A New Logic for Complexity

After a long hiatus from this project, due to the interference of that multi-splendored thing called life, I am proposing to introduce a new logic for complexity over the next couple of weeks. This logic builds upon but transcends extant logical frameworks such as Aristotelian, dialectical, paraconsistent, and related logics. The tentative name that I have been using for this logic is "Self-transcending Constructional Logic" ("STCL"). This logic has been developing out of several major sources: 1. varied complexity constructs and mathematics, especially nonlinear dynamical systems theory; 2. set theory, particularly the method of anti-diagonalization (mistakenly called "diagonalization" as I shall explain later) which is a paramount proof method used in transfinite set theory; 3. category theory, notably the work of Ehresmann and Vanbremeersch on colimits, complexification, and multifoldedness; 4. the idea of noncomputability, both as proved by Turing with great help from earlier work by Godel, as well as found in later modifications (the latter as mostly interpreted by the mathematical logician Judson Webb); 5. various hints and intimations from the complexity theorists John Holland, James Crutchfield, Jack Cohen and Ian Stewart, John von Neumann, Charles Bennett, Walter Fontana and Leo Buss, Douglas Hofstadter (not usually considered a complexity theorist but whose amazing book Godel, Escher, Bach is rightly viewed as a long and difficult thesis on emergence), and others.

Furthermore, the major philosophical influence on the development of this complexity logic has been the conceptualization of natural complexes on the part of the American philosopher Justin Buchler (whose work I've found is unfortunately little known among complexity aficionados). Finally, there is the beguiling but enlightening philosophical method of Ludwig Wittgenstein, whose influence must be assumed on whatever I aver, having studied his work for close to 45 years.

As I have stated in earlier entries to this project, what I propose as a novel logic should not be taken as a novel mathematics. To be sure, the relation between logic and mathematics is a tangled and historically changing one. The way I am viewing that relation in this project is to conceive a logic as conceptual framework of foundation underlying any specific mathematical discipline, it is what supplies the conceptual "space", so to speak, which gives the permission to or provides allowance for mathematical methods/findings. Actually, logic does more than just permit specific mathematics, it opens the way, even points the way, to certain mathematics. Thus, a logic can be said to close off certain routes while opening up other routes for mathematics to proceed.

An example is how Aristotelian logic has notoriously rendered a cogent conceptualization of change (or transformation, or transition, or transmutation, or evolution, or...) quite problematic. One result of this has been the eruption of troublesome paradoxes in typical Aristotelian models of change, the infamous identity/change paradoxes. And once paradox arises, great effort has then been provoked to account for a requisite consistency so that we can trust in what is thought and said.

Moreover, the diremption wrought by paradox has led to a corresponding antithetical logic which supposedly "solves" any inconsistency brought by paradox by embracing paradox as on equal valuation of true and false, and so forth. This can be seen in an attenuated form in dialectical logics and in a full-blown form in dialethistic logics. In my estimation though, which I hope to adumbrate later on, dialethesis, to borrow a phrase from Bertrand Russell used in a different context, has all the benefits of theft over honest toil.

The self-transcending constructional logic being offered here, though, can be said to "flirt with" paradox but not embrace it. A flirtation and an embrace are a world apart, many a slip between the cup and the lip goes on here. As I hope to explicate, the logic of stcl can flirt with paradox in what I claim to be an insightful manner without at the same time succumbing to the temptations of embrace. I have borrowed this description using the metaphor of a flirtation with paradox from the aforementioned Douglas Hofstadter who wrote, during his elucidation of the method of diagonalization mentioned above,

Although the inclusion of the word "transcending" in the name of this logic -- "self-transcending constructional" -- might sound portentous given its connotations in lofty circumstances, in actuality "transcending" as used here is a totally neutral term employed to describe the method of diagonalization on the part of the German historian and philosopher of mathematics Oskar Becker. Becker's phrase was in turn taken-up by the Austrian-American philosopher of law and mathematics, Felix Kaufmann, who believed it was an inherently absurd thing to posit since according to Kaufmann no mathematical procedure could transcend itself. I took the phrase from Kaufmann but reverted to the neutral way it was used by Becker because I think it is a much more appropriate and unbiased description of what can take place in complex systems. I will say much more on this later on.

As I hope to show, by flirting with paradox, the logic of stcl can help deal with those paradoxes that have been said to be present in conceptualizations of complex systems, specially, in ideas on bifurcations, phase transitions, self-organization, self-modification, criticalization phenomena such as self-organized criticality, and so on. Thus, it is a logic which opens a conceptual space for understanding certain very significant phenomena in complex systems.

Moreover, it needs to be stated that I am not using the term "complex" in the sense that earlier contributors to this project seemed to mean, namely, the incredible complex aka complicated and enmeshed diverse social networks all around us, i.e., networks mostly seen as bringing about unwanted, unintended consequences leading to the general decay and corruption of the social fabric of these post-modern times. Despite this pessimistic assessment (of which I am in more agreement than not), the development of the logic of stcl herein goes in a different direction since it implies that perhaps complex systems science can provide means for helping to ameliorate the general state of decline. This entry should be read as just the beginning salvo of my aim in explicating a novel logic for complexity. Of course, comments, questions, whatever, are most welcome.

I think the time has come to tighten some things up regarding this project. I want to reiterate some points made in the beginning. What I have meant by “logic” in this project does not equate to mathematics or even broad mathematical approaches. However, maths can point in certain directions that reveal the logic of complex systems or how the world appears different due to complexity constructs. Nor does the logic of complexity involve this or that “solution” to specific problems although, to be sure, any complexity construct involved in deriving a logic for complexity only attains its usefulness, confirmation, and fullest expression in relation to its encounters with specific problems which the world offers. This is where the “rubber meets the road” in laying-out which complexity constructs impact an overall logic for complexity. Thus I would expect that a logic for complexity contains complexity constructs that have been modified and refined through their applications. Consequently, a complexity construct worth its while in carving out a logic for complexity would have to demonstrate its usability in dealing with real world complex problems and not just because this or that construct is “cool” or “interesting” or whatever in terms of its pure mathematical or scientific roots. I am in the process of going through the varied comments to the projects (at least the ones I think are relevant to the goal of the project) in order to select and condense ideas from them if these ideas further the project. I welcome others to do so. I am not asking people to repeat anything, rather select and condense and even name some of the complexity logic pointers. I will be commenting as I go along. I start with something which may not have been mentioned in any specificity so far but is something I have been interested in for a long time. I hope this generates some comments: I’ve identified a kind of complexity construct associated with emergence which I don’t yet have a good name for and am seeking help. It is also related to nonlinearity in general and unpredictability/unexpectedness. One example of this complexity construct takes place in the case of low temperature superconductivity, a phenomena which is accepted as a case of emergence by the majority of solid state or condensed matter physicists as well as a number of philosophers of science. In particular, emergence in certain low temperature superconductive metals shows itself as a “unified collective quantum wave” of electric current flowing freely without resistance. Other effects are also seen relating to the magnetic field, etc. Perhaps the complexity construct have in mind is not best thought of as one construct but instead two or three or more acting together. A descriptive term for this construct might be “transformational” but I think that specific term is too general although the complexity construct I have in mind is about a radical kind of transformation. Here is a description of this construct in operation in superconductivity by viewing it as made of three interlocked phenomena: 1. The metals exhibiting superconductivity at low temperature do not sure any greater conductivity than other metals at normal temperatures whereas those metals which are better conductors of electricity at normal temperatures are not the ones which become superconductive at very low temperatures; this is curious since it means that one cannot extrapolate from the properties of conductivity at normal temperature or the structure of metals at normal temperature to what happens in the case of the metals when superconductivity emerges at low temperatures, that is, it is not any kind of linear elongation or piece-meal process but rather a radical transformation of the superconductors metals’ electron structure; 2. Superconductivity at low temperatures involves the transformation of electrons from their status as fermions which cannot occupy the same quantum state at room temperature to actually becoming a different kind of particle, the boson, which have the property of being able to occupy the same quantum state (and hence follow Bose-Einstein statistics, hence the name “boson”); the emergent quantum state is a collective integration because the superconducting electrons as bosons are in the same quantum state (hence the description of “quantum wave”); 3. Related to 2.) is that superconductivity at low temperature involves the formation of a pairing or coupling of electrons, a strange situation given the fact that electrons normal repel each other. What I am looking for here is not any more stipulated technical “mechanisms” responsible for superconductivity as an emergence. I have been studying the BCL theory and am well aware of these explanations. Instead I am looking for the complexity construct s here that are pertinent to the logic for complexity.

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Some general comments about understanding complex systems, perhaps with some significance to the development of a logic of complexity.