Early in his life, long before he met Alice and plunged down that rabbit hole into Wonderland, Lewis Carroll was given a very curious math problem. It was a simple equation, but it had no solutions in the world of real numbers (the counting numbers, with all their fractions and multiples). All its solutions, but zero, were imaginary! He looked in vain for a way to see them – an imaginary plane perhaps, on which they might be graphed. He couldn’t find one, but we can!
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]
What do we mean when we use the term ,”elementary”, in relation to a science? Does it refer to an easy introduction, as contrasted with an “advanced” treatment of the same subject? Or does it mean a solid account of the very foundations of the science? Or, for that matter, are these the same thing?
Maxwell had a tendency toward writing “elementary” texts: he wrote one on heat, and another on mechanics, both for use in classes for workingmen – a project to which he was deeply committed. Finally, at the time of his death he was at work on his “Elementary Treatise on Electricity and Magnetism, intended to serve as the Cambridge text to support a new degree in experimental natural philosophy at Cambridge University.
My sense is that Maxwell endowed each of these with earnest attention – that he regarded the “elements” not as evident, but as a topic to be approached with great care. Our decision as to what is elementary in a science has a great deal to do with our sense of the form the finished product will take – so that the most difficult issues may focus on the most elementary beginnings.
For example, Maxwell wrote his workingmen’s text in mechanics, Matter and Motion, only after he had hit on the fundamental idea, new to him, of Lagrnagian mechanics and generalized corrdinates. This would not be a mechanics in Newtonian form, in which the elements would be assumed to be hard bodies acting upon one another according to laws; rather, elements of this sort would be the least known components of the system, represented by generalized coordinates.
In this view, what we observe initially is a whole system of some sort; it is this whole which is fundamental, and truly elementary. The parts which compose it, we may never know. Our science can be complete and secure even if that question remains unresolved, or unresolvable.
This is the point of view I believe Maxwell had come to, underlying his approach to the new program at Cambridge as well. If so, must it not represent a truly revolutionary inversion of our very concept of scientific knowledge?
It fitted the primacy he – following the path of Farday – was giving to the concept of the electromagnetic field. In this view, he field would not be a secondary phenomenon, a composite or consequence of simpler “elements”, but itself both simple and whole.
If the elementary is what is primary, then in the case of the field it is the whole which is the element, from which we deduce what we can, concerning lesser components. Faraday had felt strongly that in the case of electricity, there was no “charge” lying on the surface of a charged body, but what we call a “charge” was a field, which filled the room.
Isn’t it the case that when we ask for the “explanation” of a physical system, we are asking for an account in terms of its elements? If so, then the field is itself explanatory, and we would not seek explanation in terms of the actions of some lesser parts. What will be the consequences if we extend this view to physical explanation – or explanation beyond the realm of physics -- more generally?