List, Jeff: On Apr 30, 2014, at 12:23 AM, Jeffrey Brian Downard wrote:
> John Baez: "The point is, that a category is really a generalization of a > group." (http://math.ucr.edu/home/baez/categories.html) I fully agree with Jon Baez (who I have heard lecture on several occasions.) And, I further agree that the Erlangen program continues to be a dominant feature of the mathematics of dynamics of today. What I wish to add to this view is that category theory extends far beyond the conceptualization of the relations betweens groups as representations of solutions to the relationships between geometry and polynomials and all that implies for physics, engineering and other branches of applied mathematics. As I pointed out in my post: "Category theory is a generalization of several mathematical structures: sets, groups, rings, vector spaces, topologies and numerous other structures of mathematics. So what are you really trying to say, Jeff?" While I understand what you are seeking to imply from the present-day associations within modern mathematics, your arguments are insufficient to show that CSP adopted the logic of group theory in his world view of mathematics, logic or philosophy. Furthermore, the writings of CSP point the opposite direction, that is, he believed in his own personal views of logic, mathematics (his father's "associative algebra") and his over-arching philosophies of all things reasonable and unreasonable. CSP did not endorse set theory, nor is his logic of relatives grounded on set theoretic foundations. CSP's geometric view of continuity does not cohere with the sense of continuity used in the push-outs and pull-backs of category theory. CSP's representations of triads and medads do not infer mathematical groups. CSP's representations of beta existential graphs do not infer groups or categories. CSP's representations of gamma existential graphs, in so far as I understand them, do not infer groups. (Jeff, when I use the term "group theory", the references are the axioms that define a group and group logical operations.) >From the applied viewpoint of chemical mathematics (starting with the atomic >numbers), groups are insufficient for developing the logic of "proof of >structure". Proofs of structure are a critical component of physical atomism, >one of the several necessary essences of chemical logic. Proofs of structure are necessary to connect the logic of chemistry to the logic of molecular biology and hence to the logic of biology and medicine in general. Furthermore, proofs of structure generate three dimensional mathematical objects - including the "handedness" of optical isomers, which represent electro-magnetic interactions between matter and light, that is, consequences of Maxwell's equations. (A digression. I am more than a little aware of the fact that bio-semioticians and many philosophers who post here out right reject the associative logic of proofs of structure.) Jeff, I published a long philosophical paper on the reasoning that I use to separate chemical logic from category theory. A leading mathematical category theorist responded to my views in a special issue of Axiomathes. This special issue of Axiomathes was dedicated to the general notion of mathematical biology and mind as represented within the symbols of mathematical category theory. Three final points with respect to CSP's world view: the representations of sinsigns, icons and arguments may require three separate and distinct symbol systems to transfer meaning from presentative signs to icons to the linguistic structures of well-pointed conclusions. (By "well-pointed conclusions" I mean symbolic sentences that match, point by point, presentative symbols with representative symbols that demonstrate pragmatic realism.) Cheers Jerry
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