6 ms·

I'm not a mathematician, but I find it unsatisfying that the solution to such a simply stated theorem requires 100s of pages of mathematics. Would have been co

by hmexx 13y ago

I'm not a mathematician, but I find it unsatisfying that the solution to such a simply stated theorem requires 100s of pages of mathematics. Would have been cooler if the proof was extremely simple to write out, but required outside-the-box thought process that evaded thousands of humans for centuries. I hope the same thing does not happen to physics. Discoveries that lead to simple equations like e=mc^2 are so cool!

dbaupp 13y ago
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 with a much neater proof, it just takes a while. (Proving theorems is hard work!)
- 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! :)
  - 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.
ColinWright 13y ago
> ... I find it unsatisfying that the solution to such a > simply stated theorem requires 100s of pages of mathematics. The mathematics involved is getting easier and easier to understand as more and more work is being put into it. Although Wiles originally only proved the underlying result for semi-stable elliptic curves (which was enough for FLT), the full modularity theorem (formerly called the Taniyama–Shimura–Weil conjecture) is now proven, and the field continues to grow. I have little doubt that in 100 years or so this will be within the scope of later stage undergraduate work, just as many results in number theory (and other areas) become more and more accessible as we come to understand them properly. Wiles' proof is still comparatively new, it's still messy, it needs to be enhanced, clarified, expanded, and then "re-factored" to find the cleanest path to the result. The underlying arc of the proof is relatively easy to understand, it's the details that need the long, tedious, and careful arguments that take 100s of pages. One day it will be an obvious consequence of the full Modularity Theorem, discussed in honors projects of undergraduates. > Would have been cooler if the proof was extremely simple to > write out, but required outside-the-box thought process that > evaded thousands of humans for centuries. In some sense, perhaps. That's what happened with the proof that primality testing is in P. Three undergrads produced an elementary argument that had eluded people for centuries. And yet it's provided no really new insights into anything, and seems to be effectively a dead end. In that sense it's much more exciting to have genuinely new work that wasn't just a simple argument overlooked for centuries.
- raverbashing 13y ago
  Yeas, I find it difficult to believe that it'll get easier, but then again, calculating the sine for an arbitrary number was once impossible, later very difficult (and only in the realm of 'pros') and today it's a couple of button presses away.
  - ColinWright 13y ago
    Actually, it's pretty easy to get a few places of accuracy purely by hand using the first couple of terms of the right series, and a few double (and triple) angle formulas.
fractalsea 13y ago
What I find interesting is that these proofs aren't one liners. To me, I intuitively presume that mathematical proofs should be one liners because they are explaining fundamental truths of nature; this universe and beyond. Having said that, it's quite possible that a more "elegant" proof will be discovered in the future. Maybe it requires further thinking outside the box -- it will more likely require new mathematics. If you want really non-elegant proofs, take a look at computer-assisted proofs (e.g. proof of the four color theorem). These make the subject of this thread seem extremely elegant!
- anonymous 13y ago
  It can be a one-liner, but you'll need more language. That is, you'll need terms that encode mathematical truths that currently can only be expressed as several pages. Think of this - I can use the number two and refer to it with just a single character - 2. However, properly stating all its properties and what it is starting from just the axioms of set theory, I'll need quite a lot of pages.
  - VLM 13y ago
    Maybe a good way to resonate the situation with HN would be you wanna store the number 2. Well, if you only know and are permitted to use classic IEEE 754 floating point then this is going to be a really long story. But once someone "invents" binary integers and its widely accepted that you're "allowed" to assume everyone understands them, then to store the number 2, you just squirt out a 8-bit binary integer word 00000010 and call it good. And that's how "lets store the number 2" goes from 1000 pages and ten hours of lecture to explain to CS grads, to some 5 minute Kahn academy video that any goof off the street can more or less understand. There is no proof that any simpler explanation exists. Its quite possible no one will ever teach the proof of FLT to grade school kids. But if it ever happens it'll be like the analogy above.
- dagw 13y ago
  The proof is a one liner, if you take enough of the background theorems as given. The reason the proof is so long is that Wiles had to prove and build upon a bunch of other theorems to get to a point where proving FLT was possible. If for example you simply accept everything up to and including the Taniyama–Shimura–Weil conjecture as true, then the proof of FLT gets pretty close to one line.
stiff 13y ago
Well, why exactly would you think that a simple to state problem should have a simple solution? It is of course false, as demonstrated for example by the insolvability of equations of degree higher than 5 [1] and by countless other examples, but I don't see a reason why it would even be a reasonable heuristic. If you spent just a few hours trying to think of a proof of the Fermats theorem you would very quickly see a lot of good reasons why it is difficult and why an elementary solution most likely does not exist. E=mc^2 is just a statement of a relationship, it isn't a proof or a solution to a problem. Mathematical work in modern physics is in fact just as complicated and messy. [1] http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
- Someone 13y ago
  As others said, once you have sufficient terminology, the proof is simple. For the Ruffini theorem, http://math.stackexchange.com/questions/286927/why-cant-there-be-a-quintic-formula http://math.stackexchange.com/questions/286927/why-cant-ther... states: "The reason five is special is that the group A5 of all even permutations of 5 letters is the smallest non-abelian simple group". An example from computer science that some may understand better: A monad is just a monoid in the category of endofunctors. Both are short statements that one cannot grasp without prior knowledge, but that become simple once one has 'some' prior knowledge. And in some sense, it is turtles all the way down. Even the statement 1+1=2, when studied deeply enough, requires huge amounts of prior knowledge (about 380 pages, and that excludes the definition of 'addition' (http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations http://en.wikipedia.org/wiki/Principia_Mathematica#Quotation...) :-)) Some say that there are three kinds of theorems: false, trivial and not yet trivial.
teeja 13y ago
It should be fairly simple to calculate the simultaneous gravity effects of 3 or 4 planets on each other, then plot their courses. That simple problem has not been solved yet!! http://www.math.uvic.ca/faculty/diacu/diacuNbody.pdf http://www.math.uvic.ca/faculty/diacu/diacuNbody.pdf

by hmexx 13y ago

dbaupp 13y ago

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 with a much neater proof, it just takes a while. (Proving theorems is hard work!)

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! :)
- 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.

ColinWright 13y ago

> ... I find it unsatisfying that the solution to such a > simply stated theorem requires 100s of pages of mathematics. The mathematics involved is getting easier and easier to understand as more and more work is being put into it. Although Wiles originally only proved the underlying result for semi-stable elliptic curves (which was enough for FLT), the full modularity theorem (formerly called the Taniyama–Shimura–Weil conjecture) is now proven, and the field continues to grow. I have little doubt that in 100 years or so this will be within the scope of later stage undergraduate work, just as many results in number theory (and other areas) become more and more accessible as we come to understand them properly. Wiles' proof is still comparatively new, it's still messy, it needs to be enhanced, clarified, expanded, and then "re-factored" to find the cleanest path to the result. The underlying arc of the proof is relatively easy to understand, it's the details that need the long, tedious, and careful arguments that take 100s of pages. One day it will be an obvious consequence of the full Modularity Theorem, discussed in honors projects of undergraduates. > Would have been cooler if the proof was extremely simple to > write out, but required outside-the-box thought process that > evaded thousands of humans for centuries. In some sense, perhaps. That's what happened with the proof that primality testing is in P. Three undergrads produced an elementary argument that had eluded people for centuries. And yet it's provided no really new insights into anything, and seems to be effectively a dead end. In that sense it's much more exciting to have genuinely new work that wasn't just a simple argument overlooked for centuries.

raverbashing 13y ago
Yeas, I find it difficult to believe that it'll get easier, but then again, calculating the sine for an arbitrary number was once impossible, later very difficult (and only in the realm of 'pros') and today it's a couple of button presses away.
- ColinWright 13y ago
  Actually, it's pretty easy to get a few places of accuracy purely by hand using the first couple of terms of the right series, and a few double (and triple) angle formulas.

fractalsea 13y ago

What I find interesting is that these proofs aren't one liners. To me, I intuitively presume that mathematical proofs should be one liners because they are explaining fundamental truths of nature; this universe and beyond. Having said that, it's quite possible that a more "elegant" proof will be discovered in the future. Maybe it requires further thinking outside the box -- it will more likely require new mathematics. If you want really non-elegant proofs, take a look at computer-assisted proofs (e.g. proof of the four color theorem). These make the subject of this thread seem extremely elegant!

anonymous 13y ago
It can be a one-liner, but you'll need more language. That is, you'll need terms that encode mathematical truths that currently can only be expressed as several pages. Think of this - I can use the number two and refer to it with just a single character - 2. However, properly stating all its properties and what it is starting from just the axioms of set theory, I'll need quite a lot of pages.
- VLM 13y ago
  Maybe a good way to resonate the situation with HN would be you wanna store the number 2. Well, if you only know and are permitted to use classic IEEE 754 floating point then this is going to be a really long story. But once someone "invents" binary integers and its widely accepted that you're "allowed" to assume everyone understands them, then to store the number 2, you just squirt out a 8-bit binary integer word 00000010 and call it good. And that's how "lets store the number 2" goes from 1000 pages and ten hours of lecture to explain to CS grads, to some 5 minute Kahn academy video that any goof off the street can more or less understand. There is no proof that any simpler explanation exists. Its quite possible no one will ever teach the proof of FLT to grade school kids. But if it ever happens it'll be like the analogy above.
dagw 13y ago
The proof is a one liner, if you take enough of the background theorems as given. The reason the proof is so long is that Wiles had to prove and build upon a bunch of other theorems to get to a point where proving FLT was possible. If for example you simply accept everything up to and including the Taniyama–Shimura–Weil conjecture as true, then the proof of FLT gets pretty close to one line.

stiff 13y ago

Well, why exactly would you think that a simple to state problem should have a simple solution? It is of course false, as demonstrated for example by the insolvability of equations of degree higher than 5 [1] and by countless other examples, but I don't see a reason why it would even be a reasonable heuristic. If you spent just a few hours trying to think of a proof of the Fermats theorem you would very quickly see a lot of good reasons why it is difficult and why an elementary solution most likely does not exist. E=mc^2 is just a statement of a relationship, it isn't a proof or a solution to a problem. Mathematical work in modern physics is in fact just as complicated and messy. [1] http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem

Someone 13y ago
As others said, once you have sufficient terminology, the proof is simple. For the Ruffini theorem, http://math.stackexchange.com/questions/286927/why-cant-there-be-a-quintic-formula http://math.stackexchange.com/questions/286927/why-cant-ther... states: "The reason five is special is that the group A5 of all even permutations of 5 letters is the smallest non-abelian simple group". An example from computer science that some may understand better: A monad is just a monoid in the category of endofunctors. Both are short statements that one cannot grasp without prior knowledge, but that become simple once one has 'some' prior knowledge. And in some sense, it is turtles all the way down. Even the statement 1+1=2, when studied deeply enough, requires huge amounts of prior knowledge (about 380 pages, and that excludes the definition of 'addition' (http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations http://en.wikipedia.org/wiki/Principia_Mathematica#Quotation...) :-)) Some say that there are three kinds of theorems: false, trivial and not yet trivial.

teeja 13y ago

It should be fairly simple to calculate the simultaneous gravity effects of 3 or 4 planets on each other, then plot their courses. That simple problem has not been solved yet!! http://www.math.uvic.ca/faculty/diacu/diacuNbody.pdf http://www.math.uvic.ca/faculty/diacu/diacuNbody.pdf