The Modern Muse and the Science Museum

It's great to be able to announce the arrival of a new entry to the Articles department of this website.  One of a series of studies I wrote over the years for the Encyclopaedia Britannica's Great Ideas Today, it's titled The Abode of the Modern Muse: The Science Museum. It can be reached by going to Articles on the menu bar; there, choose Great Ideas Today,; and finally, within Great Ideas Today, select the article itself.

I took the opportunity of this assignment to reflect on a long tradition beginning with the MUSEION, the grove sacred to the Muses of ancient Greece, and leading, I claim, in a way important to us today, to the role and concurrent  responsibility  of the modern science museum. Along the way the essay makes major stops, first at Alexandria, where it treats the celebrated "Library" as more truly an academy, and thus just such a meeting-ground of human minds; and finally, at our own Smithsonian Institution, regarded from its inception as a centerpiece of the scientific spirit of our nation.

One crucial role at the outset of this story is that of Aristotle, who affirmed, very much in his manner as thoughtful observer, that the human community is in essence one, and that a fundamental goal, alike of ethics and of politics, must be to realize this truth in practice. The tradition seems secure that Philip of Macedon, to free his son from the distractions of the court at Pela, hired Arisotle as tutor of Alexander, and sent the two of them off to the hills of Macedonia to focus on education.  The curriculum may have been cut short by Alexander's early ascent to the throne, but it seems clear that Aristotle's advice concerning the unity of the human community was foremost in Alexander's mind when he made the founding of Alexandria in Egypt one of his first, and most successful projects.

There were to be many more Alexandrias as Alexander carried his campaign of munification across the Middle East. Readers may have encountered a recent exhibit of Ai Khanoun, an Alexandria discovered to everyone's complete surprise under the sands of northern Afghanistan; my guide at the East Wing of the National Gallery in Washington reported there are believed to be perhaps a dozen more to be unearthed, if our own present wars might cease. But the Egyptian center was surely the best. It began, indeed, as a "library", whose mission was to collect books from the entire Mediterranean basin; copies were made at a publishing house (apparently the building close to the harbor destroyed by the legendary fire).  The copies were sent back to the sources, while the originals were stored securely at Alexandria.

The books, however, were gathered to be studied, not simply to be stored, and in this sense Alexandria is better thought of as paradigm of the universitas, than as library, fundamental as the books themselves must be. As university, Alexandria was conceived to be a new center of human learning for the entire Mediterranean world. It succeeded in that role to a remarkable extent, and we today are its beneficiaries in ways of which we aren't always aware. This was indeed a science museum, as the works of Ptolemy and Euclid, to cite just two examples, attest. Euclid's Elements is a synoptic work, a gathering of contributions from probably widespread sources. What is most exciting in that work is Euclid's own: his brilliant grasp of a profound unity arising out of these contributions. It is a true Alexandrian moment when Euclid perceives in this mathematics the pattern of the tragic trilogy: for those tragic texts were being gathered and assembled in their own unities by that single community of thinkers. It had not occurred to anyone-least of all to Aristotle!-that the human mind need be or could be, compartmentalized into separate academic domains as we have done today.  Academic labors could indeed be divided, but the human mind, as gathered at Alexandria, remained focused on the whole.

This understanding, the article claims, remained intact in the early days of our republic: it is not by chance that our corporate seal, reproduced on the dollar bill (as well as on the seal of my own college, St. John's) depicts an Egyptian pyramid and an insightful eye. Nor that the leader of the procession dedicating the new Smithsonian Institution was reportedly wearing George Washington's Masonic apron. When Smithson's benefaction was accepted as a gift to this nation, the concept of the liberal arts and the unity of learning was still very much alive, and the institution founded in his name was meant as a center of new learning very much in the Alexandrian tradition. We tend to forget this, but other science museums, here and abroad, today wear that same mantle, whether we are always aware of it or not. Most unfortunately, we forget that is not just science, conceived as domain of human endeavor separate from others, but rather science as an integral component of that spectrum of all human thought, collectively the best we can do in understanding and guiding our precarious life on this planet today.

The essay closes with a severe criticism of the abandonment by the Smithsonian, under heavy industry pressure, of a project in conjunction with an exhibit of the Enola Gay. The exhibit had been thoughtfully designed to help the public review in a social and ethical context, the decision to launch our two atomic bombs. Some readers of the essay in the past have disagreed with this judgment on my part, and in this matter, as in all others, I would welcome readers' comments.

Now more than ever we as a world community need to gather our collective wits by any means possible. Science stands at the center of many of our pressing concerns, and the science museum may still be one of the best institutions we can turn to, as the grove of our modern muse.

What Do we Mean by the Term "Elementary"?

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?

"Prometheus Unbound: Karl Marx on Human Freedom"

 It is very hard to find a space today in which to read Karl Marx with an open mind; long history, and fairly severe social bias, stand squarely in the way.  At St. John’s College, however, we read every author with an open mind, as if this work were directed to us personally.  Such an approach is generally disparaged by the academic world, but it does have the advantage of freshness, and of giving open access to original thoughts so often obscured by criticism.  This lecture, given to the college, is an outcome of such an open reading of Marx’s Capital.

What emerges is a vivid picture, grounded in a Hegelian sense of the dialectic of history, of a new stage of true human freedom – a picture which looks remarkably attractive today.   Capital is a complex work, and easily misunderstood.  It begins with a theory of the operation of capitalism, founded primarily in the traditional economic theory of Adam Smith.  What Marx brings to this, apart from a steady suggestion of irony, is a severely scientific logic: what is the source of profit?  What underlies the operation of this system, and what must happen, if these principles are indeed allowed to operate?  A fundamental law emerges, and the structure of Capital at this point is strikingly parallel to that of Newton’s Principia. 

These laws lead to a situation like that we see on a world scale today, of severe dichotomy between those in the world who have, and those who have-not.  At the same time, Marx is surprisingly full of admiration for the accomplishments of capitalism; his chapter on “Great Industry” is a paean of admiration.  He sees not only the economic disparity, but at the same time the achievement of what we would now call the accomplishments of the “global economy”: cooperative labor on a vast scale.  What is being born, he sees, is a class of workers, practiced in cooperation, who see the contradiction between the new flood of products and their own immiseration.    

This sketch cannot pretend to do justice to Marx’s argument.  What emerges, though, is important to emphasize.  Out of this contradiction arises, dialectically, a new possibility, and a new understanding of the meaning of human freedom.  In the tradition of Smith, spelled out in the historic phase of capitalism, is an individualistic, competitive conception of personal freedom.  What Marx sees emerging is a richer concept of freedom: personal freedom indeed, but enriched by the possibilities of social cooperation.  This is not contradiction, but the birth of a new paradigm of the free individual whose possibilities are expanded, not contracted, by a cooperative approach to the resources of society.  

Marx’s reasoning is carefully worked, and his conclusions ring true as we look at the world today.  I have argued elsewhere that we must learn to think in terms of holism, the whole as primary.  Marx tells us that is not suppression of the individual, but liberation from the trap we are in.  Marx is a classicist at heart: he gets his notion of society as primarily whole from Aristotle, and his sense of the birth of freedom from Aeschylus and the founding of the Athenian polis, before he draws on Hegel.  This is a line of thought which I find important and deeply persuasive, in a world and a planet being torn apart by competition and the perpetual war which we see that it breeds.  

It is time that mankind arrived at some better idea of ourselves, and of human happiness and true human freedom.  This may be a good time to be reading authors who think outside our too-limited box.    

"Faraday's Mathematics"

“Faraday’s Mathematics” is a lecture I gave at at a conference on Faraday at St. John’s College in Annapolis.  Its subtitle is “On Getting Allong Without Euclid”, for Faraday had neither studied Euclid, nor taken on board the plan of formal demonstration which most of us learn from the study of geometry.  In short, Faraday thought in his own way, following the lead of nature and experiment.  He was in effect  liberated from the presuppositions about thought and physical theory with which others in the scientific community were encumbered. 

 The result was that Faraday hit on a fundamentally new way of understanding the phenomena of electricity and magnetism – by way of the new concept of the “field”. Maxwell deeply respected Faraday’s way, and dedicated much of his own life to comprehending how Faraday worked, and what it was that Faraday had done.  The field is a fully connected system, and fields interact, not by way of their parts, but as wholes.  This was clear enough to Faraday, but it required recourse to a new sort of mathematics – Lagrangian theory – and a major reversal of conventional thinking, to articulate a formal theory in which the whole is primary, and with it a new rhetoric of explanation.   This was Maxwell’s accomplishment in his Treatise on Electricity and Magnetism, a transformation I trace as a rhetorical adventure in my book Figures of Thought.

  In the end, Maxwell emerged with the astonishing claim that of them all, it was the uneducated Faraday who was the real mathematician.  If that could be so, what is mathematics?   That’s the question pursued in this lecture, which aims to find out what Maxwell could have meant.

Maxwell was clearly in earnest, and seems to be pointing to a mathematics embodied in nature, which lies deeper than either its symbolic or its logical forms.

"The Dialectical Laboratory": A lecture on behalf of holism in the sciences

My lecture, the “Dialectical Laboratory ” (see the "lectures" section of this website) , was given as a sort of parting statement to the St. John’s College community in Santa Fe.  But though directed to the college, and expressed by way of references to certain of the “great books” of that tradition, its message is of far broader import.  The “dialectical” issue – meaning, a watershed of western thought – is between a science based on mechanical actions between disparate parts, and a holistic science in which wholeness is respected, and whole systems are regarded as fundamental, not as mere aggregations of parts.  

Each of these two very different scientific approaches has its rigorous theory, and either can be used to solve engineering problems.  But conceptually they are worlds apart, and I am convinced it’s crucial that we follow the way of holism, and learn, before it’s too late, to appreciate and work with systems – from the least living organism to the global environment – which are more than the sum of mechanical parts.  Science is moving in this direction, but there is now no time to lose! 

Comments on these remarks, as well as on the lecture itself, will be welcome in reponse to this posting. 

"Reason", Old and New

Somewhere in the course of our western history, something fundamental has been lost: we have lost track of the wholeness of the psyche, and its membership; in a world which was whole and in which it might feel at home.   

Where did this happen?  The psyche was whole in Athens – its membership in the family, the polis and the cosmos were so presupposed that there were perhaps no words to express the separation and fragmentation so vivid to us today.   I don’t think there was a word for “objective” or “subjective”, nor was there a mind which might be thought of as a blank tablet, upon which an outside world might write. In society there was work, but no word for “job”, with the radical alienation that term implies.  I’m not suggesting life was in any sense idyllic – only that for better or for worse, the psyche was intact, and seated in the world. 

I’ll leave it to others to explain how this has come about, but somehow we now find ourselves equipped with a mind which is well-furnished with knowledge, indeed, but all too easily likened to a calculative engine, with a memory bank stored with data from an “outside” world.  We understand the mind better and better – but only as a marvelously equipped machine.  

What is missing would seem to be that faculty once called “intellectual intuition” – the power to see directly and immediately, without the intervention of words, truths which are timeless and fundamental.  That old intellect -- for which the Greeks did have a word: NOUS -- was inherentlyi drawn to beauty, which it deeply loved. 

I don’t see this as an exercise in nostalgia: there are ways open to us today by which we can recover this power, which is perhaps rather hidden than lost.  Other cultures have preserved it in ways we haven’t, and we have much to learn from them.  To a large extent it is our conception of “modern science” which denies the evidence of intuitive reason, and reduces the concept of “reason” to accurate symbolic calculation.  But there is another way within modern science, equally mathematical and rigorous, but founded in a concept of wholeness, and looking to the whole rather than the parts as the ground of “explanation”.  I have spoken about this way – the “Pinciple of Least Action” -- in my lecture, “The Dialectical Laboratory”, elsewhere on this website.   

Nothing prevents, I believe, our mending this split between that classic concept of intuitive reason, seated in the world and knowing and loving truth directly -- and the concept current today of reason as a calculative engine making what it can of an “outside” world.   We need only retrace our steps and pick up that thread of truth wherever we dropped it.  Not easy to do, of course, but worth every effort!

Any suggestions as to how to begin? 

In Praise of Generalized Coordinates

I've been expressing my enthusiasm for a holistic approach to the understanding of nature -- in relation to my favorite topic, the electromagnetic field, this takes the form of the Lagrangian equations for the field as a single, connected system characterized by its energy, not by forces.  It was crucial to Maxwell's development of the equations of the field in his "Treatise on Electricity and Magnetism" that they be formulated as instances of such a connected system -- i.e., in Lagrangian terms, and NOT on the basis of Newton's laws of motion.  (The difference -- very fundamental to our understanding of nature -- is developed in "The Dialectical Laboratory", in my "Lectures" menu.) Now, the question arises: "If we start in this way, from the 'top down', how do we ever arrive at the elements?"   The answer is, "We DON'T!" We move logically "downward" by finding the dependence of the energy of the whole system upon ANY set of measurements we want to make -- provided only that it's a complete (i.e. sufficient to determine the state of the system), with each measure "independent" of the others. We find such a set of measurements by doing experiments -- and when we get them, they are called "generalized coordinates".  The important thing is that there may be many ways we can define them, each set as good as the others: and in the whole process we never get any"real,underlying elements" -- we don't need them!  Reality is founded at the top, not the bottom, of the chain of explanation.   This is Maxwell's new view of physical reality, founded upon the field.  It is the opposite of the notion of "mechanical explanation", and it is the direction which our approach to nature desperately needs to take as we approach the challenges which lie before us today.  In terms of the philosophy of science, Maxwell it seems was far ahead of his time.  I propose to call this the "Maxwellian Revolution". 

Newton on the Field

I've just returned from a gathering in New Mexico, the first, pilot workshop of the Cosmic Serpent project, in which Native Americans and others-such as myself-gathered to compare Native American views of the natural world with those of "western science". With the essential help of Jim Judson from the Sister Creek Center in San Antonio, I brought along an "open lab" on magnetism. It seemed to me that the concept of the "field"-specifically, here the (electro-) magnetic field-might prove helpful in relating these two domains of thought about nature.

For the moment, here, I just want to comment on a document that was circulating during the conference concerning the mystery of magnetism. Asking very simply "What is Magnetism?", it was written by Bruno Maddox and published in a recent edition of Discover magazine. He reports that after exploring all options, he finds no scientific explanation of the cause  of magnetism.  If it remains a mystery, as he seems to conclude, then it may well be open to interpretation in terms compatible with Native American points of view.

That's a point of view I'll want to return to in future postings.  For the moment, I want to call attention to one of Maddox's findings. He hit on a text in which Isaac Newton-looking in this case at the mystery of gravitation-opines that "the notion that one body may act upon another at a distance through a vacuum without the mediation of anything else...is to me so great an absurdity that I believe no man who has in philosophic matters a competent faculty of thinking could ever fall into it."What did Newton have in mind?

I'm confident that he is not thinking in terms of any sort of mechanical explanation. Newton was not a mechanist: in fact, he wrote the Principia essentially as a polemic against mechanism, and in particular, against Descartes. No. His aim is to reveal the role of what he called Spirit in the world: the fact that the laws of these actions are mathematical in no way implies for Newton that they are mechanical, but is fully compatible with his concept of Spirit and its operation throughout the realm of nature.

I'm not arguing that Newton "had" the idea of the field-though his "intensive" quantity of a force seems to ascribe it to space itself, and is remarkably compatible with later ideas of the "field". My point is only that as he describes the mathematical System of the World, Newton feels himself to be in the immediate presence of mystery-in his view, divine mystery in the form of the Holy Spirit as God's agent in the natural world.

Newton's thoughts along these lines, together with those on alchemy and theology, were systematically buried by his followers, and have been uncovered only in recent years. But now that we have a better sense of what he actually meant, we may be the more ready to contemplate this bridge between "spirit" as Newton intended it, and "spirit" in Indigenous accounts of the operations of the natural world. Either way, we are contemplating something which has all the feel of wonder and mystery.

While in Santa Fe, I learned that students at St. John's College there would be gathering to witness this very mystery, in an experiment which Newton himself had thought would be impossible to carry out. Just as the Sun and Earth are joined by the gravitational force, so any two bodies on Earth must attract another by a very slight, yet calculable force. The experiment can in fact be done, with lead weights suspended by a delicate metal thread. To watch them, by way of a light beam and mirror, move toward one another slowly but surely, is to be present at a solemn ceremony at the foundation of the cosmos-as much a mystery, still today, as it ever was. I wonder if others agree with this reading of Newton's text?

Why Aristotle? Why Now?

 Here’s a brief posting, not unrelated to the previous two. I spoke in the first of a “tap root” running back from what we think of as “modern science” to sources in an ancient past. Such a tap root is not just a connection to the past, something of interest to academic historians, but potentially a powerful source of nourishment today.  This may seem a strange claim to make for Aristotle in relation to modern science, but I do put it forward in earnest. Aristotle generally gets a bad rap from those who tell the story of modern science, but to a large extent it’s latter-day Aristotelians (such as Galileo’s Simplicio), not Aristotle himself, who are the targets of such criticism.  It is well-known, and widely acknowledged, that Aristotle was a serious empiricist, conducting dissections and drawing such generalizations as he could perceive. But what was his account of scientific method, that we might give it serious attention today?  I’m writing this from memory, so my references for the moment must be inexact.  But in crude summary, here is the account which culminates in his “Posterior Analytics”.  

He has said elsewhere that the objects of true knowledge which Plato calls the “forms” are “nowhere”, not in the sense that they do not exist, but that they do not exist in separation.  The forms are everywhere in the observable world. We meet them when mind grasps something as whole and true. He says somewhere that scientific inquiry, as we gather data, is like an army in retreat: first one soldier takes a stand, then another, then more - and soon, the whole column stands fast. That standing fast is the mind grasping something true: “seeing something” whole, as we say, or achieving an intellectual intuition.  Such an intuition is not the additive sum of the component data. Between such an empirical summation (which Plato calls the “all”), and the grasp of a truth, (a “whole”), lies the difference between data-processing and great science. 

 We are so concerned today to emphasize the “objectivity” of true science, that we fail to acknowledge the role of mind - a function which grasps something the data do not themselves present. In that sense, great science, serious science, cannot be reduced to objectivity.  It cannot fly in the face of the data, but it cannot be reduced to those data, either.  We live at a time when it is becoming increasingly urgent that we rise to the challenge of recognizing whole systems as such. An ecology is something more than the sum of any quantity of data.  In biology, this whole beyond the parts is termed an “organism”; perhaps Aristotle would be reminding us today that we are in danger of failing to recognize life itself when it lies before us in our laboratories or in the seas.   

Indigenous Views of Nature and the Deep Roots of Western Science

When I wrote yesterday about the "deep roots" of Western science, I intended to point to a possible relation this opens up between the domain of "science" and Indigenous views of the natural world.  If we follow that line of development which leads from Aristotle through Leibniz to the holistic mathematical physics based on the Principle of Least Action, we find ourselves in a position much closer to that of Native American thinkers than we might have expected.Modern science in its mechanical mode cuts off "science" from any sense of wholeness or, especially, of purpose. It wants to reduce all quality to quantity, all motion to the operation of laws which bind matter apart from any sense of goal or meaning, and sees "nature" exclusively as an object from which we stand apart as mere observers. None of these limitations apply to the physics in the holistic mode.  Least Action applies to whole systems, and sees systems moving directionally toward the optimization of a quantity which applies to the system as a whole.  Although this goal may be no more than the optimization of a mathematical quantity, it opens the way to thinking of systems such as organisms or ecologies as moving as wholes toward ends -- a line of thought of which the modern world is in desperate need.One more link in this line of thought: the modern computer is bridging the gap ;between "quantitative" and "qualitative" thinking.  What goes in as number typically comes out on the computer screen as a graphical image readily grasped by the intuitive mind and conducive to interpretation in terms of purposes and goals. We can see how systems are moving, and where they "are going".   Nothing stands in the way of reading these in terms of purposes, and that is what we do on a daily basis -- think for example of evidences of the consequences of global warming emerging from complex computer modeling.  Thinking in this way in terms of whole systems,  understanding their motions in terms of a mathematics of optimization, and bridging the gap between quality and quantity -- all this is yielding an approach to science at once new and old -- in a continuous thread leading from Aristotle into the age of the modern computer.  If we follow that path and think of modern science in terms like these, then it seems to me the gap between a holistic science and Indigenous relations to the natural world is not as deep as it had seemed.  Set aside mechanistic thinking, embrace the sense of nature as a whole of which we ourselves are part, admit goal as a category amenable to science -- and then the old gap between Indigenous, or simply hunan views of the world, and those of "western science", begins to dissolve.   Thus the Cosmic Serpent project, designed to consider this relationship, begins to look much more promising than it otherwise might have.   

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”!