Newton's Error

January 2, 1997

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"...we offer this work as the mathematical principles of philosophy; for all the difficulty of philosophy seems to consist in this; --- from the phaenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phaenomena;" [From Newton's Preface to the "Principia", published in 1687]

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Newton apparently sought to determine all understanding of nature via this strategem of understanding motions, an approach which seems consistent with Plato's comment

" ... and from this source [time and motion] we have derived philosophy, than which no greater good ever was or will be given by the gods to mortal man."

However, Newton insisted on putting a religious spin on his explorations, as exemplified by his following statement in a letter to Richard Bentley;

"the growth of new systems out of old ones, without the mediation of a divine power, seems to me apparently absurd."

Newton, with his Puritan religious views, was clearly out of phase with today's nonlinear sciences which see in nature, an inherent capacity for "autonomous coevolution.", "self-organization" etc. Newton's strong duallist philosophy motivated him to put some strong religious constraints on his science, quite unlike Kepler, who complained about such approaches and distanced himself from them in his introduction to "Astronomia Nova" in 1609;

"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."

... and later in 1619 in "Harmonice Mundi";

"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;"

Newton would have done well to borrow from this wisdom of Kepler, as he borrowed from his laws. Newton used Kepler's third law to derive the universal law of gravitation, and it was in this pivotal derivation that Newton's error was sown in a position to propagate throughout "the system of the world", since this was the law that unified, for the first time, astronomical and terrestrial science.

Kepler's third law, that the ratio T**2/R**3 (ratio of square of orbital period to cube of distance between orbiting and orbited bodies) was constant, dealt with time-transgressing "harmonies and legacies". It was a validation of Heraclitus' idea that nature was inherently about polar opposites unifying through harmonic motion;

"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."

... a concept that is also extant in modern particle physics.

In the case of astronomy, the "tension between the opposites" was due to the polar opposition of the attractive force of gravity and the resistive force of inertia. The latter, a legacy of history and the former the pull of the future associated with interactions and "deterministic chaos"; --- creative tension --- between centrifugal force (wanting to form new motions) and centripetal force (wanting to preserve the old motions). The nature of the motions so forming being sensitively dependent on the analogous forces betwixt the subject bodies and a multitude of others "in the background".

Newton used Kepler's third law to derive his inverse square law of gravitational attraction (F = GMm/R**2) in the manner shown in the appendix [The maths is as taken from NASA's "SpaceMath" offering on the Internet]. Since Newton eliminated time as a variable, we can say that he made Kepler's law less general, making it only applicable to an instant in time, rather than indefinitely applicable as it was in Kepler's formulation. Thus Newton's law of gravitational attraction appears to be a sub-derivative of Kepler's more general law.

But Newton's gravitational law was also an unusual type of approximation (at best) since the derivation involves involve time derivatives (velocity, acceleration). The philosopher George Berkeley termed "fluxions" or time derivatives; "the ghosts of departed quantities", and complained that; "men would hardly admit such a reasoning as this, which in mathematics is accepted for demonstration."

In other words, Newton was taking us deeper into mathematics and farther from natural reality. Newton, however, appeared to associate his mathematical propositions directly with reality, unlike Einstein who said;

"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."

But what was the real problem with the derivative?

Berkeley had complained about this business of the limit; about the time interval, "dt" going to zero, saying that it was "unreal". That is, Newton's "fluxions" or "derivatives" were engendered by the expression (f(t) - f(t-dt))/dt as dt went to zero, suggesting that the derivative was something you would get if you could freeze time. But if you could freeze time, wouldn't you freeze motion, and motion is really what we're trying to understand. So by stopping motion, we can find out more about it?

The magnitude of any error here is important, to say the least, as a good part of natural philosophy, culture and communications has been built on this base of Newtonian fluxions and fluxion-based principles. While Kepler would have had us focusing on a base of motional "harmonies and legacies", Newton has had us focusing in on "force and material", and the latter, if is pretty much what business and economy is all about today. The associated management and investment approaches, in addition, have a decidedly Newtonian mechano-mathematical flavor.

Returning to Berkeley, he opposed the fluxions from a religious as well as philosophical point of view (Berkeley became the Bishop of Cloyne, a town near Cork, Ireland). Berkeley felt that if you could get people to believe in fluxions, it would be "on faith" and that this was no different than getting them to believe in the "mysteries" or "miracles" of the church. He contended that Newton was inventing a new religion, whose high priests (natural philosophers) would swear they could "see" the natural truth in these strange formulations, which went beyond the understanding (in a mathematical philosophy sense) of the man on the street, an amazingly perceptive projection.

In the twentieth century, quantum physics raised all sorts of questions about messing around with time in the manner that Newton did in the differential calculus. As Denis Gabor noted, the Heisenberg Uncertainty principle meant that time and frequency could not be separately resolved. i.e. that (dt*df =~ 1) where t is time and f is frequency and the "d" prefix represents uncertainty. In Gabor's words;

"The identity (dt*df =~ 1) states that "t" and "f" cannot be simultaneously defined in an exact way, but only with a latitude of the order one in the product of uncertainties."

It seems clear that the process of taking a time derivative violates this quantum constraint (Gabor invented dt-df "cells" to get around this uncertainty, but unfortunately no-one listened). It is particularly relevant in the case of shifting from Kepler's world of natural harmonies (harmony is defined as the pleasing circumstance where oscillations of differing pitch tend to come into a common phase on a regular basis). Since frequency is a measure of periodic MOTION relative to time, then it is clear that one can't let "t" go to zero without implicating "f". And if the subject we're studying is motion, then we should not expect to get realistic descriptions of motion out of the time differentiation process.

This flaw does not stop the theory so derived from being self-consistent within itself. As Wittgenstein has pointed out, mathematical theory is like a grid you can lay over nature to give it some form and allow you to make inferences about it. The grid is self-consistent within itself, but nature is not obliged to be consistent with it. The point is, the grid gives us a mental picture of what is going on in nature; if the grid does not properly reflect nature's true character, which it never can in an ultimate sense, the question then becomes; how badly can it mislead us relative to nature's true behavior?

There's nothing new in the fact that Newton's formulations do not hold up when reconciled with relativity and quantum mechanics, but the extent of the damage has really not been assessed. Just as the effects of sensitive dependence on initial conditions in nonlinear dynamics, leading to "deterministic chaos" were not really focused in on for a long time (e.g. thirty years ago, the NOAA folks still believed that they would be able to accurately predict long term weather; all they needed were bigger supercomputers and more observations). So it may be with the errors in Newton's formulations and the business and social organization analogs of "deterministic chaos". It currently appears as if we are in the process of digging ourselves deeper and faster into a hole, in an attempt to get out of it. Because it takes time to see whether today's actions give you sunny weather or a hurricane, only time will tell.

The reason I am relating "Newton's Error" to business is because the whole line of thinking expressed above is has come out of explorations into the anatomy of high performing, creative teams. What I have observed, which seems to be generalized in the archetypal "Apollo Thirteen" experience (as captured in the movie), is that there is a "synergy" (nonlinear evolution of ideas and results) in high performing teams which appears to relate to team "harmony" (in communications and work practices) and a "top-down" pattern-oriented problem solving approach. Communications and work practice are strongly focused by a clear shared purpose, and harmonious behaviors, rather than out-of-context results, are rewarded. Tangible results are the natural offspring (rather than the ultimate purpose) of this mode of working; i.e. the "purpose" is bigger than the "tangible results". In the case of Apollo Thirteen, the purpose was to bring the astronauts back alive while the tangible results were a newly designed and functioning re-entry system, delivered in a couple of days. Without the "bigger purpose", it would have been impossible to deliver those results in that timeframe.

In analogous high performance companies, such as Southwest Airlines and Motorola, their CEO's are also encouraging a shift to the "purpose and behavior" model. Whether the purpose is happy customers or being the best in their field, they are rewarding and reinvesting through "role models", who, on the way to achieving this purpose, promote harmony and creativity amongst all stakeholders (team, peer teams, customer, management, investors etc.). These role models are expected to achieve their "numbers" (tangible results) as a matter of course, but more importantly, the "behavioral leaders" amongst them are identified as those who "leave a legacy" of a better way of doing things that many can benefit from (in an evolutionary model sense, this model "selects" for preferred reproduction on the basis of behaviors or group behaviors)

This contrasts with the "Newtonian model", where the focus is on "material management" ("force and matter") as measured by instantaneous distributions of material (money, products) rather than by stakeholder harmony and sustained legacies. Those who best demonstrate control over the distribution of materials, as measured at agreed upon instants of time (e.g. December 31) are both rewarded and given more responsibility (in an evolutionary model sense, this model "selects" for preferred reproduction on the basis of "organs" or "components" within the system). The question of legacy is rarely asked and almost never measured and tracked (this was a particular sore point amongst retirees who recently participated in a "Wellspring" exercise, brainstorming about the origins of business success).

Thus, the models of Newtonian physics are used as "go-bys" for business and investment, both in a force and material distribution context, and with respect to periodic, instantaneous measurements of results.

As Gabor pointed out, a nonlinear theory of communications requires accommodation of the interplay between "time and motion" (dt*df =~ 1) and this involves custom synchronization between transmitter and receiver. That is, it is an artifact of the linear theory that we depend on a constant (mechanical clockworks) time reference to which all transmitters and receivers can "play to". In organizational communications, it is this constant time reference which allows partitioning of the work by department and individual. The assumption is that plans do not change in the interval (within the agreed clock cycle) that each subdepartment does its "material management", so that the pre-planned contributions so developed will fit together as a jigsaw puzzle.

In the case of Apollo Thirteen, however, everything was changing continuously. New information was coming in ... an explosion, power loss, CO2 levels rising oddly, and in response, new ideas were being continually knocked around, a few being selected and modified and etc. etc. In other words, the environment was continuously evolving and the team had to be continuously learning ... i.e. Apollo Thirteen was a "learning organization". The timing of communications was fundamentally linked to the harmonies and motions inherent in the problem and the team's response to them. They worked, as one might say, "harmoniously" (always coming back into common phase) and each was leaving a legacy (not just satisfying an instant, clock cycle based result, but a self-sustaining contribution towards the purpose which would not turn to discord over the life-cycle of the project).

In summary, Newton's laws of gravitation and motion are in error due to their fundamental dependence on non quantum theory compliant time differentials. Newton's religious beliefs made it essential for him to strip any/all "occult" forces (i.e. "autonomous coevolution") out of his view and mathematical discription of nature. The time differentials or "fluxions" which were the building blocks of his laws of motion, reduced everything to abstract motionless snapshots, from which, by a mechanical reconstitution, a synthetic version of reality was produced. The approach he used most certainly achieved his religious purpose. Unfortunately, or rather fortunately, it appears that nature is inherently animate, thus Newton's mechanical laws can be rather poor simulators of nature, depending on the subject matter.

Self-consistent, yet erroneous frameworks (flat earth, earth as center of solar system etc.), can work out pretty well if everyone agrees to adhere to them, particularly in local environments. For example, in warfare, it was safe to use a flat, stationary earth model for firing guns at relatively short range. As the domain of operation globalizes, however, such locally acceptable self-consistent frameworks break down (e.g. a long range missile requires more sophisticated guidance). Similarly with respect to the dimension of time, many erroneous self-consistent frameworks will work well over short intervals of time, but do not hold up over longer time periods. This is perhaps the big exposure in Newton's formulations relative to Kepler's "harmonic" views. For example, the experience of companies using Newtonian principles to acquire computerized data storage technology was that they worked well over short intervals of time and space but led to incompatibility nightmares over longer intervals of time and space (i.e. there was no larger, harmonizing purpose, such as a global information management vision).

The errors in Newton's formulations (and those of Descartes and Aristotle, on which Newton's are based) "color" almost all aspects of organizational practice and communications. The time differentials also underly communications and information theory and are so well accepted that many "AI" experts have come to believe that a string of ones and zeros can mechanically reconstitute consciousness. It would seem that a quantum view of communications, as proposed by Gabor, would dispute this.

The damage which follows from Newton's error (which compound those of Descartes and Aristotle) would seem to be so significant, that this perhaps explains the general atmosphere of denial and/or reluctance to deal with it. However, damage is now accruing at such a rate, and in so many different ways, that overt discussions on this topic are urgently needed, so as to accelerate the growth of awareness and resolution.


Derivation of Newton's Law of Universal Gravitation from Kepler's Third Law

[As Copied from NASA's Space Maths Web Pages]

The statement has been made that Newton's derivation of his inverse-square law of gravity from Kepler's third law is among the most important calculations ever performed in the history of science. Kepler's third law, based on observation rather than theory, states that the squares of the periods of any two planets are to each other as the cubes of their average distances from the Sun. Derive Newton's law from Kepler's law.

Solution: If we represent the periods of any two planets by t and T and their distances from the Sun by r and R, respectively, then

T^2/t^2 = R^3/r^3


T^2 = (t^2 x R^3)/r^3

Assuming that we know the values of t and r, and substituting a constant C for the quantity t^2/r^3 the equation can be reduced to

T^2= CR^3.

Thus if we know either T or R for the second planet, we can solve for the unknown quantity. In this problem, however, we wish to use this equation to discover a new relationship, Newton's law of gravitation. For a body moving in a circular path, the acceleration toward the center is

a = v^2/r

Substituting in F = ma,

F= mv^2/r

The velocity of the body in the circular orbit is

v = 2(pi)r/T


F = mv^2/r = m/R [2piR/T]^2

Because T^2 = CR^3, we find by substitution in the previous equation that

F = [4(pi)^2m/C] x 1/R^2

= K/R^2

That is, the force holding a planet in orbit falls off as the square of the distance R to the Sun. Newton expressed the value of K and obtained his law of universal gravitation:

F = GMm/r^2.

This law applies not only to the attraction between a planet and the Sun but also to the attraction between any two bodies. G is the constant of universal gravitation, M and m are the masses of the two bodies, and r is the distance between their centers of mass.

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