Trouble arises in any logical system when we start to make statements on the system itself. As long as there's a strict separation between the target of your lo

Trouble arises in any logical system when we start to make statements on the system itself. As long as there's a strict separation between the target of your logic and the logic itself, you can make it all work out, but recursion is what kills you. To avoid this, many thinkers have come up with a sort of infinite tower of sorts. For example, in ZFC set theory, sets are ordered in cumulative ranks. In theorem proving languages like Coq, Agda, etc, there are a hierarchy of universes. Even Aristotle's four causes can be seen to be a small subset of the infinite hierarchy of cause. It's just the nature of logic. Similarly, there is a hierarchy of equality at increasing abstraction levels. There's naive equality which is that two things are literally the same. Then there's equality as to the 'class' of objects, and we can even go beyond that talking about equality of equalities (another word for analogies). Most stop there, but you can go on and on and on. I'm currently interested in homotopy type theory systematizes this notion of varying 'levels' of equality, but it too has issues. It's an open question. The article does sensationalize a bit because math is just a method of discourse. Equality means different things to different fields, and nothing about this kind of equality means that previous results are wrong.