Recently, I’ve read the book Mathematica: a secret world of intuition and curiosity, and it blew my mind.

 

Special People

Some people have made such a mark on the world, we have even decided to preserve their brains in the hope to analyse them in the future! Like for example Albert Einstein, who is often quoted as a synonym for pure intelligence. Albert was much more modest however: “I have no special talents, I am only passionately curious.”. Yet, it is not commonly known that Einstein’s curiousity had limits, especially when it comes to the usage of probability theory and statistics and quantum mechanics. “God does not play dice”, he was quoted saying and thereby influencing progress in the later stage of his life.

When hearing these statements from Einstein, especially the modest ones about him just being extremely curious, we tend to brush it off as Einstein just showing off. But what if we could actually learn how to be more like Einstein? It’s not trivial, since Einstein never really has taken the effort to really explain to us what he means, and how we can achieve and benefit from his way of thinking.

Others have done so, most notably Alexander Grothendieck. Even though he is often heralded as one of the most important mathematicians of the 20th century, it is unlikely you have heard from him. He did rise to fame in his time, but he never really made the same impression on the general public as Einstein. His foundational work on algebra, geometry and symmetrical rings is so abstract in nature, you might have been using it without knowing it. And those courses that attempt to teach it to university students are often regarded as the most difficult, the most abstract mathematical courses available. At Harvard, it’s called “Math 55A” and you can find subreddits where people discuss its difficulty.

The story goes that when Alexandre Grothendieck arrived in Nancy, he had just left the University of Montpeillier, where his academic performance wasn’t stellar. He had struggled in several classes and even failed some. Despite this, he had been working on some innovative mathematical theories.

In Nancy, Grothendieck met with two renowned professors, Dieudonné and Schwartz, who were famous for their work in mathematics. At the time, Schwartz had already won a prestigious Fields Medal. However, when Grothendieck presented his ideas to them, they weren’t impressed at first. Instead of engaging him in conversation, they sent him home with a copy of one of their recent papers on complex mathematical topics.

The paper contained 14 open questions that Dieudonné and Schwartz had been struggling to solve themselves. These questions represented some of the biggest challenges in mathematics at the time, and each one was worthy of a PhD dissertation. But what’s remarkable is that Grothendieck was able to solve all 14 questions within months. His solutions were so convincing that he was never doubted by anyone again.

Besides his contributions to mathematics and science, Grothendieck was a radical pacificist, passionately curious about many different subjects and at the same he is said to have been kind, open to any kind of question and of gentle humour. We know this from some of his colleagues, but also from a book he wrote called “Reapings and Sowings”. In this book, Grothendieck sets out to share how he thinks mathematically and how others could also achieve this. Truth be told, I haven’t read it myself and from what I know it’s not easy to digest its 1000+ pages of deeply introspective texts, but the effort to share his mental models to pull people along his path is just marvellous.

This is also highly contrasting to many other mathematicians. In the Renaissance period, having more mathematical skills than other people would be foundational to having a academic position. This lead to people protecting their newly developed methods as trade secrets, rather than communicating them as advances in science. Like here, for example. There is also this example about the elitist publication letter of Hardy along his very influential Principae Mathematica. Many mathematicians believe that that what they do is somehow superior to other pursuits, one of the highest forms of pure thought, of pure intellect. Some even attribute these perspective to Plato and Socrates, but we don’t really know where this all came from. Irregardless, this is not particularly helpful to most people trying to make sense of numbers and formulas.

Sharing and encouraging others to improve their mental models in order to understand mathematical, that is helpful. Grothendieck grasped the concept, and he was not alone in doing so - mathematicians like Bill Thurston, whose own groundbreaking work had earned him widespread recognition. Whereas Grothendieck wrote long texts filled with his reflections, Thurston generally vocalized the same message, but in a much more digestable way. I particularly love one of his famous carreer-advice answers on Mathoverflow, as discussed on Hackernews as well. Some snippets from that piece:

The product of mathematics is clarity and understanding. Not theorems, by themselves.

The world does not suffer from an oversupply of clarity and understanding (to put it mildly).

The real satisfaction from mathematics is in learning others and sharing with others. All of us have clear understanding of a few things, and murky concepts of many more. There's no way to run out of ideas in need of clarification.

We are deeply social and deeply instinctual animals, so much that our well-being depends on many things we do that are hard to explain in an intellectual way.

Bare reason is likely to lead you astray. None of us are smart and wise enough to figure it out intellectually.

 

Special People: Conclusion

Einstein, Grothendieck and Thurston are those giants on top of whose shoulders we stand. However, rather than just sharing their shoulders, they sometimes even went out to explain their thinking and how we could arrive at the same level of mental gymnastics. No arrogance, no elitist, just sheer “reaching out”.

It’s not always that obvious, and there is still some translation needed to properly understand their message. Hence, the book:Mathematica: a secret world of intuition and curiosity by David Bessis.

 

A Secret World of Intuition and Curiosity

Being passionately curious, in a humble, respectful yet opportunistic matter is only one side of the coin. In order to digest information and develop your mental model, you’ll need help of the people around you to explain theirs.

Note that it does not always help to claim to be an expert. There are some studies that show that when experts are wrong, it is more difficult for them to accept the missing link in their thinking, as their expertise is on the line. Instead, as argued by Grothendieck and Bessis, it is important to cherish a sort of childlike curiosity, free from social and societal expectations. A sort of playful figuring-out, seeing every moment and person as a opportunity to learn.

Anyhow, the other side of the puzzle is our concept of intuition. As popularized by Kahneman in his book “Thinking Fast and Slow”, we tend to think of our brain as having both a fast intuitive mode as well as rational, slower mode. Kahneman then continues that we can train our intuition by training to be aware of its cognitive biases.

For some perspectives, this would make sense. The stoïc perspective on anger development as developed by Seneca, which is still very influential in modern anger management, stresses the point of delaying an acute anger in order to better rationalise and relativise. E.g., go for a walk instead of responding directly.

But in other cases, this way of thinking does not make sense at all. It almost defines intuition as something unchangeable innate, and our slower rational thinking as only other option. Thereby that there is a grey area in between where intuition and rationality do not always agree.

Bessis mentions our idea of numbers as an example, and the fact that throughout history people would have completely different ideas of many different abstract numbers, including: negative numbers, infinity, imaginary numbers and even zero. Yes, even zero has been invented as an abstraction, and intuition about it is nowadays as widespread as being literate. But that was not always the case.

Personally, I found learning a foreign language (french) to be a similar experience. In the beginning, you are completely overwhelmed and you really have to work to understand things, but at some point you almost just feel how to express yourself without being required to actively reflect during those particular moments. Things like the gender of nouns become intuitive (except for some notorious french exceptions).

David Bessis continues to introduce a 3rd system of thinking, between rationality and intuition that bridges the gap by establishing a dialogue between the two. The idea is that when your intuition disagrees from logical argumentation, you should tune into that dissonance. And resolving that process involves reflection, meditation and introspection while self-scrutinizing heavily through some form of cartesian doubt.

It’s important to note that this is a super slow, organic process. You might even not be aware of a great part of the process, before suddenly experiencing a revelation. Those are the best. You are chilling in the shower, and suddenly a new perspective dawns on you. From then on, things just work. Trust me, I did not go as far into academics as the people I’ve been describing so far, but I’ve had plenty of those moments while studying mathematics too.

 

Personal Conclusion

What I like about all these ideas is how they fuse things like kindness, curiosity and openness into the learning process that is mathematics while at the same time putting intuition at the center. I have seen my studies for a long period as a very rational, rigid and academic process, ignoring things that really only can be explained by intuition rather than through the application of rigid scientific methods. At the same time, mathematics is built on top of rationality and logic, but in such a way that it is best understood if the intuition is there as well. In a way, it shapes the intuition.