I’m slogging through MIT 18.03, ordinary differential equations. I generally understand the concepts, but combining them to solve problems is sometimes difficult, and some of the problems I don’t understand. Even so, I enjoy learning this material and trying to solve these problems, and that whole world is fascinating to me. Eventually I’ll finish with the ordinary differential equations course, and then I’ll begin my journey with classical mechanics.
I have real weaknesses in my mathematical education, in my ability to do mathematical reasoning and problem solving. The same could be said of my geometric education. I have been wrestling with how best to overcome these weaknesses. One solution is to create courses in all these mathematical subjects. After all, the best way to learn a subject is to teach it. If I can teach these subjects well, then I will be much more solid in my grasp of them myself. Rework all of my study materials into my own words and explanations. Record the proofs, work through them, and explain them as best I can. (The course materials in MIT OCW are available for this sort of use – I can shamelessly use all of it, as long as I publish the results non-commercially if at all.)
Last night I was browsing Newton’s Principia. Two things struck me: First, back then mathematical physics was called natural philosophy. There’s something wonderful about that label, something true that “mathematical physics” doesn’t communicate.
Second, Newton had a very firm foundation in Euclidean geometry, using his Euclid to prove physical principles. There is a mathematical approach to proof, and there is a geometrical approach to proof. It is good to have facility in both approaches, because both mathematics and geometry are at play.
Today we saw an actor playing Einstein – it was one of those one-man monologue shows. Very well done, with a reasonable German accent cloaking the actor’s native east coast one. Einstein solved some of the hardest problems in physics by doing two things well: (1) He kept working at a puzzle for a long time, and (2) He suspended common sense (as the actor put it) to follow the puzzle wherever it would take him. That is how he was able to conclude that that the speed of light is absolute, that time therefore is not, and later that space as well is not absolute. Along the way he articulated a relationship between mass, energy, and the speed of light, unifying so much in the universe by that amazingly simple equation, E = mc2.
I wonder if, by suspending disbelief and meditating for a long while on a puzzle, one could find a solution to the next puzzle that the universe has for us.
Theoretical physics, it seems to me, has been mired for a good long while now – several decades – and a lot of what is being done seems like very expensive thrashing. Experimental particle physics is being conducted on increasingly expensive and enormous machines like the Large Hadron Collider (LHC) in Switzerland. Meanwhile, theoretical physicists are cataloging ever more fundamental particles to explain the data from the high energy experiments, while others are inventing ever more dimensions to explain the subtleties of strings that can never be measured or verified. Physics is like a black hat trying to crack strong encryption through brute force, by running every possible combination. The secrets of the universe is very secure, and something smarter than brute force is going to be needed to open them further.
Physics needs another dreamer with deep reflection and profound insight. The high power, high cost, vast resources methods have had their day in the sun. I wonder if the time isn’t at hand for the stars to rise on a more reflective, more insightful way of studying the physical universe. And I wonder if it can’t be worshipful and worship-prompting, giving honor to the one who deserves it above all others.
It’s interesting to ponder. Meanwhile, I have a great deal of material to master if I’m going to be able to understand and explain these things to others. I am in the process of trying to boot a very complex operating system. My hope is that the underlying principles will prove to be relatively simple and straightforward, enabling me to hold the operating system in working memory . . . and build on it.