Unification in science and mathematics
One of the wonders in the history of science and mathematics has been a continued evolution in the unification of concepts or classifications previously considered as independent. Some recent attempts at unification have been a search for the discovery or creation of a Grand Unified Theory in particle physics, and for a Theory of everything, a single, all-encompassing, coherent theoretical framework of physics.
- 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.
- Eric Temple Bell, The Development of Mathematics (1940)
- 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.
- Eric Temple Bell, The Development of Mathematics (1940)
- 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.
- Eric Temple Bell, The Development of Mathematics (1940)
- 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?
- L. Lévy-Bruhl, "Essay on the Philosophy of Descartes" (1903) The Meditations, and Selections from the Principles of René Descartes (1596-1650) Tr. John Veitch, pp. xii-xii
- 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.
- A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927) Forward
- 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.
- A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927) Forward
- 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.
- A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927) pp. 327-328
- 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.
- A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927) pp. 459-460
- 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.
- With 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...
- René Descartes, Rule IV: "Necessity of Method in the Search for Truth," "Rules for the Direction of the Mind," Part I of Discourse Upon Method in The Philosophy of Descartes: In Extracts from His Writing pp. 71-72, Tr. Henry A. P. Torrey
- 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.
- Freeman Dyson, Disturbing the Universe (1979), p. 62
- 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.
- 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.
- Albert Einstein, Science and Religion (1941)
- 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.
- Paul Karl Feyerabend, Farewell to Reason (1987) p. 100
- 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.
- 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.
- Peter Freund, "Physics and Geometry," (Aug 28, 2003) Symposium on Theoretical Physics at the University of Helsinki, Helsinki, Finland and at the Freydoon Mansouri Memorial Session of the 3rd International Symposium on Quantum Theory and Symmetries (Sept 13, 2003) University of Cincinnati, Cincinnati, OH. Report #EFI03-47.
- 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.
- Carl Friedrich Gauss as quoted in Gauss, Werke, Bd. 8, p. 298
- 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 arthropods 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.
- Stephen Jay Gould, "Brotherhood by Inversion," Leonardo's Mountain of Clams and the Diet of Worms (1998) p. 326
- 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.
- Brian Greene, The Elegant Universe (1999) Ch. 5 The Need for a New Theory: General Relativity vs. Quantum Mechanics.
- 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.
- Brian Greene, The Elegant Universe (1999) Ch. 8 More Dimensions Than Meet the Eye.
- 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.
- Malcolm Guthrie, On Mr. Spencer's Unification of Knowledge (1882) Preface
- All knowledge... is unification of the multiple.
- Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.
- David Hilbert, "Axiomatic Thought" (1918) as quoted in William Bragg Ewald, From Kant to Hilbert, Vol. 2
- 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.
- Eric Kandel, In Search of Memory (2006)
- 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.
- Eric Kandel, The Age of Insight (2012)
- 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...
- Morris Kline, Mathematics and the Physical World (1959) Ch. 16: Deductions from the Law of Graviation, p. 252.
- 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.
- Morris Kline, Mathematics and the Physical World (1959) Ch. 25: From Calculus to Cosmic Planning, p. 438.
- 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.
- Morris Kline, Mathematics and the Physical World (1959) Ch. 25: From Calculus to Cosmic Planning, p. 441.
- 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.
- Morris Kline, Mathematics for Liberal Arts (1967) republished as Mathematics for the Nonmathematician (1985)
- 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.
- Edna E. Kramer, The Nature and Growth of Modern Mathematics (1970)
- 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.
- Edna E. Kramer, The Nature and Growth of Modern Mathematics (1970)
- 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.
- 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.'
- Gottfried Wilhelm Leibniz, De Scientia Universali seu Calculo Philosophico (c. 1680) as quoted by Robert Latta, Introduction, Leibniz, The Monadology and Other Philosophical Writings (1898) Tr. Robert Latta
- 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.
- Mario Livio, Is God a Mathematician? (2009)
- 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.
- 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.
- The significance of the contention that the laws of the several sciences are discontinuous appears chiefly when you thus regard the sciences as corresponding to stages in the process of evolution. ...that means that at certain points in the evolutionary sequence matter begins to behave in essentially new ways, develops novel properties and methods of action which were in no true sense contained in or implied by its earlier characteristics and performances. If, on the other hand, all the laws of biology and chemistry are ultimately reducible to, and deducible from, the laws of some fundamental branch of physics, that means that, in a very thorough-going sense, the first morning of creation wrote what the last dawn of reckoning shall read.
- Arthur O. Lovejoy, "The Unity of Science" (1912)
- Velocity of transverse undulations in our hypothetical medium, calculated from the electromagnetic experiments of 'MM'. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.
- James Clerk Maxwell, Lecture at Kings College (1862) as quoted by F. V. Jones, "The Man Who Paved the Way for Wireless," New Scientist (Nov 1, 1979) p. 348 & Andrey Vyshedskiy, On The Origin Of The Human Mind 2nd edition
- The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.
- James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field" (1864)
- The object of the book is philosophical, in the sense now accepted by many and by divergent schools—i.e., it desires to contribute something towards a unification of thought.
- John Theodore Merz (November 1896) Preface A History of European Thought in the Nineteenth Century (1903)
- In both Newton's theory and Maxwell's theory the unification consists, partly, in showing that two different processes or phenomena can be identified, in some way, with each other—that they belong to the same class or are the same kind of thing. Celestial and terrestrial objects are both subject to the same gravitational-force law, and optical and electromagnetic processes are one and the same. Because each of these theories unifies such a diverse range of phenomena, they have traditionally been thought to possess a great deal of explanatory power.
- Margaret Morrison, Unifying Scientific Theories: Physical Concepts and Mathematical Structures (2007)
- Since the ancients made great account of the science of Mechanics in the investigation of natural things; and the moderns, laying aside substantial forms and occult qualities, have endeavoured to subject the phænomena of nature to the laws of mathematics; I have in this treatise cultivated Mathematics... The ancients considered Mechanics in a twofold respect; as rational, which proceeds accurately by demonstration, and practical. To practical Mechanics all the manual arts belong, from which Mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that Mechanics is so distinguished from Geometry, that what is perfectly accurate is called Geometrical, what is less so is called Mechanical. But the errors are not in the art, but in the artificers. ...the description of right lines and circles, upon which Geometry is founded, belongs to Mechanics. ...To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from Mechanics; and by Geometry the use of them, when so solved, is shewn. And it is the glory of Geometry that from those few principles, fetched from without, it is able to produce so many things. Therefore Geometry is founded in mechanical practice, and is nothing but that part of universal Mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that Geometry is commonly referred to their magnitudes, and Mechanics to their motion. In this sense Rational Mechanics will be the science of motions resulting from any forces whatsoever and of the forces required to produce any motions, accurately proposed and demonstrated. ...we consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces whether attractive or impulsive. And therefore we offer this work as mathematical principles of philosophy. For all the difficulty of philosophy seems to consist in this, from the phenomena of motions to investigate the forces of Nature, and then from these forces to demonstrate the other phenomena.
- The most immediate result of this unbalanced specialisation has been that to-day, when there are more "scientists" than ever, there are much less "cultured" men than, for example, about 1750. And the worst is that with these turnspits of science not even the real progress of science itself is assured. For science needs from time to time, as a necessary regulator of its own advance, a labour of reconstitution, and, as I have said, this demands an effort towards unification, which grows more and more difficult, involving, as it does, ever-vaster regions of the world of knowledge. Newton was able to found his system of physics without knowing much philosophy, but Einstein needed to saturate himself with Kant and Mach before he could reach his own keen synthesis.
- In early physical systems we have optics dealing with phenomena perceived by the eye; acoustics treating of auditory percepts, and so on. The subjective concepts of "tone" and "colour" have now been replaced by the objectified concepts of frequency of vibration; and wave-length. The object of this process of elimination is, according to Planck, the striving towards a unification of the whole theoretical system, so that it shall be equally significant for all intelligent beings.
- The unification of knowledge is the natural consequence of the intellectual and moral development of the race.
- Raymond St. James Perrin, The Religion of Philosophy or The Unification of Knowledge (1885)
- The Pythagoreans were the first who attempted a complete classification of the facts of the universe. Their effort, though feeble, was in the right direction; for the first principle of perception is analysis, or classification; and knowledge can never be unified until an ultimate or complete analysis has been performed.
- Raymond St. James Perrin, The Religion of Philosophy or The Unification of Knowledge (1885)
- All the early thinkers sought with wonderful perseverance the knowledge of the First Cause. The Four Causes of Aristotle, though they had been separately recognized, had not all been proclaimed necessary. Aristotle... gave his chief attention to the solution of the problem of First Causes. He maintained that there were four, as follows: First, the Material Cause, or Essence; second, the Substantial [or Formal] Cause; third, the Efficient Cause, or the principle of motion; fourth, the Final Cause, or the Purpose and End.
- Raymond St. James Perrin, The Religion of Philosophy or The Unification of Knowledge (1885)
- It is the opinion of all competent authorities that the fundamental principles of Kant's philosophy declare against the possibility of a unification of knowledge.
- Raymond St. James Perrin, The Religion of Philosophy or The Unification of Knowledge (1885)
- Ever since Hermann Minkowski's now infamous comments in 1908 concerning the proper way to view space-time, the debate has raged as to whether or not the universe should be viewed as a four-dimensional, unified whole wherein the past, present, and future are regarded as equally real or whether the views espoused by the possibilists, historicists, and presentests regarding the unreality of the future (and, for presentests, the past) are more accurate. Now, a century after Minkowski's proposed block universe first sparked debate, we present a new, more conclusive argument in favor of eternalism.
- Daniel Peterson and Michael Silberstein, "Relativity of Simultaneity and Eternalism: In Defense of the Block Universe" Space, Time, and Spacetime: Physical and Philosophical Implications of Minkowski's Unification of Space and Time Vesselin Petkov, Ed. (2010).
- A third stage appears between 7 and 8, which we shall call the stage of incipient cooperation. Each player now tries to win, and all, therefore, begin to concern themselves with the question of mutual control and of unification of the rules. But while a certain agreement may be reached in the course of one game, ideas about the rules in general are still rather vague. In other words, children of 7-8, who belong to the same class at school and are therefore constantly playing with each other, give, when they are questioned separately, disparate and often entirely contradictory accounts of the rules observed in playing marbles.
- Jean Piaget, The Moral Judgment of the Child Tr. Marjorie Gabain (1932) Ch. 1 : The Rules of the Game
- Now what is science? ...it is before all a classification, a manner of bringing together facts which appearances separate, though they are bound together by some natural and hidden kinship.
- Henri Poincaré, The Value of Science (1907) Ch. 11: Science and Reality, pp.137-138, Tr. George Bruce Halsted
- Maxwell got a huge bonus for understanding the unification of electricity and magnetism. He understood the nature of light! When I first heard about this in high school I thought this was the coolest thing, and I still do. It's what we're all trying to do.
- Mathematics and logic, historically speaking, have been entirely distinct studies. Mathematics has been connected with science, logic with Greek. But both have developed in modern times: logic has become more mathematical and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one. They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic. This view is resented by logicians who, having spent their time in the study of classical texts, are incapable of following a piece of symbolic reasoning, and by mathematicians who have learnt a technique without troubling to inquire into its meaning or justification. Both types are now fortunately growing rarer. So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: starting with premises which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary.
- Science itself is badly in need of integration and unification. The tendency is more and more the other way … Only the graduate student, poor beast of burden that he is, can be expected to know a little of each. As the number of physicists increases, each specialty becomes more self-sustaining and self-contained. Such Balkanization carries physics, and indeed, every science further away, from natural philosophy, which, intellectually, is the meaning and goal of science.
- There is obviously only one alternative, namely the unification of minds or consciousnesses. Their multiplicity is only apparent, in truth there is only one mind.
- Erwin Schrödinger, Mind and Matter (1958)
- In Newton's system of mechanics... there is an absolute space and an absolute time. In Einstein's theory time and space are interwoven, and the way in which they are interwoven depends on the observer. Instead of three plus one we have four dimensions.
- Willem de Sitter, "Relativity and Modern Theories of the Universe," Kosmos (1932)
- Einstein’s dissent from quantum mechanics and immersion in the search for a unified field theory were not failures but anticipations. After all, even if many string theorists would disagree with Einstein about the incompleteness of quantum mechanics, much of what goes on in string theory these days looks a lot like what Einstein was doing in his Princeton years, which was trying to find new mathematics that might extend general relativity to a unification of all the forces and particles in nature.
- Lee Smolin, "The Other Einstein," The New York Review of Books (June 14, 2007)
- In both quantum theory and general relativity, we encounter predictions of physically sensible quantities becoming infinite. This is likely the way that nature punishes impudent theorists who dare to break her unity. ...If infinities are signs of missing unification, a unified theory will have none. It will be what we call a finite theory.
- Lee Smolin, The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next (2007
- A scientific hypothesis may be defined in general terms as a provisional or tentative explanation of physical phenomena. But what is an explanation in the true scientific sense? The answers to this question which are given by logicians and men of science, though differing in their phraseology, are essentially of the same import. Phenomena are explained by an exhibition of their partial or total identity with other phenomena. Science is knowledge; and all knowledge, in the language of Sir William Hamilton is a "unification of the multiple." "The basis of all scientific explanation," says Bain, "consists in assimilating a fact to some other fact or facts. It is identical with the generalizing process." And "generalization is only the apprehension of the One in the Many." Similarly Jevons: "Science arises from the discovery of identity amid diversity," and "every great advance in science consists in a great generalization pointing out deep and subtle resemblances." ...the author just quoted in another place: "Every act of explanation consists in detecting and pointing out a resemblance between facts, or in showing that a greater or less degree of identity exists between apparently diverse phenomena."
All this may be expressed in familiar language thus: When a new phenomenon presents itself to the man of science or to the ordinary observer, the question arises in the mind of either: What is it?—and this question simply means: Of what known, familiar fact is this apparently strange, hitherto unknown fact a new presentation—of what known, familiar fact or facts is it a disguise or complication? Or, inasmuch as the partial or total identity of several phenomena is the basis of classification (a class being a number of objects having one or more properties in common), it may also be said that all explanation, including explanation by hypothesis, is in its nature classification.
Such being the essential nature of a scientific explanation of which an hypothesis is a probatory form, it follows that no hypothesis can be valid which does not identify the whole or a part of the phenomenon, for the explanation of which it is advanced, with some other phenomenon or phenomena previously observed. This first and fundamental canon of all hypothetical reasoning in science is formally resolvable into two propositions, the first of which is that every valid hypothesis must be an identification of two terms—the fact to be explained and a fact by which it is explained; and the second that the latter fact must be known to experience.
- The theories that we describe here provide the basis of progress toward a unification of macroeconomics and microeconomics.
- Joseph Stiglitz, "The Causes and Consequences of The Dependence of Quality on Price", Journal of Economic Literature, Vol. 25, No. 1 (Mar., 1987)
- Lagrange's "Mécanique analytique" is perhaps his most valuable work and still amply repays careful study. ...the full power of the newly developed analysis was applied to the mechanics of points and rigid bodies. The results of Euler, of D'Alembert, and of the other mathematicians of the Eighteenth Century were assimilated and further developed from a consistent point of view. Full use of Lagrange's own calculus of variations made the unification of the varied principles of statistics and dynamics possible...
- Dirk Jan Struik, A Concise History of Mathematics (1948) Ch. 8 The Eighteenth Century.
- In the year 1749 he first suggested his idea of explaining the phenomena of thunder-gusts and of the aurora borealis, upon electrical principles. He points out many particulars in which lightning and electricity agree; and he adduces many facts, and reasonings from facts, in support of his positions. In the same year he conceived the astonishingly bold and grand idea of ascertaining the truth of his doctrine, by actually drawing down the lightning, by means of sharp pointed iron rods raised into the region of the clouds. Even in this uncertain state, his passion to be useful to mankind displays itself in a powerful manner. Admitting the identity of electricity and lightning, and knowing the power of points in repelling bodies charged with electricity, and in conducting their fire silently and imperceptibly, he suggested the idea of securing houses, ships, &c. from being damaged by lightning, by erecting pointed rods, that should rise some feet above the most elevated part, and descend some feet into the ground or the water.
- Henry Stuber, “History of the Life and Character of Benjamin Franklin” Universal Asylum and Columbian Magazine (May, June & July, 1790) as quoted in The Life of Dr. Benjamin Franklin, Written by Himself (1823)
- We only call an elephant or a bacterium an 'organism' because, by analogy we attribute to those beings a similar unification of sensation and of consciousness to that we are conscious of in ourselves; but in human societies and in humanity this essential indication is lacking, and therefore, however many other indications we may detect that are common to humanity and to an organism, in the absence of that essential indication, the acknowledgement of humanity as an organism is incorrect.
- From Pappus it appears, however, that the early Mathematicians had at first some reluctance in admitting either the Conic Sections or superior curves in the solution of problems, considering them as not strictly geometrical; but afterwards these lines became objects of much curious investigation, even among the ancients; and in modern times ultimately were of the most extensive utility, both in abstract and in physical science.
- William Trail, Account of the Mathematical Collections of Pappus, Account of the life and writings of Robert Simson, M.D. Late Professor of Mathematics in the University Of Glasgow, ed. William Trail (1812) Appendix I.
- More than a hundred years have elapsed since Benjamin Franklin, employing a phraseology now superseded, put forth a theory of matter. It was pronounced "a delusion" by the physicists of the nineteenth century, but the scientists of the twentieth century, according to Sir Oliver Lodge, may be forced to rehabilitate it as the only means of issue from the labyrinth in which all physical study is now involved. ...the Franklin theory is that electricity and matter in combination form a neutral substance, which is the atom of matter as we know it.
The most interesting part of the problem for ourselves, says Sir Oliver, is the explanation of matter in terms of electricity, the view that electricity is, as Franklin seems to have supposed, the fundamental "substance." What we men of to-day have been accustomed to regard as an indivisible atom of matter is thus built up out of electricity. All atoms—atoms of all sorts of "substances"—are built up of the same thing. In our day... the theoretical and proximate achievement of what philosophers from Franklin's day to ours have always sought—a unification of matter—is offering itself to physical inquiry.
- We can describe general relativity using either of two mathematically equivalent ideas: curved space-time or metric field. Mathematicians, mystics and specialists in general relativity tend to like the geometric view because of its elegance. Physicists trained in the more empirical tradition of high-energy physics and quantum field theory tend to prefer the field view, because it corresponds better to how we (or our computers) do concrete calculations. ...the field view makes Einstein's theory of gravity look more like the other successful theories of fundamental physics, and so makes it easier to work toward a fully integrated, unified description of all the laws. ...I'm a field man.
- Frank Wilczek, The Lightness of Being: Mass, Ether, and the Unification of Forces (2008).
- I feel that we are so close with string theory that—in my moments of greatest optimism—I imagine that any day, the final form of the theory may drop out of the sky and land in someone's lap. But more realistically, I feel that we are now in the process of constructing a much deeper theory than anything we have had before and that well into the twenty-first century, when I am too old to have any useful thoughts on the subject, younger physicists will have to decide whether we have in fact found the final theory.