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If I prove something, does it have to be true?

I can structure the question more convolutedly: Does successfully proving something depend on the credulity of the audience, or the truth of the argument?

NOAD defines the word:

prove, v. demonstrate the truth or existence of (something) by evidence or argument

Other dictionaries are similar. I find it unclear, though, whether 'demonstrating truth' requires something be true.

The problem mainly arises in historical arguments. In matters we still don't know, the distinction doesn't matter—"She proved God exists" vs. "She proved God doesn't exist"—because we don't know the truth. Some things, though, were previously thought to be true that are no longer considered to be true (cf. science). When writing after an idea has been disproved, should we still say it was proven?

Example:

Copernicus proved that the earth rotated around the sun. We still believe this. However, he also argued that the planets rotated the sun in perfectly circular orbit and that the sun was the center of the universe. At the time it would have made unequivocal sense to say he proved that the sun was the center of the universe. Now, however, we no longer believe the sun is the center of the universe, or that planets rotate it in perfect circles.

Is it correct, today, after Kepler et al, to say Copernicus proved the sun was the center of the universe?

Why does this matter? If I read that someone proved something, should I trust that their finding is true?

The connotation probably derives from its Latin origin, which referred to showing that something was good, credibile:

Prove (v.):

• late 12c., pruven, proven "to try, test; evaluate; demonstrate," from Old French prover, pruver "show; convince; put to the test" (11c., Modern French prouver), from Latin probare "to make good; esteem, represent as good; make credible, show, demonstrate; test, inspect; judge by trial" (source also of Spanish probar, Italian probare), from probus "worthy, good, upright, virtuous".

(Etymonline)

The crux of your question seems to be:

I find it unclear ... whether “demonstrating truth” requires something be true. … If I read that someone proved something, should I trust that their finding is true?

If you consider the author reliable, then yes. When the author says someone proved something, the plain meaning is that the author considers someone to have confirmed something by means of a test:

She proved God doesn’t exist.

This means the author accepts the proof. But context is important. For example, there are many proofs of God’s existence or nonexistence, and also many counterproofs. In context, one can meaningfully write:

She proved God doesn’t exist. Her proof was discredited later when a flaw was found.

In this context, the author doesn’t accept the proof.

For a good confirming definition and numerous examples of prove in context, see the Oxford dictionary entry.¹

What is proven, what is true, and what is real are different things. When logic is used to prove a statement, that means that the premises and the arguments justify the conclusion. It does not mean that the premises are true, and so does not mean that the conclusion is true. It only means the conclusion is true when the premises are true. It also does not necessarily mean that a thing being argued about is real.

All blue cats are hyperintelligent. (True, but this does not mean there are any real hyperintelligent blue cats.)

My cat is real. (True, but this does not mean my cat is true.)

This is a true map of the territory. (Means the map does not lie, but does not mean the map tells you every real fact about the territory.)

As someone pointed out, your Copernicus sentence isn't an authentic use of the word prove.

Let's review how things work in science:

Someone proposes a hypothesis. Evidence accumulates. Eventually we find the hypothesis so well supported that we begin to call it a theory.

A theory is very likely true. There is a great deal of supporting evidence from a variety of sources. You may feel that the supporting evidence is so strong that you are comfortable believing the theory and treating it, in practice, as fact. Then we say that you accept the theory.

In other words, as more and more supporting evidence comes to light, the probability that the hypothesis is true increases.

I would not therefore recommend that you think in terms of black or white, true or untrue, but rather, that you think in terms of probabilities.

Math has special meanings for "prove" and "truth" and so on, but I don't think you're asking about that.

In general, prove does not necessitate truth.

The meaning of prove (and truth) depends on the context.

In the U.S., a person is innocent until proven guilty in a court of law. I might be proven guilty in a court of law, but that doesn't mean I am guilty. It simply means that the judge or jury was persuaded by the evidence that I am guilty. So in this case, prove does not necessitate truth, as most people would construe truth.

In mathematics, one often speaks of proofs, but prove in that case means deduced from some set of axioms. In this case, prove necessitates truth, depending on the truth of the axioms.

Prove doesn't mean anything in science. Every scientific result is subject to change based on new information.

I would say that no scientist has ever proven anything in the sense that their proof has necessitated truth, if by truth you mean "absolute truth about reality" (if there is such a thing as "absolute truth about reality"). They may have hypothesized or theorized that X is true, and a great deal of evidence supporting the hypothesis or theory that X is true may subsequently have emerged, but that doesn't make X true. It just makes X supported by compelling evidence, presumably more compelling evidence than the evidence for competing hypotheses or theories.

As you have pointed out in your question, in science, we've learned later based on new information that X isn't true (or isn't quite true, viz. Newtonian mechanics versus special relativity), and put forward new hypotheses or theories to account for the new information.