Source: Simons Foundation, Dec 2014
Gromov’s first bombshell was the homotopy principle, or “h-principle,” a general way of solving partial differential equations. “The geometric intuition behind the h-principle is something like this,” explained Larry Guth of the Massachusetts Institute of Technology. “If you had a sweater and you wanted to put it into a box, then because the sweater is soft, it is easy to put it into the box, and there are lots of ways to do it. But if you had to write a list of totally precise instructions about exactly how to put the sweater into the box, it would actually be hard and kind of complicated.”
In mathematics, the question was whether some high-dimensional object could be embedded into a given space. “And the only way to deal with high-dimensional objects, at least traditionally,” said Guth, “was to write down equations that say precisely where everything goes, and it’s hard to do that. Like the situation with the sweater, the only way that we could describe how to put the sweater into the box was to write a completely precise list of instructions about exactly how to do it, and it makes it look as if it is complicated. But Gromov found a good way of capturing the idea that the sweater is very soft, hence you can do almost anything and it will fit into the box.”
… a list of questions: “What is mathematics and how has it originated? Where does the stream of mathematical ideas flow from? What is the ultimate source of mathematics in the brain? Gromov rebuffed as absurd his own question about what mathematics is. But, he continued, “you can say, ‘How is it being created, how is it being studied, how is it being learned?’”
“The way I think of mathematics,” he said, “is as a physical and psychological process, not some abstraction.” And herein arises his notion of a “bug on a leaf,” which gives a nice sampling of his inquisitive excursions away from research mathematics proper and into biology, evolution, the structure of the brain, and the question of how scientific ideas evolve.
“A bug on a leaf displays two simple phenomena,” he said. “It always does the same thing, one leg, another leg, one leg, another leg. Just movement. Many, many things are done this way, including Euclidean geometry. That’s one point. The other point is information theoretic. And that is that the bug spends more time on the edge of the leaf compared to the interior, and more time on the tip of a leaf compared to the interior. And your eye does the same thing when looking at an image, your eye will spend more time around the image.”
Both of these processes, with the bug and the eye, in Gromov’s opinion, are run by universal mechanisms. “The logic of the world forces the way we think,” he said. “This is what I have been thinking about for the last 10 years: the basic principle underlying thinking, and specifically underlying thinking about mathematics. It’s very different from logic, it’s not logic.” He calls it “ergologic” — a reconsideration of traditional logic, encompassing the “ergo system” and the “ergo brain” and “ergo thinking.”
“Life is similar to how mathematics is organized in our brain. If you don’t accept it, life is impossible,” he said. And then one final Gromovian snippet I managed to catch was this: “If it is impossible, you try to do it anyway.”