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Mathematics of Motion: A Christmas Corollary

December 25, 1996

A dialogue between Johannes Kepler (1571-1630) and Isaac Newton (1642-1727).

We join Kepler and Newton as they tour the science classes of a modern high school, invisible and unheard by the students and teachers they observe. It will soon be Christmas, 1996 and the students are frantically cramming for exams.

* * *

Kepler: It gives me great pleasure, Sir Isaac, in having this opportunity to meet and visit with you on this Christmas tour of science and education in the Americas.

Newton: The pleasure is a mutual one, Johannes, for your book "Optics" was one of the first scientific works I studied and inspired my early interest and explorations into astronomy and motion.

Kepler: I thank you for your compliment, and though I had no opportunity, while alive, to read your great works, I have studied them in depth in more recent times. .... Do you not find the current educational scene, while changed in many respects from our day, still showing more likenesses than difference?

Newton: Indeed I do. However, one of the differences, a source of great satisfaction to me, is to see and hear how frequently, in this maths class, my propositions are written and my name is spoke.

Kepler: Yes, your "calculus" has most certainly made a "difference". Yet I feel that there is much unexploited opportunity to further apply your theories within these teachings.

Newton: How so?

Kepler: Both students and teachers use your mathematical tools much as weapons, to attack the problems of the day. Sighting along the barrel of your canons, intent on obliterating the problems which stand before them, they would do well to recall your laws of action-reaction and conservation. For these laws will hold the momentum-sum of advancing projectile and recoiling intellect to zero.

Newton: Yours is a peculiar sense of humor which seems to me out of place in men of science such as history purports yourself to be. (Newton goes on, mumbling) Perhaps the history witch the Wuerttemberg courts would have made of your mother and you, fired up as they were over the strangeness of your art, had been more apt.

Kepler: If the harmonies of the season favor a measure of frivolity and jolliness, I am for it, and so fortunately are these students. A moment ago I saw a pair of them studying calculus and making fun with their new-found "Newtonian" erudition. In their state of "nerdiness" and seasonal good humor one said to the other; "What is the entity which, when differentiated with respect to "t", gives "d-cabin" over "t"?" The other answered with a knowing grin, "log cabin". But his smile was brief as his friend retorted; "No, you forgot the "C", the constant of integration --- the correct answer is "houseboat", that is; "log cabin plus C."" Though I took my leave at this point, I could hear persisting fits of nerdy mirth from the far end of the corridor.

Newton: As we are here to review history's contributions to the mathematics of motion taught in these times, I have difficulty understanding why we fritter away this opportunity on the trivia of this epoch.

Kepler: Just for that, I shan't share with you the several superb lines of graffiti, which I captured from the girls' loo a few minutes ago.

Newton: I refuse to be drawn into the trap of asking you why you chose the girls' over the boys' loo, and as concerns the graffiti, the curiousity you raise in me is more in regard to pathologies of the adult human brain than with the sexual fantasies of twentieth century teenage girls.

Kepler: More's the pity. But I make no apologies for the associations which my mind so liberally makes between all manner of things, for it was through this process that the three laws of motion of the planets and sun came to me, and I suspect to you as well. Apart from my oversight on the nature of inertia, the true nature being recognized by Galileo and attributed to you, the primary difference which separates your laws of motion and my own discoveries, is the banishment of all harmonies in your versions.

Newton: Having been accused of ego inflation by so many, it gives me some solace to see others so obviously and similarly smitten.

Kepler: While I concede to ego as being my travelling companion, it has been, and still is, the search for the truths in nature which motivates my life's work. As I said in "Harmonice Mundi", "... my mind is never at leisure for the game of inventing new doctrines that are contrary to the true. Whatever I profess outwardly, that I believe inwardly: nothing is a worse cross for me than --- I do not say, to speak what is contrary to my thought --- to be unable to utter my inmost sentiments."

If I take pride in my achievements, I take most pride in my manner of achieving them. Though I see many of my discoveries labelled with the names of others, this grieves me in measures vanishingly small by comparison to the pleasures taken from tributes such as that given me by Bernstein;

"In addition to being one of the most original thinkers in the history of science, he had an extraordinarily open and appealing personality. Unlike Copernicus and Newton, who were extremely secretive, and unlike Galileo who was not eager to share ideas, Kepler wrote letters, books, pamphlets in reams about his discoveries and his life. He was never content just to announce his discoveries in final, polished form; he shared the process of discovery by taking his readers with him every hesitant step of the way, discoursing all the while on his own fallibility." ... "Woe to me, here I blundered." ... " [this is] irrelevant, false, and based on illegitimate assumptions" ... " the reasoning of the whole chapter is wrong" ... "This question is superfluous. ... Since there is no discrepancy, why did I have to invent one?"

As I also said in "Harmonice"; "it gives me pleasure to remember how many detours I had to make, along how many walls I had to grope in the darkness of my ignorance until I found the door which lets in the light of truth. ... In such manner did I dream of the truth."

Newton: Having recently been made the scapegoat of Heraclitus in another one of these time-transgressing tours, I hesitate to come clean with you on these issues. Yet, I will venture to say that never was I more hurt than by the statement of my assistant, William Whiston, my successor as Lucasian Professor of Mathematics at Cambridge, who said of me; "Newton was of the most fearful, cautious and suspicious temper that I ever knew."

Kepler: Well, after all, your actions have not been entirely unmeritorious of such an appraisal. In your master work Principia, you surreptitiously put the proofs of all of your propositions in "reverse-engineered" geometric form although you had come by them using your new method of fluxions. Because you did not want the reader to doubt your proofs, by letting him know that you had used a new, home-made and unfamiliar calculus to develop the propositions, you had most of the scientists in Europe scratching their heads as to how you could ever have come up with these seemingly valid propositions from such obscure geometric argument.

Newton: I was persuaded to publish the Principia by Halley, for the purpose of shedding light on the processes of nature, not for exposing the workings and visionings of the personal sanctuaries of my mind.

Kepler: This is indeed where there is a parting of minds between the two of us, and I should like to return to the "trivial" allegory of the log cabin and the houseboat, if I may, to make my point.

Newton: Since your soul-mate Heraclitus seemed to find more value in cowdung than in the teachings of his doctors, I should not be surprised at your attempt to select adolescent babblings as the prima materia for developing your scientific insights.

Kepler: Well then, do you recall the boy's reference to something often ignored when one descends into the domain of the calculus, i.e. into the domain of instantaneous behavior, of local properties of things which have been divided up into small parts? The boy referred to it as "C", the "constant of integration", associating the letter "C" with the idea of "sea". "Sea" is often used by the mind as a metaphor for consciousness, and the imagery of a log cabin is very different from the imagery of a houseboat floating on the waves of consciousness.

Newton (under his breath); I wonder if the Wuerttemberg Courts are still accepting depositions for their dossiers on "the forbidden arts"?

Kepler: It is clear that the calculus, the mathematics of instantaneous behavior, is the ultimate implementation of Descarte's "Methode" of dividing up problems into small parts so that they can be solved by part then summed to give an overall solution. In your Principia, you refer to your use of Descartes' "Methode" in the following terms;

"In this philosophy particular propositions are inferred from the phenomena, and after rendered general by induction. Thus it was that the impenetrability, the mobility, and impulsive forces of bodies, and the laws of motion and of gravitation, were discovered."

Thus did you take the phenomena of the harmonies of the planets into this domain of small local parts and differences, generalizing the forces and motions in these disconnected fragments, then rendering the general whole once again by a summing of the parts. But ..... (a shadowy sadness comes across Kepler's countenance as if the earth's shadow had passed over the moon or a cloud had suddenly obscured the sun) ... but there was a consciousness, a life in the phenomena you began with which did not survive the return trip from the calculus back into the generalization .... the "C", once banished, could no longer return.

In 1619, in "Harmonice", sixty years before your Principia, I had discussed this problem of the particular and the general, the parts and the whole, saying;

"From the celestial music to the hearer, from the Muses to Apollo the leader of the Dance, from the six planets revolving and making consonances to the Sun at the centre of all the circuits, immovable in place but rotating into itself. For although the harmony is most absolute between the extreme planetary movements, not with respect to the true speeds through the ether but with respect to the angles which are formed by joining with the centre of the sun the termini of the diurnal arcs of the planetary orbits; while the harmony does not adorn the termini, i.e., the single movements, in so far as they are considered in themselves but only in so far as by being taken together and compared with one another, they become the object of some mind; "

Isaac, do you see what you have done? ... Your calculus and your propositions have extracted the living harmony, nature's consciousness, the "Logos" from our view of nature and irrevocably banished it! For as we extract the parts from the whole, examining and "understanding" them in detail and out of natural context, thence returning them to their relative positions, we ignore the living aiether which gave them birth. And from the laws of the harmonies of the planets, from my third law which shows the two-thirds proportioning of period to radial length of orbit to be constant, it was from this law that you developed your "universal law of gravitation", F=GMm/r**2 through a sterilizing process which left it barren of any reference to time or harmony.

...And, adding injury to insult, you endowed this theory with the ability to estimate the effects amongst multiple bodies through the mechanical contrivance of a gravity "field", a computational exorcism of natural harmonies which split apart the effecting from the effected bodies. Thus did your "field" contrivance artificially inseminate the future from the past, denying conjunctio to the participants and destroying all records of inheritance through an averaging process ... an "immaculate" conception which orphaned present and future from their rightful parentage and rich traditions.

When the goddess Venus comes to me in my dreams, I could take your calculus and your propositions and map her every pore in space and time. But in looking through or beyond her charms to get to such detail, then returning to the whole by way of a summing of the parts, her vital essences would be lost. The harmonious, undulating "C" which infused her pores with the dance of life, would be no more, but in its place, a machine-like matrix, dumbly awaiting my next round of manipulations.

Newton: Mathematics cannot be the carrier of all manner of concept and explanation and thus have I qualified my propositions in Principia, by saying;

"I wish we could derive the rest of the phaenomena of nature by the same kind of reasoning from mechanical principles; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto attempted the search of nature in vain; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy."

Kepler: Well stated, Sir Isaac, but how many of the followers of your method have read and understood your Principia? It is a book so obscure and difficult that even Voltaire, one of your most committed supporters, proclaimed it a book that "many want to have read, but few want to read."

And how often are your qualifications carried forth by those who would quote you, ... in your own austere and literal manner of speech and writing? As you say, you chose not to open up the sanctuary of your mind to the public, and your writings were thus taken as the testament of a method suggesting man's ultimate mastery over nature; for my part, a nature in which we and He Himself are bound. How many eyes followed or ears pricked when, after reviewing the problem of three, rather than two bodies moving under the inverse square force of gravity, you said that an exact solution for three bodies; "...exceeds, if I am not mistaken, the force of any human mind." The difficulty you so quietly enunciated lay unheard and unseen until late in the 19th century, when it was resurrected by Henri Poincare.

Newton: You may now be in a position to understand why I focused my energies more on my work at the Mint in my last thirty years, leaving the philosophical implications and the political and religious machinations surrounding my findings, to others.

Kepler: Some would say that it was because your post at the Mint was one of the most lucrative in England, starting at 1500 pounds per year and rising substantially during your tenure as Master of the Mint. Some would say it was your fascination with Descartes religio-political justification of his "Methode", the mother of your own approach; ... in Descartes words;

"And as a multiplicity of laws often furnishes excuses for evil-doing and as a State is much better ruled, when, having but very few laws, these are most strictly observed; ...".

Your own words, born of some of your unusual theological persuasions, echoed those of Descartes;

"It is the perfection of all God's works that they are done with the greatest simplicity ... And therefore as they that would understand the frame of the world must endeavour to reduce their knowledge to all possible simplicity..."

Could this have been the reason why you attended so assiduously to the task of executing "coiners" during your tenure at the Mint? Or did your bent for executions go back to the underpinnings of Cartesianism, to Aristotle's Nichomachian ethics, a cultural conditioning which has persisted into the present;

"Virtue, then, is a kind of moderation inasmuch as it aims at the mean or moderate amount. Again, there are many ways of going wrong (for evil is infinite in nature, to use a Pythagorean figure, while good is finite), but only one way of going right; so that the one is easy and the other hard--easy to miss the mark and hard to hit it. On this account also, then, excess and deficiency are characteristic of vice, hitting the mean is characteristic of virtue: 'Goodness is simple, evil takes any shape.'"

As it has been noted by several of your biographers, including John Brooke, there are echoes of such ethics in your own biblical study; "...whereby the cherubim and seraphim became hieroglyphs of ordinary social groups. Evil spirits became mental disorders, and devils became the imaginary ghosts of the departed, whom the heathen characteristically worshipped as gods."

Newton: Yet again, as it was in my time-travels with Heraclitus, I am unfairly put on trial. I am well aware of Descartes' introductory remarks on his 1637 work "Methode ...". I am also aware of his abandonment and self-suppressing of his non-Aristotelian work "Le Monde" when he heard, in 1633, of the burning of Galileo's books in Rome and Galileo's punishment by the Inquisition. Be that as it may, it was not my intent to popularize bad or deceptive theory out of either fear or ego as is attested to by my numerous statements qualifying the limits of my work. Again in Principia, in keeping with your own thoughts on the complex and pleasing harmonies in nature, I said;

"but though these bodies may, indeed, persevere in their orbits by the mere laws of gravity, yet they could by no means have at first derived the regular position of the orbits themselves from those laws."

"but it is not to be conceived that mere mechanical causes could give birth to as many regular motions, since the comets range over all parts of the heavens in very eccentric orbits;"

"This most beautiful of system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of other like systems, these, being fomed by the like wise counsel, must be all subject to the dominion of One;"

Kepler: It is true that these statements of yours were made so as to be clearly seen, with their placement in the Preface and in the General Scholium summarizing the Principia. At the same time, the public's fascination with your mechano-mathematical treatments of motion lulls them into believing, as did Parmenides and later Aristotle, that the world is fully describable by the exterior behaviors of the smallest, most instantaneous fragments of time-space; that our reality is based on "being", rather than, as maintained by Heraclitus, an inseparable mix of "being" and "becoming". For as one tries in vain to reconstitute a Venus from an understanding of each and every of her pores --- taken separately and once again assembled, so must one conclude that the ultimate essence of nature is far more than "corpuscles and forces", ... it is an innate blending of what "is" with the creative harmonies or "logos", a magic broth capable in itself of shaping the present out of the past and giving birth to the future.

Newton: As I have walked such halls as these we now walk, over the passing centuries, I have come to be of a like mind with you. In my life, I was frightened by the "nonlinear" yet fascinated with Alchemy, Astrology, Theology and Astronomy. As I mentioned, one of the early books on my reading list was your own "Optics". But because of the many uncertainties and dangers, yes, because of the many potential evils which lurked around the fringes of what was known and true, I suscribed to the counsel of Aristotle which Descartes transformed into logical proposition number 1. in his "Methode"; "... to accept nothing as true which I did not clearly recognise to be so."

This line of thinking led me towards the development of the solid logical "bricks" or propositions which could be safely built upon, to construct answers to any question. Thus did I feel no necessity to share my private visions with the public, having done my best to assure the validity and perfection of the propositions. However, as time passed, I began to doubt that logical propositions, however solid and precise, would be sufficient to deliver a complete and consistent view of the whole of our complex reality.

This, my worst fear was slowly but relentlessly realized, as subsequent generations of philosophers and scientists surfaced one convincing argument after another; First Berkeley questioned the reality of calculus differentials themselves, calling them "the ghosts of departed quantities", and complaining that; "men would hardly admit such a reasoning as this, which in mathematics is accepted for demonstration." In the same timeframe, the 1750's, Boscovich unearthed a major conflict between my corpuscular theory and experimental realities.

By the beginning of the twentieth century, Poincare was observing, in "La Science et l'Hypothese" (1903) that; "induction applied to the physical sciences is always uncertain..."; Wittgenstein, in "Tractatus Logico-Philosophicus" (1921) pointed out that logic and mathematics were "content-free" and incapable of guaranteeing tomorrow's sunrise. Einstein said in "Geometry and Experience" (1921); "So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality." And Kurt Goedel, in 1931, put these disconcerting allegations into the most precise language of mathematics, an incontrovertible symbolic proof.

Thus did my strategy of building from the bottom, from solid mathematical propositions, a full and faithful general view of nature, crumble and crack.

Kepler: I shall pass over the inconsistencies in your remarks concerning your subscription to Descartes' propositions, while you simultaneously sought insights from alchemy and theology. What I don't understand is why you made no attempt to state that the visions of nature in your mind went far beyond what you were capable of expressing in your mathematics of motion? For we are all faced with the problem that the creative imagery of our minds goes beyond our means of expressing it. As I said in "Harmonice";

"For as the sun rotating into itself moves all the planets by means of the form emitted from itself, so too --- as the philosophers teach --- mind, by understanding itself and in itself all things, stirs up ratiocinations, and by dispersing and unrolling its simplicity into them, makes everything to be understood. And the movements of the planets around the sun at their centre and the discourses of ratiocinations are so interwoven and bound together that, unless the Earth, our domicile, measured out the annual circle, midway between the other spheres --- changing from place to place, from station to station --- never would human ratiocination have worked its way to the true intervals of the planets and to the other things dependent from them, never would it have constituted astronomy."

And as Einstein later spoke to these same issues, more clearly and succinctly than had I, yet with no less conviction;

"A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To ``visualise'' a theory, or bring it home to one's mind, therefore means to give a representation to that abundance of experiences for which the theory supplies the schematic arrangement."

Newton: Yes, the human mind is indeed capable of understanding far more complex order in nature than we are yet able to express through language, even mathematical language. At the time I wrote "Principia" as I have demonstrated with the passages I have just quoted, I neither thought nor suggested that my mathematical propositions were adequate to achieve an understanding of nature as a whole, nor of grappling with the profundity of all thoughts which come to mind in this regard.

David Bohm was right when he guessed that my insight on gravity came out of the same subtle harmonies of which you speak, as I studied the moon's orb in the time of the plague; ... "as with the order of movement of an apple in fall, so with that of the moon and so with all." But this mental notion; that all motion was influenced by all other motion, though known by the ancients but kept secret through symbolic encryption, was too profound and dangerous to be openly discussed, even in the mathematics of my day. For this reason, I made the following, false documentation of the origins of my insight on gravity;

"In the same year I began to think of gravity extending to ye orb of the Moon and (having found out how to estimate the force with wch globe revolving within a sphere presses the surface of a sphere) from Kepler's rule of the periodical times of the Planets being in sesquialternate proportion to their distances from the centres of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must [vary] reciprocally as the squares of their distances from the centres about wch they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth, and found them answer pretty nearly. All this was in the two plague years of 1665-1666... "

Several of my biographers noted that this account did not fit with historical fact. My reasoning behind this action to suppress the true nature and source of my insight on gravity .... I, uh .... I can do no better than to refer you once again to the counsel of Descartes and Aristotle, that departures from the known and true unleash the agents of evil. It is up to the priesthood and the academies to lead students of these topics in safe passage through this dangerous maze.

Kepler: It is indeed curious how fearfully we approach the unknown and how often we unconsciously cling to the current cultural "knowns", whatever their deficiencies. For example, Aristotle's tidy formulae for gravity and inertia held for two millenia until Galileo thought to run them through a few "reality checks". When Galileo tested Aristotle's claims that objects fall to earth with a velocity proportional to their weight, as Aristotle had claimed, and everyone had accepted, and that bodies return to a natural state of rest if there is nothing pushing on them, he found these claims to be entirely unfounded. And as you well know, Galileo's revision to the concept of inertia, showing that bodies persisted in motion in a straight line and unaltered velocity in the absence of an influencing force, came too late for me. I too, had fallen victim to the insidious cultural indoctrination which had one believe in flawed abstractions though they clashed with one's sensibilities. This simple proposition, F=ma, stood between me and a full reconciliation of the laws of motion and gravity. It seems so obvious to me now.

But what did you say; --- that it was up to priests and academies to guide the students? It seem you felt no need to avail yourself of such advice, for thus guided, your innovations might never have seen light of day. And such a statement is scarce consistent with your most vigorous and public efforts to prevent the admission of a Benedictine Monk into Cambridge.

My thought is that we need no stewards of truth in this world, for as Blake said, "the tygers of wrath are wiser than the horses of instruction". And as I myself say, in "Harmonice";

"For the boundary posts should not be set up in the narrow minds of a few men. 'The world is a petty thing, unless everyone finds the whole world in that which he is seeking,' as Seneca says. But the boundary posts of true speculation are the same as those of the fabric of the world; but the Christian religion has put up some fences around false speculation which is on the wrong track, in order that error may not rush headlong but may become in other respects harmless in itself. Antiquity teaches us by examples how vainly man sets up boundary posts where God has not set them up;"

....and; "As regards the academies, they are established in order to regulate the studies of the pupils and are concerned not to have the program of teaching change very often: in such places, because it is a question of the progress of the students, it frequently happens that the things which have to be chosen are not those which are most true but those which are most easy. And by that division in things which makes different people form different judgements, it so happens that certain people are in error contrary to their own opinion." ... "Accordingly, if certain subtleties which are difficult to grasp should not be laid before beginners, or if they should not be preferred to the accepted and necessary teachings, it does not follow that therefore, those things should neither be written nor read privately."

So I am not at all regretful that it was you, Newton, rather than I, who was recognized as solving the problem of motion and force, though my writings testify to the completeness of my understanding of the issues. Neither am I envious of you living out your final years in luxury and physical comfort, while I came to my end in Ratisbon, sick and begging for that which was owed me, so that I might further fund my researches into the truths of nature and living. What I do regret is that your mathematics of motion, the calculus and "field" theory, which came to eclipse my work, so thoroughly and shamelessly exorcised the living harmonies of nature from our common concepts of motion and organization.

My further regret is that the children who grace these hallways, distracted for the moment by the hubbub of the season, .. will soon be exposed to unprecedented levels of crime, violence, drugs, and material madness, which relate in no small measure to your abdication of responsibility in making the serious incompletenesses in your method known to the world.

This young woman studying at her desk, Madonna as her friends are calling her, whose attractiveness and grace you had earlier made note of, will not be joining friends and family in the traditional Christmas carollry, nor does the coming spring promise a continuance of the happiness of her youth. For the small gift-package just now passed to her contains drugs which, when found by dogs sniffing at her locker, will forever change her forward journey.

The clouds of trouble and destruction now gathering round her are born from simple facts and simple rules; pass or fail, innocent or guilty. Taken together, the outgrowth of the events of this day will be more than her youthful consciousness can support. As psychologists of this age, such as Laing have noted, the probabilities are far more "favorable" for young people to be admitted to mental hospitals than to universities.

Newton (deeply concerned): But what of the harmonies of her experience, her history .... will they not be weighed into the determining of her future?

Kepler: As Berkeley noted in his criticism of your work, and as Laing noted in his criticism of our "Newtonian" culture, your own mathematics of motion have encouraged an abandonment of any consideration of "experience" or history. For the "simple" view of reality ignores time's rich context. As the reality of the present and the reality of the past narrows in on the behavior in the shrinking interval of the present; in your "words", as (f(t) - f(t-a))/a is transformed by "a"'s passage to zero, all experience is erased. What is left is simply an instantaneous snapshot of behavior, like the image of a highjumper, body contorted and back arched as he clears an invisible bar. And like the snapshot, all vestige of experience, of what went before, is lost; ... was his approach and take-off graceful, or are we indeed looking at the victim of a terrorist bomb? The snapshot can tell no tales, and neither is her school counsellor or lawyer permitted to do so, as they attempt to moderate the judgements levied upon her.

Newton (his eyes darting about wildly as he bites on a half-crown of British origin, taken from Kepler's purse): I have no more stomach for this discussion, nor this tour of twentieth century "mathematics of motion". Though you waste no opportunity to speak of nature's harmonies, you present yourself to me as a doctor of discord, a sower of disenchantment and a revivor of sins of the past for which there is now't to be done.

Kepler: But something can be done. For the suffering which grows out of the incompleteness of your science shall linger on the longer if you do not, in your own austere and convincing way, divulge the inner workings of your mind as you sowed these ideas so deeply into the collective psyche of the western world. And by this disclosure, you may help to purge the torment in a multitude of minds, yours included.

Newton: Though I await a flood of anger which must follow such provocation, I sense none. Instead, I feel the onset of a warm and steady tide of peace and harmony, an unexpected gratitude towards you, for giving me this opportunity to unburden myself of these repressed thoughts.

Kepler: Let it flow forth, then, free of all pretence and conditioning, for as Pareto has said; "Give me truthful error any time, full of seeds, bursting with its own correction. You can keep your sterile truth for yourself."

Newton: Though your words mock my life's habitudes, I hear you, and I shall do my best. While the roots of this story are long, it is short in leaf and blossom. But beware that size is no measure of complexity, as Aristotle has warned, the devil is in the detail.

Johannes, as you already know, and as I tried to say in my publications, most of that which has been attributed to us was already known in ancient times; i.e. that the moon was to the earth, as the earth was to other celestial bodies; that there was a mutual force of gravitation that extended to all other systems, that force itself extending indefinitely in all directions; that all matter was made of atoms which were hard, solid and immutable; that gravity accrued to both atoms and the bodies they composed; and that gravity was proportional to the amount of matter in every body; that the proportion in which gravitational power decreased with distance was analogous to the law governing tension and pitch in a string instrument, and was concealed in the ancient concept of the 'harmony of the spheres', and that the relationship of tension to the period of the vibration was an inverse square relationship.

So has it been remarkable to me that the greatest of public attributions I have received is in "my" discovery of the inverse square law of gravitational attraction. In fact, Pythagorus himself had demonstrated the inverse square relationship of tension and pitch and such had been confirmed by numerous scientists, well prior to my "Principia" including Galileo and Marin Mersenne, the latter publishing the result in his "Harmonie Universelle" in 1636. Your third law and this Pythagorian notion sufficed for the rest.

So you see that what weighs largest in the public recognition of our works has not been the abstract notions themselves, for these abounded well prior to our efforts. What matters is the harmony with which new science can be melded with the thought streams of the times. So it was for scientists in our time, that the challenge was less one of discovery than of cultural appeasement, and thus t'will ever be.

Kepler: You are speaking of such things as the concept of a force acting at a distance being considered "occult" in our time and, as Kuhn has said, "representing a regression to the "mystic sympathies" and "potencies" of mythopoeic thought. It being the job of natural philosophers such as ourselves to show cause for all natural motion through direct and causal contact.

Newton: Precisely. To put together a workable "system of the world", one needed to "neutralize" and thus make usable the "occult forces" of gravity and inertia, for they constituted essential pieces in nature's jigsaw puzzle. Thus was the rehabilitation of the occult the real challenge. In opposition to this goal, as none in Europe could ignore, stood the thought-controlling alliance of Church and monarchy. For in too blunt attempt to field new thought, had the Cathars not been systematically exterminated by an alliance of the Church and French royalty over a bitter campaign of 160 years? Had a long line of scientists not suffered at the hand of the Inquisition, including, in our own time, Galileo? And had you yourself not been threatenened by a Church-influenced court in Wurtemberg, for alleged forays into "the forbidden arts", a crime punishable by burning at the stake?

While our thoughts were thus stewarded by an unholy alliance of Church and State, we were unearthing grave flaws in our enforced beliefs that our earth was at the center of the universe, that bodies fell to earth with a velocity proportional to their bulk, that moving objects slowed and came to rest if no forces acted upon them .... we were finding that these abstract notions, woven into doctrine of church and culture, were easily refuted by experiment! Yet the stewards of nature's truths had successfully preserved these falsehoods for two millenia.

But in seventeenth century England, where Henry VIII had ceased listening and obeying, where Cromwell raided and disempowered the monasteries, where Parliament had beheaded the King, people were no longer in a mood to listen and believe ... they demanded that their truths be delivered, not on Papal parchment, but on the laboratory table and the field of experiment.

Kepler: It is true that in this time, the need to bring the mysteries of nature down to earth in real and tangible form was powerfully felt amongst the populace. But the need for simplicity prevailed in a like proportion. And we both know that the most beautiful of these mysteries, the order and harmony which comes out of the unifying of the opposites, and which builds continuously as latent experience and learning, ... this mystery and beauty cannot be reduced to simple terms, to imagery of billiard balls and their collisions. How then, did this conflict resolve itself within your mind's conjectures?

Newton: In two ways; by moving those aspects of understanding as could be moved into the domain of the real and tangible, in the form of physically verifiable propositions, and by ascribing those aspects not amenable to this process, to the realm of God and his transcendent blueprints..

Your third law, published in "Harmonice" in 1619, furnished me all ingredients for the former task, ... a plentiful supply of accurate observations assuring tangible validation, and a strong established base of popular acceptance for the law throughout the scientific world. The backbone of my mission was then reduced to a restating of your law in terms of force while stripping out harmonics, delivering the former to science and leaving the latter to the domain of God. And this is where the irony, with respect to both our works, begins.

Kepler: Howso?

Newton: You have commented that only your flawed Aristotelian notion of inertia kept you from putting the whole system together as I did in Principia. The concept of inertia you were building upon was that bodies needed a force to sustain their motion, while that which I was using, from Galileo, was that bodies would continue at the same velocity in a straight line if no force acted upon them. Our concepts of inertia were thus out of step with the body of our respective philosophies.

Kepler: I do not understand.

Newton: Your theory is all about the experience and history of material bodies; harmonies of motion building on harmonies of motion, the fusion of being and becoming, while your concept of inertia was that moving bodies retained no memory of their history; that motion was a product of the present, the result of the forces acting upon bodies at a particular instant of time.

My theory, on the other hand, was all about instantaneous behavior, and ignored experience and history. Rather than being about motion in the sense of cycles and harmonies, it was about instantaneous tangents. Yet the concept of inertia which I embraced was one in which the experience and history of the body was all important, for it is prior experience which imparts to a body its velocity which in turn shapes its form of motion as manifest in the present.

Kepler: Say no more. Preserving the harmonies while expressing the system in terms of force would have required that you deal with the experience aspect of matter. This would have complicated the problem and your mission enormously.

Newton: The complications would have come in two forms; cultural acceptance would have been rendered far more difficult as would have the mathematical treatment.

With the "experience" aspect of this tale now in the picture, we can now return to the ideas of the ancients, particularly Heraclitus, who said;

"They do not comprehend how a thing agrees at variance with itself; it is an attunement turning back on itself, like that of the bow and the lyre." ... "The name of the bow is life; its work is death."

Heraclitus knew that nature was composed of opposing forces which were unified through "hidden attunement". Just as the bow string and the bow oppose each other, the "life" in the tensioned system of bow and string resolves itself through oscillatory motion, until forces and oscillations fully dissipate and the system "dies".

And just as bow and bowstring, when in tension, constitute life based on opposition unified through harmonious co-evolution, so do the centrifugal and centripetal forces of the planets in their orbit round the sun live, oppose and unify, in a harmonious way.

To Heraclitus, aitherial "fire" or motion was the motive agent for all cosmological processes; "all things are an equal exchange for fire and fire for all things, as goods are for gold and gold are for goods." ... he maintained that; "pure, or aitherial, fire has a directive capacity." For Heraclitus, understanding the hidden order in the world equated to wisdom; "Men should try to comprehend the underlying coherence of things; It is expressed in the Logos, the formula or element of arrangement of all things."

It did not escape my attention that your own statement in "Harmonice"; "Geometry existed before the Creation, is coeternal with the mind of God, 'is God himself'." paralleled the thoughts of Heraclitus; "The ordering, the same for all, no god or man has made, but it ever was and is, and will be: fire everliving, kindled in measures and in measures going out." ..."Wisdom consists in understanding the way the world works" ... "One thing, the only truly wise, does not and does consent to be called by the name of Zeus." Thus did Heraclitus, like yourself, ascribe to nature --- order and harmony, and to god --- understanding.

Thus, the legacy of the ancients was rich in abstractions which covered most, if not all of the ground of our current philosophizings, even to the epoch we now visit. For one cannot avoid seeing, in the imagery of Heraclitus, the notions of "phase trajectories" and "strange attractors", ex nihilo motion in the form of opposites self-unifying in aesthetic fractal harmonies. Who can say that the "logos" of a Heraclitus differed from, or was more obscure than the "autonomous co-evolution" of a Kauffman?

Science is but a part of the economy of ideas, and often stocks the shelves well in advance of even the beginnings of demand. That a new theory satisfies a scientific need signifies little in terms of its general acceptance, as was the case with your first two laws, and many others which, though of proven validity, have failed to capture the common imagination. To find acceptance, it must satisfy a cultural need and have at its disposal the tools needed for its conveyance.

Kepler: What you say is true, for all of my works were founded on the ideas of Pythagorus, Plato, Proclus or others who struggled to illuminate the secrets of the heavens well before me. My own contribution lay in applying their ancient wisdom to the interpretation of the new and more accurate observations of Tycho Brahe. It turned out that my third law was most well received while the first two were largely ignored, perhaps because it was so simple in concept, and tangibly demonstrable, in measures of time over distance. In fact, history has assessed that my biggest impact came from my publication of the "Rudolphine Tables" in 1627, an offering which was little more than a bookkeeping exercise to me. I would venture that the appeal of this offering was in its potential for liberating scientific thought from prescribed doctrine and channels.

This Christmas dialogue is, for me, rich with new insight beyond the clarifying of historical perspectives. For years I struggled to determine the continuous motive force of the planets, and only after my life had run its course have I come to recognize that it was inertia itself that fuelled the continuance of motion. And eluding my conscious attention until this very moment, has been the fact that the instantaneous behavior of bodies in motion are shaped by their dynamical "experience" carried into the present from their earlier adventures.

Newton: Yes, and now you can perhaps see my quandary. For had I opened the door to an accounting of "experience" for all bodies, I would have found myself, mathematically, in the domain of complex variables, phase angles and wave equations. Can you imagine the reaction of the mathematicians of the day if, when asked about motion, I asked for a qualification as to whether the real, or imaginary or both components of motion were intended?

Beyond this complication, the world of experience is the world of the collective, and once one opens the door to interactions which give rise to experience, there is no end to the matter. As Heraclitus has said; "thinking is shared by all" since thinking is most often the playing back of experience in the mind.

Kepler: So comes it now clear to me, why my harmonic theory of motion was too early for its time, for it would have introduced, or re-introduced too many new and abstract notions into the field of natural philosophy, and been too lacking in the scientific tools to make the truths in these abstractions apparent.

Newton: Yes, you will recall that even with my more limited theory, my fluxions and fields were strongly challenged by Leibnitz, Berkeley and others. Only through dealing with the "real" components of the problem could I reliably demonstrate the validity of my propositions in a form simple enough to convince the masses. To this end, the concept of a gravity "field" was as essential as the calculus to simplify the concepts and remove the complexities of harmonic interaction. For the role of the field, which ascribes a value and direction of force arising from the effects of multiple bodies, to arbitrary points in space, is simply to decouple the harmonic interactions of the masses which emerge from their dynamic histories.

Kepler: I hear your words, but see no harmony in your thoughts, for if nature is complex and you present it as being only "real", there will be a price to pay. And it seems to me, as it did to Laing, that the bulk of this price is paid by innocents such as Madonna.

For no matter how finely you partition an observed problem into "real" fragments, so as to explain the problem part by part, ... like Humpty-Dumpty, there will be no way to re-assemble the parts such that nature's intent is honored. For the harmonies which were essential to their true nature will then be lost. Give me the most detailed "factual", in other words, "part-by-part" account of any man, and a Shakespeare can give you back a thousand different characters, all of which honor these "facts". For it is the timing relative to the whole that makes all the difference in how such facts are put together; is not rape untimely embrace, or murder untimely killing? The wisdom of Solomon, 1000 years before the birth of Christ has spoken to this issue;

To every thing there is a season, and a time to every purpose under the heaven.

A time to be born, and a time to die; a time to plant, and a time to pluck up that which is planted;

A time to kill, and a time to heal;

a time to break down, and a time to build up;

A time to weep, and a time to laugh; a time to mourn, and a time to dance;

A time to cast away stones, and a time to gather stones together; a time to embrace, and a time to refrain from embracing;

A time to get, and a time to lose; a time to keep, and a time to cast away;

A time to rend, and a time to sew; a time to keep silence, and a time to speak;

A time to love, and a time to hate; a time of war, and a time of peace.

That you could cast out this ancient wisdom, as the baby with its bath water, contending that; "the 'music of the spheres' is a concealed representation of the inverse square law of gravitational attraction" both astounds and dismays me. Though you claim that you see farther by standing on the shoulders of giants, the lenses of your propositions reveal to you a desert of static force and matter where lived a garden of divine harmonies. And those harmonies, banished as ungodly "noise", linger on in the shadows of the mind as denial, repression, projection and estrangement from nature.

Stealing your laws of motion from the diverse and universal harmonies of nature, yet feigning a simpler birthright for them, you encourage all to set aside the wisdom of Proclus, that; "there is no royal road to learning", to seek wisdom through attrition, rather than appreciation of nature's embroideries.

Thus are we encouraged to look upon Madonna through these lenses which reduce live harmony to inert fact. But there exists no sufficiency of facts, without the seasons and harmonies which give them context, which can come near to illuminating Madonna's truths. Justice so constrained, shall be ill equipped to distinguish between worth and unworth in hearing her case.

Newton: As you had counselled, so did I give forth my thoughts replete with "fruitful error", in a manner I had never before attempted. And as you promised, so did the seeds contained within these errors germinate in our minds and deliver the fruits of a deeper understanding than I had previously owned. I am grateful to you for it.

A unity has come to my thinking which eluded me throughout my life, and whilst the final arbitor of my thoughts has always been the holy scriptures, it becomes apparent that in interpreting these writings, I did give them the same rough prunings as I had nature's works, and so obscured the fuller truths within them.

Kepler: We must be ever vigilant to keep the trickster within us from laundering the unconfessed selfishnesses of our minds through the external symbolry from which we seek unbiased counsel and approvals. Indeed, through this self-deception, Christ's holy image has been more oft used to justify earthly politics than to solicit heaven's harmonies. As I said in "Astronomia Nova...";

"So much for the authority of the Holy Scripture. Now as regards the opinions of the saints about these matters of nature, I answer in one word, that in theology the weight of Authority, but in philosophy the weight of Reason alone is valid. Therefore a saint was Lanctantius, who denied the earth's rotundity; a saint was Augustine, who admitted the rotundity, but denied that antipodes exist. Sacred is the Holy Office of our day, which admits the smallness of the earth but denies its motion; but to me more sacred than all these is Truth, when I, with all respect for the doctors of the Church, demonstrate from philosophy that the earth is round, circumhabited by antipodes, of a most insignificant smallness, and a swift wanderer among the stars."

Newton: What God has given us is a mind whose creative pleasures oft take the bit between their teeth and proceed at their own pace and leisure, spurred on by what is fanciful in the moment. Thus from the selfsame observations of nature, one man looks to the limit where all timing is spurned and sees that force pays homage to material accumulation, precipating this idea in an icon, F = GMm/r**2 which thence governs his perspective and behavior. Another man looks to the limit where all material accumulation and unbalanced force falls away and sees transcendent harmony and proportion, precipitating this in a mantra, T**2/r**3 = C which opens for him new states of seeing and being.

Kepler: With the synergies given us by this atunement of once opposing minds, and in the spirit of this season, can we not now turn our thoughts to Madonna, discovering some way to lift the shadow of our obsession with "facts" and denial of "experience" as now threatens her with grave and unjust injury?

Newton: My heart is truly in support of such an enterprise. My interpretation of the scriptures, suffering from the same flaw in approach as did Principia, reduced them to 15 essential propositions. This led me to refuse to listen to narrative pleadings on the part of coiners whom I prosecuted while Master of the Mint. Accepting only factual evidence and applying the rules literally and self-righteously, I may have sent many to undeserved deaths.

Kepler: Distilling complex nature into simple rules of "good" and "evil" yields a clarity whose essence is deception. For in the applying of such rules an infinity of unspecified situations present themselves. And as nature's complexity re-rears its head, so returns the questions of balance in judgement, which follow from the words of Aristotle; ... there are many ways of going wrong but only one way of going right, for "evil is infinite in nature, while good is finite".

By no means do I myself embrace this "western" ethic or any other which deems itself, by virtue of God's will in Man or other such ratiocinations, worthy to sit in judgement of itself, nature, the whole of which we are part. As my words and work are testimony, my ethic is to judge things by nature's Truths, the seasonal harmonies or disharmonies characterizing men and all nature's children, but never by frozen moments separated from the whole, snapshots for friend or foe to caption according to their current pleasures. But given that we use such simple fishnets for "entrapping evil" with a mind to "concentrating good", Aristotle's words make clear that such industry rests on character of both fisherman and fish. For "evil" being infinite, fishermen tenacious as a Javert shall surely discover "evil" where'er they look, and fish honest as a Jean ValJean shall surely admit to it whene'er asked. Thus is the ultimate legacy of this ethic a world of liars and lawyers, ... or do I repeat myself?

Newton: The wellspring of remorse from these reflective insights, joined with memories of the self-deluding actions of my past, is made the more bitter-sweet as I look upon Madonna, struggling to prepare for her exams, unaware of the rude change in her life's course, about to unfold. ... A plague on all rules which deny past harmonies safe passage through the portals of the present, on their mission to enrich our future!

Kepler: In this moment of clarity and feeling, can you, ... can we find something to say or do which could change the outcome for Madonna? I can see the officers just now parking their van and escorting the dogs towards the school entrance.

Hello, ... Isaac?

Kepler (thirty minutes later): Where did you go? The most curious thing occurred. As the officers took the dogs by Madonna's locker, they passed without a sniff or a whimper, as if the drugs had somehow vaporized.

Newton (smiling broadly): Yes I know. It really doesn't add up! (Newton now suppressing giggles as Kepler's gaze betrays concern for Newton's mental state): Let's get out of here. I could really use some pizza!

Kepler (as the penny finally drops, giving Newton a wink and a thumbs up): ... A wise and harmonious choice, my friend.

(Kepler and Newton work their way arm-in-arm through the bustling corridors, and exit)

* * *

Primary Source Documents:

Editors Fauvel, Flood, Shortland, Wilson, "Let Newton Be", Oxford University Press, 1988

Newton, Isaac, "The Principia", Prometheus Books, 1995

Kepler, Johannes, "Epitome of Copernican Astronomy & Harmonies of the World", Prometheus Books, 1995

Kirk, G. S., "The Presocratic Philosophers", Cambridge University Press, 1957

Kahn, Charles H., "The Art and Thought of Heraclitus", Cambridge University Press, 1979

Laing, R. D., "The Politics of Experience", Pantheon Books, 1967

Bernstein, Jeremy, "Experiencing Science", E. P. Dutton, 1978

Koestler, Arthur, "The Sleepwalkers", Penguin Books, 1959

Kuhn, Thomas S., "The Copernican Revolution, Harvard University Press, 1957

Frankfort, Henri et al, "The Intellectual Adventure of Ancient Man", The University of Chicago Press, 1946

Murray, Oswyn, "Early Greece", Harvard University Press, 1978