Subwiki talk:Property-theoretic organization

This discussion page is intended for a discussion on using property theory as a means of organizing material in subject wikis, as well as the underlying issues of mathematics, logic, computation and philosophy. Vipul 20:20, 25 February 2009 (UTC)

Initial Discussion
Previously on Buffer &hellip;


 * The History of Subject Wikis
 * Comments

Vipul,

From the initial sample of your writing distribution that I’ve read so far, your approach to mathematical objects by way of their properties is essentially a logical angle.

If one elects to think in functional, computational terms from the very beginning, then it is convenient to think of a property $$p\!$$ as being a mapping from a space $$X,\!$$ the universe of discourse, to a space of 2 elements, say, $$\mathbb{B} = \{ 0, 1 \}.$$ So a property is something of the form $$p : X \to \mathbb{B}.$$  That level of consideration amounts to propositional calculus. Trivial as it may seem, it’s been my experience that getting efficient computational support at this level is key to many of the things we’d like to do at higher levels in the way of mathematical knowledge management.

Well, enough for now, as I’m not even sure the above formatting will work here. Let’s look for a wiki discussion page where it will be easier to talk. Either at MyWikiBiz, one of your Subject Wikis, or if you like email archiving there is the Inquiry List that an e-friend set up for me.

Jon Awbrey, 24 Feb 2009, 20:42 UTC

VN: Jon, I agree with your basic ideas. There's of course the issue that the space $$X$$ under consideration is often not a set (it's too large to be one -- for instance, a group property is defined as a function on the collection of all groups), so we cnanot simply treat it as a set map. Still, it behaves largely like a set map. And as you point out, properties behave a lot like propositions with the parameter being from the space $$X$$.

At some point in time, I was interested in the mathematical aspects of properties. Properties form a Boolean algebra, inheriting conjunction and disjunction etc. (in set-theoretic jargon, this would just be the power set of $$X$$, except that $$X$$ isn't a set). Then, we can use various partial binary operations on $$X$$ to define corresponding binary operations on the property space. I believe this, too, has been explored by some people, who came up with structures such as quantales and gaggles. The broad idea, as far as I understand, is that the binary operation must distribute over disjunctions (ORs). Many of the operators that I discuss in the group theory wiki, such as the composition operator for subgroup properties and the join operator, are quantalic operators. Some of the terminology that I use for some property-theoretic notation is borrowed from terminology used for such operators, for instance, the left residual and right residual. Some of the basic structural theorems and approaches are also taken from corresponding ideas that already existed in logic. (I arrived at some of them before becoming aware of the corresponding structures explored in logic, and later found that much of this had been done before).

As far as I know, though, there has been no systematic application of the ideas developed in different parts of logic to this real world (?) application to understanding properties in different mathematical domains. This puzzled me since I think that using the ideas of logic and logical structure in this way can help achieve a better understanding of many disciplines, particularly those in mathematics and computer science. Perhaps it is because there are no deep theorems here?

My own use of property theory as an organizational principle seems to rely very minimally on the logical ideas beneath and rests more on the practical results that are important within the specific subject. But perhaps there are deeper theoretical logic issues that should be explored. I'd be glad to hear about your perspective, logical, philosophical, or any other.

I'd also like to hear more about your views on what you mean by supporting such a model computationally. On the group theory wiki, I'm using MediaWiki's basic tools as well as Semantic MediaWiki to help in property-theoretic exploration. This isn't smart in any sense on the machine's part, but it meets some of the practical needs. Nonetheless, if you have ideas for something more sophisticated, I'd be glad to hear. Vipul 22:13, 25 February 2009 (UTC)

JA: Sure, there is all that. My main interest at present is focused on using logic as a tool for practical knowledge management in mathematical domains. That means starting small, breaking off this or that chunk of suitably interesting or useful domains and representing them in such a way that you can not only get computers to help you reason about them but in such a way that human beings can grok what's going on.

JA: Uh-oh, I'm being called to dinner, I will have to continue later tonight. Jon Awbrey 23:08, 25 February 2009 (UTC)

VN: Well, that's interesting. I'm myself interested in the practical aspects, too, though perhaps from a different angle. I'd like to hear more. I'll also go through some of your past writings to understand your perspective. If you have comments on the way things have been done on the subject wikis specifically, I'd like to hear those as well. Vipul 00:00, 26 February 2009 (UTC)

JA: Maybe it will be quickest just to list a few links. Most of the longer pieces at MyWikiBiz are unpublished papers and project workups of mine that I'm still in the process TeXing and Wikifying and redoing the graphics that got mushed up moving between different platforms over the years, so most of them have a point where I last left off reformatting or rewriting. At any rate, here are a few serving suggestions on the basic logic side of things:


 * Propositional Equation Reasoning Systems


 * Differential Propositional Calculus


 * Differential Logic and Dynamic Systems


 * Functional Logic : Quantification Theory

JA: Jon Awbrey 03:26, 26 February 2009 (UTC)

VN: Jon, I've started looking at your work. Your language and approach are new to me, so it'll take me some time to go through it. Other work has also come up. I hope to get back to you in a couple of weeks. Vipul 18:05, 1 March 2009 (UTC)

JA: No particular rush. I went looking for some old e-lectures of mine where I worked up much more concrete examples, but they need to be converted from ASCII to HTML-TeX-Wiki formats to be readable at all, and that will take me a while. I will probably keep outlining the occasional notes here in the interval. Later, Jon Awbrey 22:40, 1 March 2009 (UTC)

Scope and Purpose of a Logic Wiki

 * Objectives
 * Maintain a focus of active methods that is parallel to the focus on static content.
 * Algorithms
 * Computational tools
 * Efficient calculi
 * Heuristics
 * Proofs
 * Here's a project that looks interesting in this connection &mdash;
 * Arnold Neumaier : Automated Mathematical Research Assistant

VN: Jon, that certainly looks interesting. Thanks for the link. Vipul 15:57, 7 March 2009 (UTC)

VN: I think the discussion has two parallel tracks here: what a logic wiki should contain and what logical paradigms can be used efficiently to organize content in all subject wikis (may be you're talking of only one of these right now).

VN: Regarding the organization of content across subject wikis. I had a quick look at the FMathL link. From what I can figure out, this is interesting, but still preliminary. What I'm interested in right now is something actionable that I can use to make the definitions and proofs easier and quicker to understand, easier to link to, etc. I'm sure projects such as FMathL can give some useful ideas in that connection, so I'd be glad to hear what ideas you've extracted from there. I wrote a recent blog post describing some of the issues that I'm thinking about, or rather, the place I'm starting from.

VN: Regarding a specific logic wiki, it seems to me that logic is an extremely big subject. Your comments and past writings give me the impression that you're interested more in the proof theory/proof calculus part of the subject. I'm assuming you're not interested so much in the model theory side of things. Okay, so how about calling it a proof theory wiki or something like that? Of course, I might be misunderstanding the scope, so more clarification from you is welcome. I'm also not sure how Neumaier's automated research assistant connects up with the content of the proposed wiki. Are you sugesting that FMathL's ideas be used to organize the wiki, or that the wiki contain information about FMathL? Vipul 23:54, 11 March 2009 (UTC)

JA: First time I've stopped by this way in a while. It will take me a while longer to work up the concrete example that I had in mind, which is currently in progress here:


 * Differential Analytic Turing Automata (DATA).

JA: I will have to leave the whole of logic to others. The most I can hope to do with my time here is maybe to grease the wheels of our practical vehicles at critical points.

JA: Lunch! Back later &hellip;  Jon Awbrey 16:02, 16 March 2009 (UTC)

JA: Just passing through. Currently reviving some old work I did on the Propositions As Types Analogy, located here:


 * Propositions As Types

Narrative Memoir
JA: I thought it might help to switch from didactic to personal narrative. Recent discussions on various lists reminded of why I first started work on "automatic theorem proving" (ATM) or "mechanized mathematical reasoning", as they called it back then. I had long been interested in the interplay between algebra, geometry, and logic &mdash; what one of my professors later called "the lambda point" &mdash; and when I started grad school in math in the late 70's this interest congealed into a study of the linkages between group theory, number theory, graph theory, and logic. On the graph theory face of it I got bitten by the Reconstruction Conjecture bug and eventually worked out a group-theoretic (orbit counting) way of looking at the problem. But it was all getting to be a bit too much for my unaided brain, and so one of my office-mates suggested that I try bringing the mainframe to bear on it. I was more than hesitant at first, but had been taking courses in Lisp, so I eventually said $$y_0?\!$$ and went to work on trying to write a theorem prover. Jon Awbrey 20:16, 16 March 2009 (UTC)

JA: To be continued &hellip;

Back In Town
I was thinking I might include the Subject Ref Wiki in a trans-site project related to Logic that I've been developing. See the "focal nodes" Inquiry Live and Logic Live for an indication of what I'm trying to do. If you ever get around to creating a subject wiki for logic then the material could be moved there. Jon Awbrey 18:24, 9 June 2010 (PDT)