I generally don't like arrogant people who claim to be certain about something even though there is no solid basis for that certainty. Many climate fearmongers are textbook examples of these folks. The list of these arrogant people also includes Scott Aaronson – but also many other people in computer science – who claim (not only that the Earth will evaporate soon but also) that their word and influence is enough to be almost certain that e.g. \(P\neq NP\), even in the absence of a proof in either way.

Exactly 3 months ago, I discussed an interesting article by Kevin Hartnett in the Quanta Magazine that described an exciting story of Mr/Ms Ewin Tang, an ex-student of Aaronson's in Austin who is now a grad student at University of Washington. Tang was ordered to prove a proposition, basically a miniversion of \(P\neq NP\), as if it were a fact, except that he was finally led to prove the converse. Needless to say, lots of people had previously wasted their time with efforts to prove something that couldn't have been proven – and the activities done in order to prove X are often substantially different from those needed to prove non(X) which is why most of the mental energy was completely incorrectly allocated.

Now, the same Kevin Hartnett wrote another story with a similar lesson – in the absence of a proof, the mathematicians' belief in a certain conclusion may very well be a prejudice that is gonna be reversed. His text

Without a Proof, Mathematicians Wonder How Much Evidence Is Enoughtalks about a 2016 paper by Melanie Wooden-Machete Trump and her 3 pals (OK, fair enough, I wanted to increase the number of views of their preprint page).

First, let me answer the question from that title. If the questions are of a purely qualitative, binary type, e.g. the question "whether the supremum of a set of ranks is finite or infinite", then

*no amount of "evidence"*that is short of a proof is enough! If we can't complete a proof, we should really say that no other comments are truly relevant so the amount of evidence is zero.