Math is a Hugely Undervalued Experience

This entry was first posted on my old blog page, on December 9 2014.

Charles Darwin once wrote that he « deeply regretted that [he] did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense ». When he asserted this, Darwin might have been too old to start learning mathematics. By too old, I mean over 70. Hopefully, dear reader, this is not your case. Allow me, then, to argue why you should learn what Darwin regretted he never did.

First, and most importantly, mathematics is fun.

Don’t get me wrong, not all mathematics is fun. I too got bored by many mathematical courses (even some that I gave!). That’s because the experience of learning mathematics in schools is mostly not fun. After all, it is mined by frightening exams, and it often boils down to the repetitions of classical calculations. More often than not, maths in schools have nothing to do with « the great leading principles of mathematics » Darwin referred to. But, if you let yourself rock by the leading math popularizers, you’ll immediately see how easy it is to enjoy mathematics.

Now, I should add, seven minutes are definitely not enough to fall in love with mathematics. You need more. You need to immerse yourself into the world of mathematical wonders. That’s why I strongly invite you to check out all the awesome mathematics available online. First, here’s my Science4All website. Next, I strongly recommend you to subscribe to Youtube channels like Numberphile, ViHart or Tipping Point Math. But most importantly, instead of blindly following some teacher or math popularizer, you should venture yourself into the messiness of mathematics on your own. As the prince of mathematics Carl Friedrich Gauss once said, « the enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it ».

Second, mathematics is all about creativity.

As Darwin said, mathematics teaches its learners the use of some extra sense. This extra sense Darwin was referring to is probably a sense of mathematical intuition. As Felix Kelin once said, « mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs ». This surely sounds surprising to anyone who’s not gone far enough in the learning of mathematics. But it is something that strongly resonates with any mathematician’s thoughts. It is not possible to do good mathematics without being guided by intuition and thinking creatively.

In this sense, mathematics is not far from artistic creation. It always boils down to looking at the world (either the real world or the mathematical one) in a different way. It’s all about talking about familiar objects from a new perspective. This is why so many mathematical explanations start by « you can see it this way », or « imagine what would happen if », or « forget about what this represents », or « let me take an example ». These mouldings or twists of our thinkings are the essence of creativity in general, and of mathematical creativity in particular. They are usually followed by some spectacular shifts of our understanding of the world. Like when Einstein once thought: « Imagine you were free falling, then you wouldn’t feel gravity« … and then derived from this thought the laws of general relativity and the bending of spacetime.

What Einstein’s example illustrates is that mathematical creativity usually leads to highly uncomfortable reasonings. Good mathematics, like any major artistic creation, takes courage and perseverance. Einstein himself is a good example of this. He had the intuition of general relativity as soon as 1907 but couldn’t navigate in the hugely complex mathematics of tensor calculus and non-Euclidean geometry to formalize his intuition (and thus to give it an actual mathematical sense). He wrote: « The years of anxious searching in the dark, with their intense longing, their alternations of confidence and exhaustion and the final emergence into the light – only those who have experienced it can understand it. » Creativity takes patience, perseverance, and the courage to carry on a search that may or may not turn out fruitful. Just ponder the fact that Andrew Wiles spent 6 years of his life on one single question (Fermat’s last theorem), only to find a mistake in his early reasonings — he later found a way to get around this mistake and finally proved Fermat’s 350-year-old theorem. Or that Georg Cantor spent his life for his theory of the infinite, persecuted by mathematicians and philosophers who wouldn’t accept the existence of several infinites. In the end, Cantor doubted his own theory (which others eventually picked up) and went a little crazy.

Third, mathematics is rigorous.

The most important insight mathematics teaches is to spot mistakes in reasonings. This is because mathematics is the only human endeavour where a proof is conclusive. When I hear scientists claim that science tells the truth, I see obvious logical gaps in arguments for this claim. At best, science gets it almost right. But in mathematics, a proof that is almost right is definitely wrong. And the great skill mathematics teaches is to distinguish what’s almost right from what’s definitely right. This ability is called rigour. It costs a lot of effort, but yields dramatical clarifications.

As suggested by the video above, science often boils down not to finding out a solution, but to finding out where it is that a reasoning goes wrong. Mathematics, more than any other fields (although computer science is also a lot like that), is precisely about noticing whatever is wrong in a reasoning. This is the unavoidable detour in the path to irrefutable proofs.

Now, rigour is not the only skill that increases your insights. Building upon the ground-breaking work of 20-year-old Évariste Galois, mathematicians of the 20th Century have constructed modern mathematics around the central concept of structures. In fact, I’d define mathematics as the rigorous study of structures.

How on earth did this field that started with the study of numbers and shapes end up becoming that of structures? In short, algebra noticed that numbers were revealing their true nature as we looked at them all as a whole, from a very distant perspective. And this gives incredible insight. Similarly, when you are studying complex systems, like free falling objects within gravitational fields, it helps to abstract details of the experiment (like where the objects are, who threw them or whether they’re going up or down) to derive the key feature that determines their trajectories: Free falling objects accelerate at the rate determined by gravity.

Finally, mathematics is simple.

Yes, you read well. As von Neumann once said, « if people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. » Maybe (some) physics is not too complicated as well, but most fields like chemistry, biology, economics, geopolitics or sociology are unbelievably complex. Think about it. Can anyone predict what the next Youtube buzz will be? This is impossible. This is too complicated. Reality — whatever that is — is too complicated. And if you want to understand anything, it’s always better to start with what’s simple enough to be understood. Like mathematics.

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