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>This may be a mathematician's bias, but Category Theory is somewhat underwhelming because you can't really do anything with it. It's actually too abstract It

by s_baby 14y ago

>This may be a mathematician's bias, but Category Theory is somewhat underwhelming because you can't really do anything with it. It's actually too abstract It seems Category Theory might one day be useful in ordering "Upper Level Ontologies". Since it's logically impossible to have one consistent ontology( Incompleteness Theorem ) the next best thing is to find morphisms across knowledge domains, no?

jkbyc 14y ago
What exactly would be the benefit of using category theory? What would you gain in comparison to doing just normal homomorphisms? And that's even assuming you would be able to find a homomorphism at all; it may well be prevented by the ontologies being mutually inconsistent, right?
- Dn_Ab 14y ago
  The benefit is its flaw. Because it is so abstract it is able to describe many, many things. It can serve as a basis for all of maths and unlike set theory really focuses on structure. Very roughly, think of it like functional programming to Set theory's imperative nature.
  - blair_strang 14y ago
    I'm not getting your analogy :/
    - Dn_Ab 14y ago
      It is not a tight analogy. But: In set theory you pay a lot of attention to the internals of collections of objects. In category theory you are more interested in capturing structure and relationships. In imperative programming you specify the internal sequence that makes up an operation, in functional programming you are more interested in the high level declarations, compositions and that the constraints on them are satisfied.
- jasomill 14y ago
  You gain proofs that apply to a wider variety of algebraic structures.
Dn_Ab 14y ago
I'm going to quote myself from 1 year ago: ""The amount of information out there is growing exponentially. The amount that needs to be known before you can contribute keeps exploding with each generation. Such that the practitioner might know just as much if not more than their predecessors but the scope of their knowledge as a fraction of the full body is many orders of magnitude smaller. It is not just a matter of will but of physics of time and the chemistry of the brain. Underutilized connections fade such that unless you spent time actively practising the wide skill sets to sufficient depth they will fade. But there is not enough time to be able to do that. I do think that more must be done in enabling bridges as a counter to this. It is a shame category theorist must wrap their material in such obtuse language as it seems that it would be just the tool for the job.""
bane 14y ago
"Since it's logically impossible to have one consistent ontology( Incompleteness Theorem )..." This pings right at an interest area of mine. Can you point to any references you know on the subject? Particularly cross-domain morphisms?
- s_baby 14y ago
  That was very much the goal of "Systems Theory"[1]. The problem is it's focused on "natural systems" with quantitative relationships. The work has fallen short in capturing phenomenological relationships. I think Wilber's AQAL ontology comes closest to being the most "well defined"[2] but also a little whacky. The cornerstone of the model is differentiation across pronouns. The assertion is perspectives are partitioned across 1st, 2nd, and 3rd person perspectives. So for example, art, ethics, and "truth" are the broad knowledge domains. He takes this further by then saying you can take a 1st/2nd/3rd person perspective on actual 1st/2nd/3rd persons. [1] http://en.wikipedia.org/wiki/Systems_theory http://en.wikipedia.org/wiki/Systems_theory [2] http://wilber.shambhala.com/html/books/kosmos/excerptC/appendix-B.cfm/ http://wilber.shambhala.com/html/books/kosmos/excerptC/appen...

by s_baby 14y ago

jkbyc 14y ago

What exactly would be the benefit of using category theory? What would you gain in comparison to doing just normal homomorphisms? And that's even assuming you would be able to find a homomorphism at all; it may well be prevented by the ontologies being mutually inconsistent, right?

Dn_Ab 14y ago
The benefit is its flaw. Because it is so abstract it is able to describe many, many things. It can serve as a basis for all of maths and unlike set theory really focuses on structure. Very roughly, think of it like functional programming to Set theory's imperative nature.
- blair_strang 14y ago
  I'm not getting your analogy :/
  - Dn_Ab 14y ago
    It is not a tight analogy. But: In set theory you pay a lot of attention to the internals of collections of objects. In category theory you are more interested in capturing structure and relationships. In imperative programming you specify the internal sequence that makes up an operation, in functional programming you are more interested in the high level declarations, compositions and that the constraints on them are satisfied.
jasomill 14y ago
You gain proofs that apply to a wider variety of algebraic structures.

Dn_Ab 14y ago

I'm going to quote myself from 1 year ago: ""The amount of information out there is growing exponentially. The amount that needs to be known before you can contribute keeps exploding with each generation. Such that the practitioner might know just as much if not more than their predecessors but the scope of their knowledge as a fraction of the full body is many orders of magnitude smaller. It is not just a matter of will but of physics of time and the chemistry of the brain. Underutilized connections fade such that unless you spent time actively practising the wide skill sets to sufficient depth they will fade. But there is not enough time to be able to do that. I do think that more must be done in enabling bridges as a counter to this. It is a shame category theorist must wrap their material in such obtuse language as it seems that it would be just the tool for the job.""

bane 14y ago

"Since it's logically impossible to have one consistent ontology( Incompleteness Theorem )..." This pings right at an interest area of mine. Can you point to any references you know on the subject? Particularly cross-domain morphisms?

s_baby 14y ago
That was very much the goal of "Systems Theory"[1]. The problem is it's focused on "natural systems" with quantitative relationships. The work has fallen short in capturing phenomenological relationships. I think Wilber's AQAL ontology comes closest to being the most "well defined"[2] but also a little whacky. The cornerstone of the model is differentiation across pronouns. The assertion is perspectives are partitioned across 1st, 2nd, and 3rd person perspectives. So for example, art, ethics, and "truth" are the broad knowledge domains. He takes this further by then saying you can take a 1st/2nd/3rd person perspective on actual 1st/2nd/3rd persons. [1] http://en.wikipedia.org/wiki/Systems_theory http://en.wikipedia.org/wiki/Systems_theory [2] http://wilber.shambhala.com/html/books/kosmos/excerptC/appendix-B.cfm/ http://wilber.shambhala.com/html/books/kosmos/excerptC/appen...