Mathematics self-proves its own Consistency (contra Gödel et. al.)

I stopped reading after the first paragraph, so I don't know what the rest of it says. You're right, he does agree with me. But he presents it with a shroud of mystery. I was turned off by his attempt at demonstrating a paradox (a la alarming title "contra Gödel et. al.) and using word play to trick the reader into believing there is one. But there is no paradox.

Why is LtU even bringing up this abstract? Are they just trying to further embarrass him?

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.

From my naive point of view I'm confused about how this argument can possibly work. Perhaps someone can explain it?

1. It seems to me that if you assume that mathematics is inconsistent, you shouldn't then consider an inconsistency in the argument sufficient grounds to throw away the assumption. I suppose that means that I don't accept proof by contradiction in the case that the assumption is the inconsistency of mathematics.

2. Sanity Check: We can create known inconsistent systems, could the same logic prove that such inconsistent systems are consistent too? It seems to me that it could, am I wrong?

3. Supposing that mathematics did prove its own consistency, the fact that inconsistent systems can also prove their own consistency means that you can't take a proof of consistency to be strong evidence that the system is consistent.

4. Gödel didn't say that mathematics couldn't prove its own consistency, he just said that if it could it was unsound, which is a bit of a concern too.

I know Carl a little from the late 90s. I figured there was a 50% chance he would come up with something historically important, and a 60% chance he was crazy. (The two options are not quite mutually exclusive.)

Indeed, most examples illustrating the Incompleteness Theorem involve self-reference. But, it seems hard to prove anything interesting in a system that precludes self-reference.

BTW, the Y Combinator is a way of adding self-reference to a non-self-referential formalism. And we're debating his argument here. Deep.