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I was watching Bill Shillito Lectures on Higher Mathematics, in the second episode he says the basic stuff about axioms and theorems - that axioms are unproved statements in which we build theorems and these theorems are proved with those axioms or with other theorems:

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So are there treelike representations of axioms and theorems (such as the one in the given image)? I've read some mathematics books but never found such a representation of the axioms/theorems/lemmas/corollaries but I believe someone may have had this idea.

If it's not too broad, It would be cool to have a list with various treelike representations for various fields. I remember I've made a similar question in the past, now I'm trying to narrow down to the level of axioms/theorems/lemmas/corollaries. I believe such thing would be a nice educational resource, at least for people like me who like treelike diagrams.

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Red Banana
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    i don't know about an existing version, but i think this could be implemented on a wiki type site (like proofwiki) as an autogenerated thing. it's an interesting idea in any case – citedcorpse Jul 25 '13 at 05:53
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    I'm guessing you are going for something more detailed than this, but there's a cool chart of some mathematical structures on the second page of this paper: http://arxiv.org/pdf/gr-qc/9704009v2.pdf – asperanz Jul 30 '13 at 03:11

2 Answers2

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I'm not an expert in this area.

There are a number of concepts related to your question. On one level, there are formal verification projects which use computers to check the logical consistency of mathematical statements as deduced from some starting point (like axioms). Once you agree that proof-checking is computationally straightforward, then the question becomes "How to enter a proof into the computer?" You could do it manually, entering the entire proof by hand using something resembling a programming language. But really you'd want the machine to help out. At the next level, you find proof assistants to help with the data entry process. In a perfect world, you'd want to enter the theorem and have the machine return a proof; you want an automated theorem prover. This is hard.

Once you have a system to encode and verify mathematics using a computer, then you would probably want start developing a large body of verified mathematics. These formally verified libraries exist. The Mizar system is one such example and you can navigate the dependency tree of their articles from their journal of formalized mathematics. Similar projects include HOL Light and Metamath. HOL Light has been used to formalize 100 theorems, whereas Metamath has a tool called the Hilbert Space Explorer.

Once you have the dependency tree, you can do neat visualizations and even determine important ideas in mathematics. The blogger/graduate student Andy Reagan entered the "theorem network" in Rudin's Principles of Mathematical Analysis into a computer and ran Google's PageRank algorithm. You can read about the results and even download the network files at the link provided. Khan Academy uses a similar idea for their Exercise Dashboard, combining a dependency tree and mastery learning to introduce new topics as only after you understand the existing material.

These last two examples are closer to your question than the formal systems I mentioned in the first half of the post. Even though I think Khan's experiments combining a dependency tree and mastery learning only scratch the surface of what's possible pedagogically, there's a huge benefit in simply taking out a pen and paper and building a knowledge tree yourself.

dls
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Corollary
↑
Theorem / Proposition
↑
Lemma
↑
Axiom / Postulate
↑
Conjecture / Claim
↑
Definition


These are not set in stone. One usually starts with a definition (on the bottom). Then they make an unproven claim or conjecture. Axioms are meant to be minimal assumptions before building up. Lemmas are milestones as we work towards proving a theorem. Proof of a theorem is the primary goal. Corollaries are short proofs that rely heavily on the theorem.

Harold Toomey
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    As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community Oct 25 '21 at 18:32