EV Review, June, 2009

Algebraic Biology: Creating Invariant Binding Relations for Biochemical and Biological Categories

Jerry LR Chandler, Ph. D. Research Professor George Mason Universsity Krasnow Institute for Advanced Study Washington Evolutionary Systems Society 837 Centrillion Drive, McLean, Virginia, USA 22102 Jerry_LR_Chandler @ Mac.com 571-296-4056

Re-Submitted to:

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Axiomathes: An International Journal in Ontology and Cognitive Systems Editor-in-Chief: Roberto Poli June 17, 2009

6/17/09 Review Draft, Submitted 5/1/09

Running title: Creating Invariant Binding Relations Type: Research Article Title: Algebraic Biology: Creating Invariant Binding Relations for Biochemical and Biological Categories Author: Jerry LR Chandler, Ph. D. Affiliations:

George Mason University Krasnow Institute for Advanced Study Washington Evolutionary Systems Society

Address: 837 Centrillion Drive, McLean, Virginia, USA 22102 Email address: Jerry_LR_Chandler @ Mac.com Cell Phone: 571-296-4056

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Abstract: The desire to understand the mathematics of living systems is increasing. The widely held presupposition that the mathematics developed for modeling of physical systems as continuous functions can be extended to the discrete chemical reactions of genetic systems is viewed with skepticism. The skepticism is grounded in the issue of scientific invariance and the role of the International System of Units in representing the realities of the apodictic sciences. Various formal logics contribute to the theories of biochemistry and molecular biology and genetics. Various paths of extension are invoked in these formal logics in order to express the information of biological apodicticism. Symbolizing the appropriate notations for invariant relations and for biological extensions of relations is fundamental to the exact generating functions of discrete algebraic biology. Aspects of philosophical perspectives of the relation scientific number systems are contrasted. The deep distinction between physical motion and biological motion is expressed in terms the roles of Aristotelian causes. The interior motion within perplex numbers is contrasted with the exterior motion of physical systems. The need for a new mathematics for biology is suggested. Key Words: algebraic biology, systems biology, invariance, memory evolutive systems, anticipatory systems, perplex number system, mathematical philosophy of science

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Algebraic Biology: Creating Invariant Binding Relations for Biochemical and Biological Categories. … the concept of an individual substance involves all its changes and all its relations, even those which are commonly called extrinsic. G. Leibniz, (1686) In a complete theory, there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. Einstein, Podolsky and Rosen, Phys. Rev. 47, 777, (1935) A pair and an apple are two fruits; a brick and a castle can barely be called two things; a brick and a musical sound are not two, a man and a truth and a taste of an apple do not make three and so on. Otto Jesperson, Philosophy of Grammar, (1934) p. 189, Prologue. Unintended Consequences. In 1795, France began to construct, for purely economic reasons, national standards for units of measure based on natural objects, the regularities of the decimal system and simple arithmetic operations. Over several decades, seven units were elevated to become the invariant basis of scientific relations. The seven units - length (meter), mass (kilogram), time (second), electricity (amperes), temperature (kelvin), luminosity (candela), and mole are postulated as natural quantities. These seven measures became the basis for standard quantity in commercial transactions. An unintended consequence was that the economic units were gradually transformed into the fundamental units of nature expressed in terms of the decimal system. A philosophy of the physical sciences developed around the coherence of the artificial economic units of distance (space), mass, and time. A further unintended consequence was that the quantity named “mass” became a substitute for individual matter. The concept of individual identity lost its face to an economic unit. That is to say, the concept of individuality was generalized to a concept of mass in order to meet the immediate economic needs of the State. As the chemical sciences advanced in the 19 th and 20 th Centuries, the irreducible nature of each chemical element absolutely necessitated a new logical notation for matter. Following Volta’s construction (in 1800) of a voltaic pile from alternating layers of zinc and silver, the facts about chemical entities were interpreted as structures composed from electrical particles. Chemical progress revealed three deep unanticipated conundrums for the simplistic view of the French system of standardized decimal units of quantity. Each chemical element was unique, but the regular order of the decimal system did not parallel the irregular order of the properties of chemical elements. Secondly, the electrical properties of a chemical element were not fixed for all time and place but rather changed by discrete units, not decimal units. Thirdly, the masses of a collation of elements did not specify a unique economic entity; numerous structural forms of molecular isomers were recognized. These families of compounds with the same mass but different structural graphs imply that the internal relations of matter determine the chemical, biological and economic identity. (For example, virtually all biological chemicals are isomers, including DNA, RNA, and proteins.) Finally, the irregularities of the changes of chemical relations can be transitive, without being decimal; arithmetic regularities do not describe changes in isomeric biological systems. A gene cannot be decimalized. Thus, life itself is peripheral, if not excluded from the international scientific system of standardized decimalized logic. The unanticipated consequences of this exclusion were simple and direct. Symbol systems for

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chemistry and genetics were invented that expressed the empirical observations of individuality without regard to the international standardized decimal system! Consequently, in the scientific literature, a natural object may manifest itself in one of several different logical systems. A scientist or philosopher must choose among symbol systems to communicate observations. Three principle choices exist. The physical symbol system usually invokes a triad of philosophical references - continuity, linearity and infinity. The biochemical symbol system invokes a pentadic referential identity system - units, integers, multisets, adjacency graphs and internal relations. A genetic symbol system refers to both an organism and to multisets of chemical structures. A defining attribute of each symbol system is that it invokes its own grammar, its own rules. The absence of an invariant logical system of scientific representation is confusing, prone to logical misunderstanding, subject to logical misinterpretation, liable to logical miscommunication, as well as a source of costly errors. The coherency of the life sciences – biochemistry, biology, physiology, anatomy, pharmacology, toxicology, epidemiology, and medicine – develops from the logical consistency of relations expressed within these new nondecimalized symbol systems. For example, the discrete character of chemical isomers encodes directly genetic information. The internationally standardized system of universal units lacks the capacity to describe the fundamental the source of biogenesis. The empirical “gene,” as a necessary component of life, is not describable in terms of the regularity of International Units. To be valid, a universal system of measure must address both the intrinsic irregularities of chemistry and life, as well as the regularities desirable for commerce. The International System of Units is not universal; it is merely physical. A further unintended consequence of the French revolution is that modern philosophy of science is based on the physical concepts of motion and momentum in time and space, rather than the quantities of the individual identities of unique species of matter and of life. Algebraic biology, in order to model the exactness of biomedical-genetic systems needs a system of exact accounting. Such a number system, the perplex number system, was recently proposed. With respect to living systems, this number system is a self-reflexive symbol system, the mathematical analogy of mirror neurons. Individual human beings that develop the capacity to calculate with this abstract number system can apply the consequences self-reflexively. The homologous empirical symbol system, the formal notation for chemistry, is routinely used to produce nutrients, drugs, therapeutic agents, genes, proteins, and numerous commercial products. Extension of the methodology to the synthesis of life itself is on the distant horizon. The perplex number system creates a rational basis for correcting the unintended consequences of the failed attempt to decimalize nature. The intent of this essay is to provide the reader a few of the relevant distinctions between the various philosophies of mathematics and the sciences germane to symbolic invariance. In light of a new formal number system for chemistry, the open question, “To what extent can traditional mathematics address the rhetoric, the grammar, and the logic of living systems?” is very briefly introduced.

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I. Introduction During the past century, the chemistry of life emerged to become the central focus of modern science. The fact that the structural logic of life is comparable to the structured logic of mathematical category theory opens unparalleled opportunities for new mathematics. Two inchoative principles for such a new mathematics are simple. First, chemical elements combine in ratios of small whole numbers. Secondly, from simple ratios among a few common substances, such as nitrogen, oxygen, carbon dioxide and water, the graphic pathways to the codes of life are demonstrated. The codes of genetics, the codes of the immune system, the codes of metabolism, secondary codes for actions of drugs and toxins and especially the tertiary codes of organic catalysis are the invariant codes invoked to describe biological motion. Algebraic biology is simply the study of life-motions. The creation of the codes of life emerged from the pragmatic accounting system of formal numerical relations rather than images of orbits or orbitals. What is it that is within biochemical mathematics that creates it biological motion? I believe that the uniqueness of chemical sciences lies in the constraint that its logic is self-reflexive with the logic of life itself. As noted in the prologue, this essay explores interrelationships between the self-reflective and self-reflexive reasoning of life and the intentionally sterile formalisms of traditional mathematics. A simple answer to the provocative question of what limits the relevance of formal mathematics to life is that symbolic codes of life are fundamentally different from symbolic codes of logic derived from the mathematical philosophy of Bertram Russell, a principle proponent of atoms of logic and set theory. Of course, Russell’s philosophy emerged from Cantor’s imaginative extension of the real number system to a hierarchy of infinities. These abstract codes are independent of biology. Recently, I proposed an abstract, albeit static, description of the numbers that encode the mathematics of life (see Chandler, 2009). The logic of the number system is constructed from three principle components - a finite list of reference symbols, the synductive logic of conjunctions (meso-syllogisms) and the necessity for regular and irregular concepts of extension. (The appendix summarizes critical features of this system.) Further, the existential logic of chemistry supposes that the absence of a thing means the absence of measurable properties; the presence of something means the present of measurable properties-this means that the concept of zero is absent from the perplex number system. Of the several possible pathways of development of this line of reasoning about the role of codes in science, I choose to approach these issues from the general perspective of human communication as encoding and decoding of symbolic (mathematical) messages. Two competing traces of threads of logic contribute to the conundrum of the self-reflexive encodings of human communication. One is a trace of fact; the other is a trace of reasoning. In the logic of the medieval Modestae, these were referred to as modi essendi and modi intelligendi. The latter trace is couched often in the conceptual independence of mathematical reasoning as a component of human consciousness. This line of logic is perhaps best exemplified by the famous ontological assertion of Russell’s student, W. V. Quine, “To be is to be a quantifiable variable.” The ontological position of Quine runs opposite to the logical trace constructed from over two hundred years of chemical experimentation, modi essendi, and the logic of the demonstrative sciences. This is a system of synductive logic for relating the dependencies between elemental units and properties as formal mathematical objects, graphs with internal relationships. The dichotomy between Quine / Russell ontology and the chemical ontology is simply stated by the

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paraphrase, “To be is to be unique.” A belief about a mathematical/scientific variable is a belief, usually nominalistic, about reasoning. A belief about a unique chemical structure creates a new symbol that commutes with experimental fact and is coherent with the extended family of relatives. These two belief systems must necessarily follow different encoding and decoding of messages, different logical operations (Chandler, 2009). Is it conceptually possible to resolve the logical tensions between these two ontologoies, “To be is to be a quantifiable variable” and “To be is to be unique”? The more immediate motivation for this essay comes from meditation on the remarkable text of Ehresmann and Vanbremeersch (hereafter, EV), Memory Evolutive Systems (MES). MES extends the frontiers of the deeply intertwined relations between French philosophy and French mathematics. It is a brilliant discourse, reminiscent of such illustrious French thinkers such as Descartes, Poincare, Bergson, Thom, Merleau-Ponty, and De Chardin. The text intertwines strands of reasoning from two of the prominent philosophical developments of the past century, philosophical phenomenology and mathematical category theory as a philosophy of mathematics. This text projects the philosophy of the mathematical categories thru general systems theory, thru cybernetics and thru representation of the dynamical systems to conscious human decisions. From a trace of pure reasoning perspective, the authors present a mathematical account of the relational mathematics that spans the domains of scientific discourse from the foundations of quantum mechanics to cultural systems theory. The flow of conceptual time is harnessed as a universal “natural glue” and concomitantly as a surrealistic image of cultural momentum itself. Thus, a general presupposition of physicalistic logic underlies the categorically oriented narrative of the MES text. I accept the philosophical position of Ehresmann and Vanbremeersch as possible physical inferences from the mathematical presuppositions. But, I find no way to connect these mathematical symbols of reasoning to the symbols of reasoning used for interpreting chemical facts. The modes of inference and the diagrammatic logic of the perplex number system differ from the modes of inference and diagrammatic logic of “natural transformations.” Consequently, these remarks merely seek to outline an approach to describing the incompatible methods of symbolic reasoning used in mathematics, physics, chemistry, and medicine. The basic issues discussed here are often addressed under different guises. For those readers interested in the roots of these issues, I refer to the following works: Solomon Bochner, The Role of Mathematics in the Rise of the Sciences (1960), Einstein, Podolsky and Rosen, Can Quantum Mechanical Descriptions of Physical Reality Be Considered Complete? (1935), H. Weyl, Philosophy of Mathematics and the Natural Science (1966), Ernest Mayr, The Autonomy of Biology (2004), Alicia Juarrero, Dynamics in Action (1999), Brian Rotman, Mathematics as Sign, (2000) and my own recent papers on the perplex number system, including An Introduction to the Perplex Number System (2009) and Ordinate Logic (2008). Given the disjoint roots of this essay, the reader should prepare for a rambling discourse in which ideas from widely disparate arenas of human activity are teased into proximity and contrasted. The nimble will find this exhilarating, others dismaying. What mathematical options exist for interpreting the signs of life? This is an ancient question; the early history of which is discussed in the work of Manetti, Theories of Sign in Classical Antiquity (1993). MES is one abstract interpretation of the signs of life. The philosophy of biological categories as generalized by Ehresmann and Vanbremeersch is a unique hierarchical approach to the signs of life as biological categories. An earlier categorical position was put forth by the mathematical biologist, Robert Rosen

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(hereafter, RR), in Life Itself (1991). The logical tensions between the two systems of physicalistic thought are palpable. Rosen proposes that the formal logic of mathematical symbol system of category theory is not representative of life itself. He pleads for a theory of natural systems. A hypothetical point of logical departure that separates physics from life is pinpointed by Rosen’s colleague, Howard Pattee. For Pattee, the absence of initial conditions necessary for physical description of equations turns the problem into a different sort of symbolic or semiotic problem. In the absence of a crisp starting point for reasoning, the concatenation of scientific reasoning wilts. In other words, the RR interpretation in Life Itself is that life itself is incommensurable with mathematical category theory. Thus, the difference that makes a difference, to cite the words of Gregory Bateson, between the MES text and the Life Itself text is a philosophical difference on the meaning of mathematical symbols. One is forced to conclude that EV and RR both use the symbolism of category theory to express deeper intuitions, beliefs about the nature of nature. Applied mathematicians are not unanimous about the role of belief in constructing models. The creator of the philosophy of mathematical intuitionism and constructivism, L. J. Brouwer (1906), asserted that “But what I bring you now is exclusively concerned with the way mathematics is rooted in life, and what therefore the starting point of any theory should be.i The issue among the categorians appears to address the converse question, How is life rooted in mathematics? Relations among facts, empirical chemical, biological and medical facts, are not the principle issue creating the tensions between EV and RR. Rather, in my opinion, the central issue is the art of mathematical reasoning about life. Both mathematicians and physicists tended, for centuries, to reason in great generality. Indeed, Alicia Juarrero asserts, “Explanation qualifies as scientific only when trafficking in the universal. As a result, episteme cannot explain the contingent and individual as such.” This claim of generality is further illustrated by a noted logician’s (Quine) ontological assertion that “to be is to be a quantifiable variable”! However, as experimental demonstrations, chemistry and biology focus on the particulars of realism. They are apodictic sciences. Indeed, the modern science of the primitive Greek “atoms” is based on an exact ordered list of atoms, each with a unique set of attributes or properties. Correspondingly, the genetic material, DNA, one of the two sources of molecular biological motion, is a unique molecule with a particular isomeric graph. The noted biologist, Ernst Mayr, contradicts the ontological claim of W. Quine, stating emphatically, biology is unique. The categorical dichotomy of the roots of these two scientific categories is as logically crisp as true and false. The conclusions of RR and EV with respect to the appropriate mathematical interpretations of biological systems are also dichotomous. EV asserts a model of conscious within the continuous dynamics of category theory while RR asserts the necessity of a separate symbol system for life itself. I will seek to weaken the categorical conundrum of reasoning about scientific categories by reasoning about facts, facts based on the individual electrical particles of chemicals. This approach is based on the dual causality of nature and nurture, the convergence of abstract codes entailed by being alive and the matching with codes from the surrounding ecosystem, the ecosis. While the scientific facts come from 21st century biochemical codes, the interpretation of these facts is rooted in the earlier “logic of relatives” of C. S. Peirce (1839-1914). Peirce’s sense of relation was explicitly stated in terms of three arbitrary logical terms, firstness, secondness and thirdness: “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.

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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.” (Letter to Lady Welby, 1904 Oct 12, CP 8.328)

The Peircian generalization of the concept of a relation does not depend on any mathematical notion. Examples of the Peircian sense of ontological relations are the relations of hydrogen and oxygen to form water or to form hydrogen peroxide. Ordinarily, the property of three nouns, firstness, secondness, and thirdness, imply the transitive order of the initial three positions of a list. A logician could construct a simple valid syllogism (of the class Barbara) from these three nouns as follows: Secondness precedes Thirdness. Firstness precedes Secondness. Therefore, Firstness precedes Thirdness. However, the grammar of the three, parallel, tightly coupled ontological propositions of Peirce does not imply this syllogism. The antecedent term for the beginning position of the list (Firstness) does not imply an extension to another position in the list, Secondness. Analogously, the ontology of Secondness does not imply a further extension to another position. Thirdness refers not to an extension of the list, but to the verb of ‘bringing’, or creating a relation. The initial proposition of Peirce requires that the basic ontology of an object (firstness) require a presence, a presence conceptualized as being without either reference or extension. Analogously, the presence of secondness does not infer yet another logical term. The deduction from the three independent but grammatically parallel propositions is an unambiguous definition of a relation as a totality within a proposition. Thus, Peirce succeeds in logically separating the concept of relation from the concept of units, integers, numbers, arithmetic operations, functions, continuity, infinity, and other terms, such as categories. Peirce’s accomplishment, based on an exact usage of relative pronouns, is an unparalleled logical tour de force. No mention of element, class, membership or containment is used, thus sharply differentiating this concept of relation from that alternative definition of membership in a set. The Quinian slogan, “to be is to be a quantifiable variable” annihilates this semantic distinction of Peirce, a semantic distinction that is integral to the unique relations of chemical structures, such as DNA. Further more, Peirce achieves the notion of a relation as a syndeton (copula) without invoking the concept of a sentence, of a verb or of a predicate or of a predication! The parallel grammars of the three Peircian propositions deftly create the syndeton. The grammar of physics is often taken to be consistent with predicate logic and set theory and category theory. This is not the case for chemistry. But predicate logic is not consistent with the logical operations on atomic numbers and chemical valences. Grammatically, the Peircian notion of a relation can be applied to individual molecules of chemical isomers (Each isomer has the same multiset of parts.) It can be applied to an individual molecule of a chemical isomer (e.g., D-tartaric acid) as well as its materially identical partner, (e.g., L-tartaric acid), both of which are represented by the same mathematical graph. This “two-ity”, to use a phrase of Brouwer, is termed a pair of optical isomers. The application of Peirce’s notion of relation to biological relationships, such as inheritance, is obvious. Peirce’s sense of relation is that of a realist, not a nominalist. From the perspective of mathematical category theory, the Peircian logic of relation is a generalization of more restrictive directed arrow diagrams.

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The abstractness of the Peircian notion of relation avoids the hegemony of the concept of efficient causality as the principle idea that relates the mathematical functions to dynamical systems theory and metaphysics. (Alicia Juarraro, in Dynamics of Action (2000), illuminates the historical philosophical roots of these concepts.) I will return to the importance of this abstract in the concluding section of this essay.

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II. Views of applied mathematics uncomplicated by biological binding relation. The coherence of the chemical sciences is ensured by the small exact set of universal reference symbols, the list of chemical elements or, abstractly, the perplex numerals. A coherent philosophy of mathematics is most noticeable by its absence. Saunders Mac Lane’s impressive list of six different philosophies of mathematic without giving priority to any one of them, communicates the murky state of affairs within the mathematical community. The coherence of philosophies of applied mathematics is even less apparent, as will be discussed. Algebraic biology needs a coherent origin to be exact. It was the recognition of the fact that chemical logic was excluded from the popular philosophies of mathematics that motivated the work to construct an abstract mathematical system of logic from the atomic numbers, the perplex number system. The atomic numbers, as integers, are ordered within the roots of empirically demonstrable facts of nature. The origins of the individual perplex numerals are known. The atomic numbers are, I believe, the natural origins of mathematics, the logical basis of the consistency between chemical molecules, metabolism, catalysis, genes, emergent biological dynamics, human consciousness, the emergence of synthetic symbol systems and the symbolic communication of mathematics itself. These consistencies are termed the self-reflexivity of the perplex number system. Subsequent study of the works of C. S. Peirce, Ernst Cassirer, and Guan Carlos Rota led to a belief that applied mathematical biology is grounded in the concepts of combinatorial operations on units and integers. At issue is the certitude of the calculations of algebraic biology. In this section, the reader’s attention is called to some provocative statements from mathematical philosophers relevant to the symbolization of biological invariance. The curious reader is invited to contrast the ontological origins of mathematical philosophy with concrete origins of the perplex numerals. Should the nominalistic beliefs in the abstract hegemony of sets, continuity and infinity negate the role of perplex numbers and electrical particles in generating the invariances of algebraic biology? The objective is to open the question of ontological relation between pure and applied mathematics in the following sections. Many historical traces of the philosophy of mathematics have been proposed. One (medieval) trace of growth of belief about the mystical powers of mathematics starts with an all powerful and perfect God as a unity, connects to the circle as a perfect geometry that connects to perfectibility of mathematics that connects to the potential perfection of nature and perhaps even to the righteousness of a perfect man. The concept of mathematical exactness can be conflated with the concept of perfection of numerical data from nature. Sadly, applications of mathematical logic to biology, epidemiology and medicine can reveal hypothetical order at the expense of obscuring the irregularity of individuality emerging from dual causality intrinsic to life itself. The works of EV and RR differ with respect to the philosophy of mathematics, particularly on the notions of nature of binding of relational algebras to living systems. EV supposes a physicalistic view. RR seeks to relate chemical reactions to physical states and physical states to categories. At issue is the potential for mathematical perfection to occlude biological perplexity. Physicalism as a basis for the relations between mathematic and all of the sciences is promoted by Weyl in “Philosophy of Mathematics and Natural Sciences” (1949). Weyl bases his projections from mathematics to physics and hence to chemistry and biology on traditional notions of mathematical relations. It appears intractable to represent either the structuralism of chemical isomers or the dynamics of biological reproduction within this logical framework. Both EV and RR develop their categorical


theories without invoking the philosophy of Weyl. This raises the generic question as to what is the appropriate basis for algebraic biology. Next, the reader is presented several general philosophical views of mathematics that are relevant to algebraic biology. The objective is to provide the reader with a range of cognitive images contributing to the origins of the tensions separating the philosophies of RR and EV from those of the author. At issue is the concept of pairing of symbols as interpreted by nominalists and realists. 1. Prior to the advent of twentieth century philosophy, mathematical truths were intertwined with philosophical truths. Benjamin Peirce (1809-1880), father of C. S. Peirce, thus conflated the science of quantity with the logic of nature when he defined it as: “Mathematics is the science of necessary conclusions.” Although his writings are relatively silent on his philosophy of mathematics, this definition of mathematics may express the approach of Robert Rosen toward the relation between mathematics and nature. 2. Albert Einstein crisply separated the independence of logical operations from both the interiority of human experience and the exteriority of metaphysics. “How can it be that mathematics, being after all, a product of human thought independent of experience, is so admirably adapted to the objects of reality?” “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” 3. In “Mathematics as Sign”, mathematics professor Brian Rotman professes to the fundamental interiority of mathematical activity. He proposes a tripartite arrangement of mental mathematical activity within consciousness that he holds to be free of the activity of muscular motions (mental exformation prior to motor expression.) For Rotman, the externalization of these thoughts as symbolic expressions is a separate mental activity, clearly and distinctly from the exformative activity of generating a message. (This view parallels that of Einstein.) “Thus, mathematics, characterized here as a discourse whose assertions are predictions about the future activities of its participants, is about – insofar as this locution makes sense – itself. The entire discourse refers to, is true about, nothing other than its own signs. And, since mathematics is entirely a human artifact, the truths it establishes – if such is what they are – are attributes of the mathematical subject; the tripartite agency of Agent/Subject/Person who reads and writes mathematical signs and suffers it persuasions. … Mathematicians believe because they are persuaded to believe; so that the what is salient about mathematical assertions is not their supposed truth about some world that proceeds them but instead the inconceivability persuasively creating a world in which they are denied. Thus, instead of a picture of logic as a form of truth – preserving inference, a semiotics of mathematics would see it as an inconceivability – preserving mode of persuasion – no mention of truth anywhere.“ p. 41(2000)

4.

A temporal root, entwined with “two-ity” is propounded by the E. L. Brouwer, the founder of intuitionist’s mathematics. In defending his natural philosophy of time as a driver for symbolization of “two-ity”. “The first act of intuitionism completely separates mathematics from mathematical language, in particular from the phenomena of language which are described by theoretical logic. It recognizes that mathematics is a language less activity of the mind having its


origin in the basic phenomenon of a move of time, which is 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 diverted of all quality there remains the common substratum of all two-ities, the mental creation of the empty two-ity. This empty two-ity and the two unities, of which it is composed, constitute the basic mathematical systems. And the basic operation of mathematical construction is the mental creation of the two-icity of two mathematical systems previously acquired and the consideration of this two-icity as a new mathematical system. (p. 523).” The intuitionist notion of time is particularly revealing. Modern biochemical studies show that biological time is generated by a duality of causality; genes that encode proteins that contribute directly to generating circadian rhythms and catalytic reactions that that are subject to temporal adjustment. One can only speculate on how Brower would have adapted his paradigm to this massive body of empirical facts. (Brouwer’s notion of two-ity strongly influence the philosophical development of the system of ordinate logics for synthetic symbol systems (Chandler, 2008.) 5. A respected analyst, Solomon Bochner, in “The Role of Mathematics in the Rise of Science”, places mathematics at the root of science yet recognizes the challenges of invariance relations between the two in general and with respect to chemistry. “Ours is an age in which scientists are the wise men and the root of this wisdom is mathematics. But, mathematics, if taken by itself, is almost a pastime only, albeit as esoteric one; or so it seems. What makes mathematics so effective when it enters science is a mystery of mysteries, and the present book wants to achieve no more than to explicate how deep that mystery is.” (preface, p. v.) “Cognitively, mathematics is not part of science, or subordinate to science as a whole. A scientist in his working mood may please himself to view mathematics as nothing more than a “handmaiden” of science; but, philosophically, this is a shallow statement. Ontologically, mathematics and science have different subject matters, and the difference cannot be bridged, unless by deliberate identification. Any natural science deals with an “external world,” whatever that may be. This even applies to the psychology of the human mind, say. In a scientifically conducted psychological introspection or self-inspection, the internal object of the inspection becomes “externalized” by the procedure itself. Mathematics, however, deals, strictly speaking, only with objects of its own “aesthetics” perception and aprioristic emanation and these objects are internally conceived, internally created and inwardly structured, even though they are tangible, substantive, selfsame and somehow communicable and sharable.” p. 46 “At this stage one may be tempted to say that if stringent conceptualization is not yet characteristic of mathematical thinking, then it is linking and pairing with symbolization is, and we mean bona fide, outwardly recognizable symbolization, in print, with which we are all familiar, and not implicit and internal symbolization of the kind we imputed to myths above. Now, the use of outward symbolization has a certain powerful pace in mathematics, but, as one can quickly see, if the manifest use of symbols is to be made into a significant part of the characterization of mathematical thinking, considerable clarifications have to be proffered. Chemical theories of valency and bond are steeped in highly conventionalized symbolism from which they could not be separated. But it is not a symbolism which is interlocked with


mathematization; in fact, in fully theoretical works on quantum chemistry the ‘typical symbolism’ is displayed less emphatically than in regular manuals of chemistry. Bochner does not mention the natural order of the atomic numbers, thus neglecting the origin of the relations of natural information in biochemical systems. He seeks to place chemical theory within the frame of quantum chemistry, illustrating no interest in the fact that the discrete logic of units and integers in valency and bonding is remote from the logic of continuity of motion assumed in formulating quantum mechanics. Paradoxically, he notes the absence of chemical symbolism from quantum chemistry! Bochner fails to recognize that the international system of units precludes the use of chemical symbolism in mechanical calculations, as was discussed in the prologue. On the relation between the regularity of mathematics and the pragmatics of applied mathematics. Applied mathematics is often highly pragmatic at the expense of scientific authenticity. In the usual practice of biomedical research, concatenation of propositions can be ordered to create correspondences to concur with nature, to concur with the implication of EPR axiom of correspondence. From the perspective of classical logic, any list of terms can be organized as a concatenated list of sorites with (n-2) syllogisms. Such concatenations of terms, propositions and syllogisms can provide scientific evidence for cause-effect relations, even in highly irregular situations. The risks of substituting nominalistic judgments for the logic of nature are substantial in the biomedical sciences where human weal is often at stake. Lest I be misunderstood, the linearity and regularity of time and space is not viewed as controversial from the perspective of chemistry and biology. What I am skeptical of is the assertion that regularity of Descartian geometry and logic of continuous functions is universally and necessarily applicable to the irregularity of chemistry and biochemistry as manifest in life itself.


III. The Binding Relation for Algebraic Biology and for “Memory Evolutive Systems” (MES) A central thesis of “Memory Evolutive Systems” (MES) is that category theory is a universal language for the systems sciences, including the neural and cognitive sciences. Category theory, in this context, is marked by a diagrammatical logic composed as directed graphs. The vocabulary describing directed graphs is technical. Translation of the abstract technical language of category theory into either technical languages of the sciences (excepting string theory) or normal conversational language cannot be done with fidelity as the critical logic terms are embedded within layers of intermediate abstractions that are seldom encountered in either science or society. Readers with some knowledge of category theory will benefit from the imaginative logical diagrams that transcend the abstract with the common place. Diligent readers of MES without knowledge of category theory but with a substantial knowledge of advanced mathematics will be able to follow the general scheme of reasoning and the philosophical perspective. Typical scientific readers will benefit substantially from the careful study of the logical diagrams and the rhetoric, as it is only rarely that deep abstractions from mathematics are translated into sentences interpretable within ordinary grammar. The methodology of MES is very general such that a diligent student of the text may identify application in neighboring applications of general systems theory and graph theory. This reviewer is not persuaded by the authors’ broad presupposition that mathematical category theory is a universal concept of symbolic binding that is equally applicable to all the sciences, including biochemistry and medicine. It is my view that the symbols and the numbers used with chemical symbols supervene on traditional mathematical logical operations, they to not supersede them. Historically, the arithmetic operations were well established before the internal transformations of relations among chemical objects became known. As noted above, the chemical symbol system is relatively new. These new facts of chemical structural relations demanded a new grammar of relations among the atomic numbers. Clearly, the arithmetic notation was not superceded by the discovery of the logic of the atoms; the new notation was merely a supervening necessity for an exact logic of matter. The two exact logics share the principles of ordered lists, units and integers. However, to supervene does not imply to supercede. From the perspective of the logic of chemistry, mathematic symbols are deprived of sufficient meaning such that they are inadequate to represent the uniqueness of individual chemical and biochemical isomers. The generic qualities of arithmetic operations correspond to the geometric qualities of distance along the real number line, restricting the concept of identity to an operation. The biochemical sciences require a concept of identity as a whole, as a multiset of perplex numbers. This multiset must be partitionable into its unique components. The parts of the multiset must be preservable. Thus, mathematical symbols deprive chemical symbols of their identity, much as the international system of units deprived chemical structures from a place in the system of scientific measurements. The deprivation is a direct consequent of the formal perplex logic of chemical structures as teridentities. Synductive logic creates new relations (in the sense of Peirce) while deductive logic reduces the number of relations (also in the sense of Peirce.) The construction of a multiset from atomic numbers creates new relations. Synductive logic extends relations such that independent parts become conjoined wholes - syndetons. These new binding relations manifest themselves physically as new properties of matter. (For example, the particular synductive operations on the units and integers of


hydrogen and oxygen to form the syndetons of either water or hydrogen peroxide.) As chemical components are electrical components, a multiset creates a new pattern of attractors and repellors by application of Coulombic rules. The antecedents of a multiset become the source of motion for the system, the source of the properties that create the identity. This “deficiency principle” between mathematical symbols and chemical symbols is a generic deficiency, expressing the insufficiency of mathematical relations to express material syndetons. The deficiency principle applies to the meaning of the symbols themselves, the logical operations on the symbols, the meaning of numbers operating on the marks for the symbols and the methods of logic used to validate the truth claims of the relational symbols. A new, more general, mathematical notion of order and part-whole relations is desperately needed to dispel the multiplicity of polysemic conundrums. From the perspective of cultural history, the mathematical symbolic deficiency principle is an indirect consequence of the deficiency of International System of Units as natural units of quantity, as noted in the prologue. .


IV. Conundrums of Creating Binding Relations for Algebraic Biology Algebraic biology presents somewhat of a philosophical conundrum for mathematics and mathematicians. These conundrums are not to be conflated with the semeosis of the deficiency principle discussed in the previous section. The irregularity, as well as the regularity, of life is obvious to every biologist and physician. The conundrums originate in the differentiation of irregular from regular relations. 1. In sharp contrast to the regularity of counting, numbers, arithmetic relations, and geometry, the irregular encoding of DNA creates uniqueness by selective irregular combinatorial graph operations. 2. In contrast to the regularity of the time-honored and culturally entrenched arithmetic operations, the metabolic operations are synductive operations on the labeled bipartite graph of perplex numbers, expressed in self-reflexivity of genetically informed catalysis. 3. In contrast to the regularity of points alone a line, irregular multisets of chemical structures occupy irregular portions of three-dimensional space (as irregular members of families of isomers). 4. In contrast to the regularity of motion of a point mass in an orbit or an orbital, living systems demonstrate the irregularity of encoded feedback and feed forward cycles (meso-orbits) over graphs, such metabolic cycles are both intrinsic in and necessary for the reproduction of life. 5. The notation of medical practice uses a linguistic system of dynamic identifiers that is foreign to the regularity of the usual mathematical structures such as groups, rings, fields, vector spaces and categories. 6. Many other irregularities could be listed. Just as logical regularity begets regularity, logical irregularity begets irregularity. Philosophical idealism can be used as a substitute for an exact solution for many applied mathematical conundrums, including the irregularity of life. A simple approach to algebraic biology would be to assert an hypothesis for life based on the universality of mathematics and the universality of the laws of nature, all conjoined by the universality of international system of units. From this idealistic view of nature, one simply concludes that these hypotheses are sufficient for explaining biology and medicine and searches for simplifying approximations. But, this fails to create a logical basis for the realism intrinsic to the irregularity of life itself, for example, the irregularity of feed forward relations from DNA to metabolism or the duality of biological causality. The dynamic irregularity of life is grounded in realism, not idealism. For example, the noted biologist, Ernest Mayr, in “What Makes Biology Unique,” challenges four physicalists’ ideals as not applicable to biology. These are one, essentialism (typology), two, determinism, three, reductionism and four, the absence of universal natural law. The logic of Mayr’s arguments is irrefutable within the evidential framework of biology and medicine. Suffice it to say that Mayr’s reasoning is analogous to the exclusion of the logic of chemistry from the international system of units, the irregularity of nature. But, I emphasis from a practical perspective, that Mayr’s fourth idea, the absence of universal natural law, conjoined with the absence of a universal system of units, posse a major obstacle to applications of algebraic biology to medicine and other disciplines. For example, what exactly is the ontological status of claims deduced from the physicalist interpretation of chemical and genetic


signs and symbols? When, where, and how can such biomedical claims bear the enormous weight of the Hippocratic oath? Semeosis is the study of signs. For the purposes of algebraic biology, the concept of the study of signs in homologous to the concept of inquiry for the apodictic sciences. Since antiquity, a central activity of biology and medicine is the study and interpretation of natural signs (Manetti, 1993). While all living systems are capable of interpreting and generating signs, algebraic biology focuses directly on the quantitative interpretation of biological signs and communicating the results to others. But, signs may generate different significations in different disciplines and these may be expressed may be expressed within any one of several symbol systems (Chandler, 2008). Algebraic biology may focus on the signs for numbers, or variables or categories. If semeosis is the art of interpreting the sensory impressions from signs, then symbolic communication can be viewed as the art of generating expressions that serve as signs for others. Symbolic communication depends on both the sender and the receiver interrelating the encoded and decoded messages within the same coding system. In rejecting the physicalist hypothesis, Mayr implies the need for a different symbol system. Consistency of arguments from chemistry, biology and medicine all support the Pattee hypothesis that a different sign system is needed for algebraic biology. For practical work, the Einsteinian criterion for correspondence between mathematics and reality necessitates such a sign system. Rosen deduces the need for a new symbol system from the concept of a “particle of a function”. A “particle of function” is dependent on the system as a whole (Life Itself, page 120) in the sense of Rosen’s interpretation of the Newtonian paradigm. Rosen argues for the necessity of an additional symbol system for the formal logic of natural systems, claiming that the binding relations of physicalistic mathematics are not sufficient, scientifically. A relation between Rosen’s “particle of function” and biochemical invariance is not apparent to this reviewer. EV asserts that MES is based on the binding relations of systems science (MES Chapter 2, page 49-71). An early attempt to address these differences of binding relations from a perspective of ordinate logics was recently published, (Chandler, 2008). Further work is needed to clarify on how the syndetic principle of an organism is related to catalysis, the operational origin of biological motion. I close this section by attempting to place algebraic biology into a grounded epistemological framework. To paraphrase the dictum of Einstein, Podolsky and Rosen, the elements of biomedical reality cannot be determined by a priori mathematical considerations but must be found by an appeal to the results of observations, experiments and measurements – the study and interpretation of the signs of life. Section V. Invariant Biological Information Necessary for Reproduction This section concerns the potential for logical relations between algebraic biology and number systems. If algebraic biology is to become an integral component of applied mathematics, it must resolve the conundrums emerging from the physicalist’s interpretation of life. Two such conundrums arise from assertions about the quantum mechanical basis of life and information theory. Delbruck, Schrödinger, Weiner, and von Neumann, among others, contributed to the hypothesis that “information”, as a mathematical quantity, is essential to life. Others assert that information is central to the theories of


physics. Futurists speculate that our minds, as information, can be harvested and uploaded into computer memory, perhaps generating the equivalent of eternal life. These are philosophical extensions of nominalism, at the expense of chemical realism, into the realm of algbraic biology. The presupposition is that the biochemical encoding of life can be reduced to strings of bits, (Shannon encodings), that can be transmitted as sequences of electrical signals. The philosophy of this physicalistic envisioning of biological information is described by Weyl in “Philosophy of Mathematics and the Natural Science.” More recently, von Weizsacher in “The Unity of Nature” propounds an informatic extension of the Weyl position. It is rooted in the regularity of mathematics and the continuous functions of thermodynamics and quantum mechanics. These are theories of motion, not of existence. The philosophical beliefs expressed by Weyl are now mainly of interest to cosmologists and physicists. I have previously addressed the relations between chemical symbols and these theories and refer the reader to these published papers. A complete mathematical theory of the relations between the concept of number and biological reproduction remains to be devised. The central issue is asymmetric. The order relation of atomic numbers is regular. Nevertheless, the order within molecules and living systems is highly irregular. Physical theories focus on the regularity of the order of motion in space and time. Chemical and biological theories seek to describe and / or explain the irregularities of the individual in terms of the regularity of order of the atomic numbers and the order of the logical operations on part-whole relations generated from such numbers. The logical conundrum faced by the physicalists is to explain the reproduction of irregular systems from presupposition of the absolute regularity of mathematical logic. A priori, this task appears to require a reconciliation of mathematical regularity with biochemical irregularity. Pattee-Rosen Hypothesis: Howard Pattee and Robert Rosen postulated that physical models are insufficient for algebraic biology. Rosen rejected, in Life Itself, the sufficiency of formal logic for describing natural systems. Howard Pattee, studied the relations among biological signs, physical laws and category theory for several decades. He points to the role of “initial conditions”. The reader interested in his technical arguments is referred to his papers, (http://ssie.binghamton.edu/faculty_pattee.html). Rosen, a categorian, argues simply. A natural system is not a formal (Newtonian) system but can be represented as a formal system. In short, this is a semeiotic argument on the interpretation of symbols. Rosen argues that a living system is closed to efficient causation. The simplicity of this argument is beguiling. One mathematician commented that it would require hundreds of years to address the issues raised by Rosen. The website (http://www.rosen-enterprises.com/RobertRosen/robert_main.html) contains further references. While the syllogistic-like formulation of Rosen arguments is clear, the unstated suppositions are questionable. Is it necessarily so that only a physicalists system of formal logic for science exists? Rosen’s argument, for example, would fail, if several formal logics existed and any one of the finite number of systems of formal logics was commutative with natural systems. Although chemical noumena were of great interest to him, Rosen did not explore the formal logic of chemistry. A formal logic for science requires an ordering principle that parallels the counting numbers such that correspondences


between external reality and internal representations satisfy the EPR axiom. In the system of real numbers, the natural integers serve this foundation along with a construction of the order by a recursive argument (such as Peano’s Postulates.) The source of ordering of chemical sciences comes from the ratio of small whole numbers, from the empirical atomic numbers, each a counting integer representing a set of material properties. The sources of the second order logical operations are the internal relational structures of individual atoms and molecules. The compositions of logical terms are the syndetic connections that bind the parts into wholes. The connection to formal logic are direct and completely within the usual enumerative habits of the laboratory sciences. This formal logic was named the perplex number system as it can be viewed as a spatial extension of the complex number system. Thus, the narrow focus of Pattee and Rosen’s notion of formal logic excludes the chemical sciences and the second order logical operations necessary to describe living systems. From a historical perspective, this is not surprising; it parallels the exclusion of synductive logic implicit in the international systems of units. The merits of Rosen’s argument with respect to physicalism are another matter, a question of the interpretation of his metaphor “a particle of a function”. It calls into focus the relations among geometry, number, algebra and science. It is separate from the parity between units and integers. One wonders about the possibility, however remote, of a natural transformation between “particles of a function” and a memory evolutive system and electrical parity of part – whole relations. It is important to note that the conceptual trace of Rosen’s scientific logic goes from chemical thermodynamics to physics to mathematics to mappings. This trace differs from the conceptual trace of the logic of Ehresmann and Vanbremeersch that goes from directly from the semantics of systems theory to an application of category theory in which transformations are temporal. It is inclusive with respect to the geometry of quantum mechanics. A practical (ontological) difficulty is this. The number system for chemistry is based on natural order. Temporal change may be either of two types – a mutative change of material relations or a positional change with respect to the embedding space. Each is subject to dual causality. While the natural order of the chemical elements was decisively fixed in the earlier part of the 20th Century, the formal logic of associating the numbering of the ordering into composites (molecules) was neglected until recently. Thermodynamic arguments also suffer a deficiency principle. A priori, classical thermodynamics lacks a variable for time or a symbolism for chemical structures. (The principle bridging equation, the law of mass action, excludes chemical structures, using the ratio of numbers of individuals per unit volume, as a source of computation.) Thus, in order to contrast the two categorical hypotheses of RR and EV, the nuances of time must also be bridged. Is it conceivable that this requires a different form of supervention between number systems? The existence of two number systems, both with formal properties, is directly related to the medieval grammatical separation of the categorimatic and syncategorimatic terms. It calls to mind the Modistae logic of the 13th Century (Thomas of Erfurt) with the differentiation of modi essendi from modi intelligendi and hence from modi significandi. (See: J. Zupko, http://plato.stanford.edu/entries/erfurt/) The recent development of the perplex number system with formal logical properties as a second order abstraction from electro-chemical measurements could facilitate a proposal for formal properties of natural systems. Such an extension from synductive logic to a formal logic of perplexity is under


investigation. Unpublished work indicates that a parallel set of logical structures loosely associate a perplex number with structures of dynamical systems. Informally, I refer to these structures as mesogroups, meso-fields, patrices and patroids, the structural names being associated by logical comparability with the corresponding traditional mathematical structures – groups, field, matrices and matroids. It seems likely that these structures may shed some light on the tensions and conundrums generated by the EV and RR theories of algebraic biology. These nascent exploratory concepts may serve as a fertile source of conjectures. Unbounded opportunities for further work await the inquiring mind.


Section VI. Algebraic Biology and Systems of Reasoning about Biology. The tensions underlying algebraic biology originate from several sources – chemical isomerism, dual causality, internal electrical structures of objects, irregular encoding of genetic messages, irregular encoding of dynamic biological messages, dynamic identity, differences in formal numbers systems, semeiotic reasoning – illustrate a few of the opportunities open to biological algebraicists. In this section, several approaches to reasoning about symbols are compared. Roughly speaking, a scientific symbol system creates the basis for informed discourse among scientists. Historically, a discipline, such as chemistry or genetics, is created when prior existing symbols systems lack the expressive power to represent the natural noumena. New symbols are deemed necessary to express the correspondences between facts in the world (esse extra anima) and interpretation of the relations among the facts by the group of corresponding scientists (esse in anima). Abstractly, the new symbolic relations are needed to express the equivalence of the EPR axiom as a consequence of the deficiency of existing symbol systems. The new symbols of chemistry and genetics introduced new nouns and verbs in order to represent the new empirical relations being demonstrated in the laboratories and in the clinics. The new symbols introduce new relations in the sense of the Peircian definition of a relation as a triad of terms, firstness, secondness and thirdness. These are antecedent to new modes of human communication about natural systems. Analysis of the history of molecular biology during the twentieth century allows one to conjecture about the logical requirements for emergence of new scientific symbol systems. Initially, the specialized usage is restricted to a small group of specialists, seeking to describe apodictic situations and circumstances. Over durations of several decades, the specialized rhetoric acquires distinctive interdependencies and naturally meaningful connections with empirical observations. It is conjectured that five commutative parts co-evolve from rhetoric expressions. Systematic inquiry into the dynamics of life is an apodictic inquiry that is only rarely an Augustinian inquiry. New nouns are constructed from inferential relations abstracted from invisible relations supposed to exist from part-whole relations and other properties. New propositions are postulated from new nouns, new verbs and new relations such that the scientific symbol system acquires the capacity for exact scientific demonstrations and communication. I suggest that the five parts co-emerge during the ever-so-gradual genesis of the body of experimental data. First, the set of terms from which messages are composed is restricted to a specified set of relatable symbols. For examples, new biochemical structures are isolated from nature and assigned names systematically. New genes are postulated to exist from the observation of new molecular behaviors of organisms. Secondly, pairings of symbols (specific extensions, transitivity, associativity) are necessary to create arrangements of symbols into components of messages. Thirdly, rules of logical operations are necessary to conjoin individual symbols and individual patterns of symbols into unique individual messages (propositions / syllogisms). Fourthly, a set of correspondence relations satisfying the EPR correspondence axiom is necessary to associate the reality of external events with internal interpretations of the events. Finally, the developers of a scientific symbol system require an expressible convention for forming coherent narratives from collations of propositions. This expressible convention may be termed a grammar or a code, such as the encoding of the connections between atoms or the encoding of connections among genetic traits. The logical operations then are expressed within the specific grammar or code. Typically, a logical grammar is associated with a diagrammatic logic or meaningful images or specialized icons. (The


astute reader will observe that this five part argument on the emergence of a scientific symbol system is a dynamic application of the tricotonomic argument of the logician C. S. Peirce with respect to the relations between signs and symbolic logic.) The ground of the argument is that we use abstract symbol systems to express our reasoning or logic such that a visual diagram is implied. Both the grammar and the logical operations must be exact if the communication with others is to be exact such that unending chains of calculations are to be exact. The logic truth of an argument is often associated with an explicit visual diagram that illustrates the Peircian relations among symbols. The following discussion focuses on graphic symbols, symbols as expressions of intentional behavior, symbols widely used in human communication about science. As noted above, the new symbol systems of biochemistry and molecular biology are apodictic. The apodictical references are the concepts of truth, certainty and necessity associated with the EPR axiom. The antecedence are embedded in the propositions of experimental protocols. The consequences are contained within the molecular biological observations. Inquiry of this nature is known traditionally as demonstrative science. Semeiotics of symbolic ontologies for algebraic biology. We can now summarize four basic approaches to the study of algebraic biology in terms of the intentional (exformative) use of symbols. As with all of mathematics, the algebraic biologist is faced with identifying exact meanings for identities and relations. (For a speculative linguistic approach that does not depend on EPR correspondence, the reader is referred to Markedness as Mathematical Principle, Chapter 3 in Markedness Theory, E. Andrews (1990).) Perhaps this symbolic challenge is best described by C. S. Peirce in terms of signs as ontological objects and interpretations. For this logician, the ontology of a symbolic argument requires three terms. These are repeated here for the reader’s convenience. “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.” (Letter to Lady Welby, 1904 Oct 12, CP 8.328)

A relational algebra, such as category theory, offers a range of possible expositions. In what sense does it satisfy the EPR correspondence relation between algebra and biology such that an ontological basis for algebraic biology is created? More specifically, how is a syndeton to be arranged? This question is addressed from four viewpoints – cultural history, Ehresmann-Vanbremeersch, Rosen-Pattee and that of perplexity theory. Historically, the Greek culture endowed Western culture with rudimentary forms of symbol systems and causal logic (syllogisms). From these healthy roots, numerous symbols spouted. Vigorous organic growth of more symbols ensued. Today, we use thousands of symbols to communicate our science, our philosophy, our cultural relations and our emotions. The introduction of the symbol system for matter by Berzelius in the early 19th Century was a very late development in the symbolizations of our mental activity. Only very recently was it recognized that the natural order of the chemical elements and the logic operations on these identities constitute a formal system of logic. A cultural connotation of algebraic biology is that it is an inquiry into the philosophy of life, expressed in symbol systems. The analysis of


these symbol systems has extraordinary deep implications for human medicine and community weal. From this perspective, algebraic biology is a substantial component of the rich symbolic milieu of our time. Rosen’s Anticipatory Systems seeks to analyze life as metabolic systems from a categorical perspective. His unique philosophy of functional systems, expressed metaphorically in terms of “hardware and software” allows him to conclude that natural systems are not formal systems but are somehow related to them. He distinguishes a component as “a particle of a function” (page 120). This distinctive definition creates an ill defined but unique dependency relation between the parts and the whole. For example, Rosen asserts that: “a particle, or any unit of structural analysis, does not (indeed cannot) acquire new properties by being associated with a larger family of such units; on the contrary, the larger family is itself endowed with precisely those attributes that are contributed individually by its members. Thus, a though going reductionistic, structural approach to the natural world must deny reality to such concepts as novelty or emergence at any fundamental level. These become merely epiphenomena, new ways of collecting the same old particles.”(page 121, Life Itself.) This assertion appears to exclude the consequences for the Coulombic (product) relations between electrical particles, a critical source of relations among the biochemical components (units of structural analysis) of a cell. This assertion lacks correspondence with the chemical logic of metabolism as well as the dual causality of biological growth and development. Indirectly, it is not consistent with the functioning of genetic systems and hence the EPR axiom. A critical re-appraisal of Rosen’s methodology is needed. Other views of mathematical relations, beyond the metaphor of “hardware and software” offer more productive routes to an invariant base for algebraic biology. Memory evolutive systems are based on the theory of algebraic relations, e.g. directed graph theory or a theory of the concatenation of “natural transformations”. The categorical notation, introduced in the 1940s, is a highly technical representation of specialized branch of mathematical philosophy. This application of category theory approaches life from a generic concept of relations within nature as directed graphs. From subatomic particles to life to consciousness to human cultures, the directed graphs are supposed to be of similar mathematical character. In the MES model, emergence from lower level organization fuels the emergence of consciousness. The relations between this unique mathematical philosophy and chemical systems theory are opaque. The roots of the philosophical conundrums lie, perhaps, in the rigor of the logical of the abstraction from noumenon to phenomenon. My own work on the perplex number system is in nascent form with respect to relational algebra, yet a few tentative conjectures may stimulate deeper inquiry. Briefly, the symbolic reference system, the perplex numerals, is regular. A perplex numeral is an associative logical structure. That is, the teridentities of the perplex numerals can be systematically transformed into transitive relations. The teridentity of a perplex number is defines an exact grammar of logical individuality of part-whole relations in Peircian terms. In this grammar, the nouns are units and integers. The verbs are relations between nouns. The optional positions for sequences of logical operations distribute the relations irregularly. Extension is irregular. Nonetheless, the direct translation of labeled bipartite graphs into chemical structures ensures that a direct correspondence with molecular biology exists at the abstract level of the substitution of units


and numbers for chemical symbols. New, more primitive structures, termed meso-groups, meso-fields, patrices and patroids, are parallel to traditional mathematical structures. The mathematical basis lays in triadic identities between multisets and partitions that necessitate part-whole relations. It satisfies the necessity for correspondence relations between symbol, structure (pattern) and the realism of tangible properties. In short, it is an invariant basis for a local theory of matter, not a universal theory of motion. Emergence within this symbol system is intrinsic in extension from the perplex numerals to perplex numbers, directly analogous to the electrical extension of atoms to molecules. In this number system, occupied space is a consequence of the presence of antecedent matter. Time is a consequence of the mutation of matter. Ontologically, as a consequence of material emergence, the perplex number system is self-reflexive, the source of itself as well as other symbol systems created by the material operations of the human mind. In this symbol system, Aristotelian causality is represented directly in the following sense. First, the perplex numerals and the perplex numbers are the same as abstract Aristotelian material causes. As these are numbers and as the logical operations on these numbers generate the formal graphs of multisets, the arrangement of parts in the whole are the same as the Aristotelian formal causes. Aristotelian efficient cause requires representation of the context of the (material-formal) situation. This is established in the apodictic sciences by the thermodynamic variables of temperature, pressure, and volume and the population sizes of the material objects (perplex numerals and numbers). Formal diagrammatic logic demonstrates the changes in relations (changes in the verbs of chemical sentences.) Telos is expressed as a constant ratio of cause to effect, or, the energy created by the causal processes. In summary, the diagrammatic grammar of the perplex number system demonstrates directly the four Aristotelian causes. This illustrates the advantage of symbolizing the noumena directly. Emergent properties are composed as multi-sets of multisets, multisets of multisets of multisets, and so forth. In summary, within a particular situational context, the antecedences of perplex numbers are the same as the material, formal and efficient causes of Aristotle. The consequences of mutations in perplex numbers, that is, the telos, are the resultant material structures and the energy created. The contrasts between the exposition of Aristotelian causality in the perplex number system and in categorical theory of MES (see p. 138 –142) are striking. Identity, as a concept, plays a different role in each of these four systematic to approaches algebraic biology. The identity of any biological organism is generated from heredity via the parent-offspring relation and from the material spatial relations of matter. Within the presuppositions of category theory, the EV hypothesis is consistent with identity arrows. However, the MES hypothesis does not demonstrate a symbolic correspondence with a particular identity, a particular genetic system, or the internal spatial construction within an individual organism. The relationship between Rosen’s “particle of a function” and identity is unclear. Thus, the EPR axiom on the necessity for correspondence between mathematics and reality is uncertain in Life Itself. The perplex number system is grounded in a sequence of identities that meet the requirements of EPR axiom for corresponding identities. The formal graphic perplex objects are organized in space as a consequence of electrical attraction and repulsion relations among the parts. This is a positive identity that can serve further for unbounded extensions in the emergence of higher order systems, including the genetic identity of a human being. Among the numerous challenges for biological algebraicists is to find the extensive relations within the perplex number system and the corresponding graphic linkages with the continuous functions of time and space that create the identity of life.


Conclusions: Algebraic biology remains in a pre-natal state, but mathematical foreplay has germinated a kernel of organized activity. A crisp view of invariance is needed to thrust development forward. Its birth will be marked with a rigorous mathematical description of the reproduction of the identity of a cell. The conjecture that a marriage between the perplex numbers and the real numbers is a necessary condition for such a reproduction appears plausible. I conjecture that such a conjunct could depend on a new generalization of invariant relations that subsumes both synductive and deductive logic.

Acknowledgements: I thank the late Robert Rosen for several intensive discussions of his works. I also thank Andree Ehresmann and Jean-Paul Vanbremeersch for innumerable discussions and their generous hospitality during my visits to Amiens. In several respects, this essay is a recapitulation and extension of the discussions over the past 15 years. While I have tried to represent their arguments to the best of my abilities, the challenges of trans-disciplinary communication were, are and will continue to be substantial. I alone am responsible for errors or misinterpretations of their often delicately subtle and highly nuanced views. References Andrews, E. (1990) Markedness Theory, Duke University Press. Bochner, S. (1966). The Role of Mathematics in the Rise of Science. Princeton University Press. Chandler, J. L. (2009). "Introduction to the Perplex Number System." Discrete Applied Mathematics, 157: 2296-2309. Chandler, J. L. (2008). Ordinate Logics of Living Systems. Simultaneity: Temporal Structures and Observer Perspectives. Ed. S. Vrobel, O. Rossler, Singapore, World Scientific Press: p. 182-194. Ehresmann, A., Vanbremeersch, J. -P., (2007). Memory Evolutive Systems, Elsevier Science. Jespersen, O. (1965). The Philosophy of Grammar. NY, Norton. Juarrero, A. (1999). Dynamics in Action, Intentional Behavior as a Complex System., MIT Press. Mac Lane, S. (1986). Mathematics: Form and Function. Berlin, Springer Verlag, Manetti, G. (1993). Theories of Sign in Classical Antiquity, IUP. Mayr, Ernest (2005) The Autonomy of Biology, Harvard University Press. Peirce, C. S. (1998). The Essential Peirce (1893- 1913). Bloomington, Indiana University Press. Quine, W. V. (1986). Philosophy of Logic, Harvard University Press. Rosen, R. (1991). Life Itself: A Comprehensive Inquiry into the Nature, Origin, and Fabrication of Life, New York, Columbia University Press. Rotman, B. (2000). Mathematics as Sign Writing Imaging Counting, Stanford University Press. Von Weizsacher, Carl F., (1980). The Unity of Nature, Farror, Straus, and Giroux, Munich. Weyl, H. (1949). Philosophy of Mathematics and the Natural Sciences. Princeton, Princeton University Press.


Appendix: The perplex numerals are defined as relational objects in the sense of Leibniz and Peirce. Logically, a perplex numeral is defined by two propositions as required by the symmetry of the equinumerosity of the units and integers: An integer infers a set of units. A set of units infers an integer. The usual notation of logicians for symbolic inference employs a horizontal bar to separate the proposition from the conclusion, the antecedent from the consequent. Again, we follow the symmetry principle of equi-numerosity. We can list the correspondence relations dually. 1* _, 1

2 ___, 1,1

3 ____, … 1,1,1

Or, alternatively (dually), the list of inference of the correspondence of units to integers. 1 _, 1*

1,1 ___, 2

1,1,1 ____ , … 3

(* Note that the symbol “1*” is used as a sign for the first integer to avoid possible ambiguity in calculations.) The information content or “the difference that makes a difference” between successive terms of the series of perplex numerals is the stepwise increase in the size of the numerator and denominator so that the ratio of the integer to the units to the number of relations is constant. The uniformity of the transitive relation between successive integers is a necessary property for parity or equi-numerosity. Graphically, each perplex numeral constitutes a teridentity, a labeled bipartite graph. The teridentity is a logical reference defined for inferences among terms. It serves as the axiomatic basis for compositions of relations among multisets of terms. A composition of a multiset of numerals preserves all the parts of each member of the multiset and extends the number of relations such that parity is preserved in the composed labeled bipartite graph. A logical composition of two or more perplex numerals breaks symmetry. Formal logical operations on the graph edges create new relations such that a new object, a new labeled bipartite graph, comes into existence. It can be isomorphic to an existential entity, a reality (esse extra anima). The perplex numerals inform the perplex numbers, just as hydrogen and oxygen inform water and hydrogen peroxide. i

Cited on page 1 of “On Brouwer” by M. Van Atten. 2004. Original source not cited.
