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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 biology
 Before 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....

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.

 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:
Carotheodry's proof of the second law of thermodynamics
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.
René Thom's  Catastrophe Theory
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.
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.
 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.
Network Thermodynamics
Networkthermodynamics and complexity: a transition to relational systems theory
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.
I 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".

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

Originally posted to Readers and Book Lovers on Mon Mar 19, 2012 at 08:59 PM PDT.

Also republished by Systems Thinking, Progressive Friends of the Library Newsletter, and Progressive Hippie.

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Comment Preferences

  •  Tip Jar (7+ / 0-)

    An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

    by don mikulecky on Mon Mar 19, 2012 at 08:59:04 PM PDT

  •  Do you know the work of Eric Zeeman? (3+ / 0-)
    Recommended by:
    Larsstephens, PeterHug, don mikulecky

    Another dusty old book I found in my student years is "Topology of 3-Manifolds and Related Topics", which includes Zeeman's essay, "Topology of The Brain and Visual Perception".

    I was in a group of mostly physics and engineering undergrads attached to the the Human Learning Center at Minnesota.  This topic animated a number of all-night bull sessions.  I take it Zeeman was more enthused about the potential for topological modeling than was Rosen.

    I've not read his work on catastrophe theory.

    Where are we, now that we need us most?

    by Frank Knarf on Mon Mar 19, 2012 at 10:33:19 PM PDT

    •  That should be Erik. n/t (2+ / 0-)
      Recommended by:
      Larsstephens, PeterHug

      Where are we, now that we need us most?

      by Frank Knarf on Mon Mar 19, 2012 at 10:35:17 PM PDT

      [ Parent ]

    •  No I dont know that work (0+ / 0-)

      The topics I taught in physiology  are very old.  The homunculus topologically mapped onto the sensory and motor cortex are not models but the actual result of experimental measurement.  The same is true of the visual cortex.  The brain has no sensation so during brain surgery it is possible to do such mappings.  I don't know what Zeeman was into.

      I take it Zeeman was more enthused about the potential for topological modeling than was Rosen.
       I don't understand why you say this?  Category theory is a sophistication of the earlier topology. and most of the math Rosen uses is directly from topology.

      The work on catastrophe theory is by Rene Thom not Rosen as the diary says.

      An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

      by don mikulecky on Tue Mar 20, 2012 at 07:32:00 AM PDT

      [ Parent ]

    •  On second thought yes I do know (0+ / 0-)

      some of Zeeman's topology.  He did work with Thom and also with Smale whose work I know well.  We went through a lot of this in our discussion group.

      Sir Erik Christopher Zeeman FRS (born 4 February 1925), is a Japanese-born British mathematician known for his work in geometric topology and singularity theory.

      from 1966 to 1967 he was a visiting professor at the University of California at Berkeley, after which his research turned to dynamical systems, inspired by many of the world leaders in this field, including Stephen Smale and René Thom, who both spent time at Warwick, Zeeman subsequently spent a sabbatical with Thom at the Institut des Hautes Études Scientifiques in Paris, where he became interested in catastrophe theory. On his return to Warwick, he taught an undergraduate course in Catastrophe Theory which became immensely popular with students; his lectures generally were "standing room only". In 1973 he gave an M.Sc. course at Warwick giving a complete detailed proof of Thom's classification of elementary catastrophes, mainly following an unpublished manuscript, "Right-equivalence" written by John Mather at Warwick in 1969. David Trotman, then aged 21, attended this course, and wrote up his notes as his M.Sc. thesis. These were then distributed in thousands of copies throughout the world and published both in the proceedings of a 1975 Seattle conference on catastrophe theory and its applications,[5] and in a 1977 collection of papers on catastrophe theory by Zeeman.[6] In 1974 Zeeman gave an invited address at the International Congress of Mathematicians in Vancouver, about applications of catastrophe theory.
       This blurb does not mention the brain work nor do I recall it.  I definitely read some of his stuff on catastrophe theory.  Memory is slow these days.

      An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

      by don mikulecky on Tue Mar 20, 2012 at 07:44:48 AM PDT

      [ Parent ]

  •  Searching for cites, it seems he is often (1+ / 0-)
    Recommended by:
    don mikulecky

    referenced by determined modelers, e.g.

    Modeling cognitive systems with
    Category Theory
    Towards rigor in cognitive sciences
    Jaime G´omez and Ricardo Sanz

    http://tierra.aslab.upm.es/...

    A mathematical explanation of how the brain is structured and how it achieves
    cognitive functions like perception, conceptualization or learning is seen for cognitive
    scientists, especially for those of humanistic background, as an extreme reductionism
    that obviates essential human capabilites like agency, intention, intuition,
    emotions or feelings.
    The so called ”hard sciences” have achieved mathematical certainty about the
    world, and although it has been always the case that formal systems fall short of
    capturing reality completely, this is not because these systems (mind included) are
    impossible to formally explain, rather it is us, as modelers who seem to have limited
    (perceptual) access to reality, so that we, unfortunately, usually only get partially
    valid descriptions.
    We claim that, the object of study of the cognitive sciences, how the brain works,
    is still in a pre-scientific stage. The progress from an immature science to a mature
    one must pass through the construction of appropriate mechanisms with explanatory
    power.
    The main objective of this paper is to set the agenda of category theory as that
    appropriate mechanism that provides the theoretical framework in order to build
    cognitive models in mathematical terms.
    In (6), E.C. Zeeman holds that a mathematical explanation of how the brain
    works has to rely o the concept of isomorphism.
    First develop a piece of mathematics X that describes the permanent structure
    (memory) and the working (thinking) of the mind, and another piece of
    mathematics Y that describes the permanent structure (anatomy) and working
    (electromechanical) of the brain; then, from hypothesis based on experimental
    evidence, prove an isomorphism X = Y

    Where are we, now that we need us most?

    by Frank Knarf on Tue Mar 20, 2012 at 08:09:56 AM PDT

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