What are some correct results discovered with incorrect (or no) proofs? | Hacker News Reader

What are some correct results discovered with incorrect (or no) proofs?

mathoverflow.net

26 pointsColinWright9 comments

Everything. But we clean up the incorrectness, sweep any inconsistencies under the rug and then publish the corrected proof as if it sprang, fully-formed, from our mind. This, much to the bewilderment and bafflement of students thereafter! mwahahaha! http://upload.wikimedia.org/wikipedia/commons/8/84/Evillaugh...

If your viewpoint is the history of mathematical proof, then the answer might be "Everything up to the early Greeks." Here's a nice link: "The History and Concept of. Mathematical Proof" by Steven G. Krantz http://www.math.wustl.edu/~sk/eolss.pdf

But if you want to really understand then take a look at the book

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being by G. Lakoff & R. Núñez. http://www.amazon.com/Where-Mathematics-Comes-Embodied-Bring...

The introduction and first four chapters [PDF] are available at

http://www.cogsci.ucsd.edu/~nunez/web/INTR-04.PDF

What about basic arithmetic? (e.g. 1 + 1 = 2) This was obviously being used correctly for thousands of years before Whitehead and Russell came along and actually constructed a formal proof that starting with much more fundamental axioms and mechanical rules of inference you could indeed show that 1 + 1 = 2.

http://en.wikipedia.org/wiki/Principia_Mathematica

Of course, this formality hasn't added anything to the practical application of arithmetic but thinking about structures in this way did eventually lead to some pretty interesting results in mathematical logic.

That proof is "easy" though: http://us.metamath.org/mpegif/pm54.43.html

It's a lot harder when you try to prove that 2+2=4: http://us.metamath.org/mpegif/mmset.html#trivia

That's right, there are 25,933 steps in that one if you trace 2+2=4 all the way back to axioms.

It's interesting to realize: that's why something like quantum mechanics can't be approached by thinking purely in axioms. We have to push forward first; it's only after we've correctly approximated nature (via some informal model) that we can go back and try to prove our assumptions.

One obvious exception to this is Einstein coming up with relativity before it was ever observed in nature and approximated by people. But if you think about it, Einstein wasn't proving aximos; he was presenting a theoretical model (a guess) which turned out to precisely correspond with nature. It was still a guess.

So my point is... "results" are subjective, and don't require proofs. Proofs are very useful for checking your theories, but probably not for discovering new phenomena.

But you're talking about physics, not math. The original question was asked in a math forum, which is where it has the example about sum(1/n^2), for one.

Physics is different. Yes, you can assume axioms and see what follows in the context of a physical situation, but that's a different game.

Speculation about physics is interesting in this case, but really, the question is about math.

Oh, you're correct, sorry. I didn't read the link, just the title.

Isn't that a different definition of 2 and 4 than he was using?

They're taking 2 and 4 to be complex numbers (2 + 0i), where as he was taking 2 to be a cardinal number.

A formal proof in the sense of Hilbert's formalist program or one produced by an automated theorem prover is very different from the proofs presented in journals. The latter are full of "it is easy to see that [...]", after which they proceed to state something which is not obvious to 99% of the population--nor to any computer, for that matter.

I submit that the Principia did not change the status of 1+1=2 at all, but only used it to illustrate the correctness of its formalization of arithmetic. It was more like a soundness check: we _know_ that 1+1=2, so if it can be proved in our system it's a good thing.

Quantized electron energy levels. Niels Bohr pretty much nailed the "what" but completely missed the "why."

https://secure.wikimedia.org/wikipedia/en/wiki/Bohr_model

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