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Comment by avip | Hacker News Reader

I think you’re confusing proving with understanding. The two are almost orthogonal.

Disagree. If you truly understand something, you must have prooven it to yourself, everything is just a mental help ("donkey bridge" in german) that allows you to remember the statement better - weather it's right or not, you can only know once you've prooven it; and before you did that, it can often happen that your intuition on what's right is actually wrong.

That makes no sense to me. Or maybe your definition of "understand" is completely unrelated to mine.

Under your thesis one cannot "understand" anything physical (as in physics, chemistry, etc) as the underlying cannot be "proven", at all, by definition.

I think a lot of people can understand simple statements in math such as Fermat’s Last Theorem or the Collatz Conjecture but proving them is an entirely different matter. While it may be the case that proving some statements is sufficient to understanding them, I would say it’s never necessary.

I think you may be confounding different kinds of understanding. There's a big difference between understanding what a statement is saying, versus understanding why a statement must hold true. And understanding why a statement holds doesn't necessarily have to be a formal proof, it just means that you have convinced yourself that some concepts that you are familiar with behave in a way to support the statement. It can actually happen that following a correct formal proof doesn't automatically help you understand why something is true. And vice-versa, it may happen that when you try to formalize your understanding, you discover that you missed some crucial details.

But usually true understanding and formal proofs go hand in hand.

P.S: Though as a disclaimer, I don't know maths beyond the undergraduate level, and I'm sure there must be very complicated concepts that mathematicians don't understand, but can reason formally about, as well as concepts that mathematicians feel that they intuitively understand but can't quite prove.

Additionaly, there's von Neuman saying "Young man, in mathematics you don't understand things. You just get used to them"

I think you're confusing understanding with intuition. Which is fair; I'd say understanding builds to intuition as it's accessed more. The line is blurry at points.

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