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Level of Abstraction

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  • Jens Doll
    When reasoning about logic and model theory I came across a stunning question about abstraction. Abstraction normally means subsuming data from the observed
    Message 1 of 7 , Jul 21, 2009
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      When reasoning about logic and model theory I came across a stunning question about abstraction. Abstraction normally means subsuming data from the observed space of data and combining the facts by some generalization. This normally leads to a lower size of the newly built abstracted space.

      If you now look the ladder upwards, one might find an upper limit for the number of levels, because the abstracted space decreases monotoneously in size.

      Do you agree with me, that there is a limit for abstraction?
      Would that mean, that human thinking might be limited by natural constants?

      Regards,
      Jens
    • Fred E.J. Linton
      ... Constants ? I don t know. Constraints ? Absolutely -- cf. I. Kant. Cheers, -- Fred Linton
      Message 2 of 7 , Jul 21, 2009
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        On Tue, 21 Jul 2009 03:32:58 AM EDT "Jens Doll" <jd@...> asked:

        > Would that mean, that human thinking might be limited by natural constants?

        "Constants"? I don't know. "Constraints"? Absolutely -- cf. I. Kant.

        Cheers, -- Fred Linton
      • arun_far
        Dear, Over abastraction will lead to chaos. Arun India
        Message 3 of 7 , Aug 12 7:51 PM
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          Dear,
          Over abastraction will lead to chaos.
          Arun
          India






          --- In univalg@yahoogroups.com, "Jens Doll" <jd@...> wrote:
          >
          > When reasoning about logic and model theory I came across a stunning question about abstraction. Abstraction normally means subsuming data from the observed space of data and combining the facts by some generalization. This normally leads to a lower size of the newly built abstracted space.
          >
          > If you now look the ladder upwards, one might find an upper limit for the number of levels, because the abstracted space decreases monotoneously in size.
          >
          > Do you agree with me, that there is a limit for abstraction?
          > Would that mean, that human thinking might be limited by natural constants?
          >
          > Regards,
          > Jens
          >
        • Jens Doll
          Hello Arun, do you mean that abstraction is like simplification? Jens
          Message 4 of 7 , Aug 18 12:22 AM
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            Hello Arun,
            do you mean that abstraction is like simplification?
            Jens

            --- "arun_far" <arun_far@...> wrote:
            > ... Over abastraction will lead to chaos.
          • Vaughan Pratt
            ... Hi Jens, It wasn t clear from your original message what you were asking. However if you were looking for a formal definition of abstract, one candidate
            Message 5 of 7 , Aug 18 8:54 AM
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              > Posted by: "Jens Doll" jd@...
              > Hello Arun,
              > do you mean that abstraction is like simplification?
              > Jens

              Hi Jens,

              It wasn't clear from your original message what you were asking.
              However if you were looking for a formal definition of "abstract," one
              candidate (in the direction of "more abstract") would be the concept of
              reduct of a structure, with its converse being expansion (the direction
              of "less abstract"), see the Wikipedia article

              http://en.wikipedia.org/wiki/Reduct

              An informative instance is that of pseudoelementary class as a reduct of
              an elementary class. The concept of "abstract" is nicely illustrated by
              Example 1 of the Wikipedia article

              http://en.wikipedia.org/wiki/Pseudoelementary_class

              while examples 2 and 3 illustrate the level of technical detail that can
              arise after suitably formalizing "abstract." (In the spirit of full
              disclosure both articles were largely written by me, so I may be biased
              there.)

              Vaughan Pratt
            • arun_far
              By abstraction we mean to reduce the constraints of the system under consideration. Arun
              Message 6 of 7 , Aug 18 9:39 AM
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                By abstraction we mean to reduce the constraints of the system under consideration.
                Arun




                -- In univalg@yahoogroups.com, "Jens Doll" <jd@...> wrote:
                >
                > Hello Arun,
                > do you mean that abstraction is like simplification?
                > Jens
                >
                > --- "arun_far" <arun_far@> wrote:
                > > ... Over abastraction will lead to chaos.
                >
              • jensd99
                ... Here I have to disagree, because abstraction does not omit something [like (Z,+,-,0) - (Z,+,0)], but replaces instances (of whatever) by generalizations,
                Message 7 of 7 , Aug 21 12:47 AM
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                  --- In univalg@yahoogroups.com, Vaughan Pratt <pratt@...> wrote:
                  > ... if you were looking for a formal definition of "abstract,"
                  > one candidate (in the direction of "more abstract") would be the
                  > concept of reduct of a structure ...

                  Here I have to disagree, because abstraction does not omit something [like (Z,+,-,0) -> (Z,+,0)], but replaces instances (of whatever) by generalizations, which would read as [(Z,+,-,0) => (Z,binop,0), where binop is a binary operator]. Reduction on the other hand removes obsolete things like [a+0 -> a].

                  Jens
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