There is no way I should take the time to write this post… but I’m going to anyway. I want to talk about Freeman Dyson and problem solving and look briefly at some interesting mathematics… but I really should be working on two technical blog posts.
I have long known of Freeman Dyson as the theoretical physicist who reconciled the disparate theories of quantum electrodynamics for which Feynman, Schwinger, and Tomonaga won the Nobel Prize.
What I hadn’t realized was that he started out as a mathematician… and more, he was that most esoteric of mathematicians (at least for his day), a number theorist.
He tells a fascinating tale of compartmentalization, and I want to share it with you.
As for what I should be doing instead…. I found myself wishing, last Sunday, that I had a reserve post in my pocket… so I put some time into one during the week, but it’s not ready yet. (There’s one question left to answer: exactly what are the odds against improving a pair by drawing three cards?)
And, of course, I have a multicollinearity post to write. Three diary posts in a row, without a technical post in between. What kind of lazy dog have I become?
The simple truth is I’m in terra incognita as far as multicollinearity goes. No one out there does it thoroughly (hmm. those may be fighting words)… and I probably won’t be able to either… but I hope I can lay it out sensibly, so you end up with a better appreciation for what you might do, and what other people are recommending, and, truth to tell, when other people are simply too vague.
I’m not even picking and choosing from different authors; I’m figuring it out for myself.
Frankly, I don’t know what I’m doing yet.
That means I’m doing research… and I can’t guarantee that results will have fallen into place in time for a weekly post. If I’d only had a post in reserve last weekend….
I stopped working on the planned multicollinearity post because two results of two calculations surprised me when I was playing around. In retrospect, one of them shouldn’t have… it didn’t affect the mathematics, but it certainly would affect my narrative. The other result, however, I’m still thinking about.
The result which affects the narrative is this. Suppose we are studying the regression of y as a function of X1, X2, X3, and X4. Suppose we find that X4 can be fitted well as a function of X1, X2, and X3. Should we compare that fit to y as a function of X1, X2, X3? Or should we use y as a function of all four?
All four. I’ll explain why in a multicollinearity post.
The upshot is that I would like to finish two posts, both a reserve post and a multicollinearity post, this weekend… but I’m going to take the time today to write about something interesting I came across during the week.
In addition to the book on alternatives to ordinary least squares, and the print-on-demand publication entitled “Multi-collinearity”, I had ordered a book called “Mathematical Omnibus” by Fuchs and Tabachnikov. Two of them have arrived; I’m still waiting for the print-on-demand.
No, it’s not the alternatives to ordinary least squares that I want to talk about… but one thing in chapter 3 of the Omnibus. Now, this is not a popular book about mathematics; at the very least, it’s an upper division math book. Perhaps I should call it cultural reading for a serious student of mathematics.
Well, in chapter 3 they talk about Euler’s function, and then they quote substantially from a lecture by Freeman Dyson about missed opportunities in mathematics.
First, let me show you Euler’s function.
Euler, as usual, started simply. (I like Euler’s writing style, not that I’ve read much of the original.) Consider defined as the product of n terms: (1-x) (1-x^2) … (1-x^n).
and — thank you, Mathematica® — the first eleven terms of are the same:
If we take the limit of as , we get the Euler function . OK, now what?
Well, Jacobi showed that its cube (!), , was given by the following infinite series:
Other people found formulas for other powers of . Dyson says the case d = 8 was found by Klein and Fricke, and the cases d = 14, 26 by Atkin; d = 10 by Winquist (and that’s what got Dyson started). I think Ramanujan had d = 24.
What Freeman Dyson found, taking a vacation from physics by doing number theory, was that for values of d in this list:
d = 3, 8, 10, 14, 15, 21, 24, 26, 28, 35, 36,….
he could find elegant formulas for the dth power of Euler’s function .
But that was it. He had taken a step forward by finding a list rather than just one more formula, but he couldn’t see anything to do with the list.
Ironically, and one of the points of his lecture, is that he of all people, as a theoretical physicist, should have recognized that list! With one exception, those are the dimensions (up to 36) of the simple Lie algebras (used in quantum mechanics).
According to my handy copy of Stillwell’s “Naive Lie Theory”, the dimension of the simple Lie algebra su(n) is n^2 -1, and of so(n) is n(n-1)/2. (su(n) and so(n) are the special unitary and special orthogonal Lie algebras, respectively.)
Let’s compute some of those dimensions.
Only one of those numbers, 14, is less than 36, and if we add just it to our list of dimensions, we get:
The list of d includes 26; the list of dimensions does not. That’s the only difference between the lists I’ve got.
As Dyson put it, “As I was, for the time being, a number theorist, they made no sense to me. My mind was so well compartmentalized that I did not remember that I had met these same numbers many times in my life as a physicist.”
That’s what I found fascinating, as did he… that he himself knew something from two wildly different perspectives but failed to connect them.
It gets even more ironic. A mathematician named MacDonald eventually worked it out… and he and Dyson were both at the Institute for Advanced Studies at the very time they were working on this problem, and they had daughters in the same class at school, so they sometimes saw each other. “But since he was a mathematician and I was a physicist, we did not discuss our work. The fact that we were thinking about the same problem while sitting so close to one another only emerged after he had gone back to Oxford.”
If you want to read Dyson’s entire lecture on missed opportunities, it is here.
In addition, he has another marvelous essay about birds and frogs – by which he means mathematicians who look for the big picture, and mathematicians who focus, as he did, on what’s in front of them.
For more information about Dyson himself, you can start with this Wikipedia article.
OK, now let me go do what I’m supposed to.