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Comment by rthomas6 | Hacker News Reader

Honestly though, as a layman in math, a lot of the most important math discoveries seem like this to me. For instance, Godel's incompleteness theorem: You mean you can't make a system prove itself within its own bounds? You don't say. Or the axiom of choice: "the product of a collection of non-empty sets is non-empty." What a surprise.

I've since accepted they're probably really important and non-obvious from a more rigorous and formal standpoint. Maybe this discovery is the same way.

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Your statement of the incompleteness theorem isn't right, and the general axiom of choice isn't accepted by constructivist mathematicians. So I think your last statement is true.

Some of the greatest mathematicians of the era were working toward what Godel's incompleteness theorems proved was impossible. That alone should make you distrust your "intuition" on the matter.

In fairness a lot of mathematics isn't accepted by constructivist mathematicians god bless their little socks.

I kind of feel that Brouwer showed mathematicians the way forward and none of them took it. Unfortunately I don't have the formal chops for the discipline so I can only look on from the sidelines and cheer for my team.

There are two issues here, I think. One, as a layman in math you end up reading about / paying attention to the discoveries you can parse and understand, thus introducing a bias by ignoring the results that require years of math training to understand. Second, the specific examples you listed are not entirely accurate or even comparable (one is a theorem, one an axiom). There is a risk of misunderstanding the results when they are presented in English, oversimplified and converted to common terms. This oversimplification may lose important details (not just formal stuff) and may end up with a common-sense sounding phrase that trivializes the result and doesn't really do it justice.

> You mean you can't make a system prove itself within its own bounds?

No, that's not what the incompleteness theorem says. "proving a system" is not even meaningful.

It says that in all but the most primitively simple systems, you can make statements that are impossible to prove right or wrong within the system.

I see your point, though thte important thing about Godel's incompleteness theorem is that it consists of a proof rather than being a emre conjecture. I do admire this paper insofar as it formally describes the problem and thus gives us the tools to use tools from domain A in domain B and vice-versa, but the actual discovery part fell a bit flat for me. Perhaps I let my expectations get raised too much by the information-poor press release.

> Or the axiom of choice: "the product of a collection of non-empty sets is non-empty." What a surprise.

Right. Since it's obvious, the Banach-Tarski "Paradox" must be a trivial corollary and intuitively obvious, correct?

http://mathworld.wolfram.com/Banach-TarskiParadox.html

Also: Any two objects in three-dimensional space which do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other. Perfectly obvious, right? This follows from Banach-Tarski, and therefore the Axiom of Choice.

A fuller development of the Axiom of Choice: http://plato.stanford.edu/entries/axiom-choice/

Some implications of the Axiom of Choice, and of not taking it: http://mathoverflow.net/questions/20882/most-unintuitive-app...

My point is that mathematics is all about taking obvious statements, such as the definition of sets and set cardinality, and using them to prove non-obvious results, such as the fact there are as many rationals as there are integers, and, in fact, there are as many rationals strictly between 0 and 1 as there are integers total.

The very best mathematical proofs are magicians, taking ordinary household items, waving and knocking a few times, and pulling a rabbit out of their cane in a way that is eminently fair but that you still didn't see coming.

So, yes, an axiom by itself is perfectly obvious. It's just a prop. The wonder is in how the props get used.

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