A dialogue in The Mathematical Experience by Davis and Hersh on what is mathematical proof and who decides what a proof is?
Let’s see how our ideal mathematician (IM) made out with a student who came to him with a strange question.
Student: Sir, what is a mathematical proof?
I.M.: You don’t know that? What year are you in?
Student: Third-year graduate.
I.M.: Incredible! A proof is what you’ve been watching me do at the board three times a week for three years! That’s what a proof is.
Student: Sorry, sir, I should have explained. I’m in philosophy, not math. I’ve never taken your course.
I.M.: Oh! Well, in that case – you have taken some math, haven’t you? You know the proof of the fundamental theorem of calculus – or the fundamental theorem of algebra?
Student: I’ve seen arguments in geometry and algebra and calculus that were called proofs. What I’m asking you for isn’t examples of proof, it’s a definition of proof. Otherwise, how can I tell what examples are correct?
I.M.: Well, this whole thing was cleared up by the logician Tarski, I guess, and some others, maybe Russell or Peano. Anyhow, what you do is, you write down the axioms of your theory in a formal language with a given list of symbols or alphabet. Then you write down the hypothesis of your theorem in the same symbolism. Then you show that you can transform the hypothesis step by step, using the rules of logic, till you get the conclusion. That’s a proof.
Student: Really? That’s amazing! I’ve taken elementary and advanced calculus, basic algebra, and topology, and I’ve never seen that done.
I.M.: Oh, of course, no one ever really does it. It would take forever! You just show that you could do
it, that’s sufficient.
Student: But even that doesn’t sound like what was done in my courses and textbooks. So mathematicians don’t really do proofs, after all.
I.M.: Of course we do! If a theorem isn’t proved, it’s nothing.
Student: Then what is a proof? If it’s this thing with a formal language and transforming formulas, nobody ever proves anything. Do you have to know all about formal languages and formal logic before you can do a mathematical proof?
I.M.: Of course not! The less you know, the better. That stuff is all abstract nonsense anyway.
Student: Then really what is a proof?
I.M.: Well, it’s an argument that convinces someone who knows the subject.
Student: Someone who knows the subject? Then the definition of proof is subjective; it depends on particular persons.Before I can decide if something is a proof, I have to decide who the experts are. What does that have to do with proving things?
I.M.: No, no. There’s nothing subjective about it! Everybody knows what a proof is. Just read some books, take courses from a competent mathematician, and you’ll catch on.
Student: Are you sure?
I.M.: Well – it is possible that you won’t, if you don’t have any aptitude for it. That can happen, too.
Student: Then you decide what a proof is, and if I don’t learn to decide in the same way, you decide I don’t have any aptitude.
I.M.: If not me, then who?