You can't prove something by example, you can only show that the pattern holds for the examples that you've checked.

Biology isn't mathematics.

Yes, in math you can't prove things by example.

But in biology that's the only way you form knowledge.

Biology doesn't have axioms. In biology, proof is by induction. However philosophically unsatisfying that may be.

Alternatively, you don’t “prove” things in biology, you just provide evidence for them.

Proof implies a level of certainty that doesn’t exist in biology.

In fact you don't prove anything in science. You merely fail to disprove something.

in the “natural sciences”

Is proving something internal to a system of ad-hoc rules that cannot (to the same standard of proof) be proven to have any correspondance with the natural world really proving something? Sincere question.

I think so, yes. You start with axioms, and the proof follows from those.

You can argue whether that’s interesting or not, but IMO it’s as much a proof as anything can be, and IMO it’s very interesting.

Do we have innatural ones?

The other category of science is called formal science.

Is computer science a science?

Of course not. It's a branch of mathematics. Always has been.

„A branch“ - so we are back to trees then?

Mathematics and computer science are both in the category of formal science.

Yes, it’s a formal science.

On the other hand, "trees" are polyphyletic, among other aspects of their broad diversity. So even if you can generalize across a species, generalizing across all trees from a handful of examples is likely to lead you astray.

On the gripping hand, trees are polyphyletic because the constraints of physics and biology push in a certain direction regardless of starting point; you can imagine there is some underlying mathematical rule to this abstract "tree" even if it isn't realized in all examples.

You can imagine, sure, but we were talking about proof.

I think that's correct for some assertions.

When a statement is "this is always true", then a counterexample is all it takes to disprove it.

On the other hand, if the statement is "this is not always true" then an example of it happening is as you said - the example does not counter anything.

Yea I misspoke. (typed)