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Obvious, but Difficult

This is a fun conversation on MathOverlflow.net about famous examples of theorems in mathematics that are “obvious” but very difficult to prove.


For example, the Jordan Curve Theorem essentially states that any closed curve in the plane divides the plane into an “inside” and an “outside”.  Obvious, right?  But very difficult to prove.

The Isoperimetric Theorem is another good example.  This theorem basically says that the most efficient way to surround area in the plane is with a circle.  Again, easier to believe than to prove.

And one of the responders notes that, after taking several hundred pages in their Principia Mathematica to prove that 1 + 1 = 2, Russell and Whitehead note that the proposition “is occasionally useful”.

“Obvious” is one of the most dangerous words in mathematics!

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  1. Roy
    September 12, 2011 at 10:00 am

    How about the proof of negative x negative = positive? We take that for granted. Is the proof for that difficult? Enlighten me 🙂

  2. Peter
    September 12, 2011 at 4:20 pm

    The reason is that (-ab) is the additive inverse of (-a)(-b) in any ring — thanks to distributivity.

    First, (-a) b = -(ab), because (-a) b + ab = (-a + a) b = 0 x b = 0.

    With this, (-a) (-b) + -(ab) = (-a)(-b) + (-a)b = (-a) (-b+b) = (-a) x 0 = 0

    Since the inverse (and inverse-inverse) is unique, (-a)(-b) = ab.

  3. September 12, 2011 at 9:52 pm

    As with many of ideas discussed in the MathOverflow thread, the proof that (-a)(-b) = ab isn’t especially enlightening, except to the person who cares deeply about the finer details of algebraic and logical systems.

    You’re definitely right that we take it for granted, and I’d wager that even most of the mathematically-minded couldn’t give the correct, technical answer.

  4. September 13, 2011 at 10:49 am

    So are you saying I’m not “mathematically-minded?” — I’m offended 😉

  5. September 13, 2011 at 11:30 am

    No, not at all! Sorry, Peter, I was responding to Roy’s comment. I think you’re right on the money–the result is just a consequence of the axioms of fields (or rings, as you say).

    I think most people look for something more intuitive, but I’m not sure it’s there.

  6. September 13, 2011 at 9:10 pm

    @MrHonner, I was, of course, kidding. Intuition is a tricky beast and I can’t blame people not to have an intuition. As a mathematician, I’d say that the intuition comes naturally when you simply encounter this fact in many different rings — be it integers or reals, polynomials or Boolean algebras such as the power set algebra. That helps build intuition.

    Another way is to take the first part of my “argument”, and do something like (-a)(-b) = -(-(ab)) — then appeal to something like “double u-turn = do nothing”. But who knows.

    In any case, I find it ridiculous whenever I meet a researcher who’s shocked that people doubt such “simple” observations (and I have met too many). I never meet a professional singer who’s shocked that some people can’t sing.

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