Riemann blew me away.

It was like the Koheleth says. no matter how wise or clever you think you are, or what amazing things you can do, you die just like any other animate! so proportion things out, and enjoy your life with your wife and your food and drink, for everything is in Elohim('h¶h)'s hand.

Very Stoical , i know, and possibly a sentiment redacted into the text, but a very ancient sentiment nonetheless.

In 1766 Newton was sent home to avoid the plague in cambridge. In the next 2 years he had the most amazing visions and insights. H e was so inundated with them that he could barely sketch them out in a set of notebooks. They were a profound geometry. derived from interacting with nature and the natural dynamic world. When he returned to college in 1768 he was an entirely different person and the world was an entirely different place from then on. He spent the rest of his life developing and vindicating his ideas. Wallis , his mentor , made sure they got the primacy they deserved, even if it meant isolating British mathematicians from the rest of Europe.

Why so drastic an approach? Politics and national promotion. Wallis was no stranger to the ambitions and feneglings of men, and in particular the royalist self opinionated sort.. Wallis was a cromwellian, and thus a puritan. No doubt Newton's religious persuasions were of similar sorts, as he joined the royal society set up by Wallis and similar thinkers. Religious affiliation was crucial at the time for many spies existed for the royalist cause, and that meant that any sentiment that was politically dangerous could bring down factional violence upon your head! But spies also stole ideas!

Wallis believed the Descartes, a Catholic royalist had stolen his ideas from another , and published these ideas without acknowledgement, and he did the scholarship to back up his assertion. Thus he was not going to let po faced Leibniz get away with claiming that he had come up with the notion of the differential calculus!

Gauss as a young man was a prodigious genius, and he too had visions of a geometrical nature , which he too wrote in his notebooks. He spent the rest of his life developing and defending them. However he was in Europe and subject to aqll the European intrigue, infighting and politicking. To maintain his standing and his seat he had to work damn hard and to see off rivals and contenders, but under the patronage of a more powerful organization who would see off any potential duelists! If you moved in social circles you had to be adept at not giving opportunity for someone to pick a fight with you. The causes for the satisfaction of honour were to say the least the slightest provocation. Galois, a promising mathematician is a case in point.

The early 1800's was a very tumultuous time in Europe. To survive was hard enough, let alone maintaining a position. French expansionist aims, ie Napoleon, through all into turmoil. To maintain a positiontherefore meant not only being of political and national use, but also socially machiavellian. Gauss i believe, rose to the occasion.

http://en.wikipedia.org/wiki/Group_theory

Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term “group” and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program famously proclaimed group theory to be the organizing principle of geometry.

Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation group. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems. Thirdly, groups were (first implicitly and later explicitly) used in algebraic number theory.

The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups.

http://en.wikipedia.org/wiki/Ring_theory#History

Commutative ring theory originated in algebraic number theory, algebraic geometry, and invariant theory. Central to the development of these subjects were the rings of integers in algebraic number fields and algebraic function fields, and the rings of polynomials in two or more variables. Noncommutative ring theory began with attempts to extend the complex numbers to various hypercomplex number systems. The genesis of the theories of commutative and noncommutative rings dates back to the early 19th century, while their maturity was achieved only in the third decade of the 20th century.

More precisely, William Rowan Hamilton put forth the quaternions and biquaternions; James Cockle presented tessarines and coquaternions; and William Kingdon Clifford was an enthusiast of split-biquaternions, which he called algebraic motors. These non-commutative algebras, and the non-associative Lie algebras, were studied within universal algebra before the subject was divided into particular mathematical structure types. One sign of re-organization was the use of direct sums to describe algebraic structure.

The various hypercomplex numbers were identified with matrix rings by Joseph Wedderburn (1908) and Emil Artin (1928). Wedderburn's structure theorems were formulated for finite-dimensional algebras over a field while Artin generalized them to Artinian rings.

Many, many bright young geniuses flowered during this period, many were wasted in the war efforts of French expansionist aims, and then Prussian expansionist aims. Many found their industrial genius was valued more than their academic genius, and they sought to make a name and a living in these fields. Gauss therefore was in a precarious position.

He sought to maintain Academia, but was under immense pressure to prove its industrial worth, and to justify the social standing and influence it gave. In particular, he had to justify his own position and salary! The 1830's and 40's seem to have been a crucial 2 decade for Gauss. He had to fend off even the genius of his friends son Bolyai to maintain his own preeminence. He had to publish ideas on complex numbers he was not totally satisfied with. He had to learn Russian to compete with Lobaschewsky, he had to defend his notions against William Rowan Hamilton, and finally, in the 1840's he had to do something about Riemann. On top of that the Prussian government expected him to complete an academically boring land survey. Like Wessel it occasioned many insights, but it required such devotion to detail that he would struggle to keep abreast of the fast changing mathematical landscape.

In 1844 Grassmann published his Ausdehnungslehre. Gauss i believe knew precisely what he held in his hands. He also knew the power he had to make or break this work of genius. But he had little influence on Grassmann, because he was not in Academia. I believe he decided to bury Grassmann under the most obscure pretext: He was too philosophical and not mathematical enough to warrant further attention!

http://en.wikipedia.org/wiki/Riemann

In the meantime, while burying Grassmann, he was encouraging Riemann to explore the exact same ground and to come up with conclusions. In this way he could maintain the primacy over this area of research and get Riemann off his back and under his wing. Grassmann was sacrificed i believe on the altar of Gauss's self preservation.

In 1853 Gauss asked his student Riemann to prepare a Habilitationsschrift on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Göttingen in 1854 entitled Über die Hypothesen welche der Geometrie zu Grunde liegen ("On the hypotheses which underlie geometry"). When it was finally published in 1868, two years after his death, the mathematical public received it with enthusiasm and it is now recognized as one of the most important works in geometry.

Riemann's Treatise title "Über die Hypothesen welche der Geometrie zu Grunde liegen" is full of geometrical punnery! Über and Hypo juxtaposed with Geo and Grunde, plus of course zu liegen! But the point is that Gauss used him as a puppet to put forward his half developed ideas, ideas he knew were more fully developed in Grassmann's Ausdehnungslehre. The squashing of the Ausdehnungslehre is a remarkable bit of social manipulation, adroitly practised by a Prussian social system that still ruled like the Holy roman Empire.

Riemann did acknowledge Grassmann, but like Darwin was given the primacy over an unknown biologist in some jungle research station so Gauss and Riemann took it from Grassmann for their own benefit.

Controversial, i know, but it makes for an interesting read, don't you think?

in his 1862 Audehnungslehre Die Wissenschaft, in the Vorrede Grassmann wrote…

Denn sie liegt nicht in einer willkurlich gewahlten Form, sondern in dem Plane, den ich vor Augen hatte: die Wissenschaft unabhangig von andern Zweigen der Mathematik von Grund aus aufzubauen. Die Ausfuhrung gerade dieses Planes, wenn gleich sie fur die Wissenschaft an sich, die forderndste sein musste, wie sie es denn auch subjektiv gewesen ist, musste bei jeder Form der Darstellung bedeutende Schwierigkeiten bieten, zumal in einer Wissenschaft, wie die Ausdehnungslehre ist, welche die sinnlichen Anschaungen der Geometrie zu allgemeinen, logischen Begriffen erweitert und vergeistigt, und welche an abstrakter Allgemeinheit es nicht nur mit jedem andern Zweige, wie der Algebra, Kombinationslehre, Funktionenlehre, aufnimmt, sondern sie durch Vereinigung aller in diesen Zweigen zu Grunde liegenden Elemente noch weit uberbietet, und so gewissermassen den Schlussstein des gesammten Gebaudes der Mathematik bildet.

Ich musste daher diesen ganzen Plan aufgeben, und habe nun fur das vorliegende Werk, die Ubrigen Zweige der Mathematik, wenigstens in ihrer elementaren Entwicklung vorausgesetzt.

sondern sie durch Vereinigung aller in diesen Zweigen zu Grunde liegenden Elemente noch weit uberbietet

Because it is not in a arbitrarily chosen form, but in the scheme I had in mind: to build "the science" independent of other branches of "mathematics" from the ground up. The execution of just this plan, if it is to be equal for "the science" by itself , the most demanding has to be, as they.. since it has undoubtedly also been subjective, must have for each form of representation provided significant difficulties, especially in a science, like the Expansion Theory is, that "extends" (and spiritualizes) the sensory Immanent Manifestations of Geometry to general, logical terms , and that "pulls up-close" right onto abstract generalities, not only along with each of the other sectors, such as algebra, combination theory, function theory, but also through a process of associating all of these branches to underlying elements, it still offers far more, so to some extent formin the keystone of the whole building of mathematics.

I therefore had to give up this whole plan, and now I have provided for the present work, the remaining branches of mathematics, at least in its basic development.

After 15 years of no help, Grassmann had to abandon his dream for a natural Philosophical Science and concentrate on the Mathematical aspect of his plan, which he could only sketch out. But he echoes Riemann's phrasing, because it was his phrasing in 1844,ten years before Riemann's speech.

Gauss really would have struggled to follow Grassmann because his explanation was aimed at creating a science, a natural philosophy that encompassed all of mathematics in a geometry derived from a philosophical spiritual apprehension. However the mathematical part was clear enough. Gauss was not sure but savvy enough to recognise that it was important. He commisioned Riemann to explore the idea from a mathematical or rather geometrical perspective. Riemann uplifted the geometry into the physical sciences even as Grassmann precipitates the geometry out tof the philosophical, metaphysical basis of science: they both came at the issue from different sides so to speak.

After Riemann examined the issue Grassmann seems to capitulate to his critics,claiming that his approach and method was too difficult, even for him. He completely rewrites and extends the Ausdehnungslehre using a new method, inspired by Riemann's approach and his own successful Arithmetic treatise, which he publishes before Riemann. This is a desperate attempt to salvage his insights from the mawl of history and to present his original thinking in a form that marks out the breadth of his thinking, and tha applicability of his insight. Had he not done so Gauss would have won, through Riemann. As it is Grassmann was able to preserve that bit of his contribution to the subject that really makes a difference.

This is not to say that his philosophical approach to science was not important, but rather to demonstrate its importance by its applications. Without fully knowing his philosophy it is hard to say, but his philosophy would have been swamped by the giants Leibniz, Descartes and Spinoza.