The imprint of the world in our minds is not photographic: all the brain knows of the outside world is a chaotic sequence of electric impulses and out of these it creates a structural entity: our perception of what we see and hear. Most of the time, an adult’s brain talks to itself and creates more and more refined structures within itself. The word “structure” means a mathematical structure, something which becomes more and more abstract and better and better logically organized in the course of this self-conversation. The mathematical ability of each person’s brain by far exceeds those of the greatest geniuses of all time. Nobody, given the input the brain starts with, could be able to arrive at such a level of abstraction, for instance, as the five-fold symmetry (for example, a starfish), which you, or rather your brain, recognizes instantaneously regardless of a particular size, shape, or color of an object.
And then at some point, this process of creation of structures by the brain gets in touch with the linguistic part of the brain that generates thoughts that can be perceived and directed by your conscious mind. Here the mathematics starts. Your brain, inherently, is driven for an unknown reason and by an unknown process, to the creation of structures that are abstractions of the inputs the brain receives. When such input reflects the structures already created by the brain from the external world, it starts to analyze these structures within structures. When this process reaches the surface (the tiny fragment of your brain activity which we call consciousness), it becomes mathematics.
We are all fascinated with structural patterns: periodicity of a musical tune, a symmetry of an ornament, self-similarity of computer images of fractals. And the structures already prepared within ourselves are the most fascinating of all. Alas, most of them are hidden from ourselves. When we can put these structures-within-structures into words, they become mathematics. They are abominably difficult to express and to make others understand. Think of a village of deaf people where the music is communicated by writing down musical scores. Little by little you learn how to hear music written in scores and your brain listening to it receives an fantastic treat; then the brain demands more and more of it. Brains are our masters, with only 2 percent of our body weight, they take 20 percent of the oxygen resources of our bodies; you cannot resist their commands. You become a mathematician, a slave of this insatiable hunger of your brain, of every body’s brain, for making structures of everything that goes into it.
My random thoughts:
Taking a GR course this quarter. Also just started reading Einstein’s biography last week because someone recommended it 😉 I’m absolutely fascinated by the originality of Einstein’s work as well as how much philosophy&math influenced the development of his revolutionary theory & how much people and organizations of people mattered to Einstein and thus to the Theory of Relativity. Relating the development of GR to the progress of the study of human intelligence, I think the current status in this exploration of intelligence is like more or less like gathering a lot of experimental data–whether in the field of AI, or psychology, etc. Very similar to people gathering experiments around the study of light before Einstein developed his ideas. Sporadic. Crucial. In need for some unifying theory.
In the computer vision class I’m taking right now, the professors have mentioned a few interesting points including 1-humans abstract images into simple lines when interpreting images 2-humans are much less susceptible to illuminations conditions. Corresponding to the idea presented in Gromov’s writing, these points are examplea of how human brains structuralize information. This is what I find the most rewarding thing I’ve got from the entire quarter’s computer vision. Most of the class content was just a collection of different tools people have developed for different tasks–no unifying theme involved. Nothing like in math/physics classes where we have some big idea, and develop very carefully tools to approach the big idea and eventually get to the children of that big idea. In fields related to the exploration of human intelligence, I find this same annoying thing–that there’s nothing unifying going on in teaching; in psychology (subfields related to the exploration of human mind), we learned about different areas of human brain, experiment methods, different theories regarding child development, or language acquisition, or sleep, or consciousness/subconsciousness; in statistics & machine learning, it was also just a collection of tools–naive attempts to simulate ourselves; in linguistics, I was also asked to memorize a lot of names (this is the most unifying field compared to other fields I listed here) that describes phonetics/semantics/pragmatics/etc; in AI related classes same if not worse.
Personally I share the same vision as Gromov (learned a lot from a few of his talks as well). And I think the next big thing in scientific exploration will be about human intelligence, just like how relativity & QM dominated the scientific news 100 years ago. Please please share your insights with me as well!