May 3, 2021

FAITH SUFFICES:

The foundations of mathematics are unproven: Philosopher and logician Kurt Gödel upended our understanding of mathematics and truth. (JONNY THOMSON, 30 April, 2021, Rightly Understood)

Imagine the whole of mathematics as a huge sack, and inside are all the possible things math can do. It's a mighty big sack, indeed. What Gödel proved is that, first, there exists in this sack a set of things which cannot be proven or disproven, such as axioms. Second, there is no possible way to prove these axioms from within that sack. It's impossible for math, on its own, to prove its own axioms.

Essentially, it's a problem of self-reference. It's an issue seen, too, in Russell's paradox about sets. More famously, the liar paradox imagines a sentence like, "This sentence is false." When you examine it closely, it creates a logical circularity. If the sentence is true, then it's false; but then if it's false, it's true. It's enough to make a robot's brain explode.

Gödel applied a similar logic to the whole system of mathematics. He took the sentence, "This statement is unproven," and converted it into a number statement about numbers (with a code system known as "Gödel numbering"). He discovered that this proposition cannot be proven within that system.

Going even further than this, Gödel concluded that in every system that's rich enough to allow for arithmetic, there will be a proposition within it that cannot be proven by it's own tools. We need some kind of "meta language" to prove the rules by which a system operates. It's a bit like how we can't see our own eyes or draw around the hand that's holding the pencil.

How postmodernists weaponized Gödel
Gödel has been misrepresented, even in his lifetime. For instance, certain postmodernist philosophers used him to say, "There is no truth! Even math is groundless!" They wanted to show how everything was meaningless, and truth amounted only to opinion.

But this isn't the point. Gödel only showed that truth does not always need to be proven.

Posted by at May 3, 2021 6:48 PM

  

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