User talk:Prime.mover

A Possible Next Direction
I was planning on starting on a project for a website/wiki about proofs, but if I could just incorporate some things into this website that would also work, and probably make things much easier.

Something that I would like to see, which this website does to some extent, is the breaking down of complex mathematical topics all the way down to the simplest axioms(regardless of which they are, and one could use multiple systems) and explicitly stating every step of the way. If one has the time, why not boil down a complex theorem as far down as it can go? Is that not basically the essence of proof anyhow?

One idea is that you could use flowcharts to state every pre-established concept that the proof makes use of. With this someone could also take a famous proof that they might find interesting, and then easily find the exact path(s) of what concepts, theorems, etc. are necessary to understand before they attempt to understand the proof in question. Attached is one 'proof of concept' I was going for with the Pythagorean Theorem from Book I of Euclid's Elements(currently excluding definitions). In this case I didn't flesh out any proposition that already had been fleshed out somewhere else in the image.

Of course, it doesn't have to be in this exact format. With a website one can use hyperlinks, or make the individual branches collapsible, or use other tricks to make things more convenient.

This site already does make use of links in many proofs, which is great. The basic framework is already pretty much here, I just think it could be expanded upon and/or organized further. There wouldn't need to be a flowchart or a similar map for every proof, but wouldn't it make things more accessible/easier to understand?--Alexelam (talk) 04:25, 8 April 2021 (UTC)


 * The philosophy that you have described is exactly what the manifesto states. In theory, every proof can (ultimately) be broken down into a chain of definitions and axioms, all the way back to ZFC and (in the case of geometry) Euclid.


 * There are very likely to be instances of where we have not gone all the way back to axioms, for example, for advanced proofs using basic arithmetic. Reminding the user of the rigorous definition of real addition based on abstract algebraical constructs in field theory backed up by the ZFC construction of the natural numbers is probably overkill -- but the structure to support has already been put in place on (it was one of the first things we did). The philosophy is indeed already there.


 * I see your diagram for Pythagoras's Theorem but it's too small, and when I try to embiggen it, I find that the contents of the nodes have lost their definition. However, I suspect that there are repeated instances of the most basic axioms and definitions. I can totally see where you are coming from, but it might be more profitable to use a directed acyclic graph structure. (Fun fact: by strange coincidence, I was in the process of implementing one of those for a project in my day job while at the same time reading Stephenson's "Anathem". The significance of this would become apparent to a reader of at least the appendices of that literary work.) --prime mover (talk) 05:24, 8 April 2021 (UTC)