List, John: Because many readers may find my posts difficult to comprehend, I am posting a copy of an abstract that provides some philosophical insights into my beliefs. The abstract is from 2009. The logical developments have continued during the intervening years, now nearly a decade.
Cheers Jerry --------------------------------------------------------------------------------------------------------------------------------- Abstract, CASYS 2009, Professor Daniel Dubois, Liege, Belgium, August 3-9, 2009. Perplexity I. Logical Structures of Irregular Mathematics Jerry LR Chandler, Research Professor, George Mason University, Fairfax, Virginia, USA. 22102. Key Words: Irregular mathematics, perplex number systems, chemical valence, catalysis, biomathematics, C. S. Peirce, L.J. Brouwer, G. Leibniz, The irregularity of natural systems creates a challenge to applied mathematics. In recent decades new mathematical theories of catastrophes, chaos, fractals and fuzzy sets developed improved approaches to some irregularities. Philosophically, these theories invoke the Aristotelian concepts of efficient and formal causality and the uniformity of the line. It was recently demonstrated that the Aristotelian concepts of material, formal and (local) final causality create a basis for a natural number system, the perplex number system (Dis. Appl. Math. 157 (2009) 2296-2309). Electrical particles enumerate the sources of material causality. The quality of electrical fields (Coulomb’s law, as attraction or repulsion) is used to enumerate relations. The enumeration of particles and relations is sufficient to represent the “categorical sketches” of chemical structures and the physical organization of the electrical particles as labeled bipartite graphs. This paper describes the logical and philosophical background supporting this scientific number system. Although the logic of the perplex number system originates in the material sciences, the roots of the mathematics of irregularity can be traced to the early Greek notions of number and to the writings of G. Leibniz, C. S. Peirce and L. Brouwer. Leibniz offers notions of substance and attributes; C. S. Peirce offers notions of relation and diagrammatic logic; and L. E. J. Brouwer offers the role of two-ity as a syndeton between mathematical semantics and philosophy. … the concept of an individual substance involves all its changes and all its relations, even those which are commonly called extrinsic. (G. Leibniz, Letter to Arnauld, 14 July 1686.) An attribute is the predicate in a universal affirmative proposition (of which the subject is the name of the thing. (G. Leibniz, in Couturat, OFI, p.241) “Firstness is the mode of being of that which is such as it is, positively and without reference to anything else. Secondness is the mode of being of that which is such as it is, with respect to a second but regardless of any third. Thirdness is the mode of being of that which is such as it is, in bringing a second and third in relation to each other.” (C. S. Peirce, Letter to Lady Welby, 1904 Oct 12, CP 8.328) Completely separating mathematics from mathematical language and hence from the phenomena of language described by theoretical logic, recognizing that intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a motion of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the twoity thus born is divested of all quality, it passes into the empty form of the common substratum of all twoities. And it is this common substratum, this empty form, which is the basic intuition of mathematics. (Brouwer, 1981, 4-5) As they are unencumbered by the current hegemony of membership, containment, functions, continuity, and infinity, these three notions contribute to the logical grounding of the mathematics of local irregularity. The perplex mathematics of irregularity requires units, integers, numerals, parity, triadic identities, multisets and labeled bipartite graphs. Parity serves as an invariant for logical operations such that quantity is preserved as a weak symmetry between units and integers. A new inductive inference termed synduction creates the graphic syndetons. Geometry and arithmetic operations are secondary attributes of spatial graphs. A mutation of relations between integers and units requires the introduction of the concept of time. The perplex number system generates exact mathematical descriptions of material situations that were previously intractable. The irregularity of chemical valence, chemical isomers and chemical catalysis are readily expressed within the perplex number system. A notation for chemical systems and chemical catalysis follow naturally from the principles of material causality. An unbounded number of homologous irregular systems are easily shown. The polysemic usage of the critical term “identity” conflates the historical discussion of material and efficient causes and artificially induces many pseudo-conundrums. In fact, the perplex number system conjoins logically with the real number system, both statically and dynamically. The same material entity generates an abstract meso-group and the x-ray diffraction pattern of a real crystal structure. The law of mass action connects the material, temporal and thermodynamic relations between perplex multi-sets and sets. Spatial motion of electrical particles is described by chemical QM. Applications of irregular mathematics and synductive logic to the emergence of life, to reproduction of an individual organism, to genomic medicine and to the origins of consciousness are in progress. Jerry LR Chandler McLean, VA Revised June 20, 2009 > On Jul 13, 2018, at 6:19 PM, Jerry LR Chandler <jerry_lr_chand...@icloud.com> > wrote: > > List, John: > > Thank you for your expressions of your well-held beliefs about extension. > But, I find no relationships to the intensional logic of chemistry, biology > or medicine. > > I am leaving for Europe shortly and do not have time to respond in detail. > Perhaps we can re-open the topic after I return. > > But, I will point out that, if one analyses a perplex sentence that > empirically, logically, and numerically represents a perplex chemical > structure, the critical propositional distinctions resides in the > relationships between subjects, copula and predicates in their usual > grammatical meaning. Of course, the issue of CSP’s triadicity is also open. > > The belief structures you list, John, tend toward reductionism. > It is totally unclear to me how such beliefs would logically relate to > emergence, evolution and the structures of natural sorts and kinds, including > mind, which are extensions of informed numbers and lexical fields. > > I changed the name of the thread to bookmark the topic. > > Cheers > > Jerry > > >> On Jul 13, 2018, at 5:28 PM, John F Sowa <s...@bestweb.net> wrote: >> >> On 7/13/2018 5:30 PM, Jerry LR Chandler wrote: >>> In your belief system, is the conceptualization of CSP’s "Beta graphs >>> (FOL)" identical in all logical respects with predicate logic? >> >> As I said, by extensional criteria -- truth values and provable >> theorems -- they are logically equivalent. That means >> >> 1. Every proposition stated in one notation can be translated >> to a proposition in the other notation that has the same >> truth value for all possible models. >> >> 2. Every theorem that can be proved in one system (EG with its >> rules of inference) can be proved in the other (predicate >> calculus with its usual rules of inference) and vice-versa. >> >> This is more than a "belief". It is a theorem that can be proved >> very simply: (1) Show how to map either notation to the other. >> (2) Show how every rule of inference of one can be proved in terms >> of the other. I do that in the egtut.pdf paper. >> >> However, EGs have significant advantages that put them at the >> forefront of ongoing research in cognitive science: >> >> 1. The rules of inference for EGs are simpler and more symmetrical. >> That makes certain kinds of proofs much simpler with EG rules >> than with other FOL rules -- for example, the solution to problem >> #24 by Larry Wos. See egintro.pdf or egtut.pdf. >> >> 2. The mapping from EGs to natural languages is simpler and more >> direct. (See the references at the end of egintro.pdf.) >> >> 3. As the psychologist Johnson-Laird observed, EGs are a better >> candidate for a "natural logic" than predicate calculus. >> (See the same references.) >> >> 4. The two-dimensional EGs make it possible to insert propositions >> stated by icons with added indexes into any area of an EG by >> using Peirce's rules of inference. Peirce did not do that, >> but he seemed to hint at that option. In any case, it is >> compatible with the many comments that Peirce did state. >> >> I am currently writing a paper that goes into more detail about >> point #4. See http://jfsowa.com/talks/ppe.pdf >> >> John >> >> ----------------------------- >> PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON >> PEIRCE-L to this message. PEIRCE-L posts should go to >> peirce-L@list.iupui.edu . To UNSUBSCRIBE, send a message not to PEIRCE-L but >> to l...@list.iupui.edu with the line "UNSubscribe PEIRCE-L" in the BODY of >> the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm . >> >> >> >> >
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