Riemann for Anti-Dummies Part 3


No mortal yet as climbed so high,
As Kepler climbed and died in need, unfed:
He only knew to please the Minds
And so, the bodies left him without bread.
         --Abraham Gotthelf Kaestner

In "On Copernicanism and the Relativity of Motion," G. W. Leibniz presents a proposition that might provoke you. "To summarize my point," Leibniz writes, "since space without matter is something imaginary, motion, in all mathematical rigor, is nothing but a change in the positions [situs] of bodies with respect to one another, and so, motion is not something absolute, but consists in a relation."

It follows, then, that it were impossible to answer the question, "Which is moved, the Earth or the Sun?"

There are several common reactions to this question, among which are:

1. From the standpoint of naive sense perception: "The Sun is moved, because I see the Sun, stars and planets, all moving around me."

2. From the standpoint of popular opinion: "The Earth moves around the Sun. Everyone knows that. We learned it in school."

3. From the standpoint of naive sense perception and popular opinion, dressed up in more sophisticated evening clothes: "The Earth moves around the Sun. This can be discovered by observing the motion of all the planets, measuring the intervals between the various periods of retrograde motion, and comparing those intervals with the Sun's apparent yearly cycle. All these observations, taken as a One, clearly indicate the Earth is moving around the Sun."

There are, of course, numerous other variants. However, as Leibniz correctly states, try as you might, there is no way you can say, whether the Sun or the Earth is moved. The best you can hope, to truly say, is that the Earth and the Sun are in motion relative to one another.

"How then," you might ask, "Could Kepler so surely state, that the Sun is at rest and the Earth is in motion?"

As usual, the problem is, that you asked the wrong question. Kepler never asked, "Is the Sun or the Earth moved?" Rather, he asked, "Is the Sun or the Earth, the mover?"

See the difference between the two questions. The former concerns only a change in place, which Leibniz shows is always relative. The latter concerns a physical ordering principle. In the first question, the verb "is", refers to something other than the Sun and the Earth, i.e., the imaginary space in which the two bodies interact. In the second question, the verb "is", refers to the physical cause of the motion, "the mover." The second question is susceptible to being answered, while the first, is only susceptible to being debated by academics and fools.

"Okay, now that you've shown me the right question, it is still answered by observing the motions of the Sun, the Earth, and the planets, discovering the anomalies, and then determining that the Sun is the mover," you might assert.

Be careful. Here things can get slippery. A universal principle is not {discovered} by observation or experiment. A universal principle is discovered by an act of cognition, and its validity is demonstrated, by the replication of that act of cognition, through a crucial experiment.

As Plato, Cusa, and Kepler showed, this is not a matter of preference between equally valid methods, but concerns an ontological principle of the Universe itself.

At the end of the {Harmonies of the World}, Kepler appended an, "Epilogue Concerning the Sun, By Way of Conjecture," where he elaborated this ontological principle.

"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 center of all the circuits, immovable in place, but rotating into itself," Kepler begins.

"... 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)...."

Here an understanding of the Greek words, can help us to a more precise grasp of the concept. Since words evoke concepts, it is an error to seek an "equivalent" word, when projecting a concept from one language to another. It were more appropriate to apply a principle similar to the Kepler/Gauss concept of congruence. This may require the use of several terms, in several languages, in order to communicate the idea, to which the word is intended to refer. In this case, the Greek words "dianoia" and "noos", (usually translated as understanding and reason, respectively) refer to concepts developed by Plato in the sixth book of the Republic, to denote the two highest principles of knowledge. The highest principle, "noos", subsumes the next highest principle, "dianoia". While these are distinct principles, the higher begets the lower, denoted in the Greek by the prefix, "dia". Kepler inserts the Greek words, not to provide an equivalent term for the Latin text, but to refer his readers to the concept developed by Plato. Even the expression of universal physical principles, must be congruent with those principles.

Plato states this principle as:

"This then is the class that I described as intelligible, it is true, but with the reservation first that the soul is compelled to employ assumptions in the investigation of it, not proceeding to a first principle because of its inability to extricate itself from and rise above its assumptions, and second, that it uses as images or likenesses the very objects that are themselves copied and adumbrated by the class below them, and that in comparison with these latter are esteemed as clear and held in honor ... and by the other section of the intelligible I mean that which the Mind itself lays hold of by the power of dialectics, treating its assumptions not as absolute beginnings but literally as hypotheses, underpinnings, footings, and springboards so to speak, to enable it to rise to that which requires no assumption and is the starting-point of all, and after attaining to that again, taking hold of the first dependencies from it, so to proceed downward to the conclusion, making no use whatever of any object of sense, but only of pure ideas moving on through ideas to ideas and ending with ideas."

With that in mind, now, back to Kepler's "Epilogue":

"But we duly subordinate the created mind of whatever excellence it may be to its Creator, and we introduce neither intelligent powers as Gods, as does Aristotle and the pagan philosophers, nor armies of innumerable planetary spirits with the Magi, nor do we propose that they are either to be adored or summoned to intercourse with us by theurgic superstitions, for we have a careful fear of that; but we freely inquire by natural reasons what sort of thing each mind is, especially if, in the heart of the world, there is any mind bound rather closely to the nature of things and performing the function of the soul of the world or if also some intelligent creatures, of a nature different from human perchance do inhabit, or will inhabit, the globe thus animated. But it is permissible, using the thread of analogy as a guide, to traverse the labyrinths of the mysteries of nature. I believe the following arguments can not be put aside. The relation of the six spheres to their common center, thereby the center of the whole world, is also the same relation, as that of unfolded Mind (dianoia) to Mind (noos). On the other hand, the relation of the single planets' revolutions from place to place around the Sun, to the unvarying rotation of the Sun in the central space of the whole system, is also the same as the relation of unfolded Mind to the Mind, which is, that of the manifold of dialectics, to the most simple cognition of the Mind. 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, stirs up dialectics, by which it understands itself and in itself all things, and by unfolding and unrolling its simplicity into those dialectics, it makes everything known. And the movements of the planets around the Sun at their center, and the unfolded dialectics 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 never would human cognition have worked its way to the true intervals of the planets, and to the other things dependent from them, and never would it have constituted astronomy."

This relationship, of the Sun to all the planets, the Sun to each planet individually, of noos to dianoia, of the word to the idea, is at the heart of Gauss' and Riemann's theory of functions. With what has been developed here, we are now in a position to begin to unfold these functions in the next installment.

We end with another poem by Kaestner written for his student Christlob Mylius. The poem was meant to accompany a copy of Kepler's {Harmonies of the World} which Kaestner sent to Mylius, who was on his way to America from Leipzig. Mylius, a cousin of Lessing, never made it. He died in London, survived by this short poem. In it, Kaestner expresses his great love for Kepler, and his oft-stated disdain for those German authorities and bread-scholars, who cowardly turned their back on their great countryman and embraced the inferior Newton instead.

To Mr. Christlob Mylius:
Along with sending over Kepler's Harmonice Mundi
Friend, your tender ear perceives the graceful art of tones,
The world-form's harmonies, your deeper thoughts explore,
In here, Newton's teacher writes of both them,
Deustchland let him starve, and remains unworthy of him.