
What are some correct results discovered with incorrect (or no) proofs? - ColinWright
http://mathoverflow.net/questions/27749/what-are-some-correct-results-discovered-with-incorrect-or-no-proofs
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giardini
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...](http://upload.wikimedia.org/wikipedia/commons/8/84/Evillaugh.ogg)

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...](http://www.amazon.com/Where-Mathematics-Comes-Embodied-
Brings/dp/0465037712/ref=sr_1_1?s=books&ie=UTF8&qid=1313945868&sr=1-1)

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

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

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arethuza
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.

~~~
Natsu
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.

~~~
palish
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.

~~~
ColinWright
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.

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

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glimcat
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>

