Carl Hewitt isn't just some random crank, he is the author of PLANNER, a breakthrough programming language in the early AI days, that gave rise to Prolog later on. This is a pretty interesting paper and it doesn't claim any big breakthrough in the sense of invalidating any great mathematical result. I can try to explain what it is about, at least the beginning of the quoted fragment:

In the first paragraph, he shows that if you allow proofs by contradiction, you can use them for every theory claimed to be inconsistent to show it indeed is consistent. This is so, because to prove by contradiction means to infer "not A" if from "A" follows "B" and "not B". So, if from "mathematics is inconsistent" follows "theorem A is true" and "theorem A is false", it follows that "mathematics is consistent". He claims that this shows consistency isn't rigorously proved in classical mathematics but is just assumed, since classical mathematics allows proofs by contradiction.

He then shows that from this consistency proof paradoxically follows the conclusion that mathematics is inconsistent, because this proof contradicts Goedel theorem "if mathematics is consistent, then it cannot infer its own consistency". He resolves the contradiction of Goedels proof by his proof by restricting the logical system in which they are done to exclude self-referential sentences.

Then, and I think this is really the gist of this work of his, he goes on to describe an approach that tries to somehow reduce the bad impact of including inconsistencies in the logical system, instead of trying to completely eliminate inconsistencies, as it was the goal traditionally.