Is a proof still valid if only the author understands it?

Question

Some time ago I was reading about the recent Shinichi Mochizuki's proof for the famous ABC conjecture. It's enormous and so incredibly difficult that at that time virtually nobody was able to comprehend it (now it seems to get better), and some reputable mathematician complained that if it doesn't change soon, scientific community may ignore and forget it, and never acknowledge it as valid.

This made me wonder, what actually is a proof. Even if it's correct, can we call it a proof if nobody besides the author understands it, let alone can verify it? Apparently not, however conscientious and rigorous it would be.

It could also be interesting to look at the opposite extremum. When I studied mathematics for some time, a professor told us about an early-medieval Indian mathematician (unfortunately I can't remember his name), who independently from the Greeks proved Pythagorean theorem. His proof consisted only of a drawing and one word: "Look". Nevertheless, it was clear enough that it's now widely accepted by mathematicians as a valid proof.

At this point I'm even tempted to say that the correctness itself is irrelevant here (as Wittgenstein noticed, we cannot prove the proof's correctness) - what matters is that we're able to use the proof to convince right amount of right people that our statement is correct. Obviously, some ways of persuasion (like e.g. some psychological or rhetorical tricks) cannot count as a proof. We expect proof to be logical, substantial and ad rem. I feel that this is hardly the case with showing us a drawing and asking to stare at it. However, it can indeed make us comprehend something as being obvious - something I like to call Descartes-style proof.

What makes proof a proof then?

(I know the formal definition of a logical proof, but this doesn't answer my question; arguably the only people who ever prove things like this are logicians, and only on special occasions.)

Answer 1

From an neo-intuitionistic point of view based in Kant's assumption that space and time are aspects of human thought rather than reality (whether or not you follow down Brouwer directly in finding all negation questionable) mathematics is not an objective description of some conceptual world, it is an exploration of shared human intuitions and how far they can be leveraged by combination among themselves. It is, in that sense, the oldest branch of rational psychology, if it is a science at all, and not a privileged branch of the sciences that happens to be 'exact'.

From that point of view, I would argue that correctness is, at base, an emotion, a feeling of accord with intuition. Especially in the case of mathematics, we are seeking to express how a set of intuitions aligns, and not any actual fact.

We codify various parts of mathematics that we have agreed capture an intuition in a checkable form. But the codification itself is only going to become a community norm if it appeals broadly enough to a certain cultivated emotional response.

We are obligated to accept combinations of things already proved into new things only because we feel we should, and because we have assented to basic logic giving the rules of combination. If we cannot get the impression that we have seen the connections between the things combined, we do not agree.

Even if someone can, we are attempting to isolate shared intuitions, so from this point of view, we want enough people to weigh in for us to trust that these are not mistaken or idiosyncratic examples. So no, a proof has to be vetted by a number of people, and found not just mechanically correct, but trustworthy.

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Only a radical Kantian, like Brouwer would maintain that basic intuitions cannot conflict. And choosing to see intuition that way leads to the decision that things like negation and completed infinity are not intuitive, even though they keep showing up in simplistic explanations. They are still intutions, but they need refinement.

The fact that when you pursue some group of strong intuitions far enough they often conflict with other equally strong ones is not disproof of the notion, only its most radical form. The later forms of neointuitionism do see that intuitions conflict. And they look at what happens with mathematicians run across those places as proof they are right about what math is really doing.

We have, for instance Russell's paradox "Does the set of all sets that do not contain themseveles contain itself?" In that we can see that when we put them together negation, universal quantification and containment result in something illogical.

But solving that problem is still the analysis of the intuitions, picking and choosing between them or replacing them with other more compelling formulations. Zermelo and Fraenkel replace the notion of unbridled universal quantification with an exploration of what we mean by making up a set, Brouwer limits negation to what can be constructively validated, Quine and Lawvere reshape the notion of containment referentially or merelogically so that it is directional and things can only contain themselves in a more trivial fashion.

So we end up with the seeds of three whole branches of math from this resolution: Set Theory, Constructive Analysis, and Category Theory. But what was the goal? The goal was to refine one or the other of these intutions so that it could be freely combined with others. This is not the proof that intuition is not the focus of mathematics, it is evidence of the process of interrogating spatial intution in progress.

No successful branch of mathematics exists that is not, or at least was not originally, driven by the need to refine a given set of intuitions. Geometry and all forms of Analysis, even non-Euclidean geometry and function spaces, works from our views of space: the fact that you want to use only part of a complex of intuitions and combine it with other options does not keep them from being your focus -- and manifolds are all locally Euclidean for a reason. Algebra work from the need to refine our ideas of factorization of complex mechanisms into simple parts -- Group theory is not about operators, it is about the classification of groups by factorization. Topology is about the problems of what is an inside and what is a boundary, something that is intuitively relevant to us even if it never really happens in nature. Etc. etc. etc.

We like to pretend that all axiomatizations are equally valid, but in fact, we choose axioms that represent intutions, and when they fail to fit, or we decide that the intuitions they eventually conflict with are stronger or more important than the ones they model, we change them. And not looking at the success of various fields is silly. We do not say 'but all that unsuccessful physics is also physics', and the Formalist contention that random axiomatizations that do not capture any relevance and ultimately simply die are still mathematics is equally silly.

Answer 2

"A language that I don't understand is no language." (Wittgenstein, MS 109)

Is a proof still valid if only the author 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.

Answer 3

This is really an interesting, open philosophical question. On the one hand, convincing is an important part of proving, yet we have logico-mathematical theories of deduction that set standards of proof, and one could think that respecting these standards should be sufficient. The question is particularly salient when it comes to computer-assisted proofs: should we trust them when no one really understands them?

I suspect that when one simply convinces by showing, it is implicit that a more rigorous proof could in principle be provided, and that's why it is convincing. But there can be surprises. At the other extreme, someone might have found a valid proof but fail to convince anyone that it is valid. But there is an epistemic uncertainty: maybe she made a mistake somewhere. So the ability to convince brings epistemic confidence. But ultimately, I would say that someone could still be right even though no one understands (yet). Although I admit things are potentially more complex, because one could ask: what exactly is this person talking about? (And is the subject matter of importance or is only the structure of the proof relevant?)

You can read this article on the subject http://m-phi.blogspot.be/2015/08/book-review-john-p-burgess-rigor-and.html

I cite:

a rigorous proof is one that convinces its audience that there exists a formally rigorous proof by providing those steps in the formally rigorous proof that it is not simply routine to provide

But there are more interesting discussions in the article.

Answer 4

In theory, a mathematical proof is a sequence of statements, where each statement is either an axiom, or is the result of combining one or more of the previous statements using accepted rules. And the last line of the proof is then taken as a "proven statement". There is no need for "understanding" here. Such a proof can be checked mechanically.

Unfortunately, that's the theory. In practice, a proof created this way would likely be incredibly large. Therefore mathematicians take shortcuts. They don't use a huge chain of simple statements, but they may use conclusions with an unstated message to other mathematicians effectively saying, "I know this isn't an exact proof, but you can trust me that I could provide an exact proof if I wanted to, but I don't want to waste my time, and don't expect you to be bored to death if you had to read it." And other mathematicians will read such a conclusion, think about it and say, "I believe that you are right." If enough other mathematicians do this, then the proof is accepted.

But there are problems: The other mathematicians might not be clever enough to be able to accept some conclusion in the proof. (I downloaded a copy of Wiles' proof of "Fermat's Last Theorem", and I'm not clever enough for almost anything in that proof.) The other mathematicians might say, "This conclusion is too great of a jump to be accepted. It needs to be split up into smaller steps so I can check if they are correct." And in the worst case, the other mathematicians might say, "I believe that I understand the conclusion, but I believe that it is wrong." This happened to Wiles the first time around and he had to fix some things in the first version of his famous proof.

We add to this that the author of a proof is obviously biased, so we can't take his or her word that the proof should be accepted. So if only the author says that he can understand the proof and that it is correct, then the proof cannot be accepted.

Answer 5

The original question is very interesting. A closer scrutiny will reveal that mathematical proof does not involve halting problem.

A mathematical proof ultimately appeals to a sense - a sense of mathematics as Russell puts it. Some people's senses of mathematics are very strong, some others' are moderate, still others have none at all. Most of those people whom we call "sensitive" probably have this sense of mathematics. The following joke may illustrate the point:

A invited B, C, D and E to a dinner party, but B didn't show up.

A said, "the one who should have come didn't."

C rose and left.

Then A said, "he who shouldn't have left left."

D rose and left.

Then E said to A, "Be careful with what you say. People might be offended."

Then A said, "I was not talking about them."

E rose and left.

C, D and E are sensitive; A is not. For C, D and E, there is a strong sense that enables them to "see" when A says this, he also means that. That is why Bertrand Russell says:

What can you learn by means of deduction? Perhaps if you were sufficiently clever, you could learn nothing ... As soon as you know the multiplication table, you have the means of multiplying any two numbers, say 24657 and 35746. You apply the rules and work it out. But if you were a calculating boy, you would "see" the answers, just as you "see" 2 and 2 are 4.

Russell, Bertrand. "The Art of Drawing Inferences." The Art Of Philosophizing. New York: Philosophical Library, 1968. 42. Print.

In order to prove A implies G, most people needs tiny intermediate steps in order to "see" the validity of this implication; a small number of people, on the other hand, can directly "see" this implication from A to G. It follows that there is no halting problem involved.

I do not know exactly what this sense of mathematics is, but I wonder if there are papers on this subject.

Is a proof valid if only one person understands it?

If we replace understand with "see", it is possible that a proof understood by only one person is still valid: Suppose you see a statement written on one of your classmates' shirt, but, upon your inquiry, no one else in the room sees what you see. As more people coming in, some acknowledge that they see exactly what you see, but they too cannot convince those who don't see any statement on the shirt. This sort of thing do happen, and almost everyone had this kind of experiences.

Take the proof of the "principle of reductio ad absurdum" for example: Starting from the principle of identity:

(1) (P v P) ⇒ P

Then by replacing P with ~P, we have

(2) (~P v ~P) ⇒ ~P

Then by the definition of material implication, we have

(3) (P ⇒ ~P) ⇒ ~P

From (1) to (2), no higher principle is called upon, we simply rely on our senses. How can we go on if people deny that substituting ~P for P in (1) results in (2)? Obviously we can't replace P with Socrates because Socrates has no truth value. There are definitely rules governing this substitution.

Answer 6

The best article I've seen on Mochizuki's proof is the one by Ivan Fesenko in Inference. Although much of it is over my head, it's worth quoting some of the conclusions:

Model theorists were the first to react to [his work]. Some of the reconstruction theorems may be understood in terms of a logical interpretation. The concept of multiradiality may be understood in terms of definability.

This shows, suprisingly, given the number theoretic origins of Mochizuki's work, that the first interest came from logicians (mathematical logicians - these are different from logicians in a philosophical sense). Whereas multi-radiality is a concept of Mochizuki, definability is a traditional one in logic. This shows people attempting to understand his new concepts by mapping them to ones they already know.

Why categories...these are questions raised in arithmetic geometry more than 40 years ago.

Categories are a fairly recent mathematical invention coming out of algebraic topology. The first paper published on categories to clarify axiomatics in the subject was expressly thought to be its last. Its detractors called it 'abstract nonsense'. (This is rich coming from a subject which everyone considers to be abstract and whose major practitioners pursue the abstract.)

Here Fesenko is raising the question, why are categories useful in arithmetic? This is because arithmetic can be interpreted geometrically - but to do this requires categories, and

this was a message imparted by Grothendieck, but Grothendieck’s work has not really appeared as manna from heaven to all number theorists. Having been deprived of manna, they are slow in digesting [his work]

It happens.

And it happened for a reason. The machinery around Grothendieck's work is immense - and that is its main difficulty. For example, the stacks project has written over three thousand pages merely to explicate Grothendick's work fully. This is a formidable amount of work. One can surely understand the reluctance of number theorists who are happy enough to graze in pastures closer to their traditional number-theory land and munch on lower hanging fruit requiring less effort.

Thus Mochizuki's natural constituency has a lot of catching up to do before they even begin to digest his own work and this takes time.

Fesenko ends with an endorsement:

[Mochizuki's work] is different in its philosophy and main ideas from anything we have known in conventional number theory. It is already changing mathematics, and as more people learn and develop [his work], this will continue.

Thus, there is a long, and because it is long, slow engagement with his work.

To answer your main question, if an author is the only person to understand his proof, is it then a proof? Well, if the author is correct in his understanding, then it is a proof; but this is not enough, because it doesn't help us to audit the correctness of the proof. We cannot enter into his mind to understand how he understood the proof. If only we could, then proofs would be so much easier to digest! Instead, all we have is the proof artifact, the text of the proof. Well, not all, since the mathematical community embodies a certain body of expertise and one can speak to the author to help understand the proof tactics and strategies. This is how a proof is also understood to be socially constructed.

Answer 7

As already pointed out, most mathematical proofs are not completely formal. For a completely formal proof, we can define correctness. We can then judge whether an informal proof is correct by whether it can be turned into a formal proof. Other mathematicians' acceptance is a strong signal of whether it is correct in this sense, but it is not a foolproof signal.

There are other issues too; if I "prove" the result by stating some lemma and then claim that the lemma is obvious and the theorem obviously follows, I may be correct about both the lemma and its implication, but can still be reasonably accused of not having done the full work.

Finally, there is the separate issue of usefulness. If my proof is correct but I haven't convinced anyone else with it, I probably haven't been of much use. And even if a proof is generally agreed to be correct, a more insightful proof is often still useful.

Answer 8

A "proof," that someone claims is valid, has to be accepted as valid, /unless/until some other person can disprove it! This is specially true if the person making the claim is well known and considered an "expert" on the subject matter.

Answer 9

There is no such thing as a valid or invalid proof. A proof is a sequence of symbols in a (mathematical) language, that connects premises with conclusions. A sequence may or may not contain errors. There is no way to "make sure" that a sequence does not contain errors. If person A proves theorem X, some person B may ask "Prove that your proof does not contain errors!". If A provides a proof that his proof of X does not contain errors, B might as well ask to prove that this second-order proof does not contain errors and so on, ad infinitum.