The network of mathematics | Hacker News Reader

The network of mathematics

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75 pointsnhamann20 comments

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?

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

That's from http://arxiv.org/pdf/gr-qc/9704009.pdf

There's some more from the author here: http://space.mit.edu/home/tegmark/crazy.html

Thank you! I love the internet :)

That's it! Thanks!

So where would categories hang on that graph? Below semigroups?

Another good resource is http://www.math.niu.edu/~rusin/known-math/index/tour_div.htm....

…which is reachable from the more memorable http://www.math-atlas.org/

There is one in the book, Mathematics: Form and Function by Saunders Mac Lane (who was one of the creators of category theory).

Not mentioned in the article is that the Stacks Project is on github 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)

Is there something like this for computer science ?

The ACM has a taxonomy

http://www.acm.org/about/class/2012

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This researchers look back (looks like Brooklyn subways

http://www.cs.man.ac.uk/~navarroe/research/map/

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http://arxiv.org/abs/1304.2681

http://mocs.cs.arizona.edu/

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

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and... all of science! http://metamodern.com/2009/05/20/a-map-of-science/

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.

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

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-mathem...

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_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.

The parent comment seems to imply that there are some absolute mathematical truths and that there are some statements true in this absolute sense that can not be proved by mathematics. Goedels theorem shows something else: that starting from an axiom system there will be statements true in this axiom system that are not provable. I anyway doubt John Baez meant mapping all true sentences from all possible axiom systems in form of a graph...

Actually, my statement is limited to mathematical truths. I am not talking about truth in general. And yes, the nit-pick was out of place. And no, I am not an art major, and yes, I understand Goedel's theorem just fine.

I said, "Not all mathematical truths can be proven to be true." I don't know how you got from there to: "there are some statements true in the absolute sense that cannot be proved by mathematics".

My statement is equivalent to your statement: "Starting from an axiom system, there will be statements true in this axiom system that are not provable."

This comment is irrelevant and has high potential for derailing the discussion off-topic.

Mr. Baez here is a world class mathematician; surely he is more than familiar with formally undecidable propositions.

This post is about the future of mathematics, and what tools might become available. It also showcases the complexity of this discipline and how much material you have to be familiar with and have in the "RAM" of your brain before you have an eureka moment.

What hypothetical complementary tools do you think would meaningfully add to a mathematician's toolkit?

You are correct. My comment was irrelevant and unhelpful.

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