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.[☆]
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° 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
☆You're in good company, though: Kant did it with the Categorical Imperative. :)