A few weeks ago, Alex Albanese interviewed me on the MIT post-doc GLIMPSE Podcast. We discussed populations, cakes, toilets, Google and many more topics. The podcast was released yesterday.

It was an awesome experience. Alex is a great interviewer. And, even though he is a biologist (I guess I can forgive that ;)), he was extremely sharp and quickly saw deep connections between the various mathematical topics I discussed. It was a real pleasure to interact with him, and I think that the podcast shows this very well. In fact, I highly recommend you to check his other GLIMPSE Podcasts.

So good. I’ve just spent most of my week-end teaching as part of the MIT splash event, where high school students came in and took classes that went way beyond the high school curriculum. These guys are way smarter than I expected; and they were very, very curious. What a pleasure it was, just to get to interact with them!

On Saturday, I taught a 4-hour class on the math foundation crisis (which I kind of regarded as a warm-up my upcoming Youtube series). The 2 first hours were awesome. Both for me, and — I think — for them. The trouble was, they were way smarter and knowledgeable than I expected, so we went through all of what we prepared in just 2 hours. Seriously. They cracked the infinite hat problem with the axiom of choice in half a minute! So the last 2 hours were improvisation — and it turns out, I’m not comfortable with math foundation to improvise a class on this topic… At least so far.

Sunday was my marathon day. I first taught a 2-hour class on the mathematics of democracy — an field I have done research in. I had a class of hundreds of students — 138 were registered. So awesome. The students were very curious. They asked insightful questions, and proposed clever answers to the numerous questions I asked them. They also elected Batman as the Condorcet winner of superheroes! At the end, I proposed the voting system I have been designing in my research (which hopefully I’ll write and talk about on the Internet soon…). To my surprise, I even seemed to have convinced them that this was the right voting system… But that’s just a detail. What matters is that, for sure, they left pondering the legitimacy of all voting systems that are widely used these days… and that’s a big deal!

Next was a 2-hour class on cryptography and the theory of complexity. This was the only class that didn’t go as well as I had hoped. Somehow, I didn’t manage to connect with most of the students. Perhaps, the long historical introduction I presented was a bit too long. Perhaps, as well, the students started to get tired after over a day of advanced classes all over MIT…

At lunch, I must say, I was a bit afraid. While students started to get tired, I had become exhausted. Would I still find the energy and the voice to present another 4-hour class? Plus, by far, this 4-hour class was the one I prepared the least. Seriously, my slides were mostly pictures of physicists… But as it turned out, this last class would be the one I was actually the most prepared for, as it was about the theories of gravity I have been talking about in my recently finished series on general relativity.

This class was awesome! Step by step, I got the students to almost figure out by themselves the most brilliant thoughts of Galileo, Newton and Einstein. The climax was when I repeatedly asked them: “If gravity is not a force — which is what Einstein claimed — why do apples fall?” It took them a while. They proposed different ideas. Mostly wrong. But that’s okay. Einstein himself got mostly wrong ideas. But, slowly, they got warmer. And warmer. Until, all of sudden, one student said half convincingly: “the ground is accelerating upwards!” Yes! Yes, yes, yes! The ground is accelerating upwards!!!

I had several students thanking me for the classes. Many even said that my classes were the best classes they had attended… which, I guess, means that I’m not such a bad teacher when I don’t have to follow some stupid curriculum (which I had to when I was teaching in Montreal), and where students are not fully focused on some upcoming exam. I guess I actually love teaching. But the conditions need to be right… and I do know that conditions will hardly ever be as good as they were in this amazing Splash event…

I guess Youtube is the next best thing :p. Come on Final Cut, let’s talk logic and math foundation!

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.

Today was the premiere of the second episode of my documentary on (French) undergrad maths. I’ve had plenty of feedbacks already, with fundamental questions about the way I popularize maths.

Roughly, while the narrative of the documentary is quite understandable, I’ve decided to include high-level interviews. For instance, in the first episode available online, one interview sequence is about abstract algebras. In particular, interviewees show great excitement regarding so-called groups, rings and fields, which are not defined (groups are defined shortly after). Thus, if you haven’t learned about these mathematical structures yet, you will be lost in this interview sequence. I know it. I planned it.

Granted, viewers may reject the whole of maths, because they feel that they will never be able to understand this section. Yet, that is the opposite of the asserted aim of the video, which is rather about making maths attractive. I know that. And that deeply bothers me. So why did I plan to have the audience confused? There are several reasons for this.

First, I wanted interviewees to get excited about maths, and, as weird as it might sound to you right now, such structures easily got (most of) them excited. And I think it’s pretty cool to hear mathematicians getting excited. Second, mathematics is hard. And mathematicians spend the overwhelming majority of their time trying to make sense of what they don’t understand. To enjoy mathematics definitely requires being comfortable with not understanding. Or at least, that’s what I think. Third, anyone who does know about these structures will greatly enjoy these sections.

But more importantly, I want to plant seeds of what I call the “Einstein effect”. Why did you read this post in the first place? I bet it’s because I mentioned “Einstein” in the title… and you’ve heard somewhere that Einstein was a big deal. In fact, back in undergrads, I distinctly remember the relativity theory course being full, as opposed to the Galois theory one. Why? Is it because relativity theory is more useful or more beautiful? I don’t think so. Relativity theory is useless for all engineers, and the Galois theory is arguably the most beautiful mathematical theory ever. So, what makes relativity theory more attractive than Galois theory?

I think it’s because people have been more exposed to physicists’ excitements about Einstein’s work than to mathematicians’ trepidations regarding Galois’. And this is essential in the learning process! Think about the most boring math course you’ve had. If, at some point, someone told you that what you were learning was a key concept for understanding Einstein’s relativity, this would probably awake you and give you a small boost. This boost is what I call the “Einstein effect”. And it may be just what you need to be willing to do what’s necessary to grasp the concept (that is, spending hours pondering).

Similarly, I think that if a student has been exposed to the importance of groups, rings and fields or Galois theory, he might have this necessary additional boost to pay attention to the professor as the professor defines these concepts… That’s why, while talking about difficult concepts in a popularization video may be negative in the short run, I believe it to be very useful in the longer run. Besides, it’s also why I think we shouldn’t only teach numbers and basic geometry to kids. We should also tell them about infinite series (like 1+2+4+8+16+…=-1), probabilities (like the Monty Hall problem) and topology (like Pacman living in a torus), even if they don’t get it all!

Here it is! The first episode of my 2-episode documentary series has been released a week ago. There are English subtitles. Check it out!

Please consider sharing it. I’ve put a lot of effort, and I’d love to reach as many people as possible. Thank you!

Dans cette série, je propose de vous révéler la magie des mathématiques des classes préparatoires aux Grandes Écoles. Ce premier épisode porte sur l’algèbre.

Chapitre 1 – Al-Jabr (1:13)
Chapitre 2 – La géométrie algébrique (12:11)
Chapitre 3 – La théorie des nombres (17:51)
Chapitre 4 – L’algèbre moderne (29:04)
Chapitre 5 – L’algèbre linéaire (40:51)

Forget about global warming, economic recession, energy and water shortage and other usual political concerns. For me, the big deal is large-scale unemployment. And reducing or preventing it is not a solution.

There’s also a great 1-hour podcast follow-up (start at 33:46). In short, unemployment will skyrocket, not because of recession, but because the overwhelming majority of today’s jobs will be automated. Most humans will be “unemployable through no fault of their own”.

What’s the solution? What will happen? How can we avoid the upcoming riots of unions?

I have a request for all of you guys who appreciate my mathematics popularization work on Science4All. As I want to do this more professionally, I need to get involved with major organizations. Starting with a TedTalk.