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What often happens is the first version of a proof is long and arduous, but the new ideas that it spawns slowly circle back so that someone else can come up wit

• assafs 13y ago
  In other cases, a very complex problem can be stated very simply, especially to a lay person. A good example is the Jordan curve theorem[1], which at first sight seems like a trivial problem but in effect requires a good deal of topology and analysis to even understand why it is so complicated in the first place. [1] https://en.wikipedia.org/wiki/Jordan%27s_theorem https://en.wikipedia.org/wiki/Jordan%27s_theorem
  • ColinWright 13y ago
    There is an obvious enhancement of the Jordan Curve Theorem, which is to say that the interior is effectively a disk, and the exterior is effectively a plane with a hole. There is then an obvious 3D version, with a (topological) sphere separating space into a topological ball[0] and a topological R^3 with a hole. This enhanced version, although natural and obvious, is false. I use this as an example of why "obvious" things aren't always true, need proving, and understanding their proofs can lead to useful insights. [0] Edit: corrected "sphere" to "ball"
    • tehwalrus 13y ago
      Ooh, OK, you've tickled my interest. Can you provide an explanation (or a link to something similar) with the disproof of the 3D version?
      • ColinWright 13y ago
        In three dimensions there is an embedding of S^2 (which is the surface of a sphere[0]) into R^3 such that the exterior is not homeomorphic to R^3 with B^3 missing. http://en.wikipedia.org/wiki/Alexander_horned_sphere http://en.wikipedia.org/wiki/Alexander_horned_sphere (search for "jordan") In essence, the "outside" gets very "tangled" and can't be smoothly converted into a "proper" exterior. [0] S^2 = { (x,y,z) : , x, y, z, in R with x^2+y^2+z^2 = 1 }
        • tehwalrus 13y ago
          Hmm, so because part of the surface is a fractal, it can't be simply connected? I'd be interested to learn whether a linear version (straight pipes instead of curved/broken torus ones) also has the same topological properties. It is clearly fractal, and clearly also homotopy identical to S^2, but obviously the two ends are no longer interlocking, and I wonder if this is as crucial to the result as both 'ends' being fractal clearly is. In any case, cool! thanks :)
          • ColinWright 13y ago
            The surface is simply connected, it's the outside that ends up not being simply connected. This works equally well with piece-wise linear embeddings. And the current version of the Alexander Horned Sphere is not actually interlocking - the embedding is contractable back to S^2. It takes a while to get your head around what this really is.
            • tehwalrus 13y ago
              OK, "proximate" rather than interlocking. I did understand the geometry of it, I just chose the wrong word. Thanks for your explanations! :)
          • deleted 13y ago
            [deleted]
    • gohrt 13y ago
      (I think) this is a case were lay intuition assumes objects are piece-wise differentiable, but the formal math definitions don't. That is, most people think of physical objects, where there is a minimum "planck length", and no fractal-ish structures with infinitely fine-scale structure. It is still abstractly interesting that even 2-D fractals satisfy the Jordan Curve therorem, whie 3-D faractals do not satisfy the 3-D version of the conjecture.
  • lmm 13y ago
    I think that's largely because mathematics abstracts more than is obvious. My favourite example is Hilbert's basis theorem, which looks like it's not even a question until you realise how general the notion of "space" that it applies to.
    • gohrt 13y ago
      Right, same reason that the Banach-Tarski paradox is intuitively baffling. It relies on fractal-structures that require uncountably-infinite fine detail.