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Comment by Konohamaru | Hacker News Reader

Tbh, the fact that mathematical logic cannot prove that e + π or π^π^π^π are irrational numbers is a huge failure of formal mathematical logic.

David Hume said that the purpose of logic is to record patterns of thinking we use, and every mathematician has a subconscious pattern of thinking that let's them look at the expressions e + π and π^π^π^π and conclude "there's no way these could be rational". So the fact that we failed to write down these patterns of thought means our system of formal logic is extremely incomplete.

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It's not a failure of mathematics. It's a failure of mathematicians. There's no reason to think it's insoluble. They just haven't.

Mathematics is incomplete, but there's no reason to think that these specific problems are insoluble. A proof that they were insoluble would be remarkable, but extremely unlikely.

It is also unlikely that e + π is rational, and nobody expects it to be. Nor would it matter if it it were. The proof itself is more interesting than the fact of the matter, one way or the other. The mechanism of the proof is a tool for mathematicians, to go on and prove other things. It's a human problem, not one about the foundations of logic.

You misinterpreted what I wrote. I did not say it was a failure of mathematics. I said it was a failure of mathematical logic. Mathematics is a very successful project. Mathematical logic is a new project (invented in the 19th century by Frege) and was killed by Gödel in 1931.

I don't know where you get this notion that it is somehow "wrong" or "a failure."

It's surprising, sure, but more for people who aren't mathematicians

If the goal of formal mathematical logic is to capture the subconscious patterns of thinking humans use, then by that standard, it is a colossal failure.

I think that's a poor characterization of the purpose of mathematical logic. The purpose of mathematical logic is not to capture the human mind, it's to overcome the limitations of the human mind.

Math is like sailing a ship during the age of sail. In the middle of the ocean, there is no point of reference. You have a vague sense of direction, but it's easy to get lost. You need some tools to navigate. Math is the ocean, your mind is the ship and mathematical logic is the navigation instruments.

Did it succeed in that regard? Gödel said that first-order logic powerful enough to capture Peano Arithmetic does not have a sound proof theory (incompleteness = not onto = unsound), meaning human imagination is still the limiting reactant.

Yes, it did succeed in that regard.

To use your example, Godel would not have been able to explore those ideas without using mathematical logic.

It just reflects the fact that how hard to construct a consistent and provable world. You can just listen to politicians if you don’t like the above.

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