When I was in college, I spent a lot of time with a certain science major who just couldn’t seem to get the Zen of mathematics. Whenever I would say something like “there is no formula for the roots of a fifth-degree polynomial”, she would reflexively correct “You mean, there’s no formula you know about.”
But no, I really mean that there is no formula. I mean it is impossible to find such a formula. And when a mathematician doesn’t say “impossible” lightly; it doesn’t mean “no one’s ever done it” or “it’s prohibitively hard” or “it seems hopeless”. It means impossible.
It might seem hard to believe that people can ever know things are impossible. How is mathematical impossibility different from garden-variety closed-mindedness?
But there is more going on here than “I’ve seen a lot of things, and if never seen something like [X], then [X] doesn’t exist.” There’s a lot more. In fact, often one can learn a lot more from proving a problem has no solution than from solving a problem.
Suppose Sketchy Dan told you that for the past few weeks he’s been trying to find seven odd numbers that add up to a million, and he invites you to work with him at the hobby.
Well, maybe you’ll try some numbers for a minute or two, but I hope you won’t spend very long! Because soon you might realize that the sum of seven odd numbers is always an odd number! And a million is not an odd number. So it is impossible. You know this, and it’s a different kind of knowing than trying for ten years and then just sort of giving up.
This is why impossibility results can be so interesting. It is not just a negative statement, it is not just the lack of information. It is finding a deeper structure to the world, finding some unexpected property that all the possibilities share.
Here’s a very old problem. Imagine a chessboard with two opposite corners removed (so there are 62 squares left). You have 31 dominos, each sized perfectly to cover two neighboring squares of the chessboard. Find a way to completely cover the chessboard, or prove that it is impossible.
I won’t post the answer here; heaven knows it’s an old chestnut, and you can find the answer at dozens of math and/or puzzle sites, or a hundred old books. If you’ve never solved it before, give it a shot. When you make that transition from merely not being able to find a solution, to seeing, clear as crystal, why there can never be a solution, you’ll know what I’m talking about.
It is not unusual in the history of mathematics for people to spend centuries trying to solve a particular problem, only to learn that in fact a solution is impossible.
I give two examples here. In each case, understanding the impossibility opens up new branches of mathematics (in these cases, the deeper insights belong to algebra and algebraic number theory).
Solvability of the Quintic, You probably remember learning the quadratic formula in school. This says that the solutions of the equation are given by a formula .
There is a formula like that for solving cubic equations , but it’s messier, involving combinations of cube roots and square roots. There’s even such a formula for quartic equations , and it’s a horrible horrible mess. Poke around online a bit, and you can find these, if you want.
But what about the quintic ? No formula is known. Amazingly, no formula will ever be known! It is impossible, in general, to expression the solutions of such an equation in terms of the coefficients, using the basic operations of algebra. (Proving this is hard… but fascinating… and educational… )
Compass and Straightedge. Since antiquity Greek geometers were very interested in solving problems by the use of a compass (device for drawing circles, not the device for navigation) and straight-edge. (Some of you probably had to learn about such things in high school geometry.) The following three problems attracted a lot of attention for centuries (perhaps milennia?), and a lot of effort went toward their solution.
- Doubling the cube. Given any line segment, construct another line segment so that, if cubes are built with these segments as sides.
- Squaring the circle. Given a circle, construct a square with the same area.
- Trisecting the angle. Given any angle, construct an angle of one-third the measure of the original.
We now know that these things are all impossible.
The reason in a nutshell: if you start with a line segments of certain lengths, than all the line lengths you could ever draw with a compsas and a straightedge, no matter what you do, have something important in common — they can be written in terms of the original lengths, addition, subtraction, multiplication, division, and square roots.
Doubling the cube requires you to make two segments in the ratio (note the cube root), and squaring the circle. But the previous paragraph tells us a lot about what kinds of ratios we can and can’t get, and these are the wrong sort of numbers. We can never produce these ratios with (we can produce approximations that are as good as we want, but that’s different). Likewise trisecting general angles would require construction “the wrong kind of number”.
Sadly, there are still “independent mathematicians” who spend years and years trying to square the circle with compass and straightedge. I guess it’s better than kicking puppies.
(More interestingly, there are also people who study what kinds of constructions you can do with different tools or techniques. For example, it is possible to trisect angles using some basic origami moves.)
Regrettably, the phrase “mathematically impossible” has slipped into public usage lately, particularly in election coverage, but often used improperly. To convey the meaning “absolutely impossible, not just really unlikely but literally impossible”, that wouldn’t be so bad. But more than once I’ve heard something like “While it’s not technically impossible for [candidate] to win the election, if you look at the numbers it’s mathematically impossible — he’d have to win almost all the rest of the states.” Apparently some people take “mathematically impossible” to mean “unlikely for reasons that have something to do with numbers”. Misleading usage for at least two reasons.
Not only the obvious, but also because, as I say so often, mathematics isn’t numbers, it’s a way of thinking about things — numbers, yes, but also patterns, shapes, symbols, etc. Mathematical doesn’t mean “about numbers” any more than it means “about apples”.