I know the answer is "no" in general due to Gödel's Theory of Incompleteness, but I mean this question in a more real-world sense (i.e. scientific sense). In other words, I am talking about empirical rather than mathematical truths. Can there be truths no evidence of which exists? I will give an example to clarify -- let's say dinosaurs existed, but they left no fossils, or any other trace. If there was no way of proving that dinosaurs existed would it still be true that they did?
The answer is a point of contention between realism and anti-realism. Truths that "do not have evidence" are termed verification-transcendent truths (coined by Dummett), and realists are committed to their existence. Anti-realists, on the other hand, hold that unverifiable in principle statements have no truth values. So if no trace of dinosaurs remains, anti-realists would not consider claiming their existence to be meaningful. This is quite counterintuitive, but they resist imagining the all-seeing eye of God to settle the truth of claims we can never, in principle, settle. And without that, what sense, exactly, does it make to say that dinosaurs existed? To Dummett, reality of the past reduces to talking about its traces in the present, nothing else, so undetectable dinosaurs are without meaning, for more see What is the Anti-Realist and Constructionist interpretation of empirical dating methods and existence of the past? In some sense, Einstein followed this line of thinking when he abolished ether, which physics of the time (Lorentz's) declared absolutely undetectable.
For these reasons, anti-realists deny the law of excluded middle. Anti-realism is often regional, one can be an anti-realist about ethics, aesthetics, and mathematics, for example, but a realist about physics and biology. Anti-realists about mathematics are called intuitionists, and realists are called platonists. Here is from Walker's Verificationism, Anti‐Realism and Idealism:
"Anti-realism, like verificationism of the traditional kind, is a theory about meaning, and as its name implies it is directed against an alternative thesis which can in this context be called realism, or metaphysical realism: the thesis that a statement can have truth-conditions which are unrelated to anyone's capacity to find out about them. Metaphysical realism asserts, while anti-realism denies, that statements can have truth-conditions that are beyond all possible verification: truth conditions that are 'verification-transcendent'. It is quite possible to be an anti-realist about one type of truth-claim and not about another. One might take an anti-realist view of morals, for example, while remaining a firm metaphysical realist about ordinary physical object statements."
By the way, Gödel's theorems do not answer the question either way, even in mathematics. Many undecidable statements are only undecidable within a specific formalism, they are not "absolutely" undecidable. For example, the Gödel's sentence of Peano arithmetic is provable with some addition of set theory, so there is "evidence" for its truth. And incompleteness theorems tell us nothing about existence of absolutely undecidable statements. But even if any evidence was beyond our reach it does not mean that such unreachable truths are not settled anyway, in the eye of God, say.
Platonists, like realists in other domains, are committed to such mathematical truths beyond any evidence, as Gödel himself was. In a footnote to his incompleteness paper, he cites our finitude, not lack of truth values, as the reason for incompleteness:
"...the true reason for the incompleteness inherent in all formal systems of mathematics is that the formation of ever higher types can be continued into the transfinite... the undecidable propositions constructed here become decidable whenever appropriate higher types are added."
Gödel was also quite optimistic about overcoming our limitations in practice, in the spirit of Hilbert's motto:"We must know - we will know". To this effect, he advocated adopting new axioms of set theory that would decide the continuum problem, for example:
"A much higher degree of verification than that, however, is conceivable. There might exist axioms so abundant in their verifiable consequences, shedding so much light upon a whole field, and yielding such powerful methods for solving problems (and even solving them constructively, as far as that is possible) that, no matter whether or not they are intrinsically necessary, they would have to be accepted at least in the same sense as any well-established physical theory."
See Feferman's Does Mathematics Need New Axioms? for a discussion.
I would say they exist because of the definition of Knowledge as Justifiable True Belief.
Now obviously the answer is that it's subjective, because we're going to have to define a lot of terms. However, the claim that knowledge is "justifiable true belief" is popular enough to be a meaningful anchor in the discussion. In particular, I point out that "justifiable" and "true" are separable requirements on knowledge. You can make a true statement that Colonel Mustard killed Professor Plum with a candlestick without any justification thereof. And I think that's important to your question because empirical evidence is typically seen as a justification of statements.
If truth and justification were confounded, there would be no reason to define knowledge in such a way. Thus, while I cannot claim this is a complete proof for every person that ever lived that what they consider to be true is separable from what they believe to be provable, but it does provide what I think is strong evidence that many philosophers consider them separable.
And besides, a stopped watch is right(true?) twice a day.
Clearly there is scope for different views on this but it is largely a matter of your choice of terminology. I prefer to consider that exactly one of the two statements "Intelligent life exists on other planets" and "Intelligent life does not exist on other planets" must be true, but we have no evidence to tell us which of the two statements is true. But if you prefer to consider that neither statement is true, on the grounds that we have no evidence either way, that's fine too: it just means you are using the word "truth" in a different way than I am.
Let's use a concrete example.
There exist supermassive black holes such that the tidal forces at the event horizon aren't strong enough to tear you apart. Suppose we were to throw an enthusiastic volunteer across the event horizon along with a red button. After 30 seconds (using their inertial reference frame), they either push the button or they don't. By the law of excluded middle, one of those two possibilities must be true, but the laws of physics prohibit us from ever knowing which it is.
Alternately, if that thought experiment fails due to weird issues of time dilation causing the black hole to evaporate before the decision would be made (which it might), we can instead send our volunteer to the edge of the observable universe. Beyond that boundary, spacetime itself is expanding at faster than the speed of light relative to us, so we can never observe anything that happens there, even in principle. The volunteer is then instructed to keep moving away from us (or just wait for the expansion to do it for them) until they fall within the unobservable universe, at which point they either push the button or they don't.
(To close off loopholes related to free will and determinism, we'll also say that the button measures the spin of an electron or something.)
So yes, unprovable true statements can exist in the physical world.
Evidence is an interpretation, connected points of view, or rather connected assumptions. Its definitive nature differs little from wittgenstein's very short statement about tautologies.
At best evidence is a form, in a platonic sense, where empirical truth is a matter of memory due to the flow of time.
You can prove that the dinosaurs existed, without physical fossils, by simply reintrpretting the world as having different possibilities.
Evidence requires a truth to be evidence of, but a truth does not require evidence to be true.
We don't know everything. Thus we are missing evidence, thus there are truths we don't have the evidence for.
But even if that evidence wouldn't exist, that wouldn't make the truths any less true.
Dinosaurs existed. We know that because of the fossils, but even if those fossils had all been destroyed by geological effects, the dinosaurs would still have been there. We'd just have been unaware, and it's possible that they would have been inconsequential.
But even inconsequential truths are true.
If there was no way of proving that dinosaurs existed would it still be true that they did?
Truth is truth regardless of who knows it, or whether nobody seems to know it.
Can there be truths no evidence of which exists?
In the sense of transient facts that are of no importance, maybe (such as instantaneous the position of every particle in the universe down to femto-meter precision, progressed into the future, with certain decisions made such that the earlier state is indistinguishable from at least one imperceptibly different alternative, it could be said that such a state is no longer evident)--but be careful not to conflate such facts with things for which merely no properly attributed evidence has yet been witnessed by a party. Lack of time or adequate measurement tools and data could be one reason for the inability to trace evidence of an extant truth. So this question is really about ignorance. It is the same as "if a tree falls in the forest and nobody hears it, did it still fall?" The whole universe provides an infinite search space, and so lacking outright omniscience on the subject, an agent cannot distinguish between something for which evidence does not exist and something for which evidence exists but he has not yet found or associated it.
A key point here: A person who is more perceptive or fortunate or diligent may have found evidence for something that is dismissed by others due to their impatience or unbelief. This is an exceedingly common phenomenon, and it occurs every time someone resists learning by pretending he already knows something is untrue or does not exist when in fact he has not performed the experimentation necessary to know for himself (or else he is being dishonest about what he does know).
It is equivalent to the following fallacy:
Proving Peter: "If you do ABC, XYZ will happen."
Dubious Duke: "Oh yeah? Well I didn't do ABC, and XYZ didn't happen, so you're wrong. XYZ never happens."
Duke is wrong. He has not done ABC, meaning he has not paid the price to know that XYZ is true. He is not in a position to dispute Peter's testimony. If the promise is that doing ABC eventually leads to XYZ, then the claim is not falsifiable (it can never be proven false), however it is "true-ifiable" or verifiable, because any person who has encountered the result can confirm that the original claim it is true, and is unable to contest the claim based on personal experience and the outcome of the experiment.
I know the answer is "no" in general due to Gödel's Theory of Incompleteness
Do you mean "yes" in the sense that an unprovable but true statement exists? Gödel's Theorem applies to certain man-made formalisms of logic, not necessarily to reality or the universe at large. It essentially states that no system of logic built on a certain set of axioms can be simultaneously consistent (no answer to a particular question is both yes and no) and complete (all questions that can be asked in the logic system can also be answered in the same logic system). However, this finding may not be sufficient to distinguish itself from the possibility that one of the axioms included in the definition of such formal logic systems is itself inconsistent, and is therefore not a robust statement on the possibility of proving things in general in the real world. Fix the problem, address the limitations, and Gödel's Theorem rightly viewed may not be a universal statement about truth, but rather a statement on the limitations of a broken system of logic.
I mean this question in a more real-world sense (i.e. scientific sense).
In other words, I am talking about empirical rather than mathematical truths
Truth is truth. The label we apply to it does nothing to change its truth value. As highlighted above, an honest view of the subject admits the possibility that one of the axioms commonly accepted in such logical systems is responsible for introducing inconsistency. By the definition of reality, any inconsistency with reality is the failure of our models, and not of reality.