two initial ways to proceed, either that mathematics causes the world to be the mathematical, or that the world causes mathematics to be the way it is. This has to be developed to make sense.

The first case can be interpreted as stating that our cognitive apparatus, having mathematics embedded in it, causes us to interpret the world accordingly. A fair analogy would be that if you put on a pair of red glasses the world will seem red to you. Not because the world is red, but because the way you look it at interprets it that way. So if our brains are somehow inherently mathematical, they will both produce mathematics as an a-priori field of study, as well as interpret the physical world mathematically.

But of course, our brains turned out the way they are for some reason. Which brings us to our second possibility, that the mathematical nature of the world caused our brains to develop accordingly. After all, a brain that is not tuned to understand the environment is not much good; evolution did the tuning.

And we can have a combination of both mechanisms, whereby evolution tunes the brain, and the brain in turn selectively interprets the world according to its nature, the causal link goes both ways, although it starts in one direction.

If we briefly look at the wikipedia article, we realize that the two solutions presented here correspond to solutions 1 and 4 proposed by Richard Hamming, not bad for our simple method!

But this is not the end of the story. If in fact the world is mathematical, or at least mathematically intelligible, and brains evolved to make sense of it, then that leaves another question. Why is the world mathematical in the first place? This is echoed in Einstein’s quote:

The most incomprehensible thing about the universe is that it is comprehensible.

I’ll leave this for my next post.