Real Genius of Math

Real math is not about how much you know or even how deeply you understand other mathematicians’ work — the current math foundation, but it’s about how differently you dare to think and how insightful you are to create your own math foundation, like the genesis of topology or algebra. This kind of unfettered creation will add a whole new dimension to the math world and create a whole new world for math. No matter how much you understand the current state of math, all that belongs to the past and other mathematicians. But creating your own foundation will be the future of math. To achieve that, we first need to be tasteful and insightful, as Isaac Newton was when he asked why an apple falls down but not up, as Albert Einstein was when he figured out “I got it. It depends on what you mean by simultaneous. Simultaneity is relative, so time is relative.”, as Steve Jobs did when he realized the graphical user interface must be the future of computing and built the Macintosh team. That’s the true genius math needs. That’s the true genius the world needs. Such genius doesn’t lie in how deeply they understood other people’s work, but in being imaginative and thinking differently, with a sense of taste and insight, with a sense of curiosity and vision, and with a sense of craziness and rebellion. Such geniuses ask why even though they are the silliest and most childish questions. Such geniuses question the status quo and challenge authority. They rebelliously break existing rules that other people take for granted and even protect. If you only have a deep understanding of current math, you might be able to graduate from college at 12 and teach Nullstellensatz without any reference on the board as a young college professor, and therefore you might be called a “math genius”. You might be able to write extremely many math papers trying to push some math fields forward to a small extent and be satisfied with your contribution. However, without the spirit of unruly creation and the real genius of math, you might never contribute something that truly adds a new dimension to the current math, and math may never truly change because of your work. Therefore, you are still a pedestrian mathematician at the core.

Suppose there are several tall buildings in an ancient city, and these buildings represent the current state of math built by previous generations of mathematicians. Some people go to these buildings and spend most of their time in one building with one goal in mind: to improve the decoration and construction of each level until they reach its highest level. They know if they reach the top level then they will be able to find some path afar to reach point A from point B that no one would find on the land. And upon entering a level, they visit each corner of this level very carefully, wanting to make sure this foundation is sound enough. And they try their best to leave some mark there decorating this level. After all that, they go on to the next level. These people are great, but they only are pedestrian mathematicians. They might be able to make each level of this building more beautiful, but that’s all, they can’t make the building taller because the building’s foundation is already there and it can’t support a taller building. So they might be able to see some beautiful scenes 50 miles away, but they can’t see anything they and all mathematicians want 1000 miles away. No one can, even if they’ve climbed the tallest building there. But a real genius of math would find a place to create their own building. They are ambitious enough to make their building higher from its foundational work. With a sense of insight and taste, they know how to find the best place for their building and how much foundation they need. With a sense of rebellion, they choose to start building even when the mainstream is busy climbing the most popular building. In order to do that, they also often visit those tall buildings there and sometimes also decorate them a bit, and go above some levels. but they know the top is not all they want, the reason they are in the building is not to reach the top but to learn and gain inspiration and experience about how past mathematicians built, designed, and decorated each level, so they will borrow and apply them to their own building. Why not? Picasso once said, “Good artists borrow, great artists steal.” These buildings are already there and they are fruits of the labor of thousands of years of math research, so why don’t they take advantage of them? To truly think out of the box, it’s important to first know what’s in the box, right? Besides, they don’t want to spend thousands of years exploring a whole new way to build buildings by avoiding those existing buildings, right? They can’t live that long. So, those real geniuses are also one of those frequent visitors to tall buildings. But again, their ambition is very different from that of pedestrian mathematicians, not better, but just different. Therefore, those real geniuses don’t care too much if most other pedestrian mathematicians have passed them and reached the top first. All they need to know is how to build their own buildings. And finally, with a much more advanced foundation, and the beautiful decoration learned from the existing buildings, their new buildings are not only much more beautiful and useful than those existing buildings but much taller than them. With this new height and different location, geniuses are able to see wonderful scenes not only 1000 miles away but 10000 miles away and connect the dots, in a way that’s never before been possible. So even when those pedestrian mathematicians reach their tops first, these geniuses were the first ones who saw the dream view, and left a fantastic tall and beautiful building for future generations to use, appreciate, and explore.

Such real genius is the single most important trait in math, and is the single most important trait for a mathematician. With this genius, if you input a certain amount of time into math by getting inspiration from the thousands of years of wisdom of the current math, then math will be totally different because of you. You might not end up being more familiar with the current math knowledge or score higher on math exams or competitions than those “math geniuses” who wrote lots of papers and are teaching a graduate-level math course without reading any textbooks, because you only treat this knowledge as the means (the means and reference to unruly create your own math) but not the end as they are used to doing. Your potential cannot be weighed. You are much more than them as a mathematician. Your contribution to math is what those pedestrian geniuses would never reach, if they are too stubborn to change their mindset. As Einstein’s biographer Walter Issaccan pointed out, “I’m not sure he (Einstein) was smarter than Max Planck or Lorenz or Poincaré, certainly wasn’t as good of a mathematician as Poincaré or Hilbert or some others.” But Einstein became the most successful physicist in the 20th century because of his creativity enabled by his real genius of thinking differently. With the same genius that Einstein had, you will take his baton and become the next real genius that truly will make a difference and give a giant leap to the math field.

Ivan Zhanhu Feng
July 13, 2023