DiBeos

If we look back at the biggest breakthroughs in science or math, they tend to come from people who are willing to challenge current beliefs. And this isn’t done out of pure arrogance (although that does play a role sometimes), but mostly because this person is thinking outside of the box. 

In my Bachelor’s I had to take a class on the philosophy of science, and, even though it wasn’t my favorite subject, one concept stuck out to me, something called paradigm shift.

To very quickly explain each node, pre-science is when a model of understanding isn’t ready to solve the major problems of a field. Normal science is when the model is found and it starts to drive a part of the field forward. Model drift is when many small anomalies start to appear, but the model still works. Model crisis is when these small anomalies become big, so big that the field, or a part of it, enters a crisis. Model revolution is when all of these new theories come forward, which explain things much better, and one of them prevails. And finally, Paradigm change is when all resistance is overcome, and the model is accepted and becomes standard.

Anyway, the point is that science, or for our purposes mathematics, is something that should be controversial from time to time. When anyone tries to do anything innovative, or give any original contribution that breaks some kind of an already established system, or in other words, do anything controversial, this person will most definitely be met with resistance. Of course, I’m not saying you should focus on controversies just for the sake of standing out, but there is no way to do anything meaningful in life if you are not fine with being misunderstood by those around you. 

Georg Cantor is one of the names that sticks out because he introduced a theorem showing that there are different sizes of infinity. The controversy was so real that there’s even a page on wikipedia just dedicated to it. Henri Poincare called it a “grave disease”, Ludwig Wittgenstein said it is “utter nonsense” that is “laughable” and “wrong”. His own teacher, Leopold Kronecker, ended up completely rejecting Cantor’s infinities and called him ‘a corrupter of youth’ because of his work, and he was not alone in loudly criticising Cantor. Cantor was actually so concerned about Kronecker’s persecution that he ended up having a total nervous breakdown. I mean, just think about it, your “mentor” that you admire so much is totally disapproving of you. It took him a month to recover from it. And even afterwards, he constantly continued to fall into depressive fits and totally undermined his worth as a mathematician, and because of that he tried to transfer from being a professor of mathematics to become a professor of philosophy. The point is: his ideas are basically the norm today, but in order to change mathematics Cantor had to do something really controversial and take the consequences for it. But discovering new things in mathematics isn’t the only kind of controversy.

Let’s take Emmy Noether. Even though she made several incredible discoveries, the controversy was in the fact that she was a woman. At the time, it was forbidden for women to teach mathematics in university, but she still decided to apply for the position. The dean, Felix Klein, wrote: “I think the female brain is unsuitable for mathematical production,” but Noether stood out as “one of the rare exceptions.” She was an unofficial lecturer for a few years, but eventually was offered a full position. The point is that, Noether did something extremely controversial at the time, which was not only looked down upon but pretty much not allowed, and with that helped to change mathematics with her research, and the idea of who can teach the subject.

Another person that we admire is the well-known Grant Sanderson, from 3Blue1Brown. His YouTube channel might not seem like much compared to Cantor and Noether. I mean it’s only YouTube videos right? But the thing is, he completely changed the way people can learn math online nowadays. And on top of it, he inspired many people to share their own ideas and explanations online, not only on YouTube. But the question is: how did he do it? He did it by going against the “usual script”. Most people learn math by grinding through symbolic manipulation, by memorizing formulas, by getting stuck in notation, and using super abstract books. But Grant Sanderson made visual intuition the starting point, and that was a huge difference in how math was traditionally taught.

And I think about this all the time when I’m working on our YouTube videos. Sophia & I are really trying to build something meaningful here. We’re trying to help change how math is taught. Now, our videos and PDFs are definitely not perfect, and I am pretty sure you are going to find mistakes now and then, but we are improving every week. I mean, just take a look at the videos we used to make a year ago. I am honestly embarrassed about some of them, and that’s ok. I hope that 5, or 10 years, from now I’ll be embarrassed by the quality of the videos we have today. Plus, we hope to bring other mathematicians and physicists to work with us, to help us create more and better content for you guys, and not only in video format, but also books and courses. And here’s something interesting that Sophia has to share.

There’s a mathematician named Michel Talagrand, who in 2024 won the Abel prize. About a year ago we made a video about him, and I got to read his autobiography when we were doing research for the script. He said something that really stayed with me. When he reached sixty, and he had no more creative juices in him, he decided to study Quantum Field Theory but couldn’t find a book that was reader-friendly. I guess we can all relate right? So, he wrote that book himself, and said it was the hardest project of his life. Afterwards, he wanted to go back and put more creative effort into his books to try and explain his own mathematics better. And said “It then occurred to me that I had never put such an effort into explaining my own mathematics. So I started reworking the material… trying to better explain the material to others, I understood it better myself, reaching at places simplifications of almost embarrassing scale…Even more importantly, things fell into place and I could prove several of my conjectures dating back to the 1990s.”

What I am trying to highlight is that, maybe all of this rigidity and formalism that is found in most math books gets in the way of progress in math. Mathematicians should put more effort into their own books. They should simplify things to an embarrassing scale, when they can, so that things can fall into places. Instead of spewing poorly, and lazily explained concepts. And that’s what we want to do, we want to help and raise the bar on advanced math books and how math is taught in general. And one last thing I want to add from Talagrand: “Few mathematicians will ever admit to this, but over my career I have written many rather insignificant papers. These weren’t wasted effort – often, what I learned in writing them proved to be a stepping stone to further significant discoveries. Besides, solving problems is fun.”

So again speaking about something that all of us can relate to, even if none of us here succeed in being Cantor, or Noether, Grant Sanderson or Talagrand, we can certainly try to question why something is the way it is in mathematics. Be it something in mathematics per se or in how we do math. Maybe you will be met with resistance, maybe you won’t be, that’s not the point. The point is that it’s good to question why things are the way they are and to have the courage to be controversial. It’s good to always keep an open mind. Anyway, let us know what you guys think about this, we’re curious to hear your opinion on controversies.