Towards a Hylozoic Calculus
It always comes as a shock to the mathematically schooled, that Leibniz’ infinitesimal calculus is not, essentially, a mathematical procedure. It is just as shocking to the unschooled, but mathematically intimidated, that it were impossible to grasp the deeper implications of Leibniz’ discovery, without digging into its mathematical expression. These two, seemingly opposite, subjective reactions, arise from a common misunderstanding about the nature of man. They are the twin pathologies of the same disease–the Aristotelean/Gnostic separation of mind from matter.
Leibniz, of course, did not suffer from the disease that afflicted his enemies. In a 1678 letter to Countess Elizabeth, he recounts how he was drawn to the study of mathematics in pursuit of greater knowledge, but not as an end in itself:
“I cherished mathematics only because I found in it the traces of the art of invention in general; and it seems to me that I discovered, in the end, that Descartes himself had not yet penetrated the mystery of this great science….
“…I can state that it is for the love of metaphysics that I have passed through all these stages. For I have recognized that metaphysics is scarcely different from the true logic, that is, from the art of invention in general; for, in fact, metaphysics is natural theology, and the same God who is the source of all goods is also the principle of all knowledge. That is because the idea of God contains within it absolute being, that is, what is simple in our thoughts, from which everything that we think draws its origin.”
And in a 1716 letter to Samuel Mason, Leibniz added, “The ancients considered mathematics as the passage from physics to metaphysics or to natural theology, and they were right.”
Leibniz’ calculus is exemplary of such a passage. It was provoked by the paradoxes brought to light by Kepler’s discovery of the elliptical orbit of Mars. Kepler had demonstrated that action in the physical universe was characterized by non- uniform motion. But, as discussed in previous installments in this series, the universe presented Kepler with a new challenge. Kepler could measure the planet’s non-uniform motion, from the standpoint of the universal principles that governed the planet’s orbit, but the inverse, to measure the universal principles that governed the orbit, from the planet’s motion, required a new discovery.
Kepler had asked the universe a question and discovered the universe’s answer. The universe, in turn, posed a question back, that revealed a hitherto unthought of paradox in Kepler’s cognitive process. This irony, that the discovery of a physical principle, in turn provokes a new discovery of mind, comes as no surprise to those healthy minds not afflicted with Aristotle’s disease, and it demonstrates, as Plato and Cusa taught us, that mind and matter are inseparable.
This irony should also come as no surprise to someone who has come to know what LaRouche has shown, i.e. that the higher principle of cognition, governs the principles of living processes, which governs the principles of non-living processes. Thus, the principle of cognition is expressed in living and non-living processes, albeit in paradoxical form. As in the method of Plato and Cusa, it is through such paradoxes (such as the Kepler challenge) that we are led to the discovery of these higher principles.
Our grasp of this method of discovery by inversion, is aided by reference to the principles of musical polyphony. Think of how the underlying characteristics of the bel-canto, well- tempered system of polyphonic musical composition, are discovered with respect to the principle of inversion. For example, the true characteristic singularity of the Lydian interval only clearly emerges in the context of the principle of inversion within a musical composition.
To get a handle on the problem, think of the planet’s motion from the standpoint of the mind of the planet. The planet’s path around the Sun is governed by the characteristics of the elliptical orbit, which, in turn, are governed by the harmonic ordering of the solar system. These characteristics are expressed at each moment of the planet’s motion, as a non-uniformly changing speed and trajectory of the planet. At each such moment, the planet’s action is ceasing to be what it was, and becoming what it will be. At each such “moment of becoming”, the planet has a definite speed and trajectory, which is changing such that the resulting interval maintains a constant proportion to the whole orbit, i.e. equal areas. The change to the planet’s speed and trajectory at each moment, is also changing from moment to moment. How does the planet know, at each “moment of becoming”, how its action must change?
The subsequent discoveries of non-uniform physical action by Fermat, Leibniz, Huygens and Bernoulli, posed a similar challenge. How does the light know how to change its trajectory in order to find the path of least-time? How does the hanging chain know how to find the curve of least-tension, i.e. the catenary curve? More importantly for Fermat and Leibniz, how can the human mind know, what the planet, light, or chain, knows.
These questions perplexed and infuriated the Aristotelean/Gnostics like Descartes. Either they rejected the idea of universal principles, or they sought some mechanistic explanation, that separated the principle of mind from matter. For Aristotle, Gallileo, Newton and Descartes, action in the universe occurs along straight-lines or perfect circles. Such uniform action has no need of cognition, as bodies moving according to straight-lines or circles, don’t change except by an “outside” force. For Aristotle, that “outside” force–cognition is not only outside the physical universe, but outside the human mind as well.
Fermat and Leibniz had no such problem. They understood that principles such as least-time and Kepler’s orbits, reflected a universal characteristic, which was present at each “moment of becoming”. The solution to the Kepler problem was, thus, to determine the universal characteristics in the “moments of becoming”.
As Leibniz expressed this with respect to light:
“Indeed, neither can the ray coming from C make a decision  about how to arrive, by the easiest way possible, at points E, D, or G, nor is this ray self-moving towards them ; on the contrary, the Architect of all things created light in such a way that this most beautiful result is born from its very nature. That is the reason why those who, like Descartes, reject the existence of Final Causes in Physics, commit a very big mistake, to say the least; because aside from revealing the wonders of divine wisdom, such final causes make us discover a very beautiful principle, along with the properties of such things whose intimate nature is not yet that clearly perceived by us, that we can have the power to explain them, and make use of their efficient causes, along with their artifacts, such as the Creator employed them in order to produce their results, and to determine their ends. It must be further understood from this that the meditations of the ancients on such matters are not to be taken lightly, as certain people think nowadays.”
This is the purpose for which Leibniz’ calculus was developed as a form of geometry of position. As we saw in the last installment of this series, Fermat began this work by generalizing the Greek concept of geometry of position (topos or locus). Now, he laid the foundations for the calculus, by inverting this concept. In effect, Fermat, and later Leibniz, were defining the completed paths from the standpoint of the “moments of becoming.” Fermat called this his “Method for Determining Maximum and Minimum and on Tangents to Curved Lines”. Leibniz extended this concept, by seeing in these “moments of becoming”, what he called, differentials.
To illustrate this method, we will begin, as Fermat and Leibniz did, with a simpler example than the Kepler challenge. (The Kepler problem proved to present paradoxes whose solution required the subsequent discoveries of Gauss and Riemann.)
Make a parabola using the wax paper method illustrated in Figure 1.9(c) on page 11 of the “Gauss” Fidelio. In this construction the parabola is formed as an envelope of the tangents, which is an inversion of forming the tangents from the parabola. Through the focus of the parabola draw an axis about which the parabola is symmetric. (Call that line A.) Now, pick one of the tangents and draw a line from the point of tangency that is perpendicular to line A. (Call the point of intersection with A, x.) Now, extend the tangent line until it intersects line A. (Call that point y.) Repeat this with several tangent lines. You should notice that as the point of tangency gets farther from the vertex, point y also gets farther from the vertex and vice versa. And, at the vertex, points x and y coincide.
Fermat showed, in the case of the parabola, that, the distance from the point x to the vertex of the parabola was always ? the distance from x to y. In this way, he could describe action along a parabola, by inversion, as that action which maintained a 1:2 ratio between points x and y. This proportion, like the equal area principle of a Keplerian orbit, reflects the change at each “moment of becoming”.
(The case of the parabola is merely an illustration for pedagogical purposes. Fermat’s method was a general one, that enabled him to solve a myriad of transcendental problems, that, by the way, made Descartes furious.)
In future installments we will work through some other examples to help solidify this method. But, to keep your mind active, look way ahead and consider where this goes.
Any problem posed from the standpoint of geometry of position, implies an assumption about the domain in which the action occurs. In the above mathematical example of the parabola, it is tacitly assumed that the parabola lies in a Euclidean plane. Yet, in a physical problem, a grave mistake would be made, if one assumed that the action, such as the refracting ray of light, or the motion of a planet, were taking place against the backdrop of a Euclidean space. But, it will still be a mistake, to merely replace the Euclidean space with some other non-Euclidean manifold. And, we would still be in error, if we adopted what we thought was an anti-Euclidean concept, but limited our investigation to non-living physical processes. In other words, in considering the geometry of position of a planet’s motion about the Sun, we cannot limit the investigation to the planet’s position with respect to non-living matter in the solar system. As stressed above, we must investigate the planet’s changing position as a function of non-living, living and cognitive processes.
Now look again at the revolutionary experimental discoveries reported in JBT’s last two pedagogical discussions giving fresh evidence of the universal principles ordering living and non- living processes. Viewed from the standpoint of LaRouche, that the principle of cognition orders living and non-living processes, the following question is provoked to a future Leibniz: How are the universal principles, of cognitive, living, and non-living processes, expressed in each “moment of becoming”? The answer requires the development of a new type of geometry of position, which might be called a, “hylozoic calculus.”