Is a proof also "evidence"?

Hypothesis: $n^2-n+41$ is prime, for all natural $n$.

Evidence: True for $n=1, 2, 3,\ldots, 40$.

That seems persuasive, but for $n=41$ the hypothesis is false.

In general, science typically uses inductive reasoning, i.e. noticing a pattern and claiming it continues forever. An improvement is the scientific method, i.e. noticing a pattern, making a prediction, and checking if the prediction is true the next time. Unfortunately, neither of these is considered a mathematical proof.

In mathematics, "evidence" is weaker than "proof".

Mathematicians use the words "proof" and "evidence" differently from the sciences. When we speak of a "proof" that something is true, we mean an irrefutable line of logical implications. When we speak of "evidence" that something is true, we typically mean indications (like many worked out examples or related theorems).

For example, as far as I know, no one has a "proof" that the Riemann Hypothesis is true. However, there is a ton of evidence that it probably is true (examples: there are infinitely many zeros on the critical line, all of the zeros must lie at least "close" to it, many of the theorems that it implies has been proven true). Again even with all of this evidence, this still does not give us "proof" that it is true.

So I guess in a kind of perverse way you could think of a "proof" as "evidence". But "evidence" is not enough to be a "proof".

Yes, science sees these terms as equals. Mathematics holds us to a higher standard. Unlike scientific results, mathematics results are true (not approximately sort of true for today).

Proposition: Every real number is less than $1,000,000$.

Evidence: The proposition seems to hold for all numbers in my test set $\{1,2,3,\ldots,100\}$.