6 ms·

The network of mathematics

75 points by nhamann 13y ago

nhamann 13y ago
Not mentioned in the article is that the Stacks Project is on github https://github.com/stacks https://github.com/stacks I've always thought that math books should in digraph rather than linear form. What would be interesting is to combine this with a wiki. You could have alternate proofs of the same lemma, or even entirely different presentations (starting from different axioms, for instance)
igravious 13y ago
I remember seeing many years ago a fairly dense one page diagram of how most of the major "bits" of mathematic hang together. I can never for the life of me dig it up again. Anyone know what I'm on about / got any pointers?
- Bakkot 13y ago
  This may have been what you're thinking of. Sadly I lack a source; it's just been kicking around my miscellaneous images folder. http://i.imgur.com/tBgQkxi.jpg http://i.imgur.com/tBgQkxi.jpg
  - friendcomputer 13y ago
    That's from http://arxiv.org/pdf/gr-qc/9704009.pdf http://arxiv.org/pdf/gr-qc/9704009.pdf There's some more from the author here: http://space.mit.edu/home/tegmark/crazy.html http://space.mit.edu/home/tegmark/crazy.html
    - GuiA 13y ago
      Thank you! I love the internet :)
  - igravious 13y ago
    That's it! Thanks!
  - pohl 13y ago
    So where would categories hang on that graph? Below semigroups?
- btilly 13y ago
  Another good resource is http://www.math.niu.edu/~rusin/known-math/index/tour_div.html http://www.math.niu.edu/~rusin/known-math/index/tour_div.htm....
  - Someone 13y ago
    …which is reachable from the more memorable http://www.math-atlas.org/ http://www.math-atlas.org/
- jonjacky 13y ago
  There is one in the book, Mathematics: Form and Function by Saunders Mac Lane (who was one of the creators of category theory).
teilo 13y ago
No, "all math" will not be linked up like this some day, because that is impossible. Not all mathematical truths can be proven to be true.
- joe_the_user 13y ago
  I don't think this is just matter of just unprovable theorems being unreachable. There isn't an equivalent "no conceptual framework is 'best' for all mathematical inquiries" theorem. Such a claim probably can't be proven. But as you say, that doesn't keep it from being true. Still, Godel's theorem on the cutting down of proofs via assume unprovable claims is worth considering. http://en.wikipedia.org/wiki/G%C3%B6del%27s_speed-up_theorem http://en.wikipedia.org/wiki/G%C3%B6del%27s_speed-up_theorem
- stiff 13y ago
  Besides this being a ridiculous nit pick, it is not even true, seems like yet another misinterpretation of Goedels theorem, the favourite theorem of liberal arts students: http://www.quora.com/Mathematics/Is-there-anything-in-mathematics-that-holds-true-but-cant-be-proven http://www.quora.com/Mathematics/Is-there-anything-in-mathem...
  - joe_the_user 13y ago
    Downvoted for your inaccessible link. And while it is kinda nit-picky, the parent's statement is literally true (see my other post also). Given any fixed axiom system, there will be true statements that aren't provable within the system (expand your axioms and you'll just have different true but provable statements in the expanded system). Now, Godel's completeness theorem shows that you construct complete mathematical system of true statements however such a system requires inserting an infinite number of arbitrarily choices among statements (and their negations) which aren't provable given the previous axioms. Since the framework of the article is finite, not infinite, I would claim the framework of the article, being finite, can't encompass all true statements of any given system, even if it an algorithm for producing axioms. https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_th... Edit: I got through the pay-wall via Google but the discussion is somewhere between confused and confusing (the large part of post mostly meaningless speculation about the term "proved in an absolute sense", that he introduces without defining). The situation is really simple. All formal proof systems have hole (at least those of any reasonable "powerfulness"). Any formal proof system can be expanded indefinitely but at any point in that expansion will still have a hole.
currywurst 13y ago
Is there something like this for computer science ?
- gtani 13y ago
  The ACM has a taxonomy http://www.acm.org/about/class/2012 http://www.acm.org/about/class/2012 ____________________________ This researchers look back (looks like Brooklyn subways http://www.cs.man.ac.uk/~navarroe/research/map/ http://www.cs.man.ac.uk/~navarroe/research/map/ ___________________________ http://arxiv.org/abs/1304.2681 http://arxiv.org/abs/1304.2681 http://mocs.cs.arizona.edu/ http://mocs.cs.arizona.edu/ http://people.cs.umass.edu/~mimno/icml100.html http://people.cs.umass.edu/~mimno/icml100.html These are clustering by different algos(sounds like SVD in the first, I'll have to read the paper later). related: extracting FAQs http://arxiv.org/abs/1203.5188 http://arxiv.org/abs/1203.5188 ______________________________ and... all of science! http://metamodern.com/2009/05/20/a-map-of-science/ http://metamodern.com/2009/05/20/a-map-of-science/

75 points by nhamann 13y ago

Not mentioned in the article is that the Stacks Project is on github https://github.com/stacks https://github.com/stacks I've always thought that math books should in digraph rather than linear form. What would be interesting is to combine this with a wiki. You could have alternate proofs of the same lemma, or even entirely different presentations (starting from different axioms, for instance)

igravious 13y ago

I remember seeing many years ago a fairly dense one page diagram of how most of the major "bits" of mathematic hang together. I can never for the life of me dig it up again. Anyone know what I'm on about / got any pointers?

Bakkot 13y ago
This may have been what you're thinking of. Sadly I lack a source; it's just been kicking around my miscellaneous images folder. http://i.imgur.com/tBgQkxi.jpg http://i.imgur.com/tBgQkxi.jpg
- friendcomputer 13y ago
  That's from http://arxiv.org/pdf/gr-qc/9704009.pdf http://arxiv.org/pdf/gr-qc/9704009.pdf There's some more from the author here: http://space.mit.edu/home/tegmark/crazy.html http://space.mit.edu/home/tegmark/crazy.html
  - GuiA 13y ago
    Thank you! I love the internet :)
- igravious 13y ago
  That's it! Thanks!
- pohl 13y ago
  So where would categories hang on that graph? Below semigroups?
btilly 13y ago
Another good resource is http://www.math.niu.edu/~rusin/known-math/index/tour_div.html http://www.math.niu.edu/~rusin/known-math/index/tour_div.htm....
- Someone 13y ago
  …which is reachable from the more memorable http://www.math-atlas.org/ http://www.math-atlas.org/
jonjacky 13y ago
There is one in the book, Mathematics: Form and Function by Saunders Mac Lane (who was one of the creators of category theory).

teilo 13y ago

No, "all math" will not be linked up like this some day, because that is impossible. Not all mathematical truths can be proven to be true.

joe_the_user 13y ago
I don't think this is just matter of just unprovable theorems being unreachable. There isn't an equivalent "no conceptual framework is 'best' for all mathematical inquiries" theorem. Such a claim probably can't be proven. But as you say, that doesn't keep it from being true. Still, Godel's theorem on the cutting down of proofs via assume unprovable claims is worth considering. http://en.wikipedia.org/wiki/G%C3%B6del%27s_speed-up_theorem http://en.wikipedia.org/wiki/G%C3%B6del%27s_speed-up_theorem
stiff 13y ago
Besides this being a ridiculous nit pick, it is not even true, seems like yet another misinterpretation of Goedels theorem, the favourite theorem of liberal arts students: http://www.quora.com/Mathematics/Is-there-anything-in-mathematics-that-holds-true-but-cant-be-proven http://www.quora.com/Mathematics/Is-there-anything-in-mathem...
- joe_the_user 13y ago
  Downvoted for your inaccessible link. And while it is kinda nit-picky, the parent's statement is literally true (see my other post also). Given any fixed axiom system, there will be true statements that aren't provable within the system (expand your axioms and you'll just have different true but provable statements in the expanded system). Now, Godel's completeness theorem shows that you construct complete mathematical system of true statements however such a system requires inserting an infinite number of arbitrarily choices among statements (and their negations) which aren't provable given the previous axioms. Since the framework of the article is finite, not infinite, I would claim the framework of the article, being finite, can't encompass all true statements of any given system, even if it an algorithm for producing axioms. https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_th... Edit: I got through the pay-wall via Google but the discussion is somewhere between confused and confusing (the large part of post mostly meaningless speculation about the term "proved in an absolute sense", that he introduces without defining). The situation is really simple. All formal proof systems have hole (at least those of any reasonable "powerfulness"). Any formal proof system can be expanded indefinitely but at any point in that expansion will still have a hole.

currywurst 13y ago

Is there something like this for computer science ?

gtani 13y ago
The ACM has a taxonomy http://www.acm.org/about/class/2012 http://www.acm.org/about/class/2012 ____________________________ This researchers look back (looks like Brooklyn subways http://www.cs.man.ac.uk/~navarroe/research/map/ http://www.cs.man.ac.uk/~navarroe/research/map/ ___________________________ http://arxiv.org/abs/1304.2681 http://arxiv.org/abs/1304.2681 http://mocs.cs.arizona.edu/ http://mocs.cs.arizona.edu/ http://people.cs.umass.edu/~mimno/icml100.html http://people.cs.umass.edu/~mimno/icml100.html These are clustering by different algos(sounds like SVD in the first, I'll have to read the paper later). related: extracting FAQs http://arxiv.org/abs/1203.5188 http://arxiv.org/abs/1203.5188 ______________________________ and... all of science! http://metamodern.com/2009/05/20/a-map-of-science/ http://metamodern.com/2009/05/20/a-map-of-science/