Riemann for Anti-Dummies, Part 26
IDEAS CAST SHADOWS, TOO
It can be a source of confusion for the naive, and a means of deception of the wicked, to restrict the meaning of Plato’s metaphor of the cave, to those objects that originate outside of one’s skin. As all great scientists have come to know, ideas cast shadows, too. A true scientist never mistakes the shadows for the idea, seeking instead to discover the idea from between the shadows. Those who merely manipulate shadows are called sophists.
This defines the clear distinction between the concept of the complex domain of Gauss and Riemann, and the sophistry of Euler, Lagrange and D’Alembert. The former understood complex numbers as a simple case of a hierarchy of multiply extended magnitudes, or as Gauss called them, “shadows of shadows.” The latter considered complex numbers, “impossible,” but susceptible to complicated, but ultimately meaningless, symbolic manipulation, whose very complexity is intended to obscure its trickery.
A passion for sophistry pervades modern academia, as exemplified by J. E. Hofmann, who penned the forward to the 1970 republication of Abraham Kaestner’s “Geschichte der Mathematik.” As LaRouche indicated in footnote 42 of his new piece, “At the End of a Delusion,” Hofmann complains that Kaestner did not show sufficient respect for the achievements of the great mathematicians of his time Euler, Lagrange, and D’Alembert. It is precisely Kaestner’s disrespect for these sophists for which he deserves our great admiration and respect today. As the history of the discovery of the complex domain demonstrates, Hofmann’s blunder is not only a matter of a lack of comprehension of the subject, it is also indicative of the illiteracy of modern academia.
Hofmann’s error is immediately exposed by examining the 1799 doctoral dissertation of Kaestner’s student Carl F. Gauss, on “A Proof of the Fundamental Theorem of Algebra.” There, the 22 year old Gauss, matriculating for his doctorate under Kaestner, openly and explicitly castigates, Euler, Lagrange, and D’Alembert as sophists on the matter of the existence of complex numbers, showing the same disregard for Euler, Lagrange, D’Alembert, for which Hofmann cricizes Kaestner.
It is revealing that all modern biographers of Gauss have gone out of their way to dismiss Gauss’ relationship to Kaestner, who Gauss called, “A poet among mathematicians and a mathematician among poets..” It was Kaestner, the passionate defender of Leibniz and Kepler, the host of America’s Benjamin Franklin, who first raised the questions leading to the development of anti-Euclidean geometry, and, who provoked the young Gauss into deciding to pursue a life of scientific investigation. Kaestner’s biting wit and sharp-tongued polemics against the sophistry of Euler, Lagrange, and D’Alembert, the fools who would fall for their methods, sticks in the craw of the his Romantic enemies to this day. While Gauss never adopted the polemical style of his teacher, he shared Kaestner’s contempt for “ivory tower” sophistry, and expressed it in his life’s work, as a plain reading of 1799 doctoral thesis shows. After Kaestner’s death in 1800 and the ensuing rise of the fascist Napoleon, Gauss became more circumspect in his public pronouncements, but his distaste for what he called, “the screeching of the Beothians” never waned.
While a fuller account of this history must still be elaborated, it can already be stated without equivocation, that those who demean Kaestner, and hold Euler, Lagrange, and D’Alembert in high esteem, do so in defense of the degraded conception of man that produced modern fascism.
The next installment will provide a pedagogical presentation of Gauss’ doctoral thesis. This week focuses on the essential pre-history of the development of complex numbers.
As discussed in “Riemann for Anti-Dummies Part 18; ‘Doing the Impossible,'” the possible existence of complex numbers was posed in a paradox by Cardan in 1545. In his Ars Magna, Cardan pointed to the existence of what he called a “subtile” magnitude through a specific problem, to wit: “Find two numbers that add up to 10 and when multiplied together equal 40.”
Cardan recognized that this problem contained the paradox that arises from the difference between a line and a surface, because addition implies linear magnitudes, while multiplication implies a surface.
Begin with a line AB which has a length of 10. Divide the line into two parts, that produce the maximum area when multiplied together, which will be two segments of 5, which when multiplied together produce an area of 25. The sought after area is 40. Subtract 40 from 25 which yields an area of -15, which is produced by (?-15)(?-15). Thus, if you add ?-15 to one of the segments of 5 and subtract it from the other, the problem is solved, since (5 + ?-15) + (5 – ?-15) = 10; and (5 + ?-15)(5 – ?-15) = 40!
“This subtility results from arithmetic of which this final point is as I have said as subtile as it is useless,” Cardan proclaimed perplexed.
The paradox arises when one limits the conception of magnitudes to the sense perception characteristics of lines and areas, resulting (in Cardan’s example) in a magnitude of negative area.
A similar paradox arises in an even similar example. Think of a line segment of length x. Now think of a different line segment of length y. Now think of adding x to y to produce the line segment z. No matter what length you choose for x and y, you will always be able to think of a line segment whose length is z. In other words, one extensible magnitude added to another extensible magnitude, produces a third extensible magnitude.
But, what happens when you try and subtract one extensible magnitude from another? No problem if you try and subtract a smaller magnitude from a larger. But, if you try and subtract a larger extended magnitude from a smaller, you get a negative length! (For this reason negative numbers were often referred to as “false” numbers.)
It’s as if subtracting a larger line from a smaller, or a larger area from a smaller, pokes us into a world, that includes objects other than lengths and areas. Or, we must recognize that lengths and areas are only shadows and should not be mistaken for the {idea} of extended magnitude. To comprehend the {idea}, we have to go behind the shadows, by “seeing” between them. Subtracting a longer line from a shorter one, shows us a world of extensible magnitudes that exist behind the visible sense perceptions of magnitudes associated with lengths, and reveals that a more general idea of magnitude must include not only length, but also direction.
The paradox arising from subtracting a larger area from a smaller one, areas, proves more subtile. As we reviewed in Part 18, Leibniz and Huygens corresponded on the implications of the existence of the square roots of negative numbers, of which Huygens would say, “there is something hidden there which is incomprehensible to us.”
To which Leibniz would reply, “The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and non-being.”
By contrast Euler, Lagrange and D’Alembert would prove adept at complicated manipulations of algebraic equations that included the square roots of negative numbers, while insisting at all times that such magnitudes were “impossible.”
This is precisely the issue that the young Gauss attacked in his proof of the fundamental theorem of algebra. These were not “impossible” magnitudes, Gauss insisted, but “shadows of shadows.” One can think of an image of such shadows by thinking of a unit circle in the complex domain divided by two perpendicular diameters, which intersect the circumference of the circle at 1, -1, ?-1, -?-1. Think of a point rotating counter-clockwise around this circle. Now think of the image of that point, as if it were observed by looking at the circle edge on. One would only see a point moving back and forth along a line from 1 to -1 and back again. In other words, the so-called “imaginary” part is always there, but you have to look behind the shadows to “see” it.
As Gauss told his friend Hansen in 1811:
“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.”