Is a proof still valid if only the writer understands it? Say that there is some conjecture that someone has just proved. 
Let's assume that this proof is correct--that it is based on deductive reasoning and reaches the desired conclusion.
However, if he/she is the only person (in the world) that understands the proof, say, because it is so complicated, conceptually, and long, does this affect the validity of the proof? Is it still considered a proof?
Essentially, what I'm asking is: does the validity of a proof depend on the articulation of the author, and whether anyone else understands it?
The reason I ask is that the idea behind a proof is to convince others that the statement is true, but what if no-one understands the proof, yet it's a perfectly legitimate proof?
 A: Cathy O'Neil doesn't think so.
(The link is to the last in a chain of three blog posts about, in some way, what a proof is to her.  I linked this one because it gives a real-life example, in addition to linking all the others.)
Her point is, basically, that proof among working mathematicians is a correct argument that can convince anyone with the appropriate background but who isn't necessarily an expert on the exact problem.  And the reason she insists on this is that it puts a heavy burden on the author to supply the details wherein lies the devil.
In the particular situation of Mochizuki's work on the ABC conjecture, she claimed that he failed even to make it comprehensible to experts (or rather, that he was the only expert), so that in a sense, the work was meaningless to mathematics at large.
My own opinion is that a proof can either be formally checkable, in which case it doesn't matter who understands it as long as a computer we trust does, or it can be humanly readable, in which case the belief that a formal analogue exists is supported by reproducibility of understanding by a control group of mathematicians who aren't biased to understand it.  The scientific analogue is very deliberate in this case: working mathematicians are concerned less with what is abstractly true than with what is usefully true.  Thus, just as good mathematics is interesting theorems, good proofs provide enough faith in the result so that it can be used to support more interesting theorems.  This may be irresponsible, but it's true.
A: 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.
A: The purpose of proof is not only to convince others.  One may not know whether a proposition is true; one may then find a proof and thus become convinced that it is true; one may then apply that to an engineering problem; if one gets it wrong then all the world's computers and bridges will crash.  Is one then behaving rationally and honestly?  I think so.  But one could make a further valuable contribution by then rearranging the proof into publishable form so that others will understand it.  Then one posts it right here in this square (or a research journal, etc. . . . . )
In the usage of mathematical logicians, proofs exist before they are known, just as in number theory (except for contructivists) facts about numbers exist before they are known.  Otherwise one would not have such a result as the theorem of logic that states that the ratio of lengths of proofs to lengths of theorems is unbounded.
And sometimes the purpose of a proof is not to convince.  One may already have read ten proofs that $\pi$ is irrational; nonetheless one finds value in a new proof because it gives further insights or has esthetic appeal.  For example, I did not need another proof that $\pi<22/7$ at the time when I first learned that
$$
\int_0^1\frac{x^4(1-x)^4}{1+x^2}\,dx = \frac{22}7 - \pi.
$$
A: I think the way you phrased one part of the question kind of gives the game away: "Is it still considered a proof?". Well, no, if only one person understands it then it isn't considered a proof except by one person who nobody else believes.
Nevertheless, "valid", "correct" and "perfectly legitimate" all mean the same thing in relation to a proof. If it is any one of them then it is all of them, regardless of who says so. It's just not accepted as a proof until people generally can agree that it's correct. Furthermore it might be correct by some standards but not others.
At some point in the future it might come to be considered a proof, once the rest of the human race catches up with the first person's genius. And those people in the future will say yes, it was a proof all along, and all the present-day people just didn't understand it. Some of the present-day people might even live long enough to join in this acceptance of their initial failure.
One can make a philosophical argument as to whether those future people have made a categorical error in ascribing the property of "correctness" to the proof itself, rather than describing it as a property of the society of mathematicians by whose standards it is either accepted or isn't. But fundamentally what would happen in that unusual situation is what I've described. Except in a very rarified philosophical context the proof doesn't "become" correct when it's accepted: rather, an objective property of the proof is confirmed by the community.
Maybe its correctness cannot ever be confirmed by the community. Let's suppose for one example that it is written, very plainly and simply, in a language that only has one remaining speaker who dies and the language is somehow never again dechiphered, so the only conceptual complexity is that language, not the mathematics. Then although it "is" a correct proof, it doesn't serve the purpose of one and is never accepted as one. Similarly you can ask, if I write out a novel proof of Cauchy's theorem and burn it, then was it a proof since it was never accepted by the community? Um, maybe. Nobody knows, maybe there was an error in it that I missed. Does anyone even care whether we say "no it wasn't" or "it may have been but we cannot ever know"?
I don't think the issue arises much, and for reasons fundamental to the process of doing mathematics. Almost all proofs, as presented by humans to humans, contain short-cuts and abbreviations that the reader is supposed to be able to fill in. That is to say, they aren't always evidently correct to all readers, and by sufficiently scrupulous standards they are incomplete and so not correct. If you write a proof and find that your readers aren't able to follow it because it's conceptually complex, then you agree with them that it's just an outline of a proof and you fill in more of the gaps until they can follow it all. The result is called "the proof". One reason you do this is to be credited with the proof, the other reason is because those gaps are generally where any errors lie in the first attempt.
Now, it may be that in future everyone is very familiar with a whole bunch of concepts that you invented in order to write your proof. Then people might start to condense your proof back down again, introduce the gaps back, and end up with something very similar to what you initially wrote. So the rarified philosophers are happy: something that initially "wasn't a proof" now "is a proof", because of a change in the readers :-) Whereas the future people who have now caught up to your genius will say, "contemporaries of alexqwx would not, of course, have accepted the proof in exactly this form since all these concepts were novel".
Then you can start arguing over whether it's "the same proof" or not, if it's substantially the same but with slight differences in what's assumed as reader knowledge/understanding.
A: Yes it is a proof.
Just like a tree that falls in a forest makes sound even if nobody's there to hear it.
The real question would be 

"If there is no way to distinguish between two scenarios, does it
  really matter what has actually happened?"

If there is no way to find out if the proof is indeed correct, would it really matter?
I don't think so.
If, for example, I claim that there is a scary bogeyman under the bed, but it is invisible and cannot be detected in any way, then would it matter its actual existance?
No, because, pratically by definition, such bogeyman cannot have any detectable interaction with our world. So why should one care about it?
In this case, just go for the easiest explanation:
There are no bogeymen under the bed, and the proof is not valid.
A: If you had a time machine and flew back in time and showed the simplest mathematical proof to a Neanderthal, he wouldn't understand it. Your Proof remains correct.
A: "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.
A: I guess your statement itself gives the answer. Assuming that the 'proof' is correct, it can be considered a proof, albeit only by the person in question. I have used here the term proof given the understanding that it is a series of logical deductions made correctly to arrive at a conclusion. Correctness does not depend on the comprehensive capabilities of people or the comprehensibility of the proof.
A: Although I see most of what I have to say already said, the issues covered do not seem to get at what I see being the heart of the matter:
What you are asking either changes throughout your post, or is not properly worded for a Stack Exchange dedicated to mathematics and mathematical reasoning. Either you mean something other than what I would expect the question to mean; or, you have a confused understanding of the question you are asking and equivocate at some point in the question.
'Is a proof still valid if only the writer understands it?'
Although 'it depends on what you mean,' is the easy way out, here. But, if you mean what I would expect someone to mean when asking this in a forum dedicated to mathematics (i.e., 'a correct, formal mathematical proof") then, yes. To say no, you will have to either: be asking something else or equivocating on at least one important term. For example, you could mean: 'an informal proof', or, 'a proof not completely based on deductive logic', or 'a proof in physics'. But, although I would agree that, for these cases, 'no' would be the right answer; I feel like it would be proper to say "but you asked in such a way as there were bound to be misunderstandings."
Your first paragraph starts:
"Let's assume that this proof is correct"
Well, so, you're accepting for the sake of your question this assertion; therefore, I lean even more towards a 'yes' being required (again, otherwise, it's not the definition of 'correct' I would expect for a question in this kind of forum.)
Okay, so, if it's 'correct,' then you seem to be coming more into line with the definition I suggested above. But again, you've not specified if you mean the same thing we usually mean; e.g. that from certain theorems, we can deductively derive the conclusion. But the second part makes it hard to be sure.
"Let's assume that this proof is correct—that it is based on deductive reasoning and reaches the desired conclusion."
The unfortunately phrasing here gives the entire thought two possible meanings: you could be adding it all together 'correct reasoning, that from certain theorems deductively derives the conclusion'; or, you could be defining "correct" reasoning as 'that which is based on deductive reasoning and reaches the desired conclusion' without referencing whether the deductive reasoning is indeed valid all the way through the conclusion. 
I'll also note that it leaves out whether you accept the premises themselves as valid—although I generally think of 'mathematical' proofs as necessarily being (please excuse the abuse of notation here) { axioms and theorems that are true } + { correct deductive reasoning from axioms to conclusion } => { conclusion would be true }. For instance, we don't say that Euclid's proofs of geometrical theorems are 'invalid'.
In other words, since mathematical proofs are reasoning about ideas, there is no need to refer to them as being, per se true or false; those are other kinds of truths; but that is unfortunately not explicit either. 
So, a charitable reading here seems to be: since you've said 'valid', 'deductive', 'proof', and asserted it to be 'correct', with what seems to be an understanding of all of those things; you seem to mean what I would have expected. So, if this were the end of your post, I could say "The answer to your question, as worded, is 'yes': even if you possibly meant to ask something different." 
And, given that, your next few sentences (while hard to reconcile as being necessary, given your preceding statements) are fairly easy to answer:
"However, If he/she is the only person (in the world) that understands the proof, say, because it is so complicated, conceptually, and long, does this affect the validity of the proof? Is it still considered a proof?"
The complexity of the proof, or length of a proof, has no bearing on it's validity (in the mathematical or logical sense); only whether it's deductive reasoning is sound; which.
And, as stated, a proof is a valid chain of deductive reasoning that leads from the premises to the conclusion (again, in the mathematical or logical sense). 
So, in answer to these questions: "No", and "Yes".
Now, it's very possible (in fact, it happens all the time) that someone has something that they think they understand, and that will be proved wrong. But that was never a valid, correct, proof.
And, finally:
"Essentially, what I'm asking is: does the validity of a proof depend on the articulation of the author, and whether anyone else understands it?"
Well, no, the essential parts of your first question, added to the clarifications you offered in the next paragraph, show that this is very much not the question you were asking. If you mean, instead, that this is what your questions in the middle were driving at; maybe so. 
But at this point, you have asked two completely distinct questions, and have done so by asking one question in the title, even offering clarifications that seemed to show you really did mean the expected thing with your question; only to offer a final "restatement" of your question that is, instead, almost diametrically opposed to your original question.[&star;]
This is where I come back to "you have a confused understanding of the question you are asking and equivocate at some point". I see absolutely no way to reconcile the fact that you accepted for the sake of argument that the argument was indeed correct, deductive, logic; only to turn by 180&deg at the last moment and say that the "restatement" of your question has to do with what happen to be entirely subjective criteria.
So, why respond if I 'largely agree' with so much of the other answers?
I just don't think that the other answers they went far enough: yes, it depends on what you mean, yes, the import of such a proof could be debated. But no one seemed to say what I see as the heart of the matter: that you did not maintain the internal self-consistency that would be necessary for this question; to keep it from being an opinion-fest. Other answers fell (more-or-less) into one of these broad categories:
Is it a proof?


*

*"A 'proof' must XYZ and this XYZs, so, "Yes.""

*"A 'proof' must ABC and and this (doesn't ABC/only XYZs)."

*"A 'proof' might mean either it must XYZ or must ABC; so, either is correct based on which you are asking."

*"A 'proof' might mean either it must XYZ or must ABC; so, neither is correct, as it's completely subjective."
As mentioned, I actually have pretty large swaths of common ground with answers 1-3 (although, none with 4, as I have it here) but if I had to summarize mine it would be something like:
TL;DR: Being per se internally contradictory, it is meaningless, and is impossible to answer. Because of several words having more than one meaning, and because of your assertion that formulations that are not equivalent, are; to answer it without pointing this out must simply end up relying on something other than a reasoned opinion.
I have a bit of an examination of what I think might be the substantial answer/solution to this question, as opposed to the trivial, e.g., "just ask two questions, get rid of the changes in meaning", answer. of, if not more substantial, it's at least a pointer to a more interesting question whereas the answer you would get to the questions behind the original question tend to be trivial, tautological, contentious, and often simply expressions of distaste or agreement; whereas, I think the basis of this question leads to some of the more interesting questions in philosophy: epistemology. (or, in this case: mathematical epistemology.)
The Aside(or, Abottom?): As mentioned, I think your question does very much speak to something that is indeed a 
hard topic, one which, in fact, makes up a very large branch of philosophy: epistemology, the theory of 
knowledge. What is knowledge, and when do we know it? Is absolute knowledge possible? If it's just that we think,
then what are the criteria we do accept, and what are the criteria we *should* accept for calling something
knowledge, vs. just reasoned opinion? And so forth. Many philosophers have dedicated their life to this topic,
and I'm sure if you look, you'll find the more specific topic of mathematical epistemology.
But to me, I think this points us in the right direction when answering a question like yours, as I think the
real answer to your question is: well, what's bothering you in the first place?
Validity is a particular logical concept regarding reasoning; and furthermore, it's a timeless one. Academic
consensus shifts over time and taken from a different direction, the answer to your question becomes not only
obvious, but the interesting path to take from there does as well.
But, feeling that something must be wrong when someone insists that someone could have, a valid proof that only 
they understand, well, it's absolutely true; and I think, obviously true. Think about it: at some point in time,
every proof that's been made, all our scientific and philosophical knowledge or best-guess at knowledge, or
whatever you wish to call it, at least that based on the kind of deductive formal proofs (some informal ones
are deductive) that we're referring to are proofs known by zero people! (In fact, there are many valid proofs 
that no one yet knows! It's why we [generally] speak of 'discovering', or 'finding' proofs, rather than
'inventing' or 'creating' them.) 

And generally, those proofs were constructed by a single person; take a single huge one from early last century,
Goedel's incompleteness theorem. I believe that when he finished it, he was the only person that understood and 
got the argument, which proved that in *any* complex logical system, mathematics included, any consistent
system was *also unavoidably an incomplete* one. And, this is now seen as having been a valid proof, but it was
a difficult one, involving entirely novel constructions that had never before been used, and, for which, the
rigorous proof was quite involved.
Because of this, and because even the *language* we use to have discussions such as this one point to this
having been a timeless truth, one beyond any particular persons' understanding of the issue, and whether or
not there was understanding, or even dissent! We can speak in the manner of proofs that "**turned out to have
been** invalid **all along**", or "valid, **all along**", but not (at least mathematical) ones that "were
valid for a time, maybe, but now they are invalid." It just doesn't make sense. It's the wrong place to push.
Rather, if you hear a claim that they do have, or could have, a proof that is valid but that no one else gets;
then you did hit some of the right places in your argument; I think you just had the wrong emphasis. You 
focused on a word, and definitions of a word, which generally are about the least productive philosophical
converstaions there are. Not that it's not necessary to talk about what things mean, but when you feel like
redefining words to mean other than their plain and normally understood meaning, you're either getting across
a metaphorical point (in this case, "a proof that's not understood by someone else, no matter *why* that is,
points to a possible problem with it, and while it may *logically* be the case that it *could* be valid,
really fairly irrelevant, since the issue isn't whether it is or is not valid: a completely valid proof
that hasn't been presented to anyone else, nor vetted by anyone else, and which no one else can learn, or
cares to learn, points to a problem with that proof or the concept its after, or even the effort put into 
finding the others that might care enough about it to put the effort in!)
And that's the right questions. That's the right direction; fighting over words gets nowhere. Accepting 
what 'valid' means, and then pointing out no one knowing that it's valid is just as bad as it actually
being invalid *as it counts towards that proof being used or useful*. Ask if they're publishing it; 
ask whether they've showed it to others in the field, or anyone else that's good at whatever type of 
logic they're using.
Again, it's not that the proof can't be valid if they're the only one that knows it. It's that any proof 
that's valid should be understandable and reproducible by those who know enough about the topic, or, 
barring that, that can learn, and care to; and if it's a *significant* proof, then the "care to" part will 
come naturally. But no need to take a timeless, even **person-less**, concept like validity and turn it on
its head, sacrificing timelessness, non-contradiction, and the like (any other definition regarding number
of people etc. will end up with times where a proof was both valid AND invalid) just speak to real issues
about the real problem, and you won't find yourself sidetracked discussing something that basically leads
to either a 'yeah I hear ya' type discussion, or a pedantic tautological one. But the fact is that a proof
with only that one person who understands (a) has no one to double-check the work and (b) isn't going to
ever do anyone any good until you figure out a way to put it to use, and that's unlikely as long as the
one that find it is the only one that understands it. 

I think this mostly covers my thoughts on the matter; if I had more time I could make it shorter. [But
'probably would end up making it longer. ;)] TL;DR, Girl! to the Rescue! much love 

&star;You're in good company, though: Kant did it with the Categorical Imperative. :)
A: 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?

A: 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.
A: 
Let's assume that this proof is correct--that it is based on deductive reasoning

I think that a little bit of deductive logic is indeed what will shed light on your question. Consider the little distinction between:

There exists a X such that for all Y, X knows Y.

And

For all Y, there exists a X such that X knows Y.

Now, consider that X are people, and Y are parts of your proof (or part of the design of a computer chip, or of a complex software).
They are far from equivalent. And something along the second line is enough to warrant a proof: if, for every part of the proof, there is someone apart from the original author that understand the part and can vouch for it, then the whole proof might be very convincing even though there isn't a single person that understand all parts of the proof.
A: In mathematics, a proof must be rigorous and logical and complete. This is somewhat different than, say, proving that Joe Smith committed the murder (that's a judgment call by the jury, given what evidence there is, which may not be rigorous or absolutely complete). 
Any crank can claim to have proved anything, and write 8000 pages dense with formulas and Big Words to support their claim, but many other people will have to read, understand, work through, and accept the claim; for the proof to be widely regarded as being valid. The proof will have to be built upon already-accepted mathematical facts, and use widely-accepted tools to reach its conclusion. Ideally, one could reduce everything to symbols and feed it to a computer logic checker that would look for unsupported claims, contradictions, false statements, etc., and accept or reject the proof, but we're not quite there yet.
A: The proof itself may be valid in reality, but if it is not apparent to others then the proof holds no real value in society. Imagine if a super genius proved that our universe was inside of an aliens hemorrhoid. This super genius can prove it using logic and reasoning, as well as mathematically but if he cannot prove it to others due to their lack of capacity, then it is of no use to society.
A: I guess there are infinitely many correct proofs and every human could read only finitely many during her or his life so there must be some proofs that none has read. For example, no person knows the whole mathematics so if I generate a theorem which combines every theorem together like "sum of two even numbers is even and x^3+y^3=z^3 has no nontrivial solutions as integers and ...", one gets such a huge theorem that none has time to read its proof during her or his life.
A: In short YES.
As a mathematical question becomes more difficult we can expect fewer people to be able to understand or answer it correctly. Likewise as a premise becomes more complex we can expect few people to be able to understand it. Even if we confine ourselves to deductive reasoning alone. We would see that if things are complex enough there will be cases were other's simply have not had the time to make n deductive inferences in a given period of time. Thus you would conclude that if a premise is obscure enough that others will not have had enough time to follow the line of reasoning.
You can argue that the prove-er has a certain head start on the proof verify-er.
