Doing the Impossible

“Nothing is fun but change,” is an apt transformation of Heraclites’ famous aphorism to convey the quality of mind required to grasp Leibniz’ calculus and its extension developed by Kaestner, Gauss, and Riemann. Inversely, one who is gripped by a bullheaded resistance to its import, and the corollary, “Without fun there is no change,” will be doomed to the dull, unchanging, “cult of Isaac Newton,” where, the only hope out, is, to change.

Like Leibniz’ original discovery of the calculus, the equally revolutionary breakthroughs of Kaestner, Gauss and Riemann were long in the making. As usual, the matter is most efficiently presented pedagogically, from the standpoint of Kepler.

Kepler’s discovery of the non-uniform motion of planetary orbits presented the paradox that while the characteristics of the planet’s orbit are knowable, the position and velocity of the planet were not susceptible to precise mathematical calculation. In his New Astronomy, Kepler introduced this paradox in the form of a challenge to future geometers. Kepler’s challenge confronted a mathematical system that confined itself to determining positions, only with respect to other positions. From this standpoint, it were impossible to determine a characteristic of change, which was always changing. But, for Kepler, Leibniz, Gauss and Riemann, change was a characteristic of mind, as well as the physical universe. Rather than measure position, as the existing mathematics insisted he do, Leibniz invented a calculus that measured the characteristics of change from which the position of the planet was produced. Gauss and Riemann extended this calculus, by measuring the characteristics of change, that produced the change, which was producing the positions. In other words, the orbit was measured by a total characteristic of change (integral) of which each momentary expression of it (differential) was a function. But, that orbit was itself a function of a characteristic of a higher process. It is to the characteristics of that process on which Gauss and Riemann focused, the which will be developed pedagogically in future weeks.

Investigations into the impossibility of mathematical solutions goes all the way back to classical Greece, as represented by the famous problems, of doubling the cube, trisecting the angle, the quadrature of the circle, and construction of the heptagon (7-gon). No methods were found by which these problems could be solved, in ways which were rigorously knowable, as Plato established the principle of “knowability” in his dialogues.

For example, as you know from the study of Cusa’s work on the quadrature of the circle (fn.1) the circle could not be measured precisely by rectilinear magnitudes. Cusa showed that the “unsovlability” of this problem was not due to an undiscovered method within the existing mathematics, rather, it was due to a deficiency in the entire system of mathematics, as long as that system did not admit of transcendental magnitudes. Such transcendental magnitudes were impossible in the domain of rectilinear magnitudes. Yet, these “impossible” magnitudes were reflected in a real principle, the principle of circular action. The existence of the circle could be known as a reflection of a distinct principle, but its measurement could not be accomplished by a mathematics that excluded that principle. There had to be a complete transformation of the system of mathematics, from a mathematics that included only one type of magnitude (rectilinear) (fn.2) to a mathematics encompassing two types of magnitudes (rectilinear and transcendental). It was not that Cusa made transcendental magnitudes possible, but that a system of mathematics without them, was proven to be impossible.

Like the quadrature of the circle, the difficulty of doubling the cube, trisecting the angle, and constructing the heptagon, resulted from the “impossibility” of constructing an appropriate magnitude, that was “knowable” within the given system of mathematics. This is most easily illustrated by the example of doubling of the cube, a problem that has been discussed in previous pedagogical discussions.

The issue involved is presented most effectively by the poetic report of the problem’s origination. It is said that the Delians were asked by the Gods to construct an altar double the size of the existing one. Plato told the Delians that Apollo posed this problem to them because, according to Kepler, “Greece would be peaceful if the Greeks turned to geometry and other philosophical studies, as these studies would lead their spirits from ambition and other forms of greed, out of which wars and other evils arise, to the love of peace and to moderation in all things.” In other words, “Change the system!”

It had been known by the Greeks how to find a magnitude that could double the area of a square, even though this magnitude was incommensurable with the side of the smaller square.(fn.3) But, they were unable to construct a magnitude that could produce a cube whose volume was doubled. Was this magnitude possible, or, was a system that could not produce it, “impossible”?

Renewed investigations into these questions emerged in the wake of Cusa’s revolution, which set the stage for the revolutionary ideas of Gauss and Riemann. This history is reported by Kaestner in his 1796, “History of Mathematics”, a much more reliable source than today’s generally available histories, which commit fraud by the fallacy of composition by lumping these investigations under the general topic of algebra. Kaestner distinguishes between the efforts to simply calculate with symbols, and those attempts to gain a deeper insight into proportions. For the former, Kaestner relates the story, from Cervantes, “Don Quixote”, about how the Aristotelean, Bachelor Sampson Carrasco, who, believing Don Quioxte was mad, pretends to be mad himself, in an effort to deceive Don Quioxte into giving up knight errantry. Having adopted Don Quioxte’s system, Carrasco gets caught in a joust with the mad knight and ends up receiving a thrashing himself. Having been left humiliated and with broken ribs, Carrasco is forced to seek out help from an algebraist. (fn.4) For the latter, Kaestner refers to the early efforts to investigate the deeper implications of the concept of powers, that had been expressed by Plato in the Theatetus dialogue. Such efforts were associated with Luca Pacioli and Girolamo Cardan (1501-1576), whose father is reported to have been a collaborator of Leonardo da Vinci.

In investigating the relations of squares and cubes, Cardan discovered magnitudes that were “impossible” according to the prevailing system of algebra. Cardan’s example was, “If someone says to you, divide 10 into two parts, one of which multiplied into the other shall produce 30 or 40 it is evident that this case is impossible. Nevertheless, we shall solve it in this fashion….” Cardan’s solution was to produce the results by the magnitudes, (5 + ?-15)(5 – ?-15). But, since the square root of a negative number were “impossible” in the algebraic system, Cardan concluded, “This subtlety results from arithmetic of which this final point is as I have said, as subtle as it is useless.”

And so, the question was posed again: were these magnitudes “impossible”, or was the system which could not produce them, “impossible”? How this question was approached, is an instructive marker that separates the true thinkers (those who know how to have fun) from the frauds.

For example, in his investigations of the same equations, Descartes maintained that the square root of a negative number was “impossible”. On the other hand, Leibniz, in a 1673 correspondence with Huygens, produced the following result:

?(1 + ?-3) + ?(1 – ?-3) = ?6

Of which he said, “I do not remember to have noted a more singular and paradoxical fact in all analysis: for I think I am the first one to have reduced irrational roots, imaginary in form, to real values….”

To which Huygens replied:

“The remark you make concerning inextractable roots and roots with imaginary magnitudes, the which, nevertheless, upon addition yield a real quantity, is surprising and completely new. One would never have believed that ?(1 + ?-3) + ?(1 – ?-3) could be equal to ?6, and there is something hidden there which is incomprehensible to us.”

Later, Leibniz would say, “The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and non-being.”

In a 1702 letter to Varignon, Leibniz further reflects on this paradox:

“Without worry one can use infinitely small and large lines as ideal concepts — even though they do not exist as real objects in the metaphysically rigorous sense — as a means to shorten calculation, just as the imaginary roots in ordinary analysis, such as for example ?-2. Irregardless of whether one calls these ‘imaginary’, they are nonetheless useful and sometimes even indispensible, in order to express real magnitudes analytically; so, for example, it is impossible, without using them, to give an analytical expression for a line segment, which divides a given angle into three equal parts. Just so, one could not elaborate our calculus of transcendental curves, without talking about differences, which are in the act of vanishing, and introducing once and for all the concept of incomparably small magnitudes….

“Also the imaginary numbers have their {foundation in reality} (fundamentum in re). When I pointed out to the late Mr. Huygens, that ?(1 + ?-3) + ?(1 – ?-3) = ?6, he was so amazed, that he answered, for him there is something incomprehensible in this. But just so, one can say, that the infinite and infinitely small have such a solid basis, that all results of geometry, and even the processes of Nature, behave as if both were complete realities … because everything obeys the Rule of Reason.”

By contrast, Euler (held in such high esteem by today’s algebraists) said, “All such expressions, as square root of -1, or square root of -2, are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible.”

It was the genius of Gauss, building on the work of Kaestner, to recognize that it was the system, not the magnitudes, which were impossible, which he demonstrated from his very earliest work. (fn.5)

Gauss’ concept of complex numbers has been treated extensively in previous pedagogicals, and will be the subject of the coming installments in this series. But, the standpoint from which he approached it is expressed in his “Second Treatise on Bi-Quadratic Residues”

“Thus we reserve for ourselves a more detailed treatment of these subjects for another opportunity. The difficulty, one has believed, that surrounds the theory of imaginary magnitudes, is based in large part to that not so appropriate designation (it has even had the discordant name impossible magnitude imposed on it). Had one started from the idea to present a manifold of two dimensions (which presents the conception of space with greater clarity), the positive magnitudes would have been called direct, the negative inverse, and the imaginary lateral, so there would be simplicity instead of confusion, clarity instead of darkness….

“It is this and nothing other, that for the true establishment of a theory of bi-quadratic residues, the field of higher arithmetic, that otherwise extends only to the real numbers, will be enlarged also to the imaginary, and these must be granted complete and equal civil rights, with the real. As soon as one considers this, these theories appear in an entirely new light, and the results attain a highly surprising simplicity.”

These concepts, however, are not limited to matters of arithmetic, as Gauss expressed in his 1811 letter to his friend Hansen:

“These investigations lead deeply into many others, I would even say, into the Metaphysics of the theory of space, and it is only with great difficulty can I tear myself away from the results that spring from it, as, for example, the true metaphysics of negative and complex numbers. The true sense of the square root of -1 stands before my mind (Seele) fully alive, but it becomes very difficult to put it in words; I am always only able to give a vague image that floats in the air.”

And, that’s where the fun begins.

NOTES

1. Cusa’s investigation and its successors by Huygens and Leibniz as reviewed by Bob Robinson in his pedagogical series a1096rar001; a1106rar001; a1146rar001; a1176rar001)

2. Here I include both rational and irrational (algebraic) magnitudes under rectilinear.

3. Euclid had given a general method for construction of such a magnitude. Draw a semi-circle and its diameter AB. Connect A and B to any point C on the circumference, forming right triangle ABC, with right angle ACB. Drop a perpendicular from C to AB, whose intersection call D. The length of line CD is the square root of the length of line DB. The reader can prove this using the Pythagorean theorem.

4. Kaestner here is pointing to a pun of Cervantes, as the Spanish word algebraist also meant “bone-mender”

5. See Gauss’ work on the division of the circle, Riemann for Anti-Dummies Parts 11 &12. Also see Gauss’ 1799 doctoral thesis, “New Proof of the Fundamental Theorem of Algebra”, in which he explicitly demolishes the mathematics of Euler, Lagarange and D’Alembert, which considered complex numbers to be “impossible”.