Posts
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.
6 comments:
"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."
This was the only defintion of proof that I had ever heard of and I really took it to heart. I would look at proofs, not understand them, declare them not to be proofs because they didn't convince me. Another attitude I had was, "I don't need to prove that it works, I know that it works already." It some effort by some very patient people to convince me that I didn't know that it worked.
Well the reason you cannot disagree with the statement that "Math is about solving problems," is because "What is the proof of this," is a problem. Proving theorems is a subset of the more general activity of "problem solving". However, I think you are vastly downplaying the role of proof in mathematics. By "mathematics", I do not mean "the mathematical sciences" which includes anything that uses math. I mean actually doing math which exapands the frontiers of the subject.
So, with that in mind, the proof is not optional. It is part of the solution. You don't give it just if you really want to nail the problem -- you give it if you want to solve it at all. Otherwise, what you get are conjectures -- perhaps even famous ones -- but not the solution to the problem which is why someone decades or even centuries later will put up a million dollar prize for anyone that actually comes along and solves it. For instance, half the millennium problems are largely facts we already "know", in that someone has conjectured them already. They are just waiting to be proven. Nevertheless, it is unfair to say something like "coming up with that proof isn't really problem solving". It is problem solving, too.
I also don't think you are capturing the intellectual issue behind the requirement of the proof. It isn't just about generality or simply being persuasive. It is more about "knowing what you 'know'". Everyone is persuaded that the Riemann Hypothesis is true. People have already used it! And conversely, I doubt the proof of Fatou's Lemma or the Dominated Convergence theorem will convince most people of the truth of these statements because they probably don't really understand the statements, themselves, in the first place, for one thing.
Mathematics really is the business of proving theorems, proving theorems being a subset of problem solving. A mathematician is really one who does math not just someone that uses math, and that is also someone that proves theorems. And, I would also say that coming up with the definitions and axioms that lead to the theorems and their proofs is part of the process of proving the theorems. So, there is a big part of mathematics proper that involves determining what the "known knowns" and the "known unknowns" are supposed to be, too. And, not just in a foundational way. Just writing a textbook on Real Analaysis, say, you have to ask yourself "Do I want to include Dedekind Cuts or not? Maybe I just treat the Least Upper Bound property as an axiom."
I agree with what's being written and am impressed with the sophistication and correctness. However I think it's hard to appreciate without some experience with higher mathematics.
Let's consider an algebra problem. Take the equation
2x = 4
The problem is to find a numerical value of x that makes the statement true. In this case, you can use "guess and check" with the proof that 2 is the right guess simply being that 4 = 4.
But what if you decide to solve the problem by some sort of process, such as dividing both sides by 2? Now you have to justify the validity of each step in the process.
With the suggested process, you would be following the rule (axiom?) that if two numerical quantities are equal, they are still equal if they are both divided by the same positive number.
I'm not sure I want to try to prove why that is true right this minute.
Sorry - haven't read the comments yet (and they may be over my head when I do) -- here are the paragraphs from Tony Falcone I was thinking of:
But beyond all this, what troubles me most is the fundamental philosophical flaw in EDM: It ignores the core beauty and power of mathematics, viz., that it is an edifice constructed out of pure reason, all of whose inferences and deductions flow logically and unarguably from more basic facts. EDM asks the students to flit willy-nilly from room to room or even floor to floor in this structure, without ever exposing them to the skeleton, the underlying architecture.
The basic premise of EDM, so much so that its part of its name, that math should be valued or appreciated only insofar as it can be applied to "everyday things," is worse than misguided, it is a lie promulgated by people who, quite frankly, don't understand the first thing about mathematics. (Example: Do we study "Everyday English Literature?” Why do we still read Shakespeare? Are people really worried about being encountered by three old women stirring a big pot, and wanting to know how to deal with them?)
My understanding of constructivist math is that all of math is to become applied math, usually applied math in the sense of statistics one might use in the social sciences.
I remember people objecting pretty strongly to the idea that math should be statistics back on ktm-1.
Trailblazers, the curriculum my school has adopted, explicitly "integrates" science and math, making math subordinate to science.
Math is the statistics you use to do science.
Trailblazers was originally called "TIMS," which stood for "Integrated Math and Science" iirc.
Well frankly who wouldn't be worried about being encountered by three old women stirring a big pot? I mean would you just say to yourself "Oh, three scary old witches -- no big deal..."? I don't think so!
good point
Post a Comment