Unification in science and mathematics
59 quotes
"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."
"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."
"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.<!--pp. 108-110-->"
"[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.<!-- pp.231-232, 1957 2nd ed.-->"
"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."
"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."
"[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."
"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."
"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.<!-- pp. 11-12-->"
"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.<!-- p. 89 -->"
"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."
"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."
"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."
"But even the distant reader must allow that Clifford's mental personality belonged to the highest possible type to say no more. The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal. And if in these modern days we are to look for any prophet or saviour who shall influence our feelings towards the universe as the founders and renewers of past religions have influenced the minds of our fathers, that prophet, if he ever come, must, like Clifford, be no mere sentimental worshipper of science, but an expert in her ways. And he must have what Clifford had in so extraordinary a degree—that lavishly generous confidence in the worthiness of average human nature to be told all truth, the lack of which in Goethe made him an inspiration to the few but a cold riddle to the many."
"Reduced to their most pregnant difference, empiricism means the habit of explaining wholes by parts, and rationalism means the habit of explaining parts by wholes. Rationalism thus preserves affinities with monism, since wholeness goes with union, while empiricism inclines to pluralistic views. No philosophy can ever be anything but a summary sketch, a picture of the world in abridgment, a foreshortened bird's-eye view of the perspective of events."
"The mathematician and versatile scientist Pierre L. M. de Maupertuis, a keen student if Newton's work on gravitation, made the next decisive step. Like Euler, Maupertuis studied under John Bernoulli. ...After having worked in the theory of light and gravitation, he announced, in 1744, a new minimum principle, the Principle of Least Action, from which he claimed he could deduce the behavior of light and masses in motion. The principle asserts that nature always behaves so as to minimize an integral known technically as action, and amounting to the integral of the product of mass, velocity, and distance traversed by a moving object. From this principle he deduced the Newtonian laws of motion. With sometimes suitable and sometimes questionable interpretation of the quantities involved, Maupertuis managed to show that optical phenomena, too, could be deduced from this principle. Hence, to an extent at least, he succeeded in uniting the optics of the eighteenth century and mechanical phenomena."