Definition talk:Letter (Mathematical Theory)

Rather than merge, I think a transclusion would be better.

Reason: "Formal Systems" is (from what I can tell) a superset of "Mathematical Theories" as defined by Bourbaki. The "Mathematical Theory" approach was my attempt to parse some coherent meaning out of the insanely difficult approach of Bourbaki, using "mathematical theory" as a specific example of the development of mathematics as a formal system.

To a certain extent this is the approach which has been aimed at for the whole of ProofWiki. The extra difficulty we have is to be sufficiently general to cover "everything", but as there are therefore so many different approaches (natural deduction, the approach of Keisler and Robbin, the Bourbaki approach etc.) this makes segregating them into their separate (parallel) threads of discourse a much more difficult process.

I'd welcome input into how we do this. A suggestion is to set up pages in which the titles of the specific instances themselves contain the name of the author of the particular formal system being described, but that may be *too* specific.

This argument is specifically interesting to me at the moment because I'm currently trying to merge the approach to number system development as presented by (who constructs the integers specifically by creating equivalence classes of ordered pairs of natural numbers) with the approach by  who takes a more general approach. However, this problem is not as tricky to resolve, as there is a limited number of approaches to this problem, whereas in expressing mathematics as a formal system there are (perhaps literally) as many approaches as there are authors. --prime mover 03:41, 16 June 2012 (EDT)


 * My idea was to well, practically, do what you suggest, creating separate branches of definitions and theorems for each of the calculi. However, I don't see the merit of transcluding them all together; that would only make pages very unwieldy and brings dilemmas on what should and what shouldn't be together. Rather, I would simply like to put on Definition:Propositional Logic (and mutatis mutandis for all other multiply axiomatised areas of research) the various formalised approaches, and then put all pages pertaining to them in separate subcategories. This seems to me the only way to keep some sort of structure in these realms. But first it will be necessary to bring the current contributions up to standard. --Lord_Farin 03:52, 16 June 2012 (EDT)
 * It thus appears, if we choose to incorporate this approach, that a category Category:Natural Deduction needs to be created, to which then most of what is now in Category:Propositional Logic has to be moved. Does this observation coincide with how you envisage the result of this exercise? --Lord_Farin 03:57, 16 June 2012 (EDT)


 * Er, probably (except I'm not sure I agree with you about transclusions, at least at the moment when pages are small). I only have a fuzzy idea of the final result; I won't recognise the sweet spot till I see it. What I'll do is leave you alone to implement your restructuring - with the caveat that if a page specifically cites an author, removing any material may no longer reflect what that author says. --prime mover 04:09, 16 June 2012 (EDT)

While I think about it, I'll just mention the book which presents a 3-axiom approach to a natural-deduction derivation of (strictly Aristotelian) PropLog which is insanely complicated but aesthetically satisfying, which I have shied away from documenting. There is also an approach which can be found somewhere on the web which uses a single (nightmarishly complicated) axiom - but I haven't found out a great deal about it. Obviously we are going to be documenting these eventually on. In due course. It can wait till after this mammoth restructuring. --prime mover 04:20, 16 June 2012 (EDT)

It may be worthwhile to introduce a proof technique of 'induction over the rules of formation' to establish uniformity among results which (effectively) use this technique (eg. WFF of PropLog is Balanced). --Lord_Farin 08:48, 16 June 2012 (EDT)