Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeJul 23rd 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexConcrete, abstract: group actions, groups; concrete categories, categories; Cartesian spaces, vector spaces; von Neumann algebras, $W^*$-alebras; material sets, structural sets; etc. At [[concrete structure]].

   Concrete, abstract: group actions, groups; concrete categories, categories; Cartesian spaces, vector spaces; von Neumann algebras, $W^*$-alebras; material sets, structural sets; etc. At concrete structure.

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 23rd 2012
   • PermaLink
   Author: Dmitri Pavlov
   Format: Markdown>and even distinguished such that a W∗-algebra comes with a representation on a Hilbert space but a von Neumann algebra does not Actually, it's the opposite. W*-algebras are "abstract", von Neumann algebras are "concrete". (Personally, though, I think it's stupid to confuse algebras and their representations.)

   and even distinguished such that a W∗-algebra comes with a representation on a Hilbert space but a von Neumann algebra does not

   Actually, it's the opposite. W*-algebras are "abstract", von Neumann algebras are "concrete". (Personally, though, I think it's stupid to confuse algebras and their representations.)

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeJul 23rd 2012
   • (edited Jul 23rd 2012)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexYes, I said that correctly twice but backwards once. Fixed.

   Yes, I said that correctly twice but backwards once. Fixed.

4. • CommentRowNumber4.
   • CommentAuthorMirco Richter
   • CommentTimeJul 28th 2012
   • (edited Jul 29th 2012)
   • PermaLink
   Author: Mirco Richter
   Format: TextI really like that entry and the distinction between concrete and abstract, because there is one. The question is, can it be made formal? ... Related to this consider the 'category of bases' on a vector space (objects are bases and morphisms are bijections). When we fully apply the principle of equivalence, this category has exactly one object. Is this "the best way" to look on it? Surely it depends and we shouldn't judge here. We can walk down the stairs of abstraction and add (or use) more and more structure and hence "brake the symmetry" i.e. make the equivalence classes smaller and smaller. In my opinion we shouldn't forget that, because we need it: In physics from general relativity, different choices of bases gives different (pseudo-)forces, so in that case it is something that we actually can feel. Just applying the full principle of equivalence rules out here all the accelerations that are not intrinsic to the gravitation field for example. (Or maybe it doesn't rule them out but just lump them together.) Moreover a huge amount of physical observables are only defined relative to a base and my conclusion is, that we simply can't describe them in a 'most abstract' theory JUST using the principle of equivalence without storing finer information about the equivalence classes of objects and morphisms. (This last example is a bit above vector spaces and bases and can be better seen in the context of the category of sections of a frame bundle together with gauge transformations)

   I really like that entry and the distinction between concrete and abstract, because there is one. The question is, can it be made formal?
   
   ...
   
   Related to this consider the 'category of bases' on a vector space (objects are bases and morphisms are bijections). When we fully apply the principle of equivalence, this category has exactly one object. Is this "the best way" to look on it? Surely it depends and we shouldn't judge here. We can walk down the stairs of abstraction and add (or use) more and more structure and hence "brake the symmetry" i.e. make the equivalence classes smaller and smaller. In my opinion we shouldn't forget that, because we need it:
   
   In physics from general relativity, different choices of bases gives different (pseudo-)forces, so in that case it is something that we actually can feel. Just applying the full principle of equivalence rules out here all the accelerations that are not intrinsic to the gravitation field for example. (Or maybe it doesn't rule them out but just lump them together.)
   Moreover a huge amount of physical observables are only defined relative to a base and my
   conclusion is, that we simply can't describe them in a 'most abstract' theory JUST using the principle of equivalence without storing finer information about the equivalence classes of objects and morphisms.
   
   (This last example is a bit above vector spaces and bases and can be better seen in the context of the category of sections of a frame bundle together with gauge transformations)