Sometime in late August or early September 1994 I had one of those quietly astounding moments on a computer. It would have been while using Maple, a program capable of doing symbolic mathematics. It was something capable not just working out what the product of two numbers is, but of holding representations of functions and working out what the product of those functions was. That’s striking enough, but more was to come: I could describe a function and have Maple do the work of symbolically integrating it. That was astounding then, and it really ought to be yet. Let me explain why.
It’s fairly easy to think of symbolic representations of functions: if f(x) equals , well, you know if I give you some value for x, you can give me back an f(x), and if you’re a little better you can describe, roughly, a plot of x versus f(x). That is, that’s the plot of all the points on the plane for which the value of the x-coordinate and the value of the y-coordinate make the statement “” a true statement.
If you’ve gotten into calculus, though, you’d like to know other things: the derivative, for example, of f(x). That is (among other interpretations), if I give you some value for x, you can tell me how quickly f(x) is changing at that x. Working out the derivative of a function is a bit of work, but it’s not all that hard; there’s maybe a half-dozen or so rules you have to follow, plus some basic cases where you learn what the derivative of x to a power is, or what the derivative of the sine is, or so on. (You don’t really need to learn those basic cases, but it saves you a lot of work if you do.) It takes some time to learn them, and what order to apply them in, but once you do it’s almost automatic. If you’re smart you might do some problems better, but, you don’t have to be smart, just indefatigable.
Integrating a function (among other interpretations, that’s finding the amount of area underneath a curve) is different, though, even though it’s kind of an inverse of finding the derivative. If you integrate a function, and then take its derivative, you get back the original function, unless you did it wrong. (For various reasons if you take a derivative and then integrate you won’t necessarily get back the original function, but you’ll get something close to it.) However, that integration is still really, really hard. There are rules to follow, yes, but despite that it’s not necessarily obvious what to do, or why to do it, and even if you do know the various rules and use them perfectly you’re not necessarily guaranteed to get an answer. Being indefatigable might help, but you also need to be smart.
So, it’s easy to imagine writing a computer program that can find a derivative; to find an integral, though? That’s amazing, and still is amazing. And that brings me at last to this tweet from @mathematicsprof:
The document linked to by this is a master’s thesis, titled Symbolic Integration, prepared by one Björn Terelius for the Royal Institute of Technology in Stockholm. It’s a fair-sized document, but it does open with a history of computers that work out integrals that anyone ought to be able to follow. It goes on to describe the logic behind algorithms that do this sor of calculation, though, and should be quite helpful in understanding just how it is the computer does this amazing thing.
(For a bonus, it also contains a short proof of why you can’t integrate , one of those functions that looks nice and easy and that drives you crazy in Calculus II when you give it your best try.)