If anyone wants to express “lesser” degrees of proof then they shlould qualify them, be it “practically true”, “true by experience” or “probabilistically provable”.

You expect that authors explain the word “provable” if they use it in a non-standard way. But my observation is that they do not, and they use it without thinking because it has become part of jargon in theoretical computer science. It is a buzzword which means nothing when used this way. And this is bad.

]]>But what are all these other uses of “provably” that you have in mind?

I can imagine many reasons to label a statement as “provable” instead of as “true”. For example, simulated annealing provably converges towards the global optimum. Here the qualification “provable” allows to separate a well defined theoretical truth from the ambiguous practical consequences of the truth. So “provable” is indeed used as an euphemism here, but who is to blame for the ambiguous practical consequences of theoretical truths (for black box optimization)?

A proof can also be part of the process to convince somebody of a certain fact. But the proof might only be convincing to the specific person to which it was delivered. So Jesus delivered proof that convinced doubting Thomas of the resurrection, but a reader of the bible cannot get the same degree of certainty from that proof as did Thomas (for obvious reasons). This sort of proof sometimes allows “the verifier” (i.e. doubting Thomas) to make the remaining doubt as small as desired (say exp(-42) to fix some order of magnitude) by repeated queries, but note that even if he could reduce the remaining doubt to 0, the fact might still not be true.

Under certain assumptions, this type of proof with arbitrary small remaining doubt can even apply to mathematical statements. Take a sufficient non-trivial identity for example (or a prime number if you prefer), which has been checked by “the verifier” by sufficiently many random tests to ensure that the remaining doubt is below exp(-421) (or maybe just below 2^-29, it is subjective to a certain degree). Only “the verifier” can know that his random tests were sufficiently random, and even he cannot be absolutely sure. Sometimes you don’t even require true randomness for this type of scenario, take for example Freeman Dyson conjecture that the reverse of the decimal representation of a power of two is never the decimal representation of a power of five.

But I didn’t really care about the actual possible meanings of “provable” too much, I just don’t want to impose arbitrary limits on the language used in mathematical papers. But I expect that authors make it clear what they mean when they use “provable” outside of a well established context, precisely because I can imagine so many possible uses of that word.

]]>