Mathematics

"One of the great mistakes in the teaching of algebra is to present it as if it were a subject unrelated to arithmetic. ...Many schoolchildren... arrive at the conclusion that algebra and arithmetic are different subjects. ...the history of mathematics teaches us that even cubic and quartic equations were successfully solved without the benefit of modern notation. ...The use of words instead of symbols can illustrate how close the formulations are to the original... problems."

"The contemporary decline in interest in geometry and its gradual disappearance from school curricula... should be deplored... Geometry is the most visual of the mathematical disciplines. It is not in principle divorced from numbers, and hence neither is it divorced from algebra. Many a pupil's understanding of algebraic proofs would be considerably reinforced by... visual geometrical proofs which were the hallmark of Greek mathematics and to some extent of Arab mathematics also. ...where a geometrical proof is clear and immediate, as... with... many algebraic identities such as (a \pm b)^2 = a^2 \pm 2ab + b^2\!, the geometry should not be forgotten. The Greeks were some of the greatest teachers of all time... [and] geometric algebra was in many ways [their] greatest achievement ..."

"Reflect on how many real problems of life require solutions which are meaningless unless they are whole numbers. The absence of Diophantine analysis from school curricula is difficult to justify. The methods... are not beyond the capabilities of schoolchildren."

"One makes sense of , whether fictional or factual, by a mental construction that is sometimes called the world of story ...[T]he imaginative effort is a standard way of understanding what people say... In order to understand connected speech about concrete things, one imagines them. ...The capacity to do this... encourages empathy, but it also allows one to do mathematics. ...This is often fun, and it is a form of playing with ideas. ...This ludic aspect of mathematics is emphasized by in his semiotic analysis of mathematics and acknowledged by David Wells in his comparison of mathematics and games. ...The ludic aspect is something that undergraduates, many of whom have decided that mathematics is either a guessing game... or the execution of rigidly defined procedures, need to be encouraged to do when they are learning new ideas. They need to fool around with them to become familiar... Changing the s and seeing what a function looks like... to learn how the function behaves. ...Mathematical research involves a good deal of fooling around, which is part of why it is a pleasurable activity... This is not competitive..."

"The usual courses in mathematics are... deceptive in a basic respect. They give an organized logical presentation which leaves the impression that mathematicians go from theorem to theorem almost naturally, that mathematicians can master almost any difficulty, and that subjects are completely thrashed out and settled. The succession of theorems overwhelms the student... The history, by contrast, teaches us the development of a subject is made bit by bit with results coming from various directions. We learn, too, that often decades and even hundreds of years of effort were required before significant steps could be made. In place of the impression that subjects are completely thrashed out one finds that what is attained is often but a start, that many gaps have to be filled, or that the really important extensions remain to be created."

"In mathematics the art of asking questions is more valuable than solving problems."

"The author holds that our school curricula, by stripping mathematics of its cultural content and leaving a bare skeleton of technicalities, have repelled many a fine mind. It is the aim of this book to restore this cultural content and present the evolution of number as the profoundly human story which it is. ...the historical method has been freely used to bring out the rôle intuition has played in the evolution of mathematical concepts. And so the story of number is here unfolded as a historical pageant of ideas, linked with the men who created those ideas and with the epochs which produced the men."

"The systematic exposition of a textbook in mathematics is based on logical continuity and not on historical sequence; but the standard high school course in mathematics fails to mention this fact, and therefore leaves the student under the impression that the historical evolution of number proceeded in the order in which the chapters of the textbook were written. This impression is largely responsible for the widespread opinion that mathematics has no human element. For here, it seems, is a structure that was erected without a scaffold: it simply rose in its frozen majesty, layer by layer! Its structure is faultless because it is founded on pure reason, and its walls are impregnable because they were reared without blunder, error or even hesitancy, for here human intuition had no part! In short the structure of mathematics appears to the layman as erected not by the erring mind of man but by the infallible spirit of God. The history of mathematics reveals the fallacy of such a notion."

"The Eudemian Summary says that "Pythagoras changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom, and investigated its theorems in an immaterial and intellectual manner." His geometry was connected closely with his arithmetic. He was especially fond of those geometrical relations which admitted of arithmetical expression."

"The student can actually carry out the mathematical tasks in an authentically historical setting. He can do long division like the ancient Egyptians, solve quadratic equations like the Babylonians, and study geometry just as the student in Euclid's day. To get involved in the same processes and problems as the ancient mathematicians and to effect solutions in the face of the same difficulties they faced is the best way to gain appreciation of the intelligence and ingenuity of the scholars of early times."

"Students enjoy... and gain in their understanding of today's mathematics through analyzing older and alternative approaches."

"Although there is no study which presents so simple a beginning as that of geometry, there is none in which difficulties grow more rapidly as we proceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear. This necessarily follows in a science which consists of reasoning from the very commencement, for it is evident that every student will feel a claim to have his objections answered, not by authority, but by argument, and that the intelligent student will perceive more readily than another the force of an objection and the obscurity arising from an unexplained difficulty, as the greater is the ordinary light the more will occasional darkness be felt. To remove some of these difficulties is the principal object of this Treatise."

"1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves. 2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra. 3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry. Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics."

"Histories make men wise; poets, witty; the mathematics, subtile; natural philosophy, deep; morals, grave; logic and rhetoric, able to contend."

"By the beginning of the seventeenth century we may say that the fundamental principles of arithmetic, algebra, theory of equations, and trigonometry had been laid down, and the outlines of the subjects as we know them had been traced. It must be, however, remembered that there were no good elementary text-books on these subjects; and a knowledge of them was therefore confined to those who could extract it from the ponderous treatises in which it lay buried. Though much of the modern algebraical and trigonometrical notation had been inroduced, it was not familiar to mathematicians, nor was it even universally accepted; and it was not until the end of the seventeenth century that the language of the subjects was definitely fixed. Considering the absence of good text-books, I am inclined... to admire the rapidity with which it came into universal use, than to cavil at the hesitation to trust to it alone which many writers showed."

"Because of white racism's ability to bludgeon us into believing that we are inferior beings and therefore incapable of learning math and the sciences, we must spend a significant amount of our learning and teaching time unlearning and unteaching. This is to say, for example, that when a brother or sister reaches the freshman college level, he or she has already been subjected to at least 17 years of conditioning that dictates: "You are too black and too ignorant to understand such lily-white and intelligent things as math, chemistry, physics, etc." Thus, a major part of the teacher's initial instructional time must be spent dealing with the psychological block against learning math—or any of the sciences."

"Those intending to continue in mathematics or science or technology... believe that a survey of the main directions along which living mathematics has developed would enable them to decide more intelligently in what particular field of mathematics, if any, they would find a lasting satisfaction. ...It is astonishing how few students entering serious work in mathematics or its applications have even the vaguest idea of the highways, the pitfalls, and the blind alleys ahead of them. Consequently, it is the easiest thing in the world for an enthusiastic teacher... to sell his misguided pupils a subject that has been dead for forty or a hundred years, under the sincere delusion that he is disciplining their minds. With only the briefest glimpse of what mathematics in this twentieth century—not in 2100 B.C.—is about, any student of normal intelligence should be able to distinguish between live teaching and dead mathematics. He will be less likely to drown in the ditch or perish in the wilderness."

"All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions."

"A finished or even a competent reasoner is not the work of nature alone... education develops faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to learn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something must be reasoned upon, it matters not much what it is, provided that it can be reasoned upon with certainty. The properties of mind or matter, or the study of languages, mathematics, or natural history may be chosen for this purpose. Now, of all these, it is desirable to choose the one... in which we can find out by other means, such as measurement and ocular demonstration of all sorts, whether the results are true or not. ..Now the mathematics are peculiarly well adapted for this purpose, on the following grounds:— 1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing. 2. The first principles are self-evident, and, though derived from observation, do not require more of it than has been made by children in general. 3. The demonstration is strictly logical, taking nothing for granted except the self-evident first principles, resting nothing upon probability, and entirely independent of authority and opinion. 4. When the conclusion is attained by reasoning, its truth or falsehood can be ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if... reason is not to be the instructor, but the pupil. 5. There are no words whose meanings are so much alike that the ideas which they stand for may be confounded. ...These are the principal grounds on which... the utility of mathematical studies may be shewn to rest, as a discipline for the reasoning powers. But the habits of mind which these studies have a tendency to form are valuable in the highest degree. The most important of all is the power of concentrating the ideas which a successful study of them increases where it did exist, and creates where it did not. A difficult position or a new method of passing from one proposition to another, arrests all the attention, and forces the united faculties to use their utmost exertions. The habit of mind thus formed soon extends itself to other pursuits, and is beneficially felt in all the business of life."

"I have tried to say to students of mathematics that they should read the classics and beware of secondary sources. This is a point which Eric Temple Bell makes repeatedly... in '... that the men of whom he writes learned their mathematics not by studying in school or by reading textbooks, but by going straight to the sources and reading the best works of the masters who preceded them. It is a point which in most fields of scholarship at most times in history would have gone without saying. ...The purpose of a secondary source is to make the primary sources accessible to you."

"Should I teach them from the point of view of the history of science, from the applications? My theory is that the best way to teach is to have no philosophy, [it] is to be chaotic and [to] confuse it in the sense that you use every possible way of doing it. ...so as to catch this guy or that guy on different hooks as you go along, [so] that during the time when the fellow who's interested in history's being bored by the abstract mathematics... the fellow who likes the abstractions is being bored another time by the history—if you can do it so you don't bore them all, all the time, perhaps you're better off. ...I don't know how to answer the question of different kinds of minds with different interests... after many, many years of trying to teach and trying all kinds of different methods, I really don't know how to do it."

"More often than we are aware, it is the jargon which is the hurdle that a student cannot overcome rather than the mathematical concepts being introduced."

"Functions are the bread and butter of modern scientists, statisticians, and economists. Once many repeated... experiments and observations produce the same functional interrelationships, those may acquire the... status of laws of nature—mathematical descriptions... Descartes' ideas... opened the door for a systematic mathematization of everything—the very essence of the notion that God is a mathematician. ...[B]y establishing the equivalence of two perspectives of mathematics (algebraic and geometric) previously considered disjoint, Descartes expanded the horizons of mathematics and paved the way to the modern era of analysis, which allows [us] to comfortably cross from one mathematical discipline to another."

"While Descartes' theory of vortices was spectacularly wrong (as Newton ruthlessly pointed out later), it was still interesting, being the first serious attempt to formulate a theory of the universe as a whole based upon the same laws that apply on the Earth's surface. In other words, to Descartes there was no difference between terrestrial and celestial phenomena—the Earth was part of a universe that obeyed uniform physical laws."

"During the 1970s Sheldon Glashow, Abdus Salam, and Steven Weinberg... came up with a theory in which there have to be three separate particles carrying the weak force... the three weak carriers are recognized as heavy photons and the weak force is seen as a modified form of electromagnetism. The obvious differences between... interactions are ascribed to the fact that the photon has no mass and the weak carriers have a lot. ... The role of the electroweak theory in the effort to streamline fundamental physics is in some respects debatable. More [three] particles are needed and the mathematical structure... is not entirely beyond reproach. But it ties two separate forces into a single theoretical device... In 1983, physicists... found the W and Z particles. The electroweak theory was thereupon deemed correct, and the number of distinct forces in the world was officially reduced to three—electroweak, strong, and gravitational. Electroweak unification has organized another corner of theoretical physics, as did the quark model before it."

"I feel that controversies can never be finished, nor silence imposed upon the Sects, unless we give up complicated reasonings in favour of simple calculations, words of vague and uncertain meaning in favour of fixed symbols [characteres]. Thus it will appear that 'every paralogism is nothing but an error of calculation. When controversies arise, there will be no more necessity for disputation between two philosophers than between two accountants. Nothing will be needed but that they should take pen in hand, sit down with their counting-tables and (having summoned a friend, if they like) say to one another: Let us calculate.'"

"Theoretical physicists today have some ideas on how to combine the strong and electroweak forces, and hope eventually to include gravity in a single unified theory of all forces. The very meaning of "unification" in this context is that one harmonizing theoretical structure, mathematical in form, should be made to accommodate formerly distinct theories under one roof. Unification is the theme, the backbone of modern physics, and for most physicists what unification means in practice is the uncovering of tidier, more comprehensive mathematical structures linking separate phenomena."

"In the late 1960s... Steven Weinberg, Sheldon Glashow, and Abdus Salam developed a theory that treats the electromagnetic force and the weak nuclear force in a unified manner. ...the electroweak theory, predicted the existence of three particles (...W+, W-, and Z bosons) that had never before been observed. The particles were unambiguously detected in 1983 in accelerator experiments... led by... Carlo Rubbia and Simon van der Meer."

"Klein showed that the Reimannian species of non-Euclidean geometry can be developed in a fashion completely analogous to the Lobachevskian type by choosing an "imaginary" absolute, that is, an "imaginary" point pair or conic, and an imaginary value of the constant k. Euclidean geometry can also be treated in the same way by choosing a "degenerate" point pair or conic."

"As long as algebra and geometry travelled separate paths their advance was slow and their applications limited. But when these two sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace towards perfection. It is to Descartes that we owe the application of algebra to geometry,—an application which has furnished the key to the greatest discoveries in all branches of mathematics."

"To define distance in their non-Euclidean geometries, Cayley and Klein proceeded by analogy with a discovery of Laguerre... who had shown that the distances and angles of ordinary Euclidean geometry can be expressed as cross ratios, in other words, that the Euclidean metric geometry is clearly a specialization of projective geometry. The concept of the "absolute" and the definition of distance unified Euclidean and non-Euclidean geometries into a single all-embracing theory."

"The more sluggish positive charges are at first of less interest,—but the behaviour of electrons cannot be fully and properly understood without a knowledge of the nature and properties of the positive constituent too. According to Larmor, positive charge must be the mirror-image of negative charge, in essential constitution. The positive electron has not, as far as I know, been as yet observed free. Some think it cannot exist in a free state, that it is in fact the rest of the atom of matter from which a negative unit charge has been removed; or, to put it crudely, that "electricity" repels "electricity," and "matter" repels "matter," but that Electricity and Matter in combination form a neutral substance which is the atom of matter as we know it. Such a statement is an extraordinary and striking return to the views expressed by that great genius, Benjamin Franklin."

"The decisive step leading to the construction of precise and verifiable scientific theories in place of vague and largely speculative accounts was the involvement of mathematics. This step was made by the Pythagoreans. ...In their philosophy of nature the Pythagoreans began with the principle that number is the essence of all substance. ...forms reduced to numbers. Since number is the essence of any object, the explanation of natural phenomena could be achieved only through number. ...Whether by a lucky stroke or by intuitive genius the Pythagoreans did hit upon two doctrines which later proved to be all important. The first is that nature is built in accordance with mathematical principles, and the second that number relationships reveal the order in nature. They underlie and unify the seeming diversity exhibited by nature."

"Group theory is the mathematical language of symmetry, and it... seems to play a fundamental role in the very structure of nature. ...In the midst of the fomenting of the new twentieth century physics was the... life of the greatest female mathematician who ever lived, Emmy Noether. ...At Göttingen, Noether achieved fame for her research into the fundamental structure of mathematics. However, she stepped briefly into the realm of theoretical physics... is a profound statement, perhaps running as deeply into the fabric of our psyche as the famous theorem of Pythagoras. Noether's theorem directly connects symmetry to physics, and vice versa. It frames our modern concepts about nature and rules modern scientific methodology. ...For scientists it is the guiding light to unraveling nature's mysteries, as they delve into the innermost fabric of matter ...To this task scientists apply ...the great s ...Emmy Noether's work interweaves our understanding of nature—through physics and mathematics—with the beauty and harmony that surrounds us... Noether's theorem provides a natural centerpiece for any discussion that unifies physics and mathematics, such as in the teaching of these... in a way that enlivens them both."

"The remarkable insight that characterized Klimpt's later work was contemporaneous with Freud's psychological studies and presaged the inward turn that would pervade all fields of inquiry in Vienna in 1900. This period... was characterized by the attempt to make a sharp break with the past and to explore new forms of expression in art, architecture, psychology, literature, and music. It spawned an ongoing pursuit to link these disciplines. ...Viennese life at the turn of the century provided opportunities in salons and coffeehouses for scientists, writers, and artists to come together in an atmosphere that was at once inspiring, optimistic, and politically engaged. ...science was no longer the narrow and restrictive province of scientists but had become an integral part of Viennese culture. ...a paradigm for how an open dialogue can be achieved."

"Galileo and Newton swept away the last traces of mysticism and superstition that had always been associated with the heavens. The heliocentric theory of Copernicus and Kepler had classed the earth among the other planets, so that there was good reason to believe that the heavens were made of the stuff of earth rather than, as Greek and medieval philosophers had maintained, of some light, perfect, indestructible substance. But the heliocentric theory... was regarded by many as a mathematical contrivance... not physically true. Moreover... the heliocentric theory created difficulties in accounting for the phenomena of motion readily observed here on earth, and hence encountered legitimate objections. The work of Galileo and Newton resolved these difficulties... and incorporated the theory of the heavenly motions in the very same physical theory that treated motions on earth. There could be no doubt now... that the substance of the other planets could be identified with the rock and clay beneath man's feet, for this affirmation is the very essence of the law of gravitation. The identification... wiped out libraries of speculation and dogma..."

"Giacomo Rizzolatti... calls these "mirror neurons" and suggests that they provide the first insight into imitation, identification, empathy, and possibly the ability to mime vocalization—the mental processes intrinsic to human interaction. Vilayanur Ramachandran has found evidence of comparable neurons in the premotor cortex of people. ...one can see a whole new area of biology opening up... that can give us a sense of what makes us social, communicating beings. An ambitious undertaking of this sort might... teach us something about the factors that give rise to tribalism, which is so often associated with fear, hatred, and intolerance of outsiders."

"But even the distant reader must allow that Clifford's mental personality belonged to the highest possible type to say no more. The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal. And if in these modern days we are to look for any prophet or saviour who shall influence our feelings towards the universe as the founders and renewers of past religions have influenced the minds of our fathers, that prophet, if he ever come, must, like Clifford, be no mere sentimental worshipper of science, but an expert in her ways. And he must have what Clifford had in so extraordinary a degree—that lavishly generous confidence in the worthiness of average human nature to be told all truth, the lack of which in Goethe made him an inspiration to the few but a cold riddle to the many."

"Reduced to their most pregnant difference, empiricism means the habit of explaining wholes by parts, and rationalism means the habit of explaining parts by wholes. Rationalism thus preserves affinities with monism, since wholeness goes with union, while empiricism inclines to pluralistic views. No philosophy can ever be anything but a summary sketch, a picture of the world in abridgment, a foreshortened bird's-eye view of the perspective of events."

"The mathematician and versatile scientist Pierre L. M. de Maupertuis, a keen student if Newton's work on gravitation, made the next decisive step. Like Euler, Maupertuis studied under John Bernoulli. ...After having worked in the theory of light and gravitation, he announced, in 1744, a new minimum principle, the Principle of Least Action, from which he claimed he could deduce the behavior of light and masses in motion. The principle asserts that nature always behaves so as to minimize an integral known technically as action, and amounting to the integral of the product of mass, velocity, and distance traversed by a moving object. From this principle he deduced the Newtonian laws of motion. With sometimes suitable and sometimes questionable interpretation of the quantities involved, Maupertuis managed to show that optical phenomena, too, could be deduced from this principle. Hence, to an extent at least, he succeeded in uniting the optics of the eighteenth century and mechanical phenomena."

"Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics."

"Science attempts to confront the possible with the actual. The price to be paid for this outlook, however, turned out to be high. It was... renouncing a unified world view. ...Most other systems of explanation—mythic, magic, or religious—generally encompass everything. They apply to every domain. They answer any possible question. They account for the origin, the present, and the end of the universe. Science proceeds differently. It operates by detailed experimentation... it looks for partial and provisional answers about those phenomena that can be isolated and well defined. ...the beginning of modern science can be dated from the time when such general questions as, "How was the universe created? What is matter made of? What is the essence of life?" were replaced by such limited questions as "How does a stone fall? how does water flow in a tube? How does blood circulate in vessels?" ...While asking general questions led to limited answers, asking limited questions turned out to provide more and more general answers."

"[I]n the nineteenth century, even the could be reduced to mechanics by the assumption that heat really consists of a complicated statistical motion of the smallest parts of matter. By combining the concepts of the mathematical theory of probability with the concepts of Newtonian mechanics Clausius, Gibbs and Boltzmann were able to show that the fundamental laws in the theory of heat could be interpreted as statistical laws following from Newton's mechanics when applied to very complicated mechanical systems."

"To the scientists of 1850, Hamilton's principle was the realization of a dream. ...from the time of Galileo scientists had been striving to deduce as many phenomena of nature as possible from a few fundamental physical principles. ...they made striking progress ...But even before these successes were achieved Descartes had already expressed the hope and expectation that all the laws of science would be derivable from a single basic law of the universe. This hope became a driving force in the late eighteenth century after Maupertuis's and Euler's work showed that optics and mechanics could very likely be unified under one principle. Hamilton's achievement in encompassing the most developed and largest branches of physical science, mechanics, optics, electricity, and magnetism under one principle was therefore regarded as the pinnacle of mathematical physics."

"My purpose is merely to illustrate the issue involved in our question about the unification of science. A complete unification... would be a unification downward, finding its ultimate and universal laws in mechanics; and it would include in its scope all the movements of human bodies. Those who assert the possibility of a rigorously complete unification thus imply a denial of all physical efficacy to thoughts and feelings as such. Those, on the contrary, who assert such efficacy deny by implication the possibility of a complete unification of even the laws of the motion of matter. They tacitly or explicitly introduce a real discontinuity into the fabric of science."

"In the history of sciences, important advances often come from... the recognition that two hitherto separate observations can be viewed from a new angle and seen to represent nothing but different facets of one phenomenon. Thus, terrestrial and celestial mechanisms became a single science with Newton's laws. Thermodynamics and mechanics were unified through statistical mechanics, as were optics and electromagnetism through Maxwell's theory of magnetic field, or chemistry and atomic physics through quantum mechanics. Similarly different combinations of the same atoms, obeying the same laws, were shown by biochemists to compose both the inanimate and animate worlds. ... Despite such generalizations, however, large gaps remain... Following the line from physics to sociology, one goes from simpler to the more complex objects... from the poorer to the richer empirical content, as well as from the harder to the softer system of hypotheses and experimentation. ...Because of the hierarchy of objects, the problem is always to explain the more complex in terms and concepts applying to the simpler. This is the old problem of reduction, emergence, whole and parts... an understanding of the simple is necessary to understand the more complex, but whether it is sufficient is questionable. ...the appearance of life and later of thought and language—led to phenomena that previously did not exist... To describe and to interpret these phenomena new concepts, meaningless at the previous level, are required. ...At the limit total reductionism results in absurdity. ...explaining democracy in terms of the structure and properties of elementary particles... is clearly nonsense."

"[A]fter a close study of the experimental work of Michael Faraday,... James Clerk Maxwell succeeded in uniting electricity and magnetism in the framework of the '. ...Beyond uniting... all... electric and magnetic phenomena in one mathematical framework, Maxwell's theory showed—quite unexpectedly, that electromagnetic disturbances travel at a fixed and never-changing speed that turns out to equal that of light. From this, Maxwell realized that visible light itself is nothing but a particular kind of electromagnetic wave... Maxwell's theory also showed that all electromagnetic waves—visible light among them—are the epitome of the peripatetic traveler. They never stop. They never slow down. Light always travels at light speed."

"The notion of a smooth spatial geometry, the central principle of general relativity, is destroyed by the violent fluctuations of the quantum world on short distance scales. ...The equations of general relativity cannot handle the rolling frenzy of the quantum foam. ...There are ...physicists ...who are deeply unsettled by the fact that the two foundational pillars of physics as we know it are at their core fundamentally incompatible, regardless of the ultramicroscopic distances that must be probed to expose the problem. This incompatibility, they argue, points to an essential flaw in our understanding of the physical universe. This opinion rests on an unprovable but profoundly felt view that the universe, if understood at its deepest and most elementary level, can be described by a logically sound theory whose parts are harmoniously united. Physicists have made numerous attempts at modifying either general relativity or quantum mechanics in some manner so as to avoid the conflict, but the attempts... have been met with failure after failure. That is, until the discovery of superstring theory."

"Geoffrey also recognized that the opposite orientations of gut and nervous system posed a problem for his claim that insects and vertebrates represent different versions of the same archetypal animal - and he proposed the first account of the inversion theory to resolve this threat to unification. ...Geoffroy pursued the... aim of establishing a "unity of type" that could generate both s and vertebrates from the same basic blueprint. ...The single grand design includes a gut in the middle and the main nerve cords somewhere on the periphery."

"Our understanding of the four basic concepts of Physics—space, time, matter and force—has undergone radical change in the course of work on unification, starting with Maxwell's unification of electricity with magnetism, all the way to present day string theory. What started as four independent concepts, with space and time postulated and the possible forms of matter and force arbitrarily chosen, now appear as different aspects of a rich and novel dynamically determined structure."