Difference between revisions of "Subwiki talk:Property-theoretic organization"

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(Prospects for a Logic Wiki)
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<p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p>
 
<p>[http://blog.subwiki.org/?p=13&cpage=1#comment-6 Jon Awbrey, 24 Feb 2009, 20:42 UTC]</p>
 
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VN: Jon, I agree with your basic ideas. There's of course the issue that the space <math>X</math> under consideration is often ''not'' a ''set'' (it's too large to be one -- for instance, a [[Groupprops:group property|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 <math>X</math>.
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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 <math>X</math>, except that <math>X</math> isn't a set). Then, we can use various partial binary operations on <math>X</math> 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 [[groupprops:composition operator for subgroup properties|composition operator for subgroup properties]] and the [[groupprops:join operator|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 [[groupprops:left residual operator for composition|left residual]] and [[groupprops:right residual operator for composition|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).
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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?
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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.
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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. [[User:Vipul|Vipul]] 22:13, 25 February 2009 (UTC)

Revision as of 22:13, 25 February 2009

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)

Prospects for a Logic Wiki

Previously on Buffer …

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 as being a mapping from a space the universe of discourse, to a space of 2 elements, say, So a property is something of the form 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 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 .

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 , except that isn't a set). Then, we can use various partial binary operations on 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)