Mathematician Claims Proof of Connection between Prime Numbers

[Angrboda]

2. My favorite economist gives a good take on this news here. I thought the discussion about 'set-theoretic foundations' in his papers was something spurious, but it's actually a big deal (whatever it is...).

His notion that axiomatic systems are not fundamental, it's what's underneath them, echoes a question I've had (and use as a frequent example). In epistemology, there are various "theories of truth". What does it mean, what is it, what are its rules. One of the modern theories of truth and logic is that there are truth bearing entities (propositions, statements, sentences...) that are both true and false at the same time. These entities are called dialetheas, and the theories of truth based on them are called Dialetheism (most of which intersect at the liar's paradox and the strengthened liar's paradox; see also, paraconsistent logic). Australian philosopher Graham Priest is a major advocate of Dialetheism, and in one example he shows how, by reconfiguring the rules of classical logic (which result in a breakdown known as logical explosion under dialetheism), he can create a system in which dialetheas occur, but the sense of the inferences is preserved (no explosion). My question has always been, since it seems that the rules of logic in some sense define the nature of truth (in whole or in part), what is that 'something' that is being preserved when the rules of logic are redrawn?