Organism vs. Mechanism: Science at the Lagrangian Divide

The Lagrangian equations are a powerful set of differential expressions describing the motion of a complex system.  With one equation for each component of the system, they would seem to offer a powerful expression of the relation of part to whole. They are, however, seriously ambivalent: they can be read in either of two opposite ways. They present, then, a stark problem for the art of interpretation, the highest branch of rhetoric, as it comes from Augustine to Bacon and Newton.  The same statement becomes a watershed; it may belong to one world, or its opposite – but not both.  Each is a containing frame, within which we picture, and live, our lives

Read in one way – the way most common today – they are seen as derived from Newton’s laws of motion, and thus adding nothing fundamentally new. From this perspective, they merely rephrase Newton in terms of the concept of energy, a mathematical convenience in certain circumstances but making no fundamental change in our understanding of the natural world. In this interpretation, they express what we today call mechanism, which sees the motion of any system as the mere aggregation of the motions of its individual parts. Causality flows upward; motions of the parts explain the motion of the whole.

Seen from the other side of the Lagrangian watershed, however, the same equations express a world of a totally different sort. Here, the same equations are derived from the Principle of Least Action – a concept which readers may recognize as one of the recurring themes of this website.  The system itself as a whole, described in terms of potential and kinetic energy, becomes the primary reality and the source of the motions of the parts. Causality arises from the  interplay of these energies, and flows in the reverse direction, from whole to part.

Within the world of mechanism – the first interpretation – there is no place for goalor purpose. These are concepts considered far too vague to meet the standard of objectivity, the signature of modern science.

Remarkably, however, Least Action reconciles purpose with quantitative objectivity. By means of the mathematical technique of variation, which considers all possible paths, this principle seeks the optimum path by which potential energy may, over he whole course of any natural motion, be transformed to kinetic. In this interpretation of Lagrange, then, our world-view is transformed. Science itself, while remaining strictly objective and quantitative, becomes at the same time goal-oriented – all at once!

More than this, however, science on the Least Action side of the Lagrangian divide becomes, at last, fundamentally organic. This arises from a further, crucial feature of Least Action: if a system as a whole moves in such a way as to minimize action,so also will, within the bounds of external constraints, every part of that system. The goal which belongs primarily to the whole, is pervasive: it is shared by every part.

It was important in stating this principle to add “within given constraints”, because a rigid part of a man-made machine has few options. By contrast, the myriad components of a leaf, or of a cell or enzyme within the system of a leaf, navigate among unimaginable options toward the common goal of turning sunlight into life, over the season of the leaf, the life of the tree, or the evolution of photosynthesis on earth.

It is this community of purpose, nested and shared, which renders a system trulyorganic – a living being, something fundamentally beyond any bio-molecular mechanism, however intricate.

It is hardly necessary to add that it is this sense of nested purpose and shared membership in natural communities which has been so lacking during the long reign of mechanism. Our so strongly-held worldview has diverted us from that other option, which has nonetheless long formed a strong alternative flow of thought and practice in science, mathematics, politics and the arts. Now in many ways, not least the earth’s biosphere itself, the demand is upon us to recognize that we do have an option of immense importance. Viewing this whole scene now, we might say, from the Lagrangian ridge-line itself, with both worldviews clearly in view, our task is truly dialectical: leaving none of the insights of the past behind, we are in a position to move forward into a new, far richer and wiser world.

That new world-view, which has appeared here as a richer interpretation of Lagrange’s equations, is the ongoing theme of this website – always with an eye to Maxwell’s turn to Lagrange as mathematical vehicle for the launch of his concept of the electromagnetic field, paradigm, if ever there was one, of that whole system of which we have been speaking.

[A brief introcution to the Principle of Least Action is given in my lecture, “The Dialectical Laboratory” .

It is important to add that in this thumbnail sketch, many nuances of the application of Least Action have been left without mention]

An Ecosystem As A Configuration Space

In my most recent posting, I've been exploring a quite classic mathematical model of an ecosystem: the Salt Marsh ecosystem model developed at Sapelo Island and described in the fascinating 1981 volume, "The Ecology of a Salt Marsh". For those of us who are devoted to grasping the "wholeness" of an ecosystem, the question arises whether matching such a system to a mathematical model helps in grasping this wholeness - or whether it may even detract. The concern would be that true unity is broken when a whole is described in terms of relationships among discrete parts: as if the "whole" were no more than a summation of parts - in Parmenides' distinction, an ‘ALL" (TO PAN), exactly the wrong approach to a true "WHOLE" (TO HOLON). An excellent guide in these matters is James Clerk Maxwell, who faced this question as he searched for equations that would characterize the electromagnetic field in its wholeness. As soon as he learned of them, he embraced Lagrange's equations of motion, and as he formulated them, his equations derive from Lagrange's equations, not from Newton's. For Lagrange, the energy of the whole system is the primary quantity, while the motions of parts derive from it by way of a set of partial differential equations. Fundamentally, it is the whole which moves, the moving entity, while the motions of the parts are quite literally, derivate.

The components of such a system may be any set of measurable variables, independent of one another and sufficient in number to characterize the state of the system as a whole. Various sets of such variables may serve to characterize the same system, and each set is thought of as representing the whole and its motions by way of a configuration space. If we have such a space with the equations of its motion, we've caught the original system in its wholeness: not as a summation of the components we happen to measure, but in that overall function in which their relationships inhere.

Now, it seems to me that a mathematical model of an ecosystem, to the extent that it is successful, is exactly such a configuration space, capturing the wholeness of the ecosystem whose states and motions it mirrors. Specifically, the authors of the Sapelo Island Marsh Model were if effect working toward just this goal, though it may not have appeared to them in just these terms. All their research on this challenging project was directed toward discovering and measuring those connections, and the integrity of the resulting mathematical system was exactly their goal.

They had chosen to construct their model in terms of carbon sinks and flows; the measures of these quantities were sufficient to characterize the state of the system and its motions, and therefore constituted a carbon-configuration space of the marsh. A different set of measures might have been chosen, and would have constituted a second configuration space for the same system: for example, they might have constructed an energy-model, which have been equivalent and represented in other terms the same wholeness of the marsh. Carbon serves in essence as a representative of the underlying energy flows through the system.

I recognize that this discussion may raise more questions than it answers, and I would be delighted to receive responses which challenged this idea. But I think it sets us on a promising track in the search for the wholeness of an ecosystem - an effort, indeed, truly compatible with the wisdom of Parmenides!

The Deep Roots of “Western Science”

I’m very excited to have been invited to participate in the Cosmic Serpent project, which will be exploring the relationships – likenesses and differences – between Indigenous views of Nature, and the world-view of “western science”.

My first thought about this is that what we are accustomed to calling “western science” is not one well-defined, monolithic structure, but rather a growing and changing, organic body including strongly contrasting strands and a deep tap root which reaches far back in history to ancient Greece and beyond.

It is this richness and diversity of our present notion of “science”, together with its vigorous signs of growth, which make the Cosmic Serpent conversation something far more than a confrontation of two distinct ideas. Whether there’s the same degree of diversity and growth within Indigenous approaches to Nature is something I don’t pretend to know, but the coming conversation may reveal.

I feel impelled to say something more about that deep “tap root” of modern science, as it lies close to my heart and has been the subject of much that I’ve thought about and written. (I wrote about one aspect of it in the lecture “The Dialectical Laboratory”, elsewhere on this website.) For me, as we look backwards from our present stance toward a distant past, it is Leibniz who’s the key. Between Leibniz and Newton lay a split far more important than the question of prioty in laying the foundations of the calculus usually referred to. In ways not always recognized, Newton was looking to Christian scripture, especially the Old Testament, when he placed the notion of “law” at the foundation of his Principia. Leibniz, by contrast, was looking to Aristotle and saw intelligible principle – not “?law” – as the basis of our approach to Nature. Two of Leibniz’ crucial terms: energy – potential and actual – come straight from Aristotle’s Physics, and remain to this day beacons of an alternative path in physics. Not forces between particles, the dominant concept of the mechanical view of Nature – but motions of whole systems guided by principles rightly thought of as holistic – set this other course. It becomes formulated mathematically as the law of least action, which evolves in turn into the equations of Lagrange and Hamilton, and in general into the Variational approach to natural motions. It is an approach inherently compatible with the notion of TELOS, or goal. In a broader arena, it is at home, for example, with Gestalt theory in psychology and the theory – at once of art and science – which Goethe sets over against that of Newton in his Farbenlehre, the Theory of Color.

For the practicing physicist, the Newtonian and the Lagrangian methods may seem convenient alternatives to be called upon as occasion demands. But in truth they reach very deep into alternative conceptions of the natural world and its ways. As I explore in Figures of Thought, it was not for convenience but out of deep conviction that Maxwell chose the Lagrangian approach in his own development of the equations of the electromagnetic field in his Treatise on Electricity and Magnetism. That this is an issue for human thought in general, and not a problem whithin mathematical physics alone, is shown beautifully by the fact that Maxwell chose the Lagrangian method as the way to express within mathematics the insights of Michael Faraday, who knew, and wanted to know, no mathamatics.

I have to acknowledge that there’s a manifest contradiction in what I’ve just written: I spoke at the outset of one “tap root” of science, but this whole discussion has been of two: one Newton’s, and the other that of Leibniz. I’m convinced these two lead back, by way of Alexandria, to one lying still deeper – but that must be the subject of another “blog”!