Sung, There are plenty of mathematical objects that are degenerate aside from category theory objects. Category theory is a very abstract form of representation that has some representational advantages, but its power is no more than that of set theory.
John From: sji.confor...@gmail.com [mailto:sji.confor...@gmail.com] On Behalf Of Sungchul Ji Sent: January 28, 2015 6:46 AM To: biosemiotics Cc: PEIRCE-L Subject: [PEIRCE-L] Re: [biosemiotics:8004] Degeneracy article (Table 1 may be disorganized on transit.) John, The paper you recommend is one of the most informative scholarly articles I have ever read. I highly recommend it to anyone who is interested in learning about the 21st-century biology. In the article, the author, Mason, defines "degeneracy" as follows: "duplicate structures that differentiate yet remain isofunctional" (012715-11) "unrelated isofunctional structures that are dispersed endogenously or exogenously" (012715-12) "variable arrangements of interacting structures that achieve the same output through multiple pathways" (012715-13) "parcellation of a structure into subunits that can still variably perform the same initial function" (012715-14) To the best of my knowledge, in physics and mathematics, the concept of degeneracy appears in the following form: "A and B are degenerate with respect to C, if the elements of A can be mapped or related (012715-15) to those of B while preserving some property or process C." This definition of degeneracy reminded me of the definition of category given in (Brown, R. and T. Porter (2006), Category Theory: an abstract setting for analogy and comparison, In: What is Category Theory?, Advanced Studies in Mathematics and Logic, Polimetrica Publisher, Italy, (2006) 257-274.2006): “We view a category as giving a fairly general abstract context for comparison. The (012715-16) objects of study are the objects of the category. Two objects, A and B, can be compared if the set C(A,B) is non-empty and various arrows A ---> B are ‘ways of comparing them’. The composition corresponds to: If we can compare A with B and B with C, we should be able to compare A with C.” In other words, Mason's "degeneracy" defined in (012715-11) through (012715-14) can be generalized as shown in (012715-15), which can then be connected to the concept of "category" in mathematics as defined in (012715-16) and to the Peircean sign which I claim can be viewed as a category. Thus, all these ideas may be viewed as different manifestations of the principle of the irreducible triad of Peirce or the commutative triangle of category theory, as summarized in Table 1 below. ________________________________________________________________________________ Table 1. The irreducible triadicity in biology, mathematics, linguistics, physics, and semiotics. ________________________________________________________________________________ Field A B C Name of the Relation ________________________________________________________________________________ 1. Biology structure 1 structure 2 function degeneracy ________________________________________________________________________________ 2. Mathematics object 1 object 2 object 3 category ________________________________________________________________________________ 3. Linguistics word 1 word 2 sentence paradigmatic relation ________________________________________________________________________________ 4. Physics particle 1 particle 2 energy degeneracy ________________________________________________________________________________ 5. Semiotics object sign/representamen interpretant semiosis _________________________________________________________________________________ It may be concluded that the first row of this table is the type called "ur-category" and the Rows 1 through 5 are its "token categories". If this conclusion turns out to be valid upon further scrutiny, "The concept of category may unify biology, mathematics, linguistics, physics, and semiotics." (012715-17) Again, if it can be proven, Statement (012715-17) may deserve to be called the "Grand Unified Theory of Sciences" (GUTS), a gutsy theory indeed. All the best. Sung ______________________________________________ Sungchul Ji, Ph.D. Associate Professor of Pharmacology and Toxicology Department of Pharmacology and Toxicology Ernest Mario School of Pharmacy Rutgers University Piscataway, N.J. 08855 732-445-4701<tel:732-445-4701> www.conformon.net<http://www.conformon.net/> On Tue, Jan 27, 2015 at 4:00 AM, John Collier <colli...@ukzn.ac.za<mailto:colli...@ukzn.ac.za>> wrote: There is a fairly good paper dealing with the issue of degeneracy in biology at http://onlinelibrary.wiley.com/doi/10.1002/cplx.21534/abstract The issue came up previously on this list. John
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