Unification in science and mathematics

"The more man inquires into the laws which regulate the material universe, the more he is convinced that all its varied forms arise from the action of a few simple principles. These principles themselves converge, with accelerating force, towards some still more comprehensive law to which all matter seems to be submitted. Simple as that law may possibly be, it must be remembered that it is only one amongst an infinite number of simple laws: that each of these laws has consequences at least as extensive as the existing one, and therefore that the Creator who selected the present law must have foreseen the consequences of all other laws."

"It appears... that the elastic theories of light, if Kelvin's gyrostatic adynamic ether be admitted, have not been wholly routed. Nevertheless the great electromagnetic theory of light propounded by Maxwell (1864, 'Treatise,' 1873) has been singularly apt not only in explaining all the phenomena reached by the older theories and in predicting entirely novel results, but in harmoniously uniting as parts of a unique doctrine, both the electric or photographic light vector of Fresnel and Cauchy and the magnetic vector of Neumann and MacCullagh. Its predictions have, moreover, been astonishingly verified by the work of Hertz (1890), and it is to-day acquiring added power in the convection theories of Lorentz (1895) and others."

"At first the mathematical disciplines were not sharply defined. As knowledge increased, individual subjects split off from the parent mass and became autonomous. Later, some were overtaken and reabsorbed in vaster generalizations of the mass from which they sprang. Thus trigonometry issued from surveying, astronomy, and geometry only to be absorbed, centuries later, in the analysis which had generalized geometry. This recurrent escape and recapture has inspired some to dream of a final, unified mathematics which shall embrace all. Early in the twentieth century it was believed by some for a time that the desired unification had been achieved in mathematical logic. But mathematics, too irrepressibly creative to be restrained by any formalism, escaped."

"Whatever its source, mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry. In the seventeenth century these two united, forming the ever-broadening river of mathematical analysis."

"If the early Greeks were cognizant of Babylonian algebra, they made no attempt to develop or even to use it, and thereby they stand convicted of the supreme stupidity in the history of mathematics. ...The ancient Babylonians had a rare capacity for numerical calculation; the majority of Greeks were either mystical or obtuse in their first approach to number. What the Greeks lacked in number, the Babylonians lacked in logic and geometry, and where the Babylonians fell short, the Greeks excelled. Only in the modern mind of the seventeenth and succeeding centuries were number and form first clearly perceived as different aspects of one mathematics."

"Science is an attempt to represent the known world as a closed system with a perfect formalism. Scientific discovery is a constant maverick process of breaking out at the ends of the system... and then hastily closing it... The act of the imagination is the opening of the system so that it shows new connections. ...every act of imagination is the discovery of likenesses between two things which were thought unlike. ...they introduce new likenesses, whether it is Shakespeare... or Newton saying that the moon in essence is exactly like a thrown apple."

"Up to this point mathematics alone appeared to Descartes worthy of being called a science. ...in order to establish the science or philosophy sought by Descartes, it was sufficient to find a method that should be to philosophy what the method of mathematical deduction is to arithmetic, algebra and geometry. ...How could one pass from these processes, which are especially adapted to particular sciences, to the general method required by general science or philosophy? Descartes would undoubtedly never have conceived such an audacious hope, had not a great discovery of his set him on this track. He had invented analytical geometry... In this way, Descartes substituted for the old methods, which were especially adapted to algebra and geometry as distinct branches, a general method, applicable to what he called the "universal mathematical science," viz., to the study of "the various ratios or proportions to be found between the objects of the mathematical sciences, hitherto regarded as distinct." Not only did this discovery mark a decisive epoch in the history of mathematics, which it provided with an instrument of incomparable simplicity and power, but it furthermore gave Descartes a right to hope for the philosophical method he was seeking. Ought not a last generalization to be possible, by means of which the method he had so happily discovered should become applicable, not only to the "universal mathematical science," but also to the systematic combination of all the truths which our finite minds may permit us to attain?"

"[A]s the great extreme of dimension is sublime, so the last extreme of littleness is in the same measure sublime... when we attend to the infinite divisibility of matter, when we pursue animal life into these excessively small, and yet organized beings... when we push our discoveries yet downward... in tracing which the imagination is lost as well as the sense; we become amazed and confounded at the wonders of minuteness; nor can we distinguish in its effects this extreme of littleness from the vast itself. For division must be infinite as well as addition; because the idea of a perfect unity can no more be arrived at, than that of an complete whole, to which nothing can be added."

"Edmund Burke, ' (1757) p. 81 of the 1898 edition."

"Copernicus had taken one course in treating the earth as virtually a celestial body in the Aristotelian sense—a perfect sphere governed by the laws which operated in the higher reaches of the skies. Galileo complemented this by taking now the opposite course—rather treating the heavenly bodies as terrestrial ones, regarding the planets as subject to the very laws which applied to balls sliding down inclined planes. There was something in all this which tended to the reduction of the whole universe to uniform physical laws, and it is clear that the world was coming to be more ready to admit such a view."

"[T]he attempt to embrace the whole course of things in time and to relate the successive epochs to one another—the transition to the view that time is actually aiming at something, that temporal succession has meaning and that the passage of ages is generative—was greatly influenced by the fact that the survey became wider than that of human history, that the mind gradually came to see geology, pre-history and history in due succession to one another. The new science and the history joined hands and each acquired a new power as a result of their mutual reinforcement. The idea of progress itself gained additional implications when there gradually emerged a wider idea of evolution."

"Let us assert, as our original postulate, that, the multiple (that is, non-being, if taken in the pure state) being the only rational form of a creatable (creabile) nothingness, the creative act is comprehensible only as a gradual process of arrangement and unification, which amounts to accepting that to create is to unite. And, indeed, there is nothing to prevent our holding that union creates. To the objection that union presupposes already existing elements, I shall answer that physics has just shown us (in the case of mass) that experientially (and for all the protests of "common sense") the moving object exists only as the product of its motion."

"The scientific spirit must then lose its present tendency to speciality, and be impelled towards a logical generality; for all the branches of natural philosophy must furnish their contingent to the common doctrine; in order to which they must first be completely condensed and co-ordinated. When the savans have learned that active life requires the habitual and simultaneous use of the various positive ideas that each of them isolates from all the rest, they will perceive that their social ascendency supposes the prior generalization of their common conceptions, and consequently the entire philosophical reformation of their present practice. Even in the most advanced sciences... the scientific character at present fluctuates between the abstract expansion and the partial application, so as to be usually neither thoroughly speculative nor completely active; a consequence of the same defect of generality which rests the ultimate utility of the positive spirit on minor services, which are as special as the corresponding theoretical habits. But this view, which ought to have been long outgrown, is a mere hindrance in the way of the true conception,—that positive philosophy contemplates no other immediate application than the intellectual and moral direction of civilized society; a necessary application, presenting nothing that is incidental or desultory, and imparting the utmost generality, elevation, unity, and consistency, to the speculative character. Under such a homogeneousness of view and identity of aim, the various positive philosophers will naturally and gradually constitute a European body, in which the dissensions that now break up the scientific world into coteries will merge; and with the rivalries of struggling interests will cease the quarrels and coalitions which are the opprobrium of science in our day."

"Prior to Newton, mathematics, chiefly in the form of geometry, had been studied as a fine art without any view to its physical applications other than in very trivial cases. But with Newton all the resources of mathematics were turned to advantage in the solution of physical problems. Thenceforth mathematics appeared as an instrument of discovery, the most powerful one known to man, multiplying the power of thought... It is this application of mathematics to the solution of physical problems, this combination of two separate fields of investigation, which constitutes the essential characteristic of the Newtonian method. Thus problems of physics were metamorphosed into problems of mathematics. ...Newton showed the mark of genius by inventing the integral calculus. As a result... problems which would have baffled Archimedes were solved with ease. ...this new departure in scientific method led to the discovery of the law of gravitation. ...the real significance in Newton's achievement lay ...in his having established the presence of law and order at least in one realm of nature ...the motions of the heavenly bodies. Nature thus exhibited rationality and was not mere blind chaos and uncertainty."

"Newton, in his application of mathematics to physics, had been concerned only with... planetary motions, mechanics, propagation of sound, etc. But when it came to applying the mathematical method to the more intricate physical problems, a considerable advance was necessary... both mathematical and empirical. Thanks to the gradual accumulation of physical data, and... to the efforts of Newton's great successors in the field of pure mathematics (Euler, Lagrange, Laplace), conditions were ripe in the first half of the nineteenth century for a systematic attack on many of nature's secrets. The mathematical theories constructed were known under the general name of theories of mathematical physics. ...they had their prototype in Newton's celestial mechanics. ...they dealt with a wide variety of physical phenomena (electric, hydrostatic, etc.) ...The most celebrated of these theories (such as those of Maxwell, Boltzmann, Lorentz and Planck) were concerned with very special classes of phenomena. But with Einstein's theory of relativity... the scope of our investigations is so widened that we are appreciably nearer than ever before to the ideal of a single mathematical theory embracing all of physical knowledge."

"The equations of gravitation... signify that whenever we recognise the existence of one of these physical magnitudes it is always accompanied by corresponding curvatures of space-time. It is usual to assume that the curvatures are produced by those concrete somethings which we call mass, momentum, energy, pressure. In this way, we must concede a duality to nature; there would exist both matter and space-time, or, better still, matter and the metrical field of space-time. Einstein... attempted to remove this duality by proving that it was possible to attribute the entire existence of the metrical field, hence of space-time, to the presence of matter. This attitude led to a matter-moulding conception of the universe... And... only when this attitude was adhered to could Mach's belief in the relativity of all motion be accepted. Eddington's attitude is just the reverse. He prefers to assume that the equations of gravitation are not equations in the ordinary sense of something being equal to something else. In his opinion they are identities. They merely tell us how our senses will recognize the existence of certain curvatures of space-time by interpreting them as matter, motion, and so on. In other words, there is no matter; there is nothing but a variable curvature of space-time. Matter, momentum, vis viva, are the names we give to those curvatures on account of the varying ways they affect our senses."

"Passing to the laws of motion, we remember that there is but one law: All free bodies (when reduced to point-masses) follow time-geodesics in space-time regardless of whether space-time be flat, as it is (at least approximately) in interstellar space, or whether it be curved by the presence of matter. If space-time is flat, the geodesics are straight and the bodies describe straight courses with constant speeds as referred to a Galilean frame. Thus Newton's law of inertia is seen to express the flatness of space-time. When space-time, and hence its geodesics, are curved by the presence of matter, the courses of free bodies appear to be curved, or else their motion to be accelerated. But whereas, under those conditions, the law of inertia was at fault in classical science, and an additional gravitational influence had to be introduced, in Einstein's theory the general law of geodesic motion still holds good. Inasmuch as the structure of space-time determines the laws of our geometry, the beatings of natural clocks (atoms) and the motion of free bodies, we see that the theory has brought about a fusion between geometry and physics."

"The age-old conflict between our notions of continuity and the scientific concept of number ended in a decisive victory for that latter. This victory was brought about by the necessity of vindicating, of legitimizing... a procedure which ever since the days of Fermat and Descartes had been an indispensable tool of analysis. ...analytic geometry ...this discipline which was born of the endeavors to subject problems of geometry to arithmetical analysis, ended by becoming the vehicle through which the abstract properties of number are transmitted to the mind. It furnished analysis with a rich, picturesque language and directed it into channels of generalization hitherto unthought of. Now, the tacit assumption on which analytic geometry operated was that it was possible to represent the points on a line, and therefore points in a plane and in space, by means of numbers. ...The great success of analytic geometry... gave this assumption an irresistible pragmatic force. ...Under such circumstances mathematics proceeds by fiat. It bridges the chasm between intuition and reason by a convenient postulate. On the one hand, there was the logically consistent concept of real number and its aggregate, the arithmetic continuum; on the other, the vague notions of the point and its aggregate, the linear continuum. All that was necessary was to declare the identity of the two, or, what amounted to the same thing, to assert that: It is possible to assign to any point on a line a unique real number, and, conversely, any real number can be represented in a unique manner by a point on a line. This is the famous Dedekind-Cantor axiom."

"[W]ith a view to summon myself to the search for a science of mathematics in general, I asked myself... what precisely was the meaning of this word mathematics, and why arithmetic and geometry only, and not also astronomy, music, optics, mechanics, and so many other sciences, should be considered as forming a part of it; for it is not enough here to know the etymology of the word. In reality the word mathematics meaning nothing but science, those which I have just named have as much right as geometry to be called mathematics; and nevertheless there is no one, however little instructed, who cannot distinguish at once what belongs to mathematics... from what belongs to the other sciences. But... all the sciences which have for their end investigations concerning order and measure, are related to mathematics, it being of small importance whether this measure be sought in numbers, forms, stars, sounds, or any other object; that, accordingly, there ought to exist a general science which should explain all that can be known about order and measure, considered independently of any application to a particular subject, and that, indeed, this science has its own proper name, consecrated by long usage, to wit, mathematics... And a proof that it surpasses in facility and importance the sciences which depend upon it is that it embraces at once all the objects to which these are devoted and a great many others besides; and consequently, if it contain any difficulties, these exist in the rest, which have themselves the peculiar ones arising from their particular subject-matter, and which do not exist for the general science."

"I think that the correct connection between quantum theory and relativity has not yet been discovered. ...I think that the present methods which theoretical physicists are using are not the correct methods. They use... a renormalization technique which involves handling infinite quantities, and this is not really a mathematically logical process. ...[I]t is just a set of working rules rather than a correct mathematical theory and I don't like this whole development at all. I think that some other important discoveries will have to be made before these questions are put into order. In particular, there is the problem of explaining the , the number 137, which plays an important role in physics, and the question is, why should it be 137 instead of some other number. That has not been explained at all, and I feel that it is necessary to get an explanation of that before one would make an important advance in understanding atomic theory. ...There is quite a different problem with the ratio of the mass of the proton to the mass of the electron, and the question is whether the ratio of these masses remains constant or whether it develops slowly with time."

"You could not imagine the sum-over-histories picture being true for a part of nature and untrue for another part. You could not imagine it being true for electrons and untrue for gravity. It was a unifying principle that would either explain everything or explain nothing. And this made me profoundly skeptical. I knew how many great scientists had chased this will-o’-the-wisp of a unified theory. The ground of science was littered with the corpses of dead unified theories. Even Einstein had spent twenty years searching for a unified theory and had found nothing that satisfied him. I admired Dick tremendously, but I did not believe he could beat Einstein at his own game."

"From the present state of theory it looks as if the electromagnetic field, as opposed to the gravitational field, rests upon an entirely new formal motif, as though nature might just as well have endowed the gravitational ether with fields of quite another type, for example, with fields of a scalar potential, instead of fields of the electromagnetic type. Since according to our present conceptions the elementary particles of matter are also, in their essence, nothing else than condensations of the electromagnetic field, our present view of the universe presents two realities which are completely separated from each other conceptually, although connected causally, namely, gravitational ether and electromagnetic field, or — as they might also be called — space and matter. Of course it would be a great advance if we could succeed in comprehending the gravitational field and the electromagnetic field together as one unified conformation. Then, for the first time, the epoch of theoretical physics founded by Faraday and Maxwell would reach a satisfactory conclusion. The contrast between ether and matter would fade away, and, through the general theory of relativity, the whole of physics would become a complete system of thought, like geometry, kinematics, and the theory of gravitation."

"The basic concepts and laws which are not logically further reducible constitute the indispensable and not rationally deducible part of the theory. ...The conception... of the purely fictitious character of the basic principles of theory was in the eighteenth and nineteenth centuries still far from being the prevailing one. But it continues to gain more and more ground because of the ever-widening logical gap between the basic concepts and laws on the one side and the consequences to be correlated with our experiences on the other—a gap which widens progressively with the developing unification of the logical structure, that is with the reduction in the number of the logically independent conceptual elements required for the basis of the whole system."

"Although it is true that it is the goal of science to discover rules which permit the association and foretelling of facts, this is not its only aim. It also seeks to reduce the connections discovered to the smallest possible number of mutually independent conceptual elements. It is in this striving after the rational unification of the manifold that it encounters its greatest successes, even though it is precisely this attempt which causes it to run the greatest risk of falling a prey to illusions."

"We have to realize that a unified theory of the physical world simply does not exist. We have theories that work in restricted regions, we have purely formal attempts to condense them into a single formula, we have lots of unfounded claims (such as the claim that all of chemistry can be reduced to physics), phenomena that do not fit into the accepted framework are suppressed; in physics, which many scientists regard as the one really basic science, we have now at least three different points of view (relativity, dealing with the very large, quantum theory for an intermediate domain and various particle models for the very small) without a promise of conceptual (and not only formal) unification; perceptions are outside of the material universe (the mind-body problem is still unsolved) - from the very beginning the salesman of a universal truth cheated people into admissions instead of clearly arguing for their philosophy. And let us not forget that it was they and not the representatives of the traditions they attacked who introduced argument as the one and only universal arbiter. They praised argument - they constantly violated its principles."

"People are always asking for the latest developments in the unification of this theory with that theory, and they don't give us a chance to tell them anything about what we know pretty well. They always want to know the things we don't know."

"Unlike the chess game... in which the rules become more complicated as you go along, in physics, when you discover new things, it looks more simple. It appears on the whole to be more complicated because we learn about a greater experience—that is, we learn more about more particles and new things—and so the laws look more complicated again. But if you realize all the time what's kind of wonderful—that is, if we expand our experience into wilder and wilder regions of experience—every once in a while we have these integrations when everything's pulled together into a unification, in which it turns out to be simpler than it was before."

"In some ways, science today is less specialized... Consider... physics and chemistry; fifty years ago they were regarded as separate fields. ...Philosophers even gave an "intelligible" reason why physics and chemistry would always be separate... Physics had to do with quantity, chemistry with quality. Then there developed the field of , later the field of . Today it would be difficult to say what the difference is between physics and chemistry... now the laws of chemistry are derived from physics, from thermodynamics, electrodynamics, and from quantum mechanics. ...The same exists between physics and biology, or between economics and anthropology. ...Today we must understand economics as a tribal custom, and tribal customs from the economic point of view. ...The disappearance of the old unity between science and philosophy can hardly be ascribed to the increasing specialization in science."

"The decisive steps toward a clear understanding of non-Euclidean geometry were taken by Riemann, Helmholtz, and Poincaré, who recognized the essential unity of geometry and physics. However, the understanding did not come into its own until Einstein showed that such a combination of geometry and physics was really necessary for the derivation of phenomena which had actually been observed."

"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."

"In general the position as regards all such new calculi is this — That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able — without the unconscious inspiration of genius which no one can command — to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."

"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."

"[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."

"In a paper he sent to Einstein in 1919, Kaluza made an astounding suggestion. He proposed that the spatial fabric of the universe might possess more than the three dimensions... it provided an elegant and compelling framework for weaving together Einstein's general relativity and Maxwell's electromagnetic theory into a single, unified conceptual framework. ...implicit in Kaluza's work and subsequently made explicit and refined by... Oskar Klein in 1926... the spatial fabric of our universe may have both extended and curled-up dimensions. ... Einstein had formulated general relativity in the familiar setting of a universe with three spatial dimensions and one time dimension. The mathematical formalism... however, could be extended fairly directly to write down analogous equations for a universe with additional space dimensions. Under the "modest" assumption of one additional space dimension, Kaluza... derived the new equations. ...Kaluza found extra equations... those Maxwell had written down in the 1880s for deriving the electromagnetic force! ...Kaluza had united Einstein's theory of gravity with Maxwell's theory of light."

"Although the first principles of a science are the first in logical order, they are generally the last in order of discovery. They are arrived at by generalisations of extended experience. They mark the attainment of true scientific inductions, and manifest their correctness by the explanations they are able to afford. They enable us to discern the coherence of large classes of facts, and give us the power to forecast a line of sequences whereby we may direct them to the accomplishment of desired ends, or shape our actions to those coming events which are beyond our control. As an instrument of discovery, first principles are of very little value, and on account of the many chances of error, and of the fascination which the idea of a completed system exercises over the imagination of great minds, the search after them has been fruitful of error. The present undertaking, therefore, is to be regarded not as an attack upon the evolutionism of Lamarck, nor as an attack upon the evolutionism of Lyell or Darwin, nor yet upon the evolutionism of Spencer as regards the development of intelligence, but as an attack upon the theory which attempts to combine all these into one continuous process."

"All knowledge... is unification of the multiple."

"[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."

"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."

"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."

"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."

"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."

"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..."

"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."

"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."

"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."

"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."