What’s in a Moment?
We are now at the point in this series, where we can begin to dig directly into that rich vein of knowledge revealed by Bernhard Riemann’ s development of complex functions. However, it is necessary, before embarking on that leg of this journey, that you first contemplate this short, but important, pedagogical exercise. Its relevance will become increasingly apparent to you.
Take the case of a Keplerian orbit. At each moment the planet is changing its speed and trajectory. That change is being guided by the underlying hyper-geometry of the solar system, which has determined the shape of the orbit. That hyper-geometry, thus, requires the planet to change its speed and trajectory, at each moment, according to a principle, derived from the characteristics of the hyper-geometry. Kepler showed that the principle governing the change in speed and trajectory was expressed by the way equal portions of the planet’s period corresponded to equal areas swept out.
Now, here’s the paradox. In any interval of the orbit, no matter how small, the planet is doing something different at the beginning of that interval than at the end, with the exception of the maximum and minimum intervals. The maximum interval is the entire orbit. There, (at least in first approximation), at the beginning and end of this interval, the planet is doing the same thing. (For example, if we consider this interval to be from, say, perihelion to perihelion.) The minimum interval is the moment of change. In that moment, the planet ceases to do what it just did, and starts becoming what it will be. Paradoxically, the beginning and end of each moment, like the entire orbit, are also equal. However, the type of change the planet is undergoing at that moment is determined by the entire orbit. Thus, the maximum interval and the minimum interval coincide.
From the standpoint of Leibniz’ calculus, the integral is the maximum as seen from the minimum, while the differential, is the minimum as seen from the maximum.
This concept was expressed by Nicholas of Cusa in “On Learned Ignorance” Book II:
“In like manner, if you consider the matter carefully: rest is oneness which enfolds motion, and motion is rest ordered serially. Hence, motion is the unfolding of rest. In like manner, the present, or the now, enfolds time. The past was the present, and the future will become the present. Therefore, nothing except an ordered present is found in time. Hence, the past and the future are the unfolding of the present. The present is the enfolding of all present times; and the present times are the unfolding, serially, of the present; and in the present times only the present it found. Therefore, the present is one enfolding of all times. Indeed the present is oneness. In like manner, identity is the enfolding of difference; equality [the enfolding] of inequality; and simplicity [the enfolding] of divisions, or distinctions.
“Therefore, there is one enfolding of all things. The enfolding of substance, the enfolding of quality or of quantity, and so on, are not distinct enfoldings. For there is not only one Maximum, with which the Minimum coincides and in which enfolded difference is not opposed to enfolding identity. Just as oneness precedes otherness, so also a point, which is a perfection, [precedes] magnitude. For what is perfect precedes whatever is imperfect. Thus, rest precedes motion, identity precedes difference, equality [precedes] inequality, and so on regarding the other perfections. These are convertible with Oneness, which is Eternity itself (for there cannot be plurality of eternal things). Therefore, God is the enfolding of all things in that all things are in Him; and He is unfolding of all things in that He is in all things.”
However, not all “moments” are the same. In the case of a planetary orbit, while at all moments the planet’s speed and trajectory are changing, there are two unique moments, in which that change is a complete transformation, specifically, aphelion and perihelion. In the former, the planet’s action changes from slowing down to speeding up, while in the latter, the action changes from speeding up to slowing down. These two moments are called singular moments, or singularities. The change at all other moments of the planet’s action, is thus determined by these two singularities, aphelion and perihelion.
As Kepler showed, these singularities are determined by a higher principle of the hyper- geometry underlying the solar system. In other words, that two singularities exist, is a characteristic of eccentric orbits; that the orbit has this specific relationship to these singularities, is a characteristic of the “more basic principle” , i.e. the harmonic principle, that governs the whole solar system.
As we work through Riemann’s discoveries in future pedagogicals, we will present more examples of this same principle. For now, think back over this one, so you get used to this way of thinking.