We have been using Rosen's "Relational Biology" models in this series without a detailed explanation of why they are so useful as a way of answering questions that can not be answered by the Cartesian-reductionist paradigm. The origin of the idea goes back to a paper in 1954 (Topology and life: In search of general mathematical principles in biology and sociology. Bulletin of Mathematical Biophysics 16 (1954): 317–348.) by Nicholas Rashevsky. Here is some of what the link says about him:
Nicolas Rashevsky (November 9, 1899 – January 16, 1972) was a Ukrainian-American theoretical biologist who pioneered mathematical biology, and is also considered the father of mathematical biophysics and theoretical biologyBefore I get into the principles Rosen developed to bring Rashevsky's general notions into a methodology, I need to set the stage by focusing on some of the reasons these ideas had to be formulated to fill a large gap left by the Cartesian-reductionist paradigm. I once got a good perspective for what this is all about by talking to a mathematician collaborator who had worked in the sub discipline called "Topology". He explained that his struggle for tenure, etc. was very different than that of his contemporaries who taught and researched in areas that were included in the analytic sub discipline that serviced the needs of Cartesian science and engineering . (Calculus and differential equations, etc.) This was just one more manifestation of the profound differences in thinking that the relational approach and the Cartesian approach exhibit. Read on below and I will explain why these two areas within mathematics have found very different applications in science and engineering.
First of all, what is Topology and why should we care?
Topology (from the Greek τόπος, “place”, and λόγος, “study”) is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation....The way Wikipedia deals with the subject should be a clue that what I said above about the perceived relative importance of parts of mathematics has some truth in it. What really needs to be seen in the above description is that rather than dealing with the kinds of things Newton did when he used calculus, Topology deals with relationships, connectivity, and related matters. Just what is needed to work with aspects of the real world that do not fit nicely into the Newtonian framework. Topology has been useful within the Newtonian framework if we merely consider but a few of the many examples:
Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics.
The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside.
Carotheodry's proof of the second law of thermodynamicsRené Thom's Catastrophe Theory
Carathéodory's "first axiomatically rigid foundation of thermodynamics" was acclaimed by Max Planck and Max Born. In his theory he simplified the basic concepts, for instance heat is not an essential concept but a derived one. He formulated the axiomatic principle of irreversibility in thermodynamics stating that inaccessibility of states is related to the existence of entropy, where temperature is the integration function. The Second Law of Thermodynamics was expressed via the following axiom: "In the neighbourhood of any initial state, there are states which cannot be approached arbitrarily close through adiabatic changes of state." In this connexion he coined the term adiabatic accessibility.This proof was a very elegant application of topology to an area that was in great need of it.
While René Thom is most known to the public for his development of catastrophe theory between 1968 and 1972, his earlier work was on differential topology. In the early 1950s it concerned what are now called Thom spaces, characteristic classes, cobordism theory, and the Thom transversality theorem. Another example of this line of work is the Thom conjecture, versions of which have been investigated using gauge theory. From the mid 50's he moved into singularity theory, of which catastrophe theory is just one aspect, and in a series of deep (and at the time obscure) papers between 1960 and 1969 developed the theory of stratified sets and stratified maps, proving a basic stratified isotopy theorem describing the local conical structure of Whitney stratified sets, now known as the Thom-Mather isotopy theorem. Much of his work on stratified sets was developed so as to understand the notion of topologically stable maps, and to eventually prove the result that the set of topologically stable mappings between two smooth manifolds is a dense set. Thom's lectures on the stability of differentiable mappings, given at Bonn in 1960, were written up by Harold Levine and published in the proceedings of a year long symposium on singularities at Liverpool University during 1969-70, edited by Terry Wall. The proof of the density of topologically stable mappings was completed by John Mather in 1970, based on the ideas developed by Thom in the previous ten years. A coherent detailed account was published in 1976 by C. Gibson, K. Wirthmuller, E. Looijenga and A. du Plessis.Thom's work is especially interesting because of the way he spent his last twenty years. He died a few years after Rosen did. While on sabbatical in France in 1992-1993 I was able to have some interesting conversations with him. We discussed his work and Rosen's and their relastionship through topology and the common phiolosophical threads.
During the last twenty years of his life Thom's published work was mainly in philosophy and epistemology, and he undertook a reevaluation of Aristotle's writings on science.
Network ThermodynamicsI am using this last abstract of a paper I wrote a few years back to link relational thinking to Cartesian mechanistic thinking with the goal of making their relationship have as much "overlap" as possible. The use of topology as the basis for network theory in the field of "network thermodynamics" (or even in electrical network theory) is a natural way to show how mechanistic aspects of a system can be combined with the more relational properties stemming from how the system is "connected together".Abstract
Most systems of interest in today's world are highly structured and highly interactive. They cannot be reduced to simple components without losing a great deal of their system identity. Networkthermodynamics is a marriage of classical and non-equilibrium thermodynamics along with network theory and kinetics to provide a practical framework for handling these systems. The ultimate result of any network thermodynamic model is still a set of state vector equations. But these equations are built in a new informative way so that information about the organization of the system is identifiable in the structure of the equations. The domain of networkthermodynamics is all of physical systems theory. By using the powerful circuit simulator, the Simulation Program with Integrated Circuit Emphasis (spice), as a general systems simulator, any highly non-linear stiff system can be simulated. Furthermore, the theoretical findings of networkthermodynamics are important new contributions. The contribution of a metric structure to thermodynamics compliments and goes beyond other recent work in this area. The application of topological reasoning through Tellegen's theorem shows that a mathematical structure exists into which all physical systems can be represented canonically. The old results in non-equilibrium thermodynamics due to Onsager can be reinterpreted and extended using these new, more holistic concepts about systems. Some examples are given. These are but a few of the many applications of networkthermodynamics that have been proven to extend our capacity for handling the highly interactive, non-linear systems that populate both biology and chemistry. The presentation is carried out in the context of the recent growth of the field of complexity science. In particular, the context used for this discussion derives from the work of the mathematical biologist, Robert Rosen.
These three examples set the stage for a more detailed description of how topolgy enters the scene with respect to Rosen's Relational Biology which will be the next installment in this series.
Herer is the last one in the series:Reading ramblings: Rosen looks at Rosen