Evidence Versus Proof

"Proof" is a much stronger claim than "Evidence". Proof is similar to mathematical proofs, of which it is either right or wrong. Evidence can be weak or gray. For example, AuthoritarianEvidence and anecdotes are useful evidence, but they should not normally be considered strong evidence. Some people say "proof" when they mean "evidence", probably because it is a shorter word (and easier to spell :-).

There isn't anything about 'proof' that has ever required it to be mathematical (or 'deductive'). 'Proof' can be inductive or abductive and remain within the boundaries of widely accepted and well proven logical reasoning. Evidence is proof towards a particular conclusion when (a) no relevant evidence is being ignored (no cherry-picking), (b) the conclusion can reasonably be drawn from the evidence, and (c) there aren't contradictory alternative conclusions that can reasonably be drawn from the evidence. Merely arguing that "evidence" is not "proof" is a fallacy; you need to also show at least one of: cherry-picking (by pointing out unreasonably ignored evidence), a reasonable alternative conclusion from the same evidence, or that the conclusion drawn from the evidence is not reasonable (i.e. it is contradictory to the evidence, or relies on too many assumptions).

What means 'reasonable' is a more open question. EvidenceEras names a few sorts of 'reasonable' that have been accepted throughout history. Scientifically, 'reasonable' means that it makes empirical predictions (qualitative, quantitative, statistical, over percept, anything - so long as it is verifiable by direct or indirect means) that are later verified. Mathematically, 'reasonable' means the argument doesn't depend upon anything outside the set of known axioms and is logically consistent. Religiously, 'reasonable' means 'consistent with the teachings, traditions, or philosophy of external authority <X>'. I, personally, consider 'reasonable' arguments that minimally (a) depend on few unverifiable assumptions (OccamsRazor), (b) are not inconsistent with observations and percept (and experience in general), (c) are not inconsistent with commonly accepted axioms of mathematics and logic (e.g. a statement of fact cannot simultaneously be true and false at the same time; a statement must not depend on equivocation or other common fallacies; etc.), and (d) allow for direct or indirect verification. Anyone who violates one of these conditions when presenting a serious argument I consider unreasonable, including people who have that 'EverythingIsRelative' mindset (which violates all four).

As a note, 'indirect verification' means using the conclusion as an axiom allows you to make other reasonable conclusions. E.g. you can't ever directly confirm that something actually is a dog just because you've observed that it looks like a dog, barks like a dog, and smells like a dog. There is no state of 'dogness' you can directly examine. But you can reasonably conclude that it is a dog, and use this to predict that it probably walks like a dog and eats like a dog, which can later be verified or shot down by future percept.

Anyhow, anecdotes and experience are fine evidence towards a point unless other people also have contradictory anecdotes - this universe has repeatedly proven that it is highly predictable and that past experiences and patterns are very useful predictors of our future experiences and patterns. Use of anecdotes and experience can even be scientific - empirical prediction over future observation (quantitative, qualitative, or otherwise). E.g. if I predict that your anecdote will, in the end, have certain properties (e.g. "you'll characterize your 'goto' based solution as 'an unmaintainable mess' after the project grows in size and number of developers") based on previous anecdotes and experience, then I've made a prediction that I can later observe and either verify or shoot down. This is what hypothesis is all about, and the theoretical science would be coming up with a theory or model as to why my predictions are true that can be applied (independently, as an indirectly verifiable axiom) to future observations in order to make more predictions. That said, it is true that basing a conclusion off of just a couple anecdotes or experiences is probably a case of premature generalization. Anecdotes, like all inductive evidence forms, are one of those cases where a bunch of little pieces of evidence add up to one very valuable bag of evidence. Concluding that the gestalt is of little value just because each individual piece has little value on its own is just as unreasonable as premature generalization.