The network of mathematics(plus.google.com) |
The network of mathematics(plus.google.com) |
There's some more from the author here: http://space.mit.edu/home/tegmark/crazy.html
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)
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://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/
http://www.quora.com/Mathematics/Is-there-anything-in-mathem...
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.
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
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?
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."
