The Significance of Precise ambiguity in Science
It is evident from all great examples of classical art, that the only way to precisely communicate an idea, is through ambiguities, in between which, one mind can say to another, “I know exactly what you mean.” It is also evidence, that cognition is an embedded principle of the universe, that, when man communicates with it, the universe speaks in similarly precise ambiguities.
When Cusa and Kepler led mankind out of the long dark age of Aristotelean slavery, the world of non-uniform motion they discovered, did not obey the precise mathematical calculations the “ivory tower” academics had deluded themselves into believing. Nevertheless, this new world was susceptible to precise measurement. All that was required was the invention of a new type of mathematics–a metaphor that reflected, rather than excluded, the principle of cognition.
This new mathematics arose from a three-way dialogue across the centuries. The participants included: Renaissance thinkers, such as Cusa, Leonardo, Kepler, Fermat, Leibniz, Kaestner, Gauss, Riemann et al.; ancient Greeks, such as, Pythagoras, Theatetus, Theodorus, Plato, Eratosthenes, Archimedes, et al.; and the universe itself. This “Great Deliberation” can still be heard today, if one works to discover its language.
It is historically accurate, and pedagogically useful, to think of the Kepler challenge as the “motivfuehrung” of this dialogue. The paradox Kepler identified is a fulcrum, whose solution requires the assimilation of the entire Socratic tradition of Greek science, while, at the same time, demanding an as yet undiscovered principle.
Subsequent discoveries of non-uniform physical action, such as Fermat’s least-time path of light; Huygens’ cycloid; Leibniz’ catenary; or Bernoulli’s brachistichrone, showed that Kepler’s challenge was exemplary of a whole class of physical problems that were characterized by non-uniform action. These discoveries posed, indisputably, the inseparability of mind and matter, as expressed by Kepler’s concept of the mind of the planet. Non-uniform physical action implies the question, “How does the planet, light, falling body, or hanging chain, or rolling wheel, know how to move, and, even more importantly, how can the human mind know what the physical universe knows?”
A paradox arises because in non-uniform action the position of, for example, the planet, is a function of a principle of change. Such positions cannot be thought of as points in space, but rather, as the effects of “moments of becoming” transformations from what the planet was, to what it will be. These transformations are themselves a function, not of the planet alone, but of the multiply-connected harmonic orderings that underlie the solar system as a whole, expressed, for example, by Kepler’s three principles, the function of the Platonic solids, and the function of the musical intervals formed by the planet’s motion. Thus, the planet’s position from moment to moment is ambiguous with respect to its previous position, or to any other position inside or outside the orbit. But, the position is less ambiguous with respect to the underlying physical characteristics that govern the “moments of becoming”. The ambiguity decreases with the discovery of new physical principles. Since the human mind must discover these principles from the inside, so to speak, these physical characteristics, not positions, are what must be measured at each “moment of becoming.”
To answer this question, Cusa, Kepler, Fermat, Leibniz, et al., turned to the Socratic tradition in Greek science, in which this hylozoic principle was a central feature. As indicated in earlier installments in this series, Kepler, and Fermat attempted to apply the Greek investigations of loci, or what today is called geometry of position, to this new class of physical problems. But, the paradoxes posed by these newly discovered physical principles required a further advance over the Greeks. The Greek investigations concerned loci that were determined by a set of conditions, with respect to a set of positions. For example, the locus of positions that are equally distant from a single position, is a circle. The locus of positions, the sum of whose distance from two points remains constant, is an ellipse. The locus of positions of the centers of circles that go through two points is a line. As these examples illustrate, the resulting locus is a function of the conditions applied to some set of positions. (See Riemann for Anti-Dummies Part 6.)
But, non-uniform motion required that the locus be thought of as determined by a set of conditions relative to a characteristic of action. For example, the positions of a planet in its orbit are not a function of the relationship of the planet to some other point or body. Rather, the position of the planet in its orbit, is a function of the harmonic ordering of the solar system as a whole. It is these principles which determine the planet’s position at each moment, not the planet’s relationship to a fixed point, such as the geometrical center of the orbit, or the equant, or even a fixed body, such as the Sun.
Kepler stated this point explicitly in his “Epilogue Concerning the Sun, By Way of Conjecture,” from the “Harmonies of the World” (see Riemann for Anti-Dummies Part 3.) Kepler adopted the conception of Pythagoras and Plato:
“…The relation of the six spheres to their common center, thereby the center of the whole world, is also the same as that of unfolded Mind (dianoia) to Mind (noos)….”
But, he warned against thinking that this ordering principle of Mind, is located in a fixed place. That it has, “a royal throne in the solar body”, as the Pythagoreans believed. Echoing Cusa, Kepler said, “But, we Christians, who have been taught to make better distinctions, we know, that the eternal and uncreated `Logos’, which was `with God’, is contained by no abode, although He is within all things, excluded by none….”
This distinction is crucial for solving the Kepler challenge. While Kepler demonstrated the Sun as the mover of the planets, the Sun itself was subordinate to a higher principle, reflected in the harmonic ordering principles, which were present at every moment of the planet’s action, and discoverable by the human mind. Thus, the need for a new mathematical metaphor that can describe position as a function of a principle of change.
That, is ultimately what Leibniz set out to create with his infinitesimal calculus, building on Fermat’s investigations. (Riemann for Anti-Dummies Parts 6 & 7.) Some examples will help construct the concept. What follows requires some work, and the introduction of concepts that might seem difficult. This is unavoidable, however, to lay the groundwork for Gauss and Riemann’s later work. The serious reader is encouraged to work through the examples, or otherwise be condemned to the dumbed down domain of Riemann for Dummies. (Due to the inability to present diagrams in this form of communication, I will refer the reader to illustrations in readily available resources.)
Look at the wax paper constructions of the conic sections on page 11 of the “Gauss” Fidelio. (If you haven’t made these constructions in a while, it would be worth your while to make a set.) In these constructions, each conic section can be thought of in two ways. One, is as a locus with respect to some fixed positions, such as a curve generated by the intersection of a plane with a cone. By inversion, each conic section can be thought of as the envelope of a set of tangents. In the first, it is the curve that determines the position of the tangents, while in the inversion, the position of the tangents determines the curve. If you begin with the curve, the positions of the tangents are precisely determined. But, if you begin without the curve, how are the tangents determined, so that their positions envelop the resulting curve?
To solve this problem, Leibniz sought not to determine the positions of the tangents, but to determine the function that governed the way the positions of the tangents changed. To do this Leibniz thought of each tangent as a “moment of becoming”, and the change between the tangents as a “differential”. The sought after function he called a “differential equation”. (Don’t get scared off by the word “equation”. In Leibniz’ sense, like Fermat’s and Kepler’s, equation is not a formula, but a set of relationships.)
To illustrate this, take the example of the parabola, beginning with Fermat’s geometry of loci. (See Riemann for Anti-Dummies Part 7.) In that example, Fermat showed that while the length and angle of each tangent to the parabola changed non-uniformly, when projected onto an axis cutting the parabola in half, a 2:1 ratio was always maintained. (This example is detailed in Part 7.) Leibniz recognized that this 2:1 ratio was the characteristic of change, differential, between each “moment of becoming”. Thus, Leibniz thought of the tangents to the parabola, as those lines, produced by a sequence of differentials, that changed according to this 2:1 ratio. The parabola, in turn, was the “integral” of this differential equation.
To put this in less technical sounding terms, the “sense certainty” approach generates the parabola as a set of positions, which in turn determines the positions of a set of tangents, which in turn determines a characteristic change from one tangent to the next. From the standpoint of Leibniz’ calculus, the characteristic change (differential equation) determines the positions of the tangents, which in turn determine the parabola.
Leibniz, of course, did not limit the application of his calculus to conic sections, or even to known curves. Instead, he and Jean Bernoulli, showed that even hitherto unknown curves could be discovered from their corresponding differential equations. The example of Bernoulli’s discovery of the brachistichrone curve illustrates this point. (The reader is referred to Bernoulli’s essay on the subject. An English translation is available in Smith’s Source Book of Mathematics p. 644.) In future installments, we will work through this example in more detail, but, for pedagogical reasons, we summarize the relevant questions here.
Bernoulli had thrown out a challenge to determine the curve on which a body falling under the influence of gravity, travelled in the least time. No one but Leibniz presented a solution. As will become evident later, Bernoulli’s challenge concerned not simply this particular problem, but was directed towards illustrating his and Leibniz’ method for discovering universal principles, and exposing the bankruptcy of Newton and Gallileo.
In summary, Bernoulli’s ingenious solution is as follows: Fermat had demonstrated that light, refracted by a change in the density of the medium, travelled the least time path. Were light travelling through a medium, whose density was continuously changing, its path would be a curve of least time. If the medium were changing according to the same proportion, by which a body fell according to gravity, then the resulting path of the light, would correspond to the least-time curve of the falling body.
To restate this problem in the language we’ve developed in this series. The locus of positions of a body falling in least time is unknown to us. However, we know that light travelling through a medium of continuously changing density will find the least-time path. We want to know what the light knows. The position of the light at each “moment of becoming” is ambiguous, but the change from moment to moment (the differential) is known precisely, and defines a differential equation. The sought for least-time curve is the integral of that differential equation.