The half-derivative

When I was in college, I once had this crazy notion of a half-derivative. We’d been taking nth-derivatives in physics, and I wondered “why stick to integers?”. Well, as it turned out, others had been there before me. At the time it looked like complete non-sense, and even now, if I had to start from scratch, I doubt I’d be able to generate a formal and operational definition from which you could actually calculate the thing. Last weekend I mentioned this this to my dad’s buddy Captain Smiley during my trip to San Diego for his change of command ceremony.

Also, during that trip I visited Wahrenbrocks Book House (a delightful 3 story used book store (that smells great!) with little piles of books on the old wooden staircase), and purchased a whole box of books (cost $180). Among the many books I purchased was Feynman’s Rainbow, which had this funny tale:

Feynman is answering Mlodinow’s question “Is it foolish to become mature?”

I’m not sure. But an important part of the creative process is play. At least for some scientists. It is hard to maintain as you get older. You get less playful. But you shouldn’t, of course.

I have a large number of entertaining mathematical type of problems, little worlds of this kind that I play in and that I work in from time to time. For example, I first heard about calculus when I was in high school and I saw the formula for the derivative of a function. And the second derivative, and the third… Then I noticed a pattern that worked for the nth derivative, no matter what the integer n was — one, two, three, and so forth.

But then I asked, what about a “half” derivative? I wanted an operation that when you do it to a function gives you a new function, and if you do it twice you get the ordinary first derivative of the function. Do you know that operation? I invented it when I was in high school. But I didn’t know how to calculate it in those days. I was only in high school, so I could only define it. I couldn’t compute anything. And I didn’t know how to do anything to check it or anything. I just defined it. Only later, when I was in the university, did I start over again. And I had a lot of fun with it. And found out that my original definition that I thought up in high school was right. It would work.

Then when I was in Los Alamos working on the atomic bomb, I saw some people doing a complicated equation. And I realized that the form they had corresponded to my half derivative. Well, I had invented a numerical operation for solving it, so I did it, and it worked. We checked it by doing it twice, which is just the ordinary derivative. So I did a nifty numerical method for solving their equation. Everything, well, not everything, but lots of fun stuff turns out to be useful. You just play it out.

So, alright, I’m no Feynman. Though I remember considering nth derivatives in high school (during my calculus class), I never actually performed them until we did it during physics class in college. And it was only then that I thought of the “half” derivative in the same definitional sense that Feynman did when he was in high school. I was never actually (still do not consider myself) capable of independently coming up with a useful operational definition that can actually be calculated.