Geometry

JohnEB

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Dec 12, 2011, 12:42:59 AM12/12/11

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It is absolutely amazing that most real physics (non-QM) is geometric. Einstein's Special Relativity is hyperbolic geomitry and Einstein's Gerneral Relativity is eliptlic geomitry.

Dr. Mills' theory as exemplified by Vol 2 of the GUT-CP and the Millsian software is largely geometric.

The following is from the book Is God Mathematician by Mario Livio:

Chapter 6

Geometers: Future Shock

In his famous book Future Shock, author Alvin Toffler defined the term in the title as "the shattering stress and disorientation that we induce in individuals by subjecting them to too much change in too short a time." In the nineteenth century, mathematicians, scientists, and philosophers experienced precisely such a shock. In fact, the millennia-old belief that mathematics offers eternal and immutable truths was crushed. This unexpected intellectual upheaval was caused by the emergence of new types of geometries, now known as non-Euclidean geometries. Even though most nonspecialists may have never even heard of non-Euclidean geometries, the magnitude of the revolution in thought introduced by these new branches of mathematics has been likened by some to that inaugurated by the Darwinian theory of evolution.

To fully appreciate the nature of this sweeping change in worldview, we have first to briefly probe the historical-mathematical backdrop.

Euclidean "Truth"

Until the beginning of the nineteenth century, if there was one branch of knowledge that had been regarded as the apotheosis of truth and certainty, it was Euclidean geometry, the traditional geometry we learn in school. Not surprisingly, therefore, the great Dutch Jewish philosopher Baruch Spinoza (1632-77) entitled his bold attempt to unify science, religion, ethics, and reason Ethics, Demonstrated in Geometrical Order. Moreover, in spite of the clear distinction between the ideal, Platonic world of mathematical forms and physical reality, most scientists regarded the objects of Euclidean geometry simply as the distilled abstractions of their real, physical counterparts. Even staunch empiricists such as David Hume (1711-76), who insisted that the very foundations of science were far less certain than anyone had ever suspected, concluded that Euclidean geometry was as solid as the Rock of Gibraltar. In An Enquiry Concerning Human Understanding, Hume identified "truths" of two types:

All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact. Of the first kind are ... every affirmation which is either intuitively or demonstratively certain ... Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would forever retain their certainty and evidence. Matters of fact are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction ... That the sun will not rise tomorrow is no less intelligible a proposition, and implies no more contradiction than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood.

In other words, while Hume, like all empiricists, maintained that all knowledge stems from observation, geometry and its "truths" continued to enjoy a privileged status.

The preeminent German philosopher Immanuel Kant (1724-1804) did not always agree with Hume, but he also exalted Euclidean geometry to a status of absolute certainty and unquestionable validity. In his memorable Critique of Pure Reason, Kant attempted to reverse in some sense the relationship between the mind and the physical world. Instead of impressions of physical reality being imprinted on an otherwise entirely passive mind, Kant gave the mind the active function of "constructing" or "processing" the perceived universe. Turning his attention inward, Kant asked not what we can know, but how we can know what we can know. He explained that while our eyes detect particles of light, these do not form an image in our awareness until the information is processed and organized by our brains. A key role in this construction process was assigned to the human intuitive or synthetic a priori grasp of space, which in turn was taken to be based on Euclidean geometry. Kant believed that Euclidean geometry provided the only true path for processing and conceptualizing space, and that this intuitive, universal acquaintance with space was at the heart of our experience of the natural world. In Kant's words:

Space is not an empirical concept which has been derived from external experience ... Space is a necessary representation a priori, forming the very foundation of all external intuitions ... On this necessity of an a priori representation of space rests the apodictic certainty of all geometrical principles, and the possibility of their construction a priori. For if the intuition of space were a concept gained a posteriori, borrowed from general external experience, the first principles of mathematical definition would be nothing but perceptions. They would be exposed to all the accidents of perception, and there being but one straight line between two points would not be a necessity, but only something taught in each case by experience.

To put it simply, according to Kant, if we perceive an object, then necessarily this object is spatial and Euclidean.

Hume's and Kant's ideas bring to the forefront the two rather different, but equally important aspects that had been historically associated with Euclidean geometry. The first was the statement that Euclidean geometry represents the only accurate description of physical space. The second was the identification of Euclidean geometry with a firm, decisive, and infallible deductive structure. Taken together, these two presumed properties provided mathematicians, scientists, and philosophers with what they regarded as the strongest evidence that informative, inescapable truths about the universe do exist. Until the nineteenth century these statements were taken for granted. But were they actually true?

The foundations of Euclidean geometry were laid around 300 BC by the Greek mathematician Euclid of Alexandria. In a monumental thirteen-volume opus entitled The Elements, Euclid attempted to erect geometry on a well-defined logical base. He started with ten axioms assumed to be indisputably true and sought to prove a large number of propositions on the basis of those postulates by nothing other than logical deductions.

The first four Euclidean axioms were extremely simple and exquisitely concise. For instance, the first axiom read: "Between any two points a straight line may be drawn." The fourth one stated: "All right angles are equal." By contrast, the fifth axiom, known as the "parallel postulate," was more complicated in its formulation and considerably less self-evident: "If two lines lying in a plane intersect a third line in such a way that the sum of the internal angles on one side is less than the two right angles, then the two lines inevitably will intersect each other if extended sufficiently on that side." Figure 39 demonstrates graphically the contents of this axiom.

While no one doubted the truth of this statement, it lacked the compelling simplicity of the other axioms. All indications are that even Euclid himself was not entirely happy with his fifth postulate — the proofs of the first twenty-eight propositions in The Elements do not make use of it. The equivalent version of the "fifth" most cited today appeared first in commentaries by the Greek mathematician Proclus in the fifth century, but it is generally known as the "Playfair axiom," after the Scottish mathematician John Playfair (1748-1819). It states: "Given a line and a point not on the line, it is possible to draw exactly one line parallel to the given line through that point" (see figure 40). The two versions of the axiom are equivalent in the sense that Playfair's axiom (together with the other axioms) necessarily implies Euclid's original fifth axiom and vice versa.

Over the centuries, the increasing discontent with the fifth axiom resulted in a number of unsuccessful attempts to actually prove it from the other nine axioms or to replace it by a more obvious postulate. When those efforts failed, other geometers began trying to answer an intriguing "what if" question—what if the fifth axiom did, in fact, not prove true? Some of those endeavors started to raise nagging doubts on whether Euclid's axioms were truly self-evident, rather than being based on experience. The final, surprising verdict eventually came in the nineteenth century: One could create new kinds of geometry by choosing an axiom different from Euclid's fifth. Furthermore, these "non-Euclidean" geometries could in principle describe physical space just as accurately as Euclidean geometry did!

Let me pause here for a moment to allow for the meaning of the word "choosing" to sink in. For millennia, Euclidean geometry had been regarded as unique and inevitable—the sole true description of space. The fact that one could choose the axioms and obtain an equally valid description turned the entire concept on its ear. The certain, carefully constructed deductive scheme suddenly became more similar to a game, in which the axioms simply played the role of the rules. You could change the axioms and play a different game. The impact of this realization on the understanding of the nature of mathematics cannot be overemphasized.

Quite a few creative mathematicians prepared the ground for the final assault on Euclidean geometry. Particularly notable among them were the Jesuit priest Girolamo Saccheri (1667-1733), who investigated the consequences of replacing the fifth postulate by a different statement, and the German mathematicians Georg Klugel (17391812) and Johann Heinrich Lambert (1728-1777), who were the first to realize that alternative geometries to the Euclidean could exist. Still, somebody had to put the last nail in the coffin of the idea of Euclidean geometry being the one and only representation of space. That honor was shared by three mathematicians, one from Russia, one from Hungary, and one from Germany.

Strange New Worlds

The first to publish an entire treatise on a new type of geometry—one that could be constructed on a surface shaped like a curved saddle (figure 41a)—was the Russian Nikolai Ivanovich Lobachevsky (1792-1856; figure 42).

In this kind of geometry (now known as hyperbolic geometry), Euclid's fifth postulate is replaced by the statement that given a line in a plane and a point not on this line, there are at least two lines through the point parallel to the given line. Another important difference between Lobachevskian geometry and Euclidean geometry is that while in the latter the angles in a triangle always add up to 180 degrees (figure 41b), in the former the sum is always less than 180 degrees. Because Lobachevsky's work appeared in the rather obscure Kazan Bulletin, it went almost entirely unnoticed until French and German translations started to appear in the late 1830s. Unaware of Lobachevsky's work, a young Hungarian mathematician, Janos Bolyai (1802-60), formulated a similar geometry during the 1820s. Bursting with youthful enthusiasm, he wrote in 1823 to his father (the mathematician Farkas Bolyai; figure 43): "I have found things so magnificent that I was astounded ... I have created a different new world out of nothing."

By 1825, Janos was already able to present to the elder Bolyai the first draft of his new geometry. The manuscript was entitled The Science Absolute of Space. In spite of the young man's exuberance, the father was not entirely convinced of the soundness of Janos's ideas. Nevertheless, he decided to publish the new geometry as an appendix to his own two-volume treatise on the foundations of geometry, algebra, and analysis (the supposedly inviting title of which read Essay on the Elements of Mathematics for Studious Youths). A copy of the book was sent in June 1831 to Farkas's friend Carl Friedrich Gauss (1777-1855; figure 44), who was not only the most prominent mathematician of the time, but who is also considered by many, along with Archimedes and Newton, to be one of the three greatest of all time.

That book was somehow lost in the chaos created by a cholera epidemic, and Farkas had to send a second copy. Gauss sent out a reply on March 6, 1832, and his comments were not exactly what the young Janos expected:

If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it, would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years. So I remained quite stupefied. So far as my own work is concerned, of which up till now I have put little on paper, my intention was not to let it be published during my lifetime.

Let me parenthetically note that apparently Gauss feared that the radically new geometry would be regarded by the Kantian philosophers, to whom he referred as "the Boetians" (synonymous with "stupid" for the ancient Greeks), as philosophical heresy. Gauss then continued:

On the other hand it was my idea to write down all this later so that at least it should not perish with me. It is therefore a pleasant surprise with me that I am spared this trouble, and I am very glad that it is the son of my old friend, who takes the precedence of me in such a remarkable manner.

While Farkas was quite pleased with Gauss's praise, which he took to be "very fine," Janos was absolutely devastated. For almost a decade he refused to believe that Gauss's claim to priority was not false, and his relationship with his father (whom he suspected of prematurely communicating the results to Gauss) was seriously strained. When he finally realized that Gauss had actually started working on the problem as early as 1799, Janos became deeply embittered, and his subsequent mathematical work (he left some twenty thousand pages of manuscript when he died) became rather lackluster by comparison.

There is very little doubt, however, that Gauss had indeed given considerable thought to non-Euclidean geometry. In a diary entry from September 1799 he wrote: "In principiis geometriae egregios progressus fecimus" ("About the principles of geometry we obtained wonderful achievements"). Then, in 1813, he noted: "In the theory of parallel lines we are now no further than Euclid was. This is the partie honteuse [shameful part] of mathematics, which sooner or later must get a very different form." A few years later, in a letter written on April 28, 1817, he stated: "I am coming more and more to the conviction that the necessity of our [Euclidean] geometry cannot be proved." Finally, and contrary to Kant's views, Gauss concluded that Euclidean geometry could not be viewed as a universal truth, and that rather "one would have to rank [Euclidean] geometry not with arithmetic, which stands a priori, but approximately with mechanics." Similar conclusions were reached independently by Ferdinand Schweikart (1780-1859), a professor of jurisprudence, and the latter informed Gauss of his work sometime in 1818 or 1819. Since neither Gauss nor Schweikart actually published his results, however, the priority of first publication is traditionally credited to Lobachevsky and Bolyai, even though the two can hardly be regarded as the sole "creators" of non-Euclidean geometry.

Hyperbolic geometry broke on the world of mathematics like a thunderbolt, dealing a tremendous blow to the perception of Euclidean geometry as the only, infallible description of space. Prior to the Gauss-Lobachevsky-Bolyai work, Euclidean geometry was, in effect, the natural world. The fact that one could select a different set of axioms and construct a different type of geometry raised for the first time the suspicion that mathematics is, after all, a human invention, rather than a discovery of truths that exist independently of the human mind. At the same time, the collapse of the immediate connection between Euclidean geometry and true physical space exposed what appeared to be fatal deficiencies in the idea of mathematics as the language of the universe.

Euclidean geometry's privileged status went from bad to worse when one of Gauss's students, Bernhard Riemann, showed that hyperbolic geometry was not the only non-Euclidean geometry possible. In a brilliant lecture delivered in Gottingen on June 10, 1854 (figure 45 shows the first page of the published lecture), Riemann presented his views "On the Hypotheses That Lie at the Foundations of Geometry." He started by saying that "geometry presupposes the concept of space, as well as assuming the basic principles for constructions in space. It gives only nominal definitions of these things, while their essential specifications appear in the form of axioms." However, he noted, "The relationship between these presuppositions is left in the dark; we do not see whether, or to what extent, any connection between them is necessary, or a priori whether any connection between them is even possible." Among the possible geometrical theories Riemann discussed elliptic geometry, of the type that one would encounter on the surface of a sphere (figure 41c). Note that in such a geometry the shortest distance between two points is not a straight line, it is rather a segment of a great circle, whose center coincides with the center of the sphere. Airlines take advantage of this fact— flights from the United States to Europe do not follow what would appear as a straight line on the map, but rather a great circle that initially bears northward. You can easily check that any two great circles meet at two diametrically opposite points. For instance, two meridians on Earth, which appear to be parallel at the Equator, meet at the two poles. Consequently, unlike in Euclidean geometry, where there is exactly one parallel line through an external point, and hyperbolic geometry, in which there are at least two parallels, there are no parallel lines at all in the elliptic geometry on a sphere. Riemann took the non-Euclidean concepts one step further and introduced geometries in curved spaces in three, four, and even more dimensions. One of the key concepts expanded upon by Riemann was that of the curvature —the rate at which a curve or a surface curves. For instance, the surface of an eggshell curves more gently around its girth than along a curve passing through one of its pointy edges. Riemann proceeded to give a precise mathematical definition of curvature in spaces of any number of dimensions. In doing so he solidified the marriage between algebra and geometry that had been initiated by Descartes. In Riemann's work equations in any number of variables found their geometrical counterparts, and new concepts from the advanced geometries became partners of equations.

Euclidean geometry's eminence was not the only victim of the new horizons that the nineteenth century opened for geometry. Kant's ideas of space did not survive much longer. Recall that Kant asserted that information from our senses is organized exclusively along Euclidean templates before it is recorded in our consciousness. Geometers of the nineteenth century quickly developed intuition in the non-Euclidean geometries and learned to experience the world along those lines. The Euclidean perception of space turned out to be learned after all, rather than intuitive. All of these dramatic developments led the great French mathematician Henri Poincare (1854-1912) to conclude that the axioms of geometry are "neither synthetic a priori intuitions nor experimental facts. They are conventions [emphasis added]. Our choice among all possible conventions is guided by experimental facts, but it remains free." In other words, Poincare regarded the axioms only as "definitions in disguise."

Poincare's views were inspired not just by the non-Euclidean geometries described so far, but also by the proliferation of other new geometries, which before the end of the nineteenth century seemed to be almost getting out of hand. In projective geometry (such as that obtained when an image on celluloid film is projected onto a screen), for instance, one could literally interchange the roles of points and lines, so that theorems about points and lines (in this order) became theorems about lines and points. In differential geometry, mathematicians used calculus to study the local geometrical properties of various mathematical spaces, such as the surface of a sphere or a torus. These and other geometries appeared, at first blush at least, to be ingenious inventions of imaginative mathematical minds, rather than accurate descriptions of physical space. How then could one still defend the concept of God as a mathematician? After all, if "God ever geometrizes" (a phrase attributed to Plato by the historian Plutarch), which of these many geometries does the divine practice?

The rapidly deepening recognition of the shortcomings of the classical Euclidean geometry forced mathematicians to take a serious look at the foundations of mathematics in general, and at the relationship between mathematics and logic in particular. We shall return to this important topic in chapter 7. Here let me only note that the very notion of the self -evidency of axioms had been shattered. Consequently, while the nineteenth century witnessed other significant developments in algebra and in analysis, the revolution in geometry probably had the most influential effects on the views of the nature of mathematics.

On Space, Numbers, and Humans

Before mathematicians could turn to the overarching topic of the foundations of mathematics, however, a few "smaller" issues required immediate attention. First, the fact that non-Euclidean geometries had been formulated and published did not necessarily mean that these were legitimate offspring of mathematics. There was the ever-present fear of inconsistency—the possibility that carrying these geometries to their ultimate logical consequences would produce unresolvable contradictions. By the 1870s, the Italian Eugenio Beltrami (1835-1900) and the German Felix Klein (1849-1925) had demonstrated that as long as Euclidean geometry was consistent, so were non-Euclidean geometries. This still left open the bigger question of the solidity of the foundations of Euclidean geometry. Then there was the important matter of relevance. Most mathematicians regarded the new geometries as amusing curiosities at best. Whereas Euclidean geometry derived much of its historical power from being seen as the description of real space, the non-Euclidean geometries had been perceived initially as not having any connection whatsoever to physical reality. Consequently, the non-Euclidean geometries were treated by many mathematicians as Euclidean geometry's poor cousins. Henri Poincare was a bit more accommodating than most, but even he insisted that if humans were to be transported to a world in which the accepted geometry was non-Euclidean, then it was still "certain that we should not find it more convenient to make a change" from Euclidean to non-Euclidean geometry. Two questions therefore loomed large: (1) Could geometry (in particular) and other branches of mathematics (in general) be established on solid axiomatic logical foundations? and (2) What was the relationship, if any, between mathematics and the physical world?

Some mathematicians adopted a pragmatic approach with respect to the validation of the foundations of geometry. Disappointed by the realization that what they regarded as absolute truths turned out to be more experience-based than rigorous, they turned to arithmetic —the mathematics of numbers. Descartes' analytic geometry, in which points in the plane were identified with ordered pairs of numbers, circles with pairs satisfying a certain equation (see chapter 4), and so on, provided just the necessary tools for the re-erection of the foundations of geometry on the basis of numbers. The German mathematician Jacob Jacobi (1804-51) presumably expressed those shifting tides when he replaced Plato's "God ever geometrizes" by his own motto: "God ever arithmetizes." In some sense, however, these efforts only transported the problem to a different branch of mathematics. While the great German mathematician David Hilbert (1862-1943) did succeed in demonstrating that Euclidean geometry was consistent as long as arithmetic was consistent, the consistency of the latter was far from unambiguously established at that point.

On the relationship between mathematics and the physical world, a new sentiment was in the air. For many centuries, the interpretation of mathematics as a reading of the cosmos had been dramatically and continuously enhanced. The mathematization of the sciences by Galileo, Descartes, Newton, the Bernoullis, Pascal, Lagrange, Quetelet, and others was taken as strong evidence for an underlying mathematical design in nature. One could clearly argue that if mathematics wasn't the language of the cosmos, why did it work as well as it did in explaining things ranging from the basic laws of nature to human characteristics?

To be sure, mathematicians did realize that mathematics dealt only with rather abstract Platonic forms, but those were regarded as reasonable idealizations of the actual physical elements. In fact, the feeling that the book of nature was written in the language of mathematics was so deeply rooted that many mathematicians absolutely refused even to consider mathematical concepts and structures that were not directly related to the physical world. This was the case, for instance, with the colorful Gerolamo Cardano (1501-76). Cardano was an accomplished mathematician, renowned physician, and compulsive gambler. In 1545 he published one of the most influential books in the history of algebra—the Ars Magna (The Great Art). In this comprehensive treatise Cardano explored in great detail solutions to algebraic equations, from the simple quadratic equation (in which the unknown appears to the second power: x^2) to pioneering solutions to the cubic (involving x^3), and quartic (involving x^4) equations. In classical mathematics, however, quantities were often interpreted as geometrical elements. For instance, the value of the unknown x was identified with a line segment of that length, the second power x^2 was an area, and the third power x^3 was a solid having the corresponding volume. Consequently, in the first chapter of the Ars Magna, Cardano explains:

We conclude our detailed consideration with the cubic, others being merely mentioned, even if generally, in passing. For as positio [the first power] refers to a line, quadratum [the square] to a surface, and cubum [the cube] to a solid body, it would be very foolish for us to go beyond this point. Nature does not permit it. Thus, it will be seen, all those matters up to and including the cubic are fully demonstrated, but the others which we will add, either by necessity or out of curiosity, we do not go beyond barely setting out.

In other words, Cardano argues that since the physical world as perceived by our senses contains only three dimensions, it would be silly for mathematicians to concern themselves with a higher number of dimensions, or with equations of a higher degree.

A similar opinion was expressed by the English mathematician John Wallis (1616-1703), from whose work Arithmetica In finitorum Newton learned methods of analysis. In another important book, Treatise of Algebra, Wallis first proclaimed: "Nature, in propriety of Speech, doth not admit more than three (local) dimensions." He then elaborated:

A Line drawn into a Line, shall make a Plane or Surface; this drawn into a Line, shall make a Solid. But if this Solid be drawn into a Line, or this Plane into a Plane, what shall it make? A Plano-Plane? This is a Monster in Nature, and less possible than a Chimera [a fire-breathing monster in Greek mythology, composed of a serpent, lion, and goat] or a Centaure [in Greek mythology, a being having the upper portion of a man and the body and legs of a horse]. For Length, Breadth and Thickness, take up the whole of Space. Nor can our Fansie imagine how there should be a Fourth Local Dimension beyond these Three.

Again, Wallis's logic here was clear: There was no point in even imagining a geometry that did not describe real space.

Opinions eventually started to change. Mathematicians of the eighteenth century were the first to consider time as a potential fourth dimension. In an article entitled "Dimension," published in 1754, the physicist Jean D'Alembert (1717-83) wrote:

I stated above that it is impossible to conceive of more than three dimensions. A man of parts, of my acquaintance, holds that one may however look upon duration as a fourth dimension, and that the product of time and solidity is in a way a product of four dimensions. This idea may be challenged but it seems to me to have some merit other than that of mere novelty.

The great mathematician Joseph Lagrange went even one step further, stating more assertively in 1797:

Since a position of a point in space depends upon three rectangular coordinates these coordinates in the problems of mechanics are conceived as being functions of t [time]. Thus we may regard mechanics as a geometry of four dimensions, and mechanical analysis as an extension of geometrical analysis.

These bold ideas opened the door for extensions of mathematics that had previously been considered inconceivable geometries in any number of dimensions—which totally ignored the question of whether they had any relation to physical space.

Kant may have been wrong in believing that our senses of spatial perception follow exclusively Euclidean molds, but there is no question that our perception operates most naturally and intuitively in no more than three dimensions. We can relatively easily imagine how our three-dimensional world would look in Plato's two-dimensional universe of shadows, but going beyond three to a higher number of dimensions truly requires a mathematician's imagination.

Some of the groundbreaking work in the treatment of n-dimensional geometry—geometry in an arbitrary number of dimensions—was carried out by Hermann Gunther Grassmann (1809-77). Grassmann, one of twelve children, and himself the father of eleven, was a schoolteacher who never had any university mathematical training. During his lifetime, he received more recognition for his work in linguistics (in particular for his studies of Sanskrit and Gothic) than for his achievements in mathematics. One of his biographers wrote: "It seems to be Grassmann's fate to be rediscovered from time to time, each time as if he had been virtually forgotten since his death." Yet, Grassmann was responsible for the creation of an abstract science of "spaces," inside which the usual geometry was only a special case. Grassmann published his pioneering ideas (originating a branch of mathematics known as linear algebra) in 1844, in a book commonly known as the Ausdehnungslehre (meaning Theory of Extension; the full title read: Linear Extension Theory: A New Branch of Mathematics).

In the foreword to the book Grassmann wrote: "Geometry can in no way be viewed ... as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I also had realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to those of geometry."

This was a radically new view of the nature of mathematics. To Grassmann, the traditional geometry — the heritage of the ancient Greeks — deals with physical space and therefore cannot be taken as a true branch of abstract mathematics. Mathematics to him was rather an abstract construct of the human brain that does not necessarily have any application to the real world.

It is fascinating to follow the seemingly trivial train of thought that set Grassmann on the road to his theory of geometric algebra. He started with the simple formula AB + BC = AC, which appears in any geometry book in the discussion of lengths of line segments (see figure 46a). Here, however, Grassmann noticed something interesting. He discovered that this formula remains valid irrespective of the order of the points A, B, C as long as one does not interpret AB, BC, and so on merely as lengths, but also assigns to them "direction," such that BA = —AB. For instance, if C lies between A and B (as in Figure 46b), then AB = AC + CB, but since CB = —BC, we find that AB = AC — BC and the original formula AB + BC = AC is recovered simply by adding BC to both sides.

This was quite interesting in itself, but Grassmann's extension contained even more surprises. Note that if we were dealing with algebra instead of geometry, then an expression such as AB usually would denote the product A x B. In that case, Grassmann's suggestion of BA = —AB violates one of the sacrosanct laws of arithmetic—that two quantities multiplied together produce the same result irrespective of the order in which the quantities are taken. Grassmann faced up squarely to this disturbing possibility and invented a new consistent algebra (known as exterior algebra) that allowed for several processes of multiplication and at the same time could handle geometry in any number of dimensions.

By the 1860s n-dimensional geometry was spreading like mushrooms after a rainstorm. Not only had Riemann's seminal lecture established spaces of any curvature and of arbitrary numbers of dimensions as a fundamental area of research, but other mathematicians, such as Arthur Cayley and James Sylvester in England, and Ludwig Schlafli in Switzerland, were adding their own original contributions to the field. Mathematicians started to feel that they were being freed from the restrictions that for centuries had tied mathematics only to the concepts of space and number. Those ties had historically been taken so seriously that even as late as the eighteenth century, the prolific Swiss mathematician Leonhard Euler (1707-83) expressed his view that "mathematics, in general, is the science of quantity; or, the science that investigates the means of measuring quantity." It was only in the nineteenth century that the winds of change started to blow.

First, the introduction of abstract geometric spaces and of the notion of infinity (in both geometry and the theory of sets) had blurred the meaning of "quantity" and of "measurement" beyond recognition. Second, the rapidly multiplying studies of mathematical abstractions helped to distance mathematics even further from physical reality, while breathing life and "existence" into the abstractions themselves.

Georg Cantor (1845-1918), the creator of set theory, characterized the newly found spirit of freedom of mathematics by the following "declaration of independence"; "Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established." To which algebraist Richard Dedekind (1831-1916) added six years later: "I consider the number concept entirely independent of the notions or intuitions of space and time ... Numbers are free creations of the human mind." That is, both Cantor and Dedekind viewed mathematics as an abstract, conceptual investigation, constrained only by the requirement of consistency, with no obligations whatsoever toward either calculation or the language of physical reality. As Cantor has summarized it: "The essence of mathematics lies entirely in its freedom."

By the end of the nineteenth century most mathematicians accepted Cantor's and Dedekind's views on the freedom of mathematics. The objective of mathematics changed from being the search for truths about nature to the construction of abstract structures —systems of axioms —and the pursuit of all the logical consequences of those axioms.

One might have thought that this would put an end to all the agonizing over the question of whether mathematics was discovered or invented. If mathematics was nothing more than a game, albeit a complex one, played with arbitrarily invented rules, then clearly there was no point in believing in the reality of mathematical concepts, was there?

Surprisingly, the breaking away from physical reality infused some mathematicians with precisely the opposite sentiment. Rather than concluding that mathematics was a human invention, they returned to the original Platonic notion of mathematics as an independent world of truths, whose existence was as real as that of the physical universe. The attempts to connect mathematics with physics were treated by these "neo-Platonists" as dabbling in applied mathematics, as opposed to the pure mathematics that was supposed to be indifferent to anything physical. Here is how the French mathematician Charles Hermite (1822-1901) put it in a letter written to the Dutch mathematician Thomas Joannes Stieltjes (1856-94) on May 13,1894: "My dear friend," he wrote, I feel very happy to find you inclined to transform yourself into a naturalist to observe the phenomena of the arithmetical world. Your doctrine is the same as mine; I believe that numbers and the functions of analysis are not arbitrary products of our mind; I think that they exist outside of us with the same necessary characteristics as the things of objective reality, and that we encounter them or discover them, and study them, just as the physicists, the chemists and the zoologists.

The English mathematician G. H. Hardy, himself a practitioner of pure mathematics, was one of the most outspoken modern Platonists. In an eloquent address to the British Association for the Advancement of Science on September 7, 1922, he pronounced:

Mathematicians have constructed a very large number of different systems of geometry. Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians' observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics ... The function of a mathematician, then, is simply to observe the facts about his own hard and intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics.

Clearly, even with the contemporary evidence pointing to the arbitrary nature of mathematics, the die-hard Platonists were not about to lay down their arms. Quite the contrary, they found the opportunity to delve into, in Hardy's words, "their reality," even more exciting than to continue to explore the ties to physical reality. Irrespective, however, of the opinions on the metaphysical reality of mathematics, one thing was becoming obvious. Even with the seemingly unbridled freedom of mathematics, one constraint remained unchanging and unshakable—that of logical consistency. Mathematicians and philosophers were becoming more aware than ever that the umbilical cord between mathematics and logic could not be cut. This gave birth to another idea: Could all of mathematics be built on a single logical foundation? And if it could, was that the secret of its effectiveness? Or conversely, could mathematical methods be used in the study of reasoning in general? In which case, mathematics would become not just the language of nature, but also the language of human thought.

JohnEB

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Dec 18, 2011, 1:33:58 PM12/18/11

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"The value of Euclid's work as a masterpiece of logic has been very grossly exaggerated." -- Bertrand Russell

Euclid's Elements stood for about two millenia as the primary example of of "truth" and "logical thought."
http://en.wikipedia.org/wiki/Euclid's_Elements

But Euclid's fifth postulate (denoted by Euclid V) was controversial from the start. Many tried to prove Euclid V, but did not succeed.
Legendre published 20 different proofs, all of them flawed.
http://en.wikipedia.org/wiki/Parallel_postulate

The problems with Euclid V led to the development of non-Euclidian geometries which are the foundation of Einstein's relativity.

JohnEB

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Dec 20, 2011, 8:47:09 AM12/20/11

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Non-Euclidian geometries began with hyperbolic geometry:

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

JohnEB

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Dec 21, 2011, 8:23:25 AM12/21/11

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David Hilbert attempted a modern treatment of Euclidean geometry:

Hilbert's axioms are a set of 20 (originally 21) assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry), as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.

http://en.wikipedia.org/wiki/Hilbert's_axioms

JohnEB

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Dec 23, 2011, 7:12:14 AM12/23/11

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Projective Geometry, where science and art meet:

http://en.wikipedia.org/wiki/Projective_geometry
http://www.geometer.org/mathcircles/projective.pdf

JohnEB

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Dec 28, 2011, 1:27:35 AM12/28/11

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Graphs are one of the prime objects of study in discrete mathematics. Refer to glossary of graph theory for basic definitions in graph theory.

Topology unifies geometry with discrete mathematics:

Graph Theory

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

JohnEB

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Dec 28, 2011, 4:02:09 PM12/28/11

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János Bolyai

János Bolyai (pronounced [ˈjaː.noʃ ˈboː.jɒ.i]) (December 15, 1802 – January 27, 1860) was a Hungarian mathematician, known for his work in non-Euclidean geometry.

Bolyai was born in the Transylvanian town of Kolozsvár (Klausenburg), then part of the Habsburg Empire (now Cluj-Napoca in Romania), the son of Zsuzsanna Benkő and the well-known mathematician Farkas Bolyai.

http://en.wikipedia.org/wiki/J%C3%A1nos_Bolyai