Sufficient Harmony:

An Introduction to the Scientific Method of Carl Friedrich Gauss, as a Continuation of that of Johannes Kepler

In Chapter 32 of the New Astronomy, Kepler begins his long stretch towards his discovery that the planets move in elliptical, not circular orbits around the Sun.  And at first glance, his method for making this discovery might appear to be confusing, or even to lack rigor.  At one point he swaps the arithmetic for the geometric mean, saying that they are “almost equal,” and again, later, treats the physical and optical equations, two clearly different angles as equal, and continues in such a fashion until he ends up with a larger error than the one he set out to try to remove.  He then declares a battle won, and in fact proceeds to win the war, discovering what is now called “Kepler’s first law.”  Confusion about this method has led some to call it “sleepwalking,” or to declare that Kepler made his discovery clumsily, or by accident!   But a look at what Kepler’s method actually was––and how it is in complete conformity with what Gottfried Wilhelm Leibniz (1646-1716) called the principle of sufficient reason, where a formally “rigorous” mathematical/mechanical treatment would not have been successful––will shed an indispensable light on the method of discovery which underlies the true genius of Carl Friedrich Gauss. 

The conditions in which Carl Friedrich Gauss was operating during the period straddling the end of the eighteenth and beginning of the nineteenth century were those of a serious conflict over the nature of the future of the human species.  A new nation had just been formed across the ocean, the United States of America , which was the first ever in human history to be based entirely on the principle of republican humanism.  The intellectual environment in which Gauss was raised was shaped by vocal supporters and organizers of this revolution, followers of the work of Gottfried Leibniz and Johannes Kepler.[1]    But it was also the center of a nightmarish counterattack by the oligarchical feudal interests who were intent on destroying that conception of man and its political expression across the sea, by first destroying any possibility of its taking hold politically in the nations of Europe.[2]   As a result, almost the entirety of Gauss’ scientific work was accomplished under conditions of occupation.  Because of this, Gauss became an expert at appearing to replace the a priori methods of Kepler, based on the “worthiness and eminence” of truthfulness of physical principle, with what Kepler called “rather long induction.”  Because of this, any discussion of Gauss’ work will have to draw largely from his private, unpublished documents, and a thorough understanding of the philosophical tradition in which he was raised, and with which he identified.  A preliminary application of that approach, in preparation for a more thorough treatment some months from now, will be given here.  We will start with the epistemlogical framework set down by Gauss’ great predecessor, Kepler, and systematized by Kepler’s successor, Gottfried Wilhelm Leibniz.

Sufficient Reason

The great foundation of mathematics is the principle of contradiction, or identity, that is, that a proposition cannot be true and false at the same time; and that therefore A is A, and cannot be not A.  This single principle is sufficient to demonstrate every part of arithmetic and geometry, that is, all mathematical principles.  But in order to proceed from mathematics to natural philosophy, another principle is requisite, as I have observed in my Theodicy: I mean, the principle of a sufficient reason, viz. that nothing happens without a reason why it should be so, rather than otherwise . . . if there be a balance, in which everything is alike on both sides, and if equal weights are hung on the two ends of that balance, the whole will be at rest . . . because no reason can be given, why one side should weigh down, rather than the other.”[3]

[T]hat God wills something, without any sufficient reason for his will . . . [is] contrary to the wisdom of God, as if he could operate without acting by reason . . . [however] I maintain that God has the power of choosing, since I ground that power upon the reason of a choice agreeable to his wisdom.  And ‘tis not this fatality, (which is only the wisest order of providence) but a blind fatality or necessity, void of all wisdom and choice, which we ought to avoid.[4]

That “nothing happens without a reason why it should be so, rather than otherwise,” seems like a simple enough idea to anyone who gives it just a little thought: if something falls, we think, for instance, that we can be sure that we can attribute some cause to its falling.  Maybe it slipped; maybe it was pushed; maybe a million particles of air moved around each other in just the right way and a  breeze blew it over.  Even if we don’t know directly what the reason was, we can be assured there was a reason.  This single fact accounts for the efficacy (and, not incidentally, as we will see below, the name) of human reason.  If any one thing in all the world could occur absent a cause, there would be no surety in knowledge, because all knowledge that is, is a knowledge of causes.

With this, there are few people who would argue.  However, by accepting this we are presented with one most interesting question: why did anything ever happen at all?  Put perhaps less modestly, the same question might be “what’s the reason for everything?”  We won’t pretend to answer that question directly here, but we will answer another question, by analogy, and in so doing touch upon the topic of this entire report: the scientific tradition initiated by Nicholas of Cusa, reified by the work of Johannes Kepler, defended and developed by the ideas of Gottfried Leibniz and Abraham Kästner, and culminating in the successive work of Carl F. Gauss and Bernard Riemann, only to decline sharply thereafter and limp along haltingly to the present day, awaiting its renaissance in the revolutionary activities of Lyndon LaRouche and the LaRouche Youth Movement today.  So, to that end, we’ll start not with nothing, but rather with an empty page.

Euclid, in his Elements, begins all of geometry with what he calls a “point:” that which has no width, breadth, or depth.  The astute reader quickly recognizes that this is nothing other than nothing at all and, as A. G. Kästner says elsewhere in this report, there is no number of nothings which can be combined to obtain a something.[5]   So if we start with Euclid, we don’t start with anything at all, which is fine.  [animation: Euclid.swf]  So, say we start in geometry with nothing; presuming that we must have something (which is indeed a presumption), for what sort of something would there be sufficient reason for its existence?  We have a million things to choose from: the square?  The triangle?  The pentagon?  We can add sides to polygons forever without any limit . . . in fact, the triangle, having the least amount of sides, seems to stand out the greatest of all of them. 

There is something significant about this state of being the least.  It would seem that in order to have sufficient reason to be selected out from the vast sea of possibilities, a thing would have to be either the greatest or the least of the entire range of choices––the maximum or the minimum.  And the triangle is indeed the least polygon.  But, how did we come to speak of polygons?

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