Is a proof still valid if only the writer understands it? [closed]

There only appears to be a problem because we are using the same word for closely-related but distinct concepts (not an uncommon situation in philosophy), namely

• "proof" as in formal proof, which Wikipedia defines as

  a finite sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference

• "proof" as in "any argument that the listener finds sufficiently convincing"

The situation you describe contains a proof according to the first meaning, but not the second. Conundrum resolved.

"Is a proof still valid if only the writer understands it?"

I do not think so.

See Yuri Manin, A Course in Mathematical Logic for Mathematicians (2010), page 45 :

A proof becomes a proof only after the social act of “accepting it as a proof.” This is as true for mathematics as it is for physics, linguistics, or biology. The evolution of commonly accepted criteria for an argument’s being a proof is an almost untouched theme in the history of science. In any case, the ideal for what constitutes a mathematical demonstration of a “nonobvious truth” has remained unchanged since the time of Euclid: we must arrive at such a truth from “obvious” hypotheses, or assertions that have already been proved, by means of a series of explicitly described, “obviously valid” elementary deductions.

Thus, the method of deduction is a method of mathematics par excellence.

[...] Every proof that is written must be approved and accepted by other mathematicians, sometimes by several generations of mathematicians. In the meantime, both the result and the proof itself are liable to be refined and improved.

The historical "stability" of the criteria for an "acceptable" proof does not imply that mathematics and proofs are supra-human : they are human (and social) activities.

I'm not sure. I suppose so.

However, the following quote from The Mathematical Experience deserves a place here. It's between an "ideal mathematician" (I.M.) and a student.

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 and 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 formulae, 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?

Your question is indeed a philosophical one. And as such it is (imo) borderline off-topic here.

That said, as with many philosophical questions, it comes down to the meaning of words. You ask whether the proof is valid when only the author understands the proof. The question is how to determine whether or not a given proof is valid.

If we mean that the proof is strictly correct in the sense that one can fill in details and complete the proof, then I would say that the proof is valid. That is, one could say that as long as there are no mistakes the the argument, then it can be considered valid. That is, in some ideal world the string of arguments in the proof might actually be valid.

Just because you can't see it, doesn't mean that it doesn't exist.

But again, how do you actually determine that the proof is valid? The problem is that if no one understands the proof, it would be impossible to label the proof as "valid". It might be valid in a theoretical sense, but how can put the proof in the box containing all "valid" proofs if that can't be determined.

So I agree with the general tone of other answer by Mauro saying that

the mathematical community must be able to verify that the proof is correct for the proof to have any value.

As with physics where an experiment gets it validity from being repeatable, I would say that a proof should (as you suggest) be able to convince.

Maybe it is a bad analogy, but in society (at least in the US) things get there truth value from being tested in a court room. A person is only guilty when the evidence has convinced a jury of the guilt of the person.

Bear in mind thought that it is not uncommon in mathematics that even published results turn out to contain mistakes. That is "proof" of the fact that even accepted "truth" can turn out to be false.