Modern philosophy and it's sidekick Mathematics owes its current status to several pairs or triples of debaters. It was ever thus from the Ancient Greek times, Zeno nd Parmenides, Leucippius and Democritus, Pythagoras and Hipparchus Socrates and Plato to name a few.
Tartaglia and Cardano, Bombelli, Descartes and De Fermat, Leibniz and Spinoza, Kant and Schelling, Grassmann and Gauss, Hermann Grassmann and Hamilton to name a few more.
The greatest philosophical mystery is motion.
The profoundest part of that mystery is circular and spiral motion, and yet we pay it scant attention thinking that its simplicity is anything but man made! We have made it simple because it is so complex!
The decade that began so auspiciously with the Disquisitiones and Ceres was decisive for Gauss. Scientifically it was mainly a period of exploiting the ideas piled up from the previous decade (see Figure 1). It ended with Theoria motus corporum coelestium in sectionibus conicis solem ambientium (1809), in which Gauss systematically developed his methods of orbit calculation, including the theory and use of least squares.
Professionally this was a decade of transition from mathematician to astronomer and physical scientist. Although Gauss continued to enjoy the patronage of the duke, who increased his stipend from time to time (especially when Gauss began to receive attractive offers from elsewhere), subsidized publication of the Disquisitiones, promised to build an observatory, and treated him like a tenured and highly valued civil servant, Gauss felt insecure and wanted to settle in a more established post. The most obvious course, to become a teacher of mathematics, repelled him because at this time it meant drilling ill-prepared and unmotivated students in the most elementary manipulations. Moreover, he felt that mathematics itself might not be sufficiently useful. When the duke raised
his stipend in 1801. Gauss told Zimmermann: “But I have not earned it. I haven’t yet done anything for the nation.”
Astronomy offered an attractive alternative. A strong interest in celestial mechanics dated from reading Newton, and Gauss had begun observing while a student at Göttingen. The tour de force on Ceres demonstrated both his ability and the public interest, the latter being far greater than he could expect in mathematical achievements. Moreover, the professional astronomer had light teaching duties and, he hoped, more time for research. Gauss decided on a career in astronomy and began to groom himself for the directorship of the Göttingen observatory. A systematic program of theoretical and observational work, including calculation of the orbits of new planets as they were discovered soon made him the most obvious candidate. When he accepted the position in 1807, he was already well established professionally, as evidenced by a job offer from St. Petersburg (1802) and by affiliations with the London Royal Society and the Russian and French academies.
During this decisive decade Gauss also established personal and professional ties that were to last his lifetime. As a student at Göttingen he had enjoyed a romantic friendship with Wolfgang Bolyai, and the two discussed the foundations of geometry. But Bloyai returned to Hungary to spend his life vainly trying to prove Euclidi’s parallel postulate. Their correspondence soon practically ceased, to be revived again briefly only when Bolyai sent Gauss his son’s work on non-Euclidean geometry. Pfaff was the only German mathematician with whom Gauss could converse, and even then hardly on an equal basis. From 1804 to 1807 Gauss exchanged a few letters on a high mathematical level with Sophie Germain in Paris, and a handful of letters passed between him and the mathematical giants in Paris, but he never visited France or collaborated with them. Gauss remained as isolated in mathematics as he had been since boyhood. By the time mathematicians of stature appeared in Germany (e.g., Jacobi, Plücker, Dirichlet), the uncommunicative habit was too ingrained to change. Gauss inspired Dirichlet, Riemann, and others, but he never had a collaborator, correspondent, or student working closely with him in mathematics.
In other scientific and technical fields things were quite different. There he had students, collaborators, and friends. Over 7,000 letters to and from Gauss are known to be extant, and they undoubtedly represent only a fraction of the total. His most important astronomical collaborators, friends, and correspondents were F. W. Bessel, C. L. Gerling, M. Olbers, J. G. Repsold, H. C. Schumacher. His friendship and correspondence with A. von Humboldt and B. von Lindenau played an important part in his professional life and in the development of science in Germany. These relations were established during the period 1801–1810 and lasted until death. Always Gauss wrote fewer letters, gave more information, and was less cordial than his colleagues, although he often gave practical assistance to his friends and to deserving young scientists.
Also in this decade was established the pattern of working simultaneously on many problems in different fields. Although he never had a second burst of ideas equal to his first, Gauss always had more ideas than he had time to develop. His hopes for leisure were soon dashed by his responsibilities, and he acquired the habit of doing mathematics and other theoretical investigations in the odd hours (sometimes, happily, days) that could be spared. Hence his ideas matured rather slowly, in some cases merely later than they might have with increased leisure, in others more felicitously with increased knowledge and meditation.
This period also saw the fixation of his political and philosophical views. Napoleon seemed to Gauss the personification of the dangers of revolution. The duke of Brunswick, to whom Gauss owed his golden years of freedom, personified the merits of enlightened monarchy. When the duke was humiliated and killed while leading the Prussian armies against Napoleon in 1806, Gauss’s conservative tendencies were reinforced. In the struggles for democracy and national unity in Germany, which continued throughout his lifetime, Gauss remained a staunch nationalist and royalist. (He published in Latin not from internationalist sentiments but at the demands of his publishers. He knew French but refused to publish in it and pretended ignorance when speaking to Frenchmen he did not know.) In seeming contradiction, his religious and philosophical views leaned toward those of his political opponents. He was an uncompromising believer in the priority of empiricism in science. He did not adhere to the views of Kant, Hegel and other idealist philosophers of the day. He was not a churchman and kept his religious views to himself. Moral rectitude and the advancement of scientific knowledge were his avowed principles.
Finally, this decade provided Gauss his one period of personal happiness. In 1805 he married a young woman of similar family background, Johanna Osthoff, who bore him a son and daughter and created around him a cheerful family life. But in 1809 she died soon after bearing a third child, which did not long survive her. Gauss “closed the angel eyes in which for five years I have found a heaven” and was plunged into a loneliness from which he never fully recovered. Less than a year later he married Minna Waldeck, his deceased wife’s best friend. She bore him two sons and a daughter, but she was seldom well or happy. Gauss dominated his daughters and quarreled with his younger sons, who immigrated to the United States. He did not achieve a peaceful home life until the younger daughter, Therese, took over the household after her mother’s death (1831) and became the intimate companion of his last twenty-four years…….
But Gauss continued to find results in the new geometry and was again considering writing them up, possibly to be published after his death, when in 1831 came news of the work of János Bolyai. Gauss wrote to Wolfgang Bolyai endorsing the discovery, but he also asserted his own priority, thereby causing the volatile János to suspect a conspiracy to steal his ideas. When Gauss became familiar with Lobachevsky’s work a decade later, he acted more positively with a letter of praise and by arranging a corresponding membership in the Göttingen Academy. But he stubbornly refused the public support that would have made the new ideas mathematically respectable. Although the friendships of Gauss with Bartels and W. Bolyai suggest the contrary, careful study of the plentiful documentary evidence has established that Gauss did not inspire the two founders of non-Euclidean geometry. Indeed, he played at best a neutral, and on balance a negative, role, since his silence was considered as agreement with the public ridicule and neglect that continued for several decades and were only gradually overcome, partly by the revelation, beginning in the 1860’s, that the prince of mathematicians had been an underground non-Euclidean……
It seems strange to call culturally narrow a man with a solid classical education, wide knowledge, and voracious reading habits. Yet outside of science Gauss did not rise above petit bourgeois banality. Sir Walter Scott was his favorite British author, but he did not care for Byron or Shakespeare. Among German writers he liked Jean Paul, the best-selling humorist of the day, but disliked Goethe and disapproved of Schiller. In music he preferred light songs and in drama, comedies. In short, his genius stopped short at the boundaries of science and technology, outside of which he had little more taste or insight than his neighbors.
The contrast between knowledge and impact is now understandable. Gauss arrived at the two most revolutionary mathematical ideas of the nineteenth century non-Euclidean geometry and noncommutative algebra. The first he disliked and suppressed. The second appears as quaternion calculations in a notebook of about 1819 (Werke, VIII, 357–362) without having stimulated any further activity. Neither the barycentric calculus of his own student Moebius (1827), nor Grassmann’s Ausdenunglehre (1844), nor Hamilton’s work on quaternions (beginning in 1843) interested him, although they sparked a fundamental shift in mathematical thought. He seemed unaware of the outburst of analytic and synthetic projective geometry, in which C. von Staudt, one of his former students, was a leading participant. Apparently Gauss was as hostile or indifferent to radical ideas in mathematics as in politics……..
Today we have a new "Socrates and Plato " in the form of Grinder and Bandler, who take the basic playfulness of the Socratic–Platonic method to its newest form in a Format called Neuro Linguistic Programming. Many age old assumptions are effectively placed in the cupboard of history and a new broom sweeps away the cobwebs and dust is blown off the existential questions. New paradigms are provided and a new pragmatic philosophy of utilitarianism is laid out.
From this basis new solutions flow like liquid gold from the Alchemists bubbling pot, and modern iconography is put in place of ancient and medieval ones. The nature of existential experience is then encoded by these new icons and characterised as representational systems in an overall cybernetic paradigm. The thing is this philosophy works as a model of an infinite number of possible models, and structurally encapsulates the Schelling Kant Grassmann debate.
I can thoroughly recommend Bandler and Grinders philosophical contribution to anyone who wants to really understand there experiential continuum.
The real question is : how did it happen that by the early nineteenth century Kant could place Philosophy in contradistinction to Mathematics, when in ancient Greek society and particularly the Italian Pythagorean school commune it was the role of the Mathematikos to Philosophise?.
The answer is the propagandising effect of the christian church which appropriated philosophy to its own Theosophy and Christology