Abstract
Algebraic/topological descriptions of living processes are indispensable to the understanding of both biological and cognitive functions. This paper presents a fundamental algebraic description of living/cognitive processes and exposes its inherent ambiguity. Since ambiguity is forbidden to computation, no computational description can lend insight to inherently ambiguous processes. The impredicativity of these models is not a flaw, but is, rather, their strength. It enables us to reason with ambiguous mathematical representations of ambiguous natural processes. The noncomputability of these structures means computerized simulacra of them are uninformative of their key properties. This leads to the question of how we should reason about them. That question is answered in this paper by presenting an example of such reasoning, the demonstration of a topological strategy for understanding how the fundamental structure can form itself from within itself.
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Notes
It may seem odd to say that life is a “function”. But this is in fact a central theme of relational biology. Life is a dynamic process rather than a static thing. In view of this, the Schrödinger question “What is life?” is somewhat ill-posed, because of its implicit assumption of the latter. A more correctly posed question is “What distinguishes a living system from a non-living one?” This is the question Rosen discusses and answers in Rosen (1991, 2000). The three-word question “What is life?”, because of its terseness and immediacy, will continue to be used as an abbreviation.
The analogy here is “organism:life :: nervous system:mind”, where the brain is a functional component of the nervous system. The term “mind/brain problem” is, however, part of the established lexicon, so we shall continue to use “mind and brain” when referring to the more proper “mind and nervous system”. The reason for making the analogy is that both relationships are the result of a hierarchical closed loop of efficient causes. The relationship of “organism:life” in Rosen’s work has the same structure of causation in cognition, as established in Kercel (2004a, b).
The essential point of Kercel’s endogenous brain paper (Kercel 2004b) was to show the circular hierarchy of synaptic, diffusive, and glial communications circuits. A critical detail that was omitted from that paper was the character of the hierarchy. It is a hierarchy of containment. In other words, B is lower than A in the hierarchy if and only if B is contained in A. Thus, one of the three communications processes is seen to be contained in the interaction of the other two. Similarly, in the circular hierarchy of the (M,R)-system (which we shall discuss in Sect. 7), there is a hierarchy of containment and no privileged position. Any of the three functions is contained within the other two. Thus we see that there is a direct analogy between the metabolism-repair paradigm in cells and the “endogenous” character of information transmission in the brain.
One might, indeed, consider final cause to be ambiguous. “The end”, i.e., the telos, may be considered either as the object entailed or as the entailment of the object, or both. In fact, an inherent ambiguity in final cause may be the underlying reason that it is inadmissible as a property of mechanisms, but an indispensable property of organisms. As a rational alternative to the claim that to speak of final cause is to “preach religion”, this inherent ambiguity may serve as a means to readmitting teleology back into science. We shall have more to say on the topics of ambiguity and impredicativity, and recast this in algebraic-topological terms, in Sect. 8.
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Louie, A.H., Kercel, S.W. Topology and Life Redux: Robert Rosen’s Relational Diagrams of Living Systems. Axiomathes 17, 109–136 (2007). https://doi.org/10.1007/s10516-007-9014-z
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DOI: https://doi.org/10.1007/s10516-007-9014-z