Problem Solving and Writing Proofs

in this post, Catherine asked:

What I read over and over and over again is that the pinnacle of mathematics achievement is the "ability to solve problems."

Isn't that wrong?

How does that square with "the ability to write proofs"?

I'm asking because I don't know.

From the perspective of a working mathematician, the pinnacle of math achievement is, in fact, the ability to solve problems. For many mathematicians, problems come from the real world; for others, they come from tiny slivers of the rareified math world, but in both cases, a mathematician needs a problem. A mathematician without a problem to solve is a fish out of water.

So what's a problem? Well, it's something that can be solved.

A solution is, basically, a Single Answer. A truth. A statement of fact. It might involve a lot of generalities, so that various cases break down into more answers, but if you've solved the problem, then you've solved ALL of those cases, however many there are. And if you've solved them all, then you've found a Truth--a solitariness that made them all part of your singular solution.

A proper solution is not an opinion. It's not a work of art that's open to interpretation.

A proof is a certain way of showing that your solution isn't just an opinion, isn't open to interpretation.

Basically, it's the "show your work" step. In calculus, or arithmetic, we show our work by doing calculations. But at some point, problem solving involves REASONING. And "Show your work" means explaining your reasoning.

A math problem is a kind of special problem where Everything Is Well Defined, and the goal is to Generalize As Much As Possible, But No More. In math problems, you know what all of your known knowns are, and all of your known unknowns. Those unknown unknowns aren't here--they've been defined away. That's where the proof comes in: the proof tells you what you assumed, and what you predicated your reasoning on. You don't need to say "because 3 * 5 is 15", but you do need to say "because we're working with the rational numbers" or "because 7 is prime".

Generally, a mathematician is trying to find a specific Truth that applies to a whole lot of instances of problems, because to him, that's all One Problem to Solve, and he'll be better at "problem solving" if he can show a solution to a LOT of problems at once. A proof is a really good method for doing that.

Here's how a mathematician actually solves a problem: First, he does problem solving. And after a lot of problem solving, he starts to distill what is true and common to all of his solvings. And then, if he really wants to nail the problem, so that all problem of the future problems of the same type can be solved, he is going to find the proof.

Here's a math problem: I've got a triangle. One angle is a right angle. I know the lengths of both sides of the right angle. What is the length of the hypotenuse?

The way a mathematician would solve this problem would be to evaluate what they know. They might start by trying to solve it for some specific triangles, with specific lengths. But while this doesn't seem like a lot of information, to a mathematician it is, because a lot of information here is hidden. But to a mathematician, Everything Is Well Defined. So by knowing you've got a triangle, you know a lot. And by knowing you've got a right angle, you've got a lot.

So the mathematician solves the problem first for a specific triangle. And then for a few more. And a few more. And at this point, he's sure he's onto something. But then he thinks about if he can generalize his answer, so it's true for any triangle. Well, no, he can't, he finds out. It's not true for ANY triangle, but it is true for every RIGHT triangle. And then he realizes what's true: that he can turn that "solve the length of the hypotenuse" into a proof, and the proof will ALWAYS tell him the length of the hypotenuse. So the problem solving then becomes "form this thing into a proof: a proof that the square of the length of the hypotenuse is the sum of the squares of the legs of the right angle in a right triangle."

Writing proofs is a form of problem solving where you are trying to find the Truth in a way that is convincing to those who don't know the answer just by looking at it. It's a convenient way to solve a whole set of problems at once. To a mathematician skilled at proofs, solving it for a whole set seems like less work than finding the answer for a specific instance. But you've got to know enough, and believe enough in the answer to be able to turn "solving a problem" into a proof.

Update: if you look carefully at this, you might be horrified to see something like Constructivism here. The difference is that the mathematician has an exceptional amount of mastery, and so the constructivism that they undertake is not aimless, confused, misguided, or pointless. It is carefully aimed in certain directions by their extensive (in both breadth and depth) mathematical experience.