⚛On Physics: On Paths

"The state of a system at a given moment depends on two things - its initial state, and the law according to which that state varies. If we know both this law and this initial state, we have a simple mathematical problem to solve, and we fall back upon our first degree of ignorance. Then it often happens that we know the law and do not know the initial state. It may be asked, for instance, what is the present distribution of the minor planets? We know that from all time they have obeyed the laws of Kepler, but we do not know what was their initial distribution. In the kinetic theory of gases we assume that the gaseous molecules follow rectilinear paths and obey the laws of impact and elastic bodies; yet as we know nothing of their initial velocities, we know nothing of their present velocities. The calculus of probabilities alone enables us to predict the mean phenomena which will result from a combination of these velocities. This is the second degree of ignorance. Finally it is possible, that not only the initial conditions but the laws themselves are unknown. We then reach the third degree of ignorance, and in general we can no longer affirm anything at all as to the probability of a phenomenon. It often happens that instead of trying to discover an event by means of a more or less imperfect knowledge of the law, the events may be known, and we want to find the law; or that, instead of deducing effects from causes, we wish to deduce the causes." (Henri Poincaré, "Science and Hypothesis", 1902)

"We frequently find that nature acts in such a way as to minimize certain magnitudes. The soap film will take the shape of a surface of smallest area. Light always follows the shortest path, that is, the straight line, and, even when reflected or broken, follows a path which takes a minimum of time. In mechanical systems we find that the movements actually take place in a form which requires less effort in a certain sense than any other possible movement would use. There was a period, about 150 years ago, when physicists believed that the whole of physics might be deduced from certain minimizing principles, subject to calculus of variations, and these principles were interpreted as tendencies - so to say, economical tendencies of nature. Nature seems to follow the tendency of economizing certain magnitudes, of obtaining maximum effects with given means, or to spend minimal means for given effects." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"Experimental physicists [...] walk a narrow path with pitfalls on either side. If we spend all our time developing equipment, we risk the appellation of 'plumber', and if we merely use the tools developed by others, we risk the censure of our peers for being parasitic." (Luis W Alvarez, "Recent Developments in Particle Physics", [Nobel] 1968)

"Although we expect to find eddies in turbulent flow, we do not know when any specific eddy will come into being or die away . We cannot yet predict how eddies interact. Similarly, we know as a general rule that any particle within a turbulent flow gets knocked about in an aimless fashion by the swirls, so that it describes an erratic meandering path, but at any given moment we cannot predict the precise location or velocity of the particle." (Peter B Stevens, "Patterns in Nature", 1974)

"The Feynman method has the virtue that it provides us with a vivid picture of nature’s quantum trickery at work. The idea is that the path of a particle through space is not generally well defined in quantum mechanics. […] So when an electron arrives at a point in space - say a target screen - many different histories must be integrated together to create this one event. Feynman’s so-called path-integral, or sum-over-histories approach to quantum mechanics, set this remarkable concept out as a mathematical procedure. It remained more or less a curiosity for many years, but as physicists pushed quantum mechanics to its limits - applying it to gravitation and even cosmology - so the Feynman approach turned out to offer the best calculational tool for describing a quantum universe. History may well judge that, among his many outstanding contributions to physics, the path-integral formulation of quantum mechanics is the most significant." (Paul Davies , [Introduction to Richard P Feynman's "Six Easy Pieces"] 1994)

"What was clearly useful was the use of diagrams to prove certain results either in algebraic topology, homological algebra or algebraic geometry. It is clear that doing category theory, or simply applying category theory, implies manipulating diagrams: constructing the relevant diagrams, chasing arrows by going via various paths in diagrams and showing they are equal, etc. This practice suggests that diagram manipulation, or more generally diagrams, constitutes the natural syntax of category theory and the category-theoretic way of thinking. Thus, if one could develop a formal language based on diagrams and diagrams manipulation, one would have a natural syntactical framework for category theory. However, moving from the informal language of categories which includes diagrams and diagrammatic manipulations to a formal language based on diagrams and diagrammatic manipulations is not entirely obvious." (Jean-Pierre Marquis, "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory", 2009)

⚛️On Physicists (Unsourced)

"Chemistry has been termed by the physicist as the messy part of physics, but that is no reason why the physicists should be permitted to make a mess of chemistry when they invade it." (Frederick Soddy [attributed])

"Empirical evidence can never establish mathematical existence - nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful." (Richard Courant)

"Every physicist knows exactly what the first and the second law mean, but [...] no two physicists agree about them." (Clifford Truesdell)

"I believe that numbers and functions of Analysis are not the arbitrary result of our minds; I think that they exist outside of us, with the same character of necessity as the things of objective reality, and we meet them or discover them, and study them, as do the physicists, the chemists and the zoologists." (Charles Hermite)

"Nothing in physics seems so hopeful to as the idea that it is possible for a theory to have a high degree of symmetry was hidden from us in everyday life. The physicist's task is to find this deeper symmetry." (Steven Weinberg)

"Symmetry does mean something different for physicists than for members of the public. It means that an object or a theory does not change when you make some transformation - either rotating or moving it or doing something to the equations." (Lawrence M Krauss)

"The difference between mathematicians and physicists is that after physicists prove a big result they think it is fantastic but after mathematicians prove a big result they think it is trivial." (Lucien Szpiro)

"The universe of Eastern mysticism is an illusion, A physicist who attempts to link it with his own work has abandoned physics." (Stephen Hawking)

"There is no drawing the line between physics and metaphysics. If you examine every day facts at all closely, you are a physicist; but if you press your physics at all home, you become a metaphysician; if you press your metaphysics at all home, you are in a fog." (Samuel Butler)

"When confronted with the order and beauty of the universe and the strange coincidences of nature, it's very tempting to take the leap of faith from science into religion. I am sure many physicists want to. I only wish they would admit it." (Tony Rothman)

⚛️On Physicists (1940-1949)

"[…] there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, [...] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real', but [...] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"At the present time it is of course quite customary for physicists to trespass on chemical ground, for mathematicians to do excellent work in physics, and for physicists to develop new mathematical procedures […] Trespassing is one of the most successful techniques in science." (Wolfgang Köhler, "Dynamics in Psychology", 1940)

"Physicists who are trying to understand nature may work in many different fields and by many different methods; one may dig, one may sow, one may reap. But the final harvest will always be a sheaf of mathematical formulae. These will never describe nature itself, hut only our observations on nature. Our studies can never put us into contact with reality; we can never penetrate beyond the impressions that reality implants in our minds." (James H Jeans, "Physics and Philosophy", 1942)

"Although we can never devise a pictorial representation which shall be both true to nature and intelligible to our minds, we may still be able to make partial aspects of the truth comprehensible through pictorial representations or parables. As the whole truth does not admit of intelligible representation, every such pictorial representation or parable must fail somewhere. The physicist of the last generation was continually making pictorial representations and parables, and also making the mistake of treating the half-truths of pictorial representations and parables as literal truths." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"Any good architect is by nature a physicist as a matter of fact, but as a matter of reality, as things are, he must be a philosopher and a physician." (Frank L Wright, "Frank Lloyd Wright: An Autobiography", 1943)

"In time they [physicists] hoped to devise a model which would reproduce all the phenomena of physics, and so make it possible to predict them all. […] To-day we not only have no perfect model, but we know that it is of no use to search for one - it could have no intelligible meaning for us. For we have found out that nature does not function in a way that can be made comprehensible to the human mind through models or pictures. […] Although we can never devise a pictorial representation which shall be both true to nature and intelligible to our minds, we may still be able to make partial aspects of the truth comprehensible through pictorial representations or parables. As the whole truth does not admit of intelligible representation, every such pictorial representation or parable must fail somewhere. The physicist of the last generation was continually making pictorial representations and parables, and also making the mistake of treating the half-truths of pictorial representations and parables as literal truths." (James H Jeans, "Physics and Philosophy" 3rd Ed., 1943)

"It is to be hoped that in the future more and more theoretical physicists will command a deep knowledge of mathematical principles; and also that mathematicians will no longer limit themselves so exclusively to the aesthetic development of mathematical abstractions." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)

"A permanent state is reached, in which no observable events occur. The physicist calls this the state of thermodynamical equilibrium, or of ‘maximum entropy’. Practically, a state of this kind is usually reached very rapidly. Theoretically, it is very often not yet an absolute equilibrium, not yet the true maximum of entropy. But then the final approach to equilibrium is very slow. It could take anything between hours, years, centuries […]." (Erwin Schrödinger, "What is Life?", 1944)

"[…] analogy [is] an important source of conjectures. In mathematics, as in the natural and physical sciences, discovery often starts from observation, analogy, and induction. These means, tastefully used in framing a plausible heuristic argument, appeal particularly to the physicist and the engineer." (George Pólya, "How to solve it", 1945) 

⚛️On Physicists (1960-1969)

"The mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Communications in Pure and Applied Mathematics 13 (1), 1960)

"The word model is used as a noun, adjective, and verb, and in each instance it has a slightly different connotation. As a noun 'model' is a representation in the sense in which an architect constructs a small-scale model of a building or a physicist a large-scale model of an atom. As an adjective 'model' implies a degree of perfection or idealization, as in reference to a model home, a model student, or a model husband. As an adjective 'model' implies a degree or perfection or idealization, as in reference to a model home, a model student, or a model husband. As a verb 'to model' means to demonstrate, to reveal, to show what a thing is like." (Russell L Ackoff, "Scientific Method: optimizing applied research decisions", 1962) 

"For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created." (Freeman J Dyson, "Mathematics in the Physical Sciences", Scientific American, 1964)

"There is one metaphor in the physicist’s account of space-time which one would expect anyone to recognize as such, for metaphor is here strained far beyond the breaking point, i.e., when it is said that time is ‘at right angles to each of the other three dimensions’. Can anyone really attach any meaning to this - except as a recipe for drawing diagrams?" (Clement W K Mundle, "The Space-Time World", Mind, 1967)

"[...] we are essentially viewing the purpose of physics as a scientific discipline as invention rather than discovery. [...] the term 'invention' implies that the physicist uses not only observation but his imaginative powers to construct points of view that identify with experience." (Robert B Lindsay, Arbitrariness in Physics, Physics Today Vol. 120 (12), 1967)

"Experimental physicists [...] walk a narrow path with pitfalls on either side. If we spend all our time developing equipment, we risk the appellation of 'plumber', and if we merely use the tools developed by others, we risk the censure of our peers for being parasitic." (Luis W Alvarez, "Recent Developments in Particle Physics", [Nobel] 1968)

"Modern science is characterized by its ever-increasing specialization, necessitated by the enormous amount of data, the complexity of techniques and of theoretical structures within every field. Thus science is split into innumerable disciplines continually generating new subdisciplines. In consequence, the physicist, the biologist, the psychologist and the social scientist are, so to speak, encapusulated in their private universes, and it is difficult to get word from one cocoon to the other." (Ludwig von Bertalanffy, "General System Theory", 1968)

"Most of us who become experimental physicists do so for two reasons; we love the tools of physics because to us they have intrinsic beauty, and we dream of finding new secrets of nature as important and as exciting as those uncovered by our scientific heroes." (Luis W Alvarez, "Recent Developments in Particle Physics", [Nobel] 1968)

"Let us consider, for a moment, the world as described by the physicist. It consists of a number of fundamental particles which, if shot through their own space, appear as waves, and are thus [...] of the same laminated structure as pearls or onions, and other wave forms called electromagnetic which it is convenient, by Occam’s razor, to consider as travelling through space with a standard velocity. All these appear bound by certain natural laws which indicate the form of their relationship." (G Spencer-Brown, "Laws of Form", 1969)

⚛️On Physicists (1950-1959)

"To seek not for end but for antecedents is the way of the physicist, who finds causes" in what he has learned to recognise as fundamental properties, or inseparable concomitants, or unchanging laws, of matter and of energy." (Sir D’Arcy W Thompson, "On Growth and Form", 1951)

"The older physicist believed in Nature and thought of himself as making experiments to see what She was like. She was there whether he could observe her or not. But the modern physicist thinks first of all of what he observes in his experiments and is not interested in anything that he cannot possibly observe. He looks for relations between his observations and ignores everything else. But he still expresses his results as though they were discoveries of the essence of Nature, because he is so used to this way of speaking that he does not realise that his discoveries no longer conform to it. When they are expressed as the characteristics of a world existing outside us and independently of us, which causes our experience by its impact on our sense organs, these discoveries require such a world to have contradictory properties. Hence, by retaining this form of expression, the physicist finds himself presenting his perfectly rational achievements as though they were nonsensical." (Herbert Dingle, "The Scientific Adventure", British Journal for the Philosophy of Science, 1952)

"We secure our mathematical knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning. A mathematical proof is demonstrative reasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning." (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"The mathematicians know a great deal about very little and the physicists very little about a great deal." (Stanislaw Ulam, "On the Ergodic Behavior of Dynamical Systems", 1955)

"Mathematicians who build new spaces and physicists who find them in the universe can profit from the study of pictorial and architectural spaces conceived and built by men of art." (György Kepes, "The New Landscape In Art and Science", 1956)

"We frequently find that nature acts in such a way as to minimize certain magnitudes. The soap film will take the shape of a surface of smallest area. Light always follows the shortest path, that is, the straight line, and, even when reflected or broken, follows a path which takes a minimum of time. In mechanical systems we find that the movements actually take place in a form which requires less effort in a certain sense than any other possible movement would use. There was a period, about 150 years ago, when physicists believed that the whole of physics might be deduced from certain minimizing principles, subject to calculus of variations, and these principles were interpreted as tendencies - so to say, economical tendencies of nature. Nature seems to follow the tendency of economizing certain magnitudes, of obtaining maximum effects with given means, or to spend minimal means for given effects." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"With the advent of special and general relativity, the symmetry laws gained new importance. Their connection with the dynamic laws of physics takes on a much more integrated and interdependent relationship than in classical mechanics, where logically the symmetry laws were only consequences of the dynamical laws that by chance possess the symmetries. Also in the relativity theories the realm of the symmetry laws was greatly enriched to include invariances that were by no means apparent from daily experience. Their validity rather was deduced from, or was later confirmed by complicated experimentation. Let me emphasize that the conceptual simplicity and intrinsic beauty of the symmetries that so evolve from complex experiments are for the physicists great sources of encouragement. One learns to hope that Nature possesses an order that one may aspire to comprehend." (Chen-Ning Yang, "The Law of Parity Conservation and Other Symmetry Laws of Physics", [Nobel lecture] 1957)

"Even for the physicist the description in plain language will be a criterion of the degree of understanding that has been reached." (Werner K Heisenberg, "Physics and Philosophy", 1958)

"Physicists do not start from hypotheses; they start from data. By the time a law has been fixed into an H-D [hypothetico-deductive] system, really original physical thinking is over." (Norwood R Hanson, "Patterns of Discovery", 1958)

⚛️On Physicists (1980-1989)

"The prevailing trend in modern physics is thus much against any sort of view giving primacy to [...] undivided wholeness of flowing movement. Indeed, those aspects of relativity theory and quantum theory which do suggest the need for such a view tend to be de-emphasized and in fact hardly noticed by most physicists, because they are regarded largely as features of the mathematical calculus and not as indications of the real nature of things." (David Bohm, "Wholeness and the Implicate Order?", 1980)

"The physicist […] engages in complex and difficult calculations, involving the manipulating of ideal, mathematical quantities that, at first glance, are wholly lacking in the music of the living world and the beauty of the resplendent cosmos. It would seem as if there exists no relationship between these quantities and reality. Yet these ideal numbers that cannot be grasped by one's senses, these numbers that only are meaningful from within the system itself, only meaningful as part of abstract mathematical functions, symbolize the image of existence. […] As a result of scientific man's creativity there arises an ordered, illumined, determined world, imprinted with the stamp of creative intellect, of pure reason and clear cognition. From the midst of the order and lawfulness we hear a new song, the song of the creature to the Creator, the song of the cosmos to its Maker." (Joseph B Soloveitchik, "Halakhic Man", 1983)

"Until now, physical theories have been regarded as merely models with approximately describe the reality of nature. As the models improve, so the fit between theory and reality gets closer. Some physicists are now claiming that supergravity is the reality, that the model and the real world are in mathematically perfect accord." (Paul C W Davies, "Superforce", 1984)

"Simple rules can have complex consequences. This simple rule has such a wealth of implications that it is worth examining in detail. It is the far from self-evident guiding principle of reductionism and of most modern investigations into cosmic complexity. Reductionism will not be truly successful until physicists and cosmologists demonstrate that the large-scale phenomena of the world arise from fundamental physics alone. This lofty goal is still out of reach. There is uncertainty not only in how physics generates the structures of our world but also in what the truly fundamental rules of physics are. (William Poundstone, "The Recursive Universe", 1985)

"As glimpsed by physicists, Nature's rules are simple, but also intricate: Different rules are subtly related to each other. The intricate relations between the rules produce interesting effects in many physical situations. [...] Nature's design is not only simple, but minimally so, in the sense that were the design any simpler, the universe would be a much duller place." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"It is positively spooky how the physicist finds the mathematician has been there before him or her." (Steven Weinberg, "Lectures on the Applicability of Mathematics" , Notices of the American Mathematical Society, 1986)

"Physicists dream of a unified description of Nature. Symmetry, in its power to tie together apparently unrelated aspects of physics, is linked closely to the notion of unity." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The beauty that Nature has revealed to physicists in Her laws is a beauty of design, a beauty that recalls, to some extent, the beauty of classical architecture, with its emphasis on geometry and symmetry. The system of aesthetics used by physicists in judging Nature also draws its inspiration from the austere finality of geometry." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The physicist's problem is the problem of ultimate origins and ultimate natural laws. The biologist's problem is the problem of complexity." (Richard Dawkins, "The Blind Watchmaker", 1986)

"Toward the end of the last century, many physicists felt that the mathematical description of physics was getting ever more complicated. Instead, the mathematics involved has become ever more abstract, rather than more complicated. The mind of God appears to be abstract but not complicated. He also appears to like group theory." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"Unlike an architect, Nature does not go around expounding on the wondrous symmetries of Her design. Instead, theoretical physicists must deduce them. Some symmetries, such as parity and rotational invariances, are intuitively obvious. We expect Nature to possess these symmetries, and we are shocked if She does not. Other symmetries, such as Lorentz invariance and general covariance, are more subtle and not grounded in our everyday perceptions. But, in any case, in order to find out if Nature employs a certain symmetry, we must compare the implications of the symmetry with observation." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"When it comes to very highly organized systems, such as a living cell, the task of modeling by approximation to simple, continuous and smoothly varying quantities is hopeless. It is for this reason that attempts by sociologists and economists to imitate physicists and describe their subject matter by simple mathematical equations is rarely convincing." (Paul C W Davies, "The Cosmic Blueprint: New Discoveries in Nature’s Creative Ability to Order the Universe", 1987)

"Where chaos begins, classical science stops. For as long as the world has had physicists inquiring into the laws of nature, it has suffered a special ignorance about disorder in the atmosphere, in the fluctuations of the wildlife populations, in the oscillations of the heart and the brain. The irregular side of nature, the discontinuous and erratic side these have been puzzles to science, or worse, monstrosities." (James Gleick, "Chaos", 1987)

"All reality is a game. Physics at its most fundamental, the very fabric of our universe, results directly from the interaction of certain fairly simple rules, and chance; the same description may be applied to the best, most elegant and both intellectually and aesthetically satisfying games. By being unknowable, by resulting from events which, at the sub-atomic level, cannot be fully predicted, the future remains malleable, and retains the possibility of change, the hope of coming to prevail; victory, to use an unfashionable word. In this, the future is a game; time is one of its rules." (Iain Banks, "The Player of Games", 1988)

"Physicists are all too apt to look for the wrong sorts of generalizations, to concoct theoretical models that are too neat, too powerful, and too clean. Not surprisingly, these seldom fit well with data. To produce a really good biological theory, one must try to see through the clutter produced by evolution to the basic mechanisms. What seems to physicists to be a hopelessly complicated process may have been what nature found simplest, because nature could build on what was already there." (Francis H C Crick, "What Mad Pursuit?: A Personal View of Scientific Discovery", 1988)

"Theoretical physicists are accustomed to living in a world which is removed from tangible objects by two levels of abstraction. From tangible atoms we move by one level of abstraction to invisible fields and particles. A second level of abstraction takes us from fields and particles to the symmetry-groups by which fields and particles are related. The superstring theory takes us beyond symmetry-groups to two further levels of abstraction. The third level of abstraction is the interpretation of symmetry-groups in terms of states in ten-dimensional space-time. The fourth level is the world of the superstrings by whose dynamical behavior the states are defined." (Freeman J Dyson, "Infinite in All Directions", 1988)

⚛️On Physicists (1900-1924)

"So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the initimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality." (Henri Poincaré, "The Value of Science", 1905)

"The laws of nature are drawn from experience, but to express them one needs a special language: for, ordinary language is too poor and too vague to express relations so subtle, so rich, so precise. Here then is the first reason why a physicist cannot dispense with mathematics: it provides him with the one language he can speak […]. Who has taught us the true analogies, the profound analogies which the eyes do not see, but which reason can divine? It is the mathematical mind, which scorns content and clings to pure form." (Henri Poincaré, "The Value of Science", 1905)

"A symbolical representation of a method of calculation has the same significance for a mathematician as a model or a visualisable working hypothesis has for a physicist. The symbol, the model, the hypothesis runs parallel with the thing to be represented. But the parallelism may extend farther, or be extended farther, than was originally intended on the adoption of the symbol. Since the thing represented and the device representing are after all different, what would be concealed in the one is apparent in the other." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)

"Imagine the forehead of a bull, with the protuberances from which the horns and ears start, and with the collars hollowed out between these protuberances; but elongate these horns and ears without limit so that they extend to infinity; then you will have one of the surfaces we wish to study. On such a surface geodesics may show many different aspects. There are, first of all, geodesics which close on themselves. There are some also which are never infinitely distant from their starting point even though they never exactly pass through it again; some turn continually around the right horn, others around the left horn, or right ear, or left ear; others, more complicated, alternate, in accordance with certain rules, the turns they describe around one horn with the turns they describe around the other horn, or around one of the ears. Finally, on the forehead of our bull with his unlimited horns and ears there will be geodesics going to infinity, some mounting the right horn, others mounting the left horn, and still others following the right or left ear. [...] If, therefore, a material point is thrown on the surface studied starting from a geometrically given position with a geometrically given velocity, mathematical deduction can determine the trajectory of this point and tell whether this path goes to infinity or not. But, for the physicist, this deduction is forever useless. When, indeed, the data are no longer known geometrically, but are determined by physical procedures as precise as we may suppose, the question put remains and will always remain unanswered." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"Indeed, a mathematical deduction is of no use to the physicist so long as it is limited to asserting that a given rigorously true proposition has for its consequence the rigorous accuracy of some such other proposition. To be useful to the physicist, it must still be proved that the second proposition remains approximately exact when the first is only approximately true. And even that does not suffice. The range of these two approximations must be delimited; it is necessary to fix the limits of error which can be made in the result when the degree of precision of the methods of measuring the data is known; it is necessary to define the probable error that can be granted the data when we wish to know the result within a definite degree of approximation." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"The physicist can never subject an isolated hypothesis to experimental test, but only a whole group of hypotheses." (Pierre Duhem, "The Aim and Structure of Physical Theory", 1906) 

"From the point of view of the physicist, a theory of matter is a policy rather than a creed; its object is to connect or co-ordinate apparently diverse phenomena, and above all to suggest, stimulate and direct experiment. It ought to furnish a compass which, if followed, will lead to observer further and further into previously unexplored regions." (Sir Joseph J Thomson, "The Corpuscular Theory of Matter", 1907)

"[...] just as the astronomer, the physicist, the geologist, or other student of objective science looks about in the world of sense, so, not metaphorically speaking but literally, the mind of the mathematician goes forth in the universe of logic in quest of the things that are there; exploring the heights and depths for facts - ideas, classes, relationships, implications, and the rest; observing the minute and elusive with the powerful microscope of his Infinitesimal Analysis; observing the elusive and vast with the limitless telescope of his Calculus of the Infinite; making guesses regarding the order and internal harmony of the data observed and collocated; testing the hypotheses, not merely by the complete induction peculiar to mathematics, but, like his colleagues of the outer world, resorting also to experimental tests and incomplete induction; frequently finding it necessary, in view of unforeseen disclosures, to abandon one hopeful hypothesis or to transform it by retrenchment or by enlargement: - thus, in his own domain, matching, point for point, the processes, methods and experience familiar to the devotee of natural science." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"[...] physics makes progress because experiment constantly causes new disagreements to break out between laws and facts, and because physicists constantly touch up and modify laws in order that they may more faithfully represent the facts." (Pierre-Maurice-Marie Duhem, "The Aim and Structure of Physical Theory", 1908)

"Much of the skill of the true mathematical physicist and of the mathematical astronomer consists in the power of adapting methods and results carried out on an exact mathematical basis to obtain approximations sufficient for the purposes of physical measurements. It might perhaps be thought that a scheme of Mathematics on a frankly approximative basis would be sufficient for all the practical purposes of application in Physics, Engineering Science, and Astronomy, and no doubt it would be possible to develop, to some extent at least, a species of Mathematics on these lines. Such a system would, however, involve an intolerable awkwardness and prolixity in the statements of results, especially in view of the fact that the degree of approximation necessary for various purposes is very different, and thus that unassigned grades of approximation would have to be provided for. Moreover, the mathematician working on these lines would be cut off from the chief sources of inspiration, the ideals of exactitude and logical rigour, as well as from one of his most indispensable guides to discovery, symmetry, and permanence of mathematical form. The history of the actual movements of mathematical thought through the centuries shows that these ideals are the very life-blood of the science, and warrants the conclusion that a constant striving toward their attainment is an absolutely essential condition of vigorous growth. These ideals have their roots in irresistible impulses and deep-seated needs of the human mind, manifested in its efforts to introduce intelligibility in certain great domains of the world of thought." (Ernest W Hobson, [address] 1910)

"The goal is nothing other than the coherence and completeness of the system not only in respect of all details, but also in respect of all physicists of all places, all times, all peoples, and all cultures." (Max Planck, Acht Vorlesungen", 1910)

"The development of mathematics is largely a natural, not a purely logical one: mathematicians are continually answering questions suggested by astronomers or physicists; many essential mathematical theories are but the reflex outgrowth from physical puzzles." (George A L Sarton, "The Teaching of the History of Science", The Scientific Monthly, 1918)

"The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them."(Albert Einstein, "Principles of Research", 1918)

"[...] the future of thought and therefore of history lies in the hands of physicists, and therefore the future historian must seek his education in the world of mathematical physics." (Henry Adams, "The Degradation of the Democratic Dogma", 1920)

"First, the physicists in the persons of Faraday and Maxwell, proposed the 'electromagnetic field' in contradistinction to matter, as a reality of a different category. Then, during the last century, the mathematicians, […] secretly undermined belief in the evidence of Euclidean Geometry. And now, in our time, there has been unloosed a cataclysm which has swept away space, time, and matter hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope and entailing a deeper vision. This revolution was promoted essentially by the thought of one man, Albert Einstein." (Hermann Weyl," Space, Time, Matter", 1922)

⚛️On Physicists (2010-2019)

"Another feature of Bourbaki is that it rejects intuition of any kind. Bourbaki books tend not to contain explanations, examples, or heuristics. One of the main messages of the present book is that we record mathematics for posterity in a strictly rigorous, axiomatic fashion. This is the mathematician’s version of the reproducible experiment with control used by physicists and biologists and chemists. But we learn mathematics, we discover mathematics, we create mathematics using intuition and trial and error. We draw pictures. Certainly, we try things and twist things around and bend things to try to make them work. Unfortunately, Bourbaki does not teach any part of this latter process." (Steven G Krantz, "The Proof is in the Pudding: The Changing Nature of Mathematical Proof", 2010)

"The objectivist view is that probabilities are real aspects of the universe - propensities of objects to behave in certain ways - rather than being just descriptions of an observer’s degree of belief. For example, the fact that a fair coin comes up heads with probability 0.5 is a propensity of the coin itself. In this view, frequentist measurements are attempts to observe these propensities. Most physicists agree that quantum phenomena are objectively probabilistic, but uncertainty at the macroscopic scale - e.g., in coin tossing - usually arises from ignorance of initial conditions and does not seem consistent with the propensity view." (Stuart J Russell & Peter Norvig, "Artificial Intelligence: A Modern Approach", 2010)

"There are actually two sides to the success of mathematics in explaining the world around us (a success that Wigner dubbed ‘the unreasonable effectiveness of mathematics’), one more astonishing than the other. First, there is an aspect one might call ‘active’. When physicists wander through nature’s labyrinth, they light their way by mathematics - the tools they use and develop, the models they construct, and the explanations they conjure are all mathematical in nature. This, on the face of it, is a miracle in itself. […] But there is also a ‘passive’ side to the mysterious effectiveness of mathematics, and it is so surprising that the 'active' aspect pales by comparison. Concepts and relations explored by mathematicians only for pure reasons - with absolutely no application in mind - turn out decades (or sometimes centuries) later to be the unexpected solutions to problems grounded in physical reality!" (Mario Livio, "Is God a Mathematician?", 2011)

"The standard view among most theoretical physicists, engineers and economists is that mathematical models are syntactic" (linguistic) items, identified with particular systems of equations or relational statements. From this perspective, the process of solving a designated system of" (algebraic, difference, differential, stochastic, etc.) equations of the target system, and interpreting the particular solutions directly in the context of predictions and explanations are primary, while the mathematical structures of associated state and orbit spaces, and quantity algebras – although conceptually important, are secondary." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"Order is not universal. In fact, many chaologists and physicists posit that universal laws are more flexible than first realized, and less rigid - operating in spurts, jumps, and leaps, instead of like clockwork. Chaos prevails over rules and systems because it has the freedom of infinite complexity over the known, unknown, and the unknowable." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

"One of the most crucial developments in theoretical physics was the move from theories dependent on fixed, non-dynamical background space-time structures to background-independent theories, in which the space-time structures themselves are dynamical entities. [...] Even today, many physicists and philosophers do not fully understand the significance of this development, let alone accept it in practice. One must assume that, in an empty region of space-time, the points have no inherent individuating properties - nor indeed are there any spatio-temporal relations between them - that do not depend on the presence of some metric tensor field. [...] Thus, general relativity became the first fully dynamical, background- independent space-time theory." (John Stachel, "The Hole Argument", 2014)

"Ironically, conventional quantum mechanics itself involves a vast expansion of physical reality, which may be enough to avoid Einstein Insanity. The equations of quantum dynamics allow physicists to predict the future values of the wave function, given its present value. According to the Schrödinger equation, the wave function evolves in a completely predictable way. But in practice we never have access to the full wave function, either at present or in the future, so this 'predictability' is unattainable. If the wave function provides the ultimate description of reality - a controversial issue!" (Frank Wilczek, "Einstein’s Parable of Quantum Insanity", 2015)

"The goal of physics is to explain the workings of the nonliving world. At first, philosophers described the properties of real objects: the wandering of planets across the night sky, the formation of ice, or the sound of a lyre. When attention turned to things that couldn’t be seen or measured so easily, physicists invented mechanical models to take the place of real things." (Hans C von Baeyer, "QBism: The future of quantum physics", 2016)

"Today, Euler’s formula is a tool as basic to electrical engineers and physicists as the spatula is to short-order cooks. It’s arguable that the formula’s ability to simplify the design and analysis of circuits contributed to the accelerating pace of electrical innovation during the twentieth century." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

⚛On Physics: On Simultaneity IX

"By means of a revision of the concept of simultaneity in a shapable form I arrived at the special relativity theory.” (Albert Einstein, 1924)

"[...] even in a temporal description of nature given by a relational theory of time. However, a theory, like the special theory of relativity, that denies the existence of an infinitely fast causal chain, deprives the concept of absolute simultaneity of its physical meaning even within a single inertial system. [...]  But since the metrical concept of velocity presupposes that we know the meaning of a transit time and since such a time, in turn, depends on a prior criterion of clock synchronization or simultaneity, we must first formulate the limiting property of electromagnetic chains [the fastest causal chain] without using the concept of simultaneity of noncoincident events." (Adolf Grünbaum, "Logical and philosophical foundations of the special theory of relativity", American Journal of Physics 23, 1955)

"The ‘relativity’ of the new theory - one of the most solidly verified theories in the entire range of physics - is chiefly, therefore, a relativity of simultaneity." (Ernan McMullin, "Simultaneity", 1967)

"By far the most important consequence of the conceptual revolution brought about in physics by relativity and quantum theory lies not in such details as that meter sticks shorten when they move or that simultaneous position and momentum have no meaning, but in the insight that we had not been using our minds properly and that it is important to find out how to do so." (Percy W Bridgman, "Quo Vadis",[in Gerald Holton (ed.), "Science and the Modern Mind"] 1971)

"According to the special theory there is a finite limit to the speed of causal chains, whereas classical causality allowed arbitrarily fast signals. Foundational studies […] soon revealed that this departure from classical causality in the special theory is intimately related to its most dramatic consequences: the relativity of simultaneity, time dilation, and length contraction. By now it had become clear that these kinematical effects are best seen as consequences of Minkowski space-time, which in turn incorporates a nonclassical theory of causal structure. However, it has not widely been recognized that the converse of this proposition is also true: the causal structure of Minkowski space-time contains within itself the entire geometry (topological and metrical structure) of Minkowski space-time." (John A. Winnie," The Causal Theory of Space-Time", 1977)

"It is hard to overestimate the impact of Einstein’s definition of distant simultaneity on philosophy in this century, set, as the words were, in the context of a highly successful theory of physics." (Graham Nerlich, 1982)

📓On Literature: On Physics (From Fiction to Science-Fiction)

"The mathematicians and physics men have their mythology; they work alongside the truth, never touching it; their equations are false But the things work. Or, when gross error appears, they invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers, "The Beginning and the End and Other Poems, The Great Wound", 1963) 

"Their minds sang with the ecstatic knowledge that either what they were doing was completely and utterly and totally impossible or that physics had a lot of catching up to do." (Douglas Adams, "So Long, and Thanks for All the Fish", 1985)

"All reality is a game. Physics at its most fundamental, the very fabric of our universe, results directly from the interaction of certain fairly simple rules, and chance; the same description may be applied to the best, most elegant and both intellectually and aesthetically satisfying games. By being unknowable, by resulting from events which, at the sub-atomic level, cannot be fully predicted, the future remains malleable, and retains the possibility of change, the hope of coming to prevail; victory, to use an unfashionable word. In this, the future is a game; time is one of its rules." (Iain Banks, "The Player of Games", 1988)

"Physics is the basic science. One can easily argue that all other sciences are specialized aspects of physics." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"It does highlight the irony that everything you go to see is changed by the very action of going to see it, which is the sort of problem which physicists have been wrestling with for most of this century." (Douglas Adams, "Last Chance to See", 1990)

"The world isn’t fair, the universe isn’t fair. Physics, chemistry and mathematics, they aren’t fair. Or unfair, for that matter. Fairness is an idea, and  only conscious creatures have ideas. That’s us. We have ideas about right and wrong. We invent the idea of justice so that we can judge whether something is good or bad. We develop morality. We create rules to live by and call them laws, all to make life more fair." (Iain M Banks, "The Business", 1999)

"Almost every credible physicist will tell you there’s nothing in physics that says time travel can’t happen [...]" (Peter Clines, "The End of the Experiment", [in J W Schnarrv (ed), "Timelines"] 2010)

"Physics admits of a lovely unification, not just at the level of fundamental forces, but when considering its extent and implications. Classifications like "optics" or "thermodynamics" are just straitjackets, preventing physicists from seeing countless intersections." (Ted Chiang, "Arrival: Film tie-in", 2016)

"'Do you really believe in physics?'  'I dont know what that means. Physics tries to draw a numerical picture of the world. I dont know that it actually explains anything. You cant illustrate the unknown. Whatever that might mean.'" (Cormac McCarthy, "Stella Maris", 2022)

⚛On Physics: On Electromagnetism (2010-)

"First, without a physical interpretation, a purely mathematical structure, here a Lie algebra, would have no empirical content. Second, if we interpret the Lie algebra in terms of physical structures, taking electromagnetic and weak currents as its representations, then we have physical content, but only at the phenomenological level. In order to understand the physical structures (the currents) properly, we have to move deeper onto the level of their constituents (hadrons or quarks) and their dynamics so that we can have a dynamic understanding of the behavior of the currents, and thus of many features of current algebra and of reasons why current algebra is so successful." (Tian Yu Cao,"From Current Algebra to Quantum Chromodynamics: A Case for Structural Realism", 2010)

"The underlying idea of current algebra, light-cone current algebra included, was to exploit the broken symmetry of strong interactions. The idea was pursued through abstracting physical predictions, as the consequences of the symmetry and in the form of certain algebraic relations obeyed by weak and electromagnetic currents of hadrons to all orders in strong interactions, from some underlying mathematical field theoretical models of hadrons and their interactions, more specifically from models of quark fields universally coupled to a neutral gluon field." (Tian Yu Cao,"From Current Algebra to Quantum Chromodynamics: A Case for Structural Realism", 2010)

"Rotations and translations are global symmetries: they apply uniformly across the whole of space and time. A rotation about some axis rotates every point in space through the same angle. Gauge symmetries are different: they are local symmetries, which can vary from point to point in space. In the case of electromagnetism, these local symmetries are changes of phase. A local oscillation of the electromagnetic field has both an amplitude (how big it is) and a phase (the time at which it reaches its peak)." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"The classical theories of space, time, and matter were brought to their peak in James Clerk Maxwell’s equations for electromagnetism. This elegant system of equations unified two of nature’s forces, previously thought to be distinct. In place of electricity and magnetism, there was a single electromagnetic field. A field pervades the whole of space, as if the universe were filled with some kind of invisible fluid. At each point of space we can measure the strength and direction of the field, as if that fluid were flowing in mathematical patterns. For some purposes the electromagnetic field can be split into two components, the electric field and the magnetic field. But a moving magnetic field creates an electric one, and conversely, so when it comes to dynamics, both fields must be combined into a single more complex one." (Ian Stewart, "Visions of Infinity", 2013)

"The invariance of physical laws with respect to position or orientation (i.e., the symmetry of space) gives rise to conservation laws for linear and angular momentum. Sometimes the implications of symmetry invariance are far more complicated or sophisticated than might at first be supposed; the invariance of the forces predicted by electromagnetic theory when measurements are made in observation frames moving uniformly at different speeds (inertial frames) was an important clue leading Einstein to the discovery of special relativity. With the advent of quantum mechanics, considerations of angular momentum and spin introduced new symmetry concepts into physics. These ideas have since catalyzed the modern development of particle theory." (George B Arfken et al, "Mathematical Methods for Physicists: A comprehensive guide", 2013)

"Symmetry is not enough by itself. In electromagnetism, for example, if you write down all the symmetries we know, such as Lorentz invariance and gauge invariance, you don’t get a unique theory that predicts the magnetic moment of the electron. The only way to do that is to add the principle of renormalisability - which dictates a high degree of simplicity in the theory and excludes these additional terms that would have changed the magnetic moment of the electron from the value Schwinger calculated in 1948." (Steven Weinberg, CERN Courier, [interview with Matthew Chalmers] 2017)

"Electromagnetism is the theory describing the interactions of electric and magnetic fields with matter based upon charges and currents. [...] Once delving a bit deeper into the theory of electromagnetism, it will become apparent that the use of tensors will help us significantly. In particular, when formulating Maxwell’s equations in the framework of special relativity, it will become clear that the electric and magnetic fields are different components of a rank two anti-symmetric tensor in four-dimensional space-time rather than vector fields in space that depend on time." (Mattias Blennow, "Mathematical Methods for Physics and Engineering", 2018)

"Euler’s formula - although deceptively simple - is actually staggeringly conceptually difficult to apprehend in its full glory, which is why so many mathematicians and scientists have failed to see its extraordinary scope, range, and ontology, so powerful and extensive as to render it the master equation of existence, from which the whole of mathematics and science can be derived, including general relativity, quantum mechanics, thermodynamics, electromagnetism and the strong and weak nuclear forces! It’s not called the God Equation for nothing. It is much more mysterious than any theistic God ever proposed." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

⚛On Physics: On Electromagnetism (2000-2009)

"For [...] small things, there must be something else. There is. We call it the electric interaction (more generally, the electromagnetic interaction), and it arises from an endowment of matter known as the electric charge. Standing still, an electrically charged particle throws up an electric potential to which other charged particles can respond.Electric or magnetic, charge gives rise to both. Whether we say 'electric potential' (because we perceive a charge to be at rest) or "magnetic potential" (because we perceive a charge to be in motion), the difference lies solely in our point of view. The source is one. From the world of mass we descend [...] into the world of charge, ready to see our most familiar surroundings in a new light. Let there be electric charge." (Michael Munowitz, "Knowing: The Nature of Physical Law", 2005)

"We divide math up into separate areas (analysis, mechanics, algebra, geometry, electromagnetism, number theory, quantum mechanics, etc.) to clarify the study of each part; but the equally valuable activity of integrating the components into a working whole is all too often neglected. Without it, the stated aim of ‘taking something apart to see how it ticks’ degenerates imperceptibly into ‘taking it apart to ensure it never ticks again’." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"In formal terms, the ground state energy (vacuum energy) of the electromagnetic quantum field is infinite. This causes mathematical trouble in quantum electrodynamics." Eberhard Zeidler, "Quantum Field Theory II: Quantum Electrodynamics", 2006)

"Topology is rooted in Maxwell’s theory on the electromagnetic field." (Eberhard Zeidler, "Quantum Field Theory I: Basics in Mathematics and Physics", 2006)

"In a Newtonian view, space and time are separate and different. Symmetries of the laws of physics are combinations of rigid motions of space and an independent shift in time. But... these transformations do not leave Maxwell's equations invariant. Pondering this, the mathematicians Henri Poincaré and Hermann Minkowski were led to a new view of the symmetries of space and time, on a purely mathematical level. If they had described these symmetries in physical terms, they would have beaten Einstein to relativity, but they avoided physical speculations. They did understand that symmetries in the laws of electromagnetism do not affect space and time independently but mix them up. The mathematical scheme describing these intertwined changes is known as the Lorentz group, after the physicist, Hendrik Lorentz." (Ian Stewart, "Why Beauty Is Truth: The History of Symmetry", 2008)

"The abstractions of Einstein's curved space and time gave rise to analogies and pictures that played a new explanatory role. Space and time gave way to space-time, visible light was augmented by images across the rest of the electromagnetic spectrum, and we realise d that we could see back towards the apparent beginnings of time." (John D Barrow,"Cosmic Imagery: Key Images in the History of Science", 2008)

"There’s matter, like the electron; antimatter, like the positron; and then there are things that are neither matter nor antimatter. The most familiar example of something that is beyond substance is electromagnetic radiation. All electromagnetic radiation, from gamma rays through X-rays and ultra-violet to visible light, infra red, and radio waves, consists of photons of different energies. Matter and antimatter can cancel one another out, their annihilation leaving non-substance in the form of photons; if the conditions are right this sequence can happen in reverse where photons turn into pieces of matter and antimatter." (Frank Close, "Antimatter", 2009)

⚛On Physics: On Electromagnetism (1975-1999)

"Electromagnetic fields [...] are neither charged nor magnetized and thus cannot contribute to their own source. The general theory does reveal a new feature of electromagnetic fields... Since they have energy, they produce gravitational effects and thus affect the structure of space-time, which, in turn, means that an electromagnetic field can affect the motion of non-charged matter." (Nancy Nersessian, "Faraday to Einstein: Constructing Meaning in Scientific Theories", 1984)

"The [...] weak force [...] couples to both quarks and leptons, and is very short-ranged due to the large rest mass of the messenger quanta involved. Its effective strength is usually many orders of magnitude weaker than electromagnetism, and its action can cause particles to change identity, as when a neutron decays. Unlike the electromagnetic and strong forces, the weak force violates parity conservation." (Paul Davies, "Forces of Nature", 1986)

"If grand unified theories are correct, we ought to be able to derive the relative power of the strong, weak, and electromagnetic interactions at accessible energies from their presumed equality at much higher energies. When this is attempted, a wonderful result emerges. …in the form first calculated by Howard Georgi, Helen Quinn, and Steven Weinberg […] The couplings of strong-interaction gluons decrease, those of the [weak interaction] W bosons stay roughly constant, and those of the [electromagnetic interaction] photons increase at short distances [or high energies] - so they all tend to converge, as desired." (Frank Wilczek, "Longing for the Harmonies: Themes and Variations from Modern Physics", 1987)

"Strong, weak and electromagnetic interaction are evidently part of a grand unified theory. These temperatures are today quite inaccessible. They were achieved only in the earliest moments of the Big Bang. Since then, the universe has congealed, losing its symmetry." (Sheldon L Glashow, "The Charm of Physics", 1991)

"Einstein was thus faced with the following apparent problem. Either give up the principle of relativity, which appears to make physics possible by saying that the laws of physics are independent of where you measure them, as long as you are in a state of uniform motion; or give up Maxwell’s beautiful theory of electromagnetism and electromagnetic waves. In a truly revolutionary move, he chose to give up neither. [...] It is a testimony to his boldness and creativity not that he chose to throw out existing laws that clearly worked, but rather that he found a creative way to live within their framework. So creative, in fact, that it sounds nuts." (Lawrence M Krauss, "Fear of Physics: A Guide for the Perplexed", 1993)

"The problems associated with the initial singularity of the universe bring us to what is called the theory of everything. It is an all-encompassing theory that would completely explain me origin of the universe and everything in it. It would bring together general relativity and quantum mechanics, and explain everything there is to know about the elementary particles of the universe, and the four basic forces of nature (gravitational, electromagnetic, weak, and strong nuclear forces). Furthermore, it would explain the basic laws of nature and the fundamental constants of nature such as the speed of light and Planck's constant." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"The reason general relativity cannot be unified with electromagnetic theory seems to be related to its nonlinearity. To unify the two fields properly we have to construct a "quantized" version of relativity; in other words, we have to quantize it, and so far no one has." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

On Physics: On Electromagnetism (1925-1949)

"The velocity of light is one of the most important of the fundamental constants of Nature. Its measurement by Foucault and Fizeau gave as the result a speed greater in air than in water, thus deciding in favor of the undulatory and against the corpuscular theory. Again, the comparison of the electrostatic and the electromagnetic units gives as an experimental result a value remarkably close to the velocity of light – a result which justified Maxwell in concluding that light is the propagation of an electromagnetic disturbance. Finally, the principle of relativity gives the velocity of light a still greater importance, since one of its fundamental postulates is the constancy of this velocity under all possible conditions." (A.A. Michelson, "Studies in Optics", 1927)

"The spirit of seeking which animates us refuses to regard any kind of creed as its goal. It would be a shock to come across a university where it was the practice of the students to recite adherence to Newton's laws of motion, to Maxwell's equations and to the electromagnetic theory of light. We should not deplore it the less if our own pet theory happened to be included, or if the list were brought up to date every few years. We should say that the students cannot possibly realise the intention of scientific training if they are taught to look on these results as things to be recited and subscribed to. Science may fall short of its ideal, and although the peril scarcely takes this extreme form, it is not always easy, particularly in popular science, to maintain our stand against creed and dogma." (Arthur S Eddington, "Science and the Unseen World", 1929)

"Classical mechanics has been developed continuously from the time of Newton and applied to an ever-widening range of dynamical systems, including the electromagnetic field in interaction with matter. The underlying ideas and the laws governing their application form a simple and elegant scheme, which one would be inclined to think could not be seriously modified without having all its attractive features spout. Nevertheless it has been found possible to set up a new scheme, called quantum mechanics, which is more suitable for the description of phenomena on the atomic scale and which is in some respects more elegant and satisfying than the classical scheme. This possibility is due to the changes which the new scheme involves being of a very profound character and not clashing with the features of the classical theory that make it so attractive, as a result of which all these features can be incorporated in the new scheme." (Paul Dirac, "The Principles of Quantum Mechanics", 1930)

"In electromagnetism [...] the law of the inverse square had been supreme, but, as a consequence of the work of Faraday and Maxwell, it was superseded by the field. And the same change took place in the theory of gravitation. By and by the material particles, electrically charged bodies, and magnets which are the things that we actually observe come to be looked upon only as 'singularities' in the field. So far this transformation from the force to the potential, from the action at a distance to the field, is only a purely mathematical operation." (Willem de Sitter,"Kosmos", 1932)

"Two points should be specially emphasized in connection with the general theory of relativity. First, it is a purely physical theory, invented to explain empirical physical facts, especially the identity of gravitational and inertial mass, and to coordinate and harmonize different chapters of physical theory, especially mechanics and electromagnetic theory. It has nothing metaphysical about it. Its importance from a metaphysical or philosophical point of view is that it aids us to distinguish in the observed phenomena what is absolute, or due to the reality behind the phenomena, from what is relative, i.e. due to the observer.S econd, it is a pure generalization, or abstraction, like Newton's system of mechanics and law of gravitation. It contains no hypothesis, as contrasted with the atomic theory or the theory of quanta, which are based on hypothesis. It may be considered as the logical sequence and completion of Newton's Principia. The science of mechanics was founded by Archimedes, who had a clear conception of the relativity of motion, and may be called the first relativist. Galileo, who was inspired by the reading of the works of Archimedes, took the subject up where his great predecessor had left it. His fundamental discovery is the law of inertia, which is the backbone of Newton's classical system of mechanics, and retains the same central position in Einstein's relativistic system. Thus one continuous line of thought can be traced through the development of our insight into the mechanical processes of nature... characterized by the sequence [...] Archimedes, Galileo, Newton, Einstein." (Willem de Sitter, "The Astronomical Aspect of the Theory of Relativity", 1933)

"The electromagnetic field theory is, for the modern physicist, as real as the chair on which he sits." (Leopold Infeld, "The Evolution of Physics, Field and Ether", 1938)

"Faraday, Maxwell, Larmor and a great number of others tried to explain electromagnetic action on these lines, but all attempts failed, and it began to seem impossible that any properties of ether could explain the observed pattern of events." (James Jeans, "Physics and Philosophy", 1942)

"Then the theory of relativity came and explained the cause of the failure. Electric action requires time to travel from one point of space to another, the simplest instance of this being the finite speed of travel of light […] Thus electromagnetic action may be said to travel through space and time jointly. But by filling space and space alone [excluding time] with an ether, the pictorial representations had all supposed a clear-cut distinction between space and time." (James H Jeans, "Physics and Philosophy", 1942)

"It is as dangerous for people unaccustomed to handling words and unacquainted with their technique to tinker about with these heavily-charged nuclei of emotional power as it would be for me to burst into a laboratory and play about with a powerful electromagnet or other machine highly charged with electrical force." (Dorothy L Sayers, "Unpopular Opinions", 1946)

"In the development of electromagnetic theory, there has been a continual and significant trend, which in a way has set the pattern for the development of all of theoretical physics. This has been the trend away from the concept known as "action at a distance" toward the concept of field theory." (John C Slater & Nathaniel H Frank, "Electromagnetism", 1947)

"The first forces brought into mathematical formulation were gravitational forces, as seen in planetary motion. Next were elastic forces. Then followed electric and magnetic forces... Their [electric and magnetic forces] study was mostly a product of the nineteenth century. [...] electromagnetic forces are of a far wider application than was first supposed. It has become evident that, instead of being active only in electrostatic and electromagnetic applications such as the telegraph, dynamo, and radio, the forces between the nuclei and electrons of single atoms, the chemical forces between atoms and molecules, the forces of cohesion and elasticity holding solids together, are all of an electric nature. [...] electromagnetic theory [...] carries us rather far into the structure of matter [...] The equations underlying the theory, Maxwell's equations, are relatively simple, but not nearly so simple as Newton's Newton's laws of motion." (John C Slater & Nathaniel H Frank, "Electromagnetism", 1947)

"This fact, that all charges are integral multiples of a fundamental unit, is still one of the unexplained puzzles of fundamental physics. It does not in any way contradict electromagnetic theory, but it is not predicted by it, and until we have a more fundamental theory that explains it, we shall not feel that we really understand electromagnetic phenomena thoroughly. Presumably its explanation will not come until we understand quantum theory more thoroughly than we do at present." (John C Slater & Nathaniel H Frank, "Electromagnetism", 1947)

"Although the Special Theory of Relativity does not account for electromagnetic phenomena, it explains many of their properties. General Relativity, however, tells us nothing about electromagnetism. In Einstein's space-time continuum gravitational forces are absorbed in the geometry, but the electromagnetic forces are quite unaffected. Various attempts have been made to generate the geometry of space-time so as to produce a unified field theory incorporating both gravitational and electromagnetic forces." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

On Physics: On Electromagnetism (1950-1974)

"[...] even in a temporal description of nature given by a relational theory of time. However, a theory, like the special theory of relativity, that denies the existence of an infinitely fast causal chain, deprives the concept of absolute simultaneity of its physical meaning even within a single inertial system. [...]  But since the metrical concept of velocity presupposes that we know the meaning of a transit time and since such a time, in turn, depends on a prior criterion of clock synchronization or simultaneity, we must first formulate the limiting property of electromagnetic chains [the fastest causal chain] without using the concept of simultaneity of noncoincident events." (Adolf Grünbaum, "Logical and philosophical foundations of the special theory of relativity", American Journal of Physics 23, 1955)

"There is nothing in the world except empty curved space. Matter, charge, electromagnetism, and other fields are only manifestations of the curvature of space." (John A Wheeler, 1957)

"In essence, the curvature in space created by the electromagnetic field is the electromagnetic field; and this curvature can in principle be detected by purely geometric measurements." (Kip Thorne, "The Dynamics of Space-Time", 1963) 

"It requires a much higher degree of imagination to understand the electromagnetic field than to understand invisible angels. […] I speak of the E and B fields and wave my arms and you may imagine that I can see them […] [but] I cannot really make a picture that is even nearly like the true waves." (Richard Feynman, "The Feynman Lectures on Physics", 1964)

"Thermostatics, which even now is usually called thermodynamics, has an unfortunate history and a cancerous tradition. It arose in a chaos of metaphysical and indeed irrational controversy, the traces of which drip their poison even today. As compared with the older science of mechanics and the younger science of electromagnetism, its mathematical structure is meager. Though claims for its breadth of application are often extravagant, the examples from which its principles usually are inferred are most special, and extensive mathematical developments based on fundamental equations, such as typify mechanics and electromagnetism, are wanting. The logical standards acceptable in thermostatics fail to meet the criteria of other exact sciences [...]." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"Let us consider, for a moment, the world as described by the physicist. It consists of a number of fundamental particles which, if shot through their own space, appear as waves, and are thus [...] of the same laminated structure as pearls or onions, and other wave forms called electromagnetic which it is convenient, by Occam’s razor, to consider as travelling through space with a standard velocity. All these appear bound by certain natural laws which indicate the form of their relationship." (G Spencer-Brown, "Laws of Form", 1969)

On Physics: On Electromagnetism (1900-1924)

"By laying down the relativity postulate from the outset, sufficient means have been created for deducing henceforth the complete series of Laws of Mechanics from the principle of conservation of energy (and statements concerning the form of the energy) alone." (Hermann Minkowski, "The Fundamental Equations for Electromagnetic Processes in Moving Bodies", 1907)

"It would be very unsatisfactory if the new way of looking at the time-concept, which permits a Lorentz transformation, were to be confined to a single part of Physics." (Hermann Minkowski, "The Fundamental Equations for Electromagnetic Processes in Moving Bodies", 1907)

"Electric and magnetic forces. May they live for ever, and never be forgot, if only to remind us that the science of electromagnetics, in spite of the abstract nature of its theory, involving quantities whose nature is entirely unknown at the present, is really and truly founded on the observations of real Newtonian forces, electric and magnetic respectively." (Oliver Heaviside, "Electromagnetic Theory", 1912)

"I cannot but regard the ether, which can be the seat of an electromagnetic field with its energy and its vibrations, as endowed with a certain degree of substantiality, however different it may be from all ordinary matter." (Hendrik Lorentz, "The Theory of Electrons and Its Applications to the Phenomena of Light and Radiant Heat", 1916)

"A new theory by the author has been added, which draws the physical inferences consequent on the extension of the foundations of geometry beyond Reimann... and represents an attempt to derive from world-geometry not only gravitational but also electromagnetic phenomena. Even if this theory is still only in its infant stage, I feel convinced that it contains no less truth than Einstein's Theory of Gravitation—whether this amount of truth is unlimited or, what is more probable, is bounded by the Quantum Theory." (Albert Einstein, "The Meaning of Relativity", 1922)

"Before the development of the theory of relativity it was known the principle of energy and momentum could be expressed in a differential form for the electromagnetic field. The four-dimensional formulation of these principles leads to an important conception, that of the energy tensor, which is important of the further development of the theory of relativity." (Albert Einstein, "The Meaning of Relativity", 1922)

"First, the physicists in the persons of Faraday and Maxwell, proposed the 'electromagnetic field' in contradistinction to matter, as a reality of a different category. Then, during the last century, the mathematicians, […] secretly undermined belief in the evidence of Euclidean Geometry. And now, in our time, there has been unloosed a cataclysm which has swept away space, time, and matter hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope and entailing a deeper vision. This revolution was promoted essentially by the thought of one man, Albert Einstein." (Hermann Weyl," Space, Time, Matter", 1922)

"When Faraday filled space with quivering lines of force, he was bringing mathematics into electricity. When Maxwell stated his famous laws about the electromagnetic field it was mathematics. The relativity theory of Einstein which makes gravity a fiction, and reduces the mechanics of the universe to geometry, is mathematical research." (James B Shaw, "The Spirit of Research", The Monist No. 4, 1922)

On Physics: On Electromagnetism (-1899)

"My theory of electrical forces is that they are called into play in insulating media by slight electric displacements, which put certain small portions of the medium into a state of distortion which, being resisted by the elasticity of the medium, produces an electromotive force [...] I suppose the elasticity of the sphere to react on the electrical matter surrounding it, and press it downwards. From the determination by Kohlrausch and Weber of the numerical relation between the statical and magnetic effects of electricity, I have determined the elasticity of the medium in air, and assuming that it is the same with the luminiferous ether I have determined the velocity of propagation of transverse vibrations." (James C Maxwell, [Letter to Michael Faraday] 1861) 

"The general equations are next applied to the case of a magnetic disturbance propagated through a non-conductive field, and it is shown that the only disturbances which can be so propagated are those which are transverse to the direction of propagation, and that the velocity of propagation is the velocity v, found from experiments such as those of Weber, which expresses the number of electrostatic units of electricity which are contained in one electromagnetic unit. This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws." (James C Maxwell, "A Dynamical Theory of the Electromagnetic Field", 1865)

"The theory I propose may therefore be called a theory of the Electromagnetic Field because it has to do with the space in the neighbourhood of the electric or magnetic bodies, and it may be called a Dynamical Theory, because it assumes that in the space there is matter in motion, by which the observed electromagnetic phenomena are produced." (James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", 1865)

"What, then, is light according to the electromagnetic theory? It consists of alternate and opposite rapidly recurring transverse magnetic disturbances, accompanied with electric displacements, the direction of the electric displacement being at the right angles to the magnetic disturbance, and both at right angles to the direction of the ray." (James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", 1865)

"Faraday is, and must always remain, the father of that enlarged science of electromagnetism." (James C Maxwell, "Faraday", Nature, 1873)

"The experimental investigation by which Ampere established the law of the mechanical action between electric currents is one of the most brilliant achievements in science. The whole theory and experiment, seems as if it had leaped, full grown and full armed, from the brain of the 'Newton of Electricity'. It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics." (James C Maxwell, "A Treatise on Electricity and Magnetism", 1873)

"One of the chief peculiarities of this treatise is the doctrine that the true electric current, on which the electromagnetic phenomena depend, is not the same thing as the current of conduction, but that the time-variation of the electric displacement must [also] be taken into account [...]" (James C Maxwell, "A Treatise on Electricity and Magnetism" 1881)

"This maze of symbols, electric and magnetic potential, vector potential, electric force, current, displacement, magnetic force, and induction, have been practically reduced to two, electric and magnetic force." (George F Fitzgerald, The Electrician, [bookmreview] 1893) 

On Physics: On Gravity (1850-1899)

"It is no valid objection that science as yet throws no light on the far higher problem of the essence or origin of life. Who can explain gravity? No one now objects to following out the results consequent on this unknown element of attraction" (Charles Darwin, "The Origin of Species", 1859)

"Newton's law is nothing but the statistics of gravitation, it has no power whatever. Let us get rid of the idea of power from law altogether. Call law tabulation of facts, expression of facts, or what you will; anything rather than suppose that it either explains or compels."(Florence Nightingale, "Suggestions for Thought", 1860)

"Any opinion as to the form in which the energy of gravitation exists in space is of great importance, and whoever can make his opinion probable will have, made an enormous stride in physical speculation. The apparent universality of gravitation, and the equality of its effects on matter of all kinds are most remarkable facts, hitherto without exception; but they are purely experimental facts, liable to be corrected by a single observed exception. We cannot conceive of matter with negative inertia or mass; but we see no way of accounting for the proportionality of gravitation to mass by any legitimate method of demonstration. If we can see the tails of comets fly off in the direction opposed to the sun with an accelerated velocity, and if we believe these tails to be matter and not optical illusions or mere tracks of vibrating disturbance, then we must admit a force in that direction, and we may establish that it is caused by the sun if it always depends upon his position and distance." (James C Maxwell, [Letter to William Huggins] 1868)

"Economic laws are statements with regard to the tendencies of man's action under certain conditions. They are hypothetical only in the same sense as are the laws of the physical sciences: for those laws also contain or imply conditions. But there is more difficulty in making the conditions clear, and more danger in any failure to do so, in economics than in physics. The laws of human action are not indeed as simple, as definite or as clearly ascertainable as the law of gravitation; but many of them may rank with the laws of those natural sciences which deal with complex subject-matter." (Alfred Marshall, "Principles of Economics", 1890)

"Everything in physical science, from the law of gravitation to the building of bridges, from the spectroscope to the art of navigation, would be profoundly modified by any considerable inaccuracy in the hypothesis that our actual space is Euclidean. The observed truth of physical science, therefore, constitutes overwhelming empirical evidence that this hypothesis is very approximately correct, even if not rigidly true." (Bertrand Russell, "Foundations of Geometry", 1897)

On Physics: On Gravity (1925-1949)

"The essence of Riemann's discoveries consists in having shown that there exist a vast number of possible types of spaces, all of them perfectly self-consistent. When, therefore, it comes to deciding which one of these possible spaces real space will turn out to be, we cannot prejudge [...] Experiment and observation alone can yield us a clue. To a first approximation, experiment and observation prove space to be Euclidean, and this accounts for our natural belief [...] merely by force of habit. But experiment is necessarily innacurate, and we cannot foretell whether our opinions will not have to be modified when our experiments are conducted with greater accuracy. Riemann's views thus place the problem of space on an empirical basis excluding all a priori assertions on the subject [...] the relativity theory is very intimately connected with this empirical philosophy; for. [...] Einstein is compelled to appeal to a varying non-Euclideanism of four-dimensional space-time in order to account with extreme simplicity for gravitation. [...] had the extension of the universe been restricted on a priori grounds [...] to three dimensional Euclidean space, Einstein's theory would have been rejected on first principles. [...] as soon as we recognise that the fundamental continuum of the universe and its geometry cannot be posited a priori [...] a vast number of possibilities are thrown open. Among these the four-dimensional space-time of relativity, with its varying degrees of non-Euclideanism, finds a ready place." (Aram D'Abro,  "The Evolution of Scientific Thought from Newton to Einstein", [Forward] 1927)

"Light-waves in passing a massive body such as the sun are deflected through a small angle. This is additional evidence that the Newtonian picture of gravitation as a tug is inadequate. You cannot deflect waves by tugging at them, and clearly another representation of the agency which deflects them must be found." (Arthur Eddington, "The Nature of the Physical World", 1928)

"The force acting on the pendulum is proportional to its active mass, its inertia is proportional to its passive mass, so that the period will depend on the ratio of the passive and the active mass. Consequently the fact that the period of all these different pendulums was the same, proves that this ratio is a constant, and can be made equal to unity by a suitable choice of units, i. e., the inertial and the gravitational mass are the same." (Willem de Sitter, "The Astronomical Aspect of the Theory of Relativity", 1933)

"In electromagnetism [...] the law of the inverse square had been supreme, but, as a consequence of the work of Faraday and Maxwell, it was superseded by the field. And the same change took place in the theory of gravitation. By and by the material particles, electrically charged bodies, and magnets which are the things that we actually observe come to be looked upon only as 'singularities' in the field. So far this transformation from the force to the potential, from the action at a distance to the field, is only a purely mathematical operation." (Willem de Sitter,"Kosmos", 1932)

"Two points should be specially emphasized in connection with the general theory of relativity. First, it is a purely physical theory, invented to explain empirical physical facts, especially the identity of gravitational and inertial mass, and to coordinate and harmonize different chapters of physical theory, especially mechanics and electromagnetic theory. It has nothing metaphysical about it. Its importance from a metaphysical or philosophical point of view is that it aids us to distinguish in the observed phenomena what is absolute, or due to the reality behind the phenomena, from what is relative, i.e. due to the observer.S econd, it is a pure generalization, or abstraction, like Newton's system of mechanics and law of gravitation. It contains no hypothesis, as contrasted with the atomic theory or the theory of quanta, which are based on hypothesis. It may be considered as the logical sequence and completion of Newton's Principia. The science of mechanics was founded by Archimedes, who had a clear conception of the relativity of motion, and may be called the first relativist. Galileo, who was inspired by the reading of the works of Archimedes, took the subject up where his great predecessor had left it. His fundamental discovery is the law of inertia, which is the backbone of Newton's classical system of mechanics, and retains the same central position in Einstein's relativistic system. Thus one continuous line of thought can be traced through the development of our insight into the mechanical processes of nature... characterized by the sequence [...] Archimedes, Galileo, Newton, Einstein." (Willem de Sitter, "The Astronomical Aspect of the Theory of Relativity", 1933)

"There are necessities and impossibilities in reality which do not obtain in fiction, any more than the law of gravity to which we are subject controls what is represented in a picture. [...] It is the same with pure good; for a necessity as strong as gravity condemns man to evil and forbids him any good, or only within the narrowest limits and laboriously obtained and soiled and adulterated with evil. [...] The simplicity which makes the fictional good something insipid and unable to hold the attention becomes, in the real good, an unfathomable marvel." (Simone Weil, "Morality and Literature", cca. 1941)

"Any region of space-time that has no gravitating mass in its vicinity is uncurved, so that the geodesics here are straight lines, which means that particles move in straight courses at uniform speeds (Newton's first law). But the world-lines of planets, comets and terrestrial projectiles are geodesics in a region of space-time which is curved by the proximity of the sun or earth. […] No force of gravitation is […] needed to impress curvature on world-lines; the curvature is inherent in the space […]" (James H Jeans," The Growth of Physical Science", 1947)

"The first forces brought into mathematical formulation were gravitational forces, as seen in planetary motion. Next were elastic forces. Then followed electric and magnetic forces... Their [electric and magnetic forces] study was mostly a product of the nineteenth century. [...] electromagnetic forces are of a far wider application than was first supposed. It has become evident that, instead of being active only in electrostatic and electromagnetic applications such as the telegraph, dynamo, and radio, the forces between the nuclei and electrons of single atoms, the chemical forces between atoms and molecules, the forces of cohesion and elasticity holding solids together, are all of an electric nature. [...] electromagnetic theory [...] carries us rather far into the structure of matter [...] The equations underlying the theory, Maxwell's equations, are relatively simple, but not nearly so simple as Newton's Newton's laws of motion." (John C Slater & Nathaniel H Frank, "Electromagnetism", 1947)

"Although the Special Theory of Relativity does not account for electromagnetic phenomena, it explains many of their properties. General Relativity, however, tells us nothing about electromagnetism. In Einstein's space-time continuum gravitational forces are absorbed in the geometry, but the electromagnetic forces are quite unaffected. Various attempts have been made to generate the geometry of space-time so as to produce a unified field theory incorporating both gravitational and electromagnetic forces." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

On Physics: On Gravity (2010-)

"Starting with Einstein’s general relativity, differential geometry has started playing a major role in physics. General relativity describes the gravitational fields as a metric property of the spacetime manifold. More precisely, spacetime (i.e., the manifold the points of which are events; we may intuitively say that an event is ‘something that happens in a given point in space at a certain time’) is supposed to be endowed with a Lorentzian metric. This means that spacetime has pointwise the same structure as the Minkowski space of special relativity but in general is not flat, as on the contrary Minkowski space is. Indeed, out of the metric tensor one can construct another tensor field, the curvature field, which measures how far the geometry of spacetime is from that of a flat space. The celebrated Einstein equations prescribe how the matter in our universe determines the curvature of spacetime, and in turn the curvature determines how matter (particles, light rays, extended bodies…) moves." (Claudio Bartocci & Ugo Bruzzo, [Claudio Bartocci et al (Eds), "New Trends in Geometry: Their tole in the natural and social sciences"], 2011)

"The Newtonian universe is material in the sense that the world was viewed as being made up of stuff - tangible, real objects. It was argued that even forces like gravity that appear to act across empty stretches of space are conveyed by tiny particles, or corpuscles. Moreover, since the universe is material, its behavior can be predicted or understood. Things are they way they are for a reason or a cause. The Newtonian world is mathematical, in that it was viewed that the regularities or laws that describe or govern the world are mathematical in nature." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)

"[…] there’s atomic physics - electrons and protons and neutrons, all the stuff of which atoms are made. At these very, very, very small scales, the laws of physics are much the same, but there is also a force you ignore, which is the gravitational force. Gravity is present everywhere because it comes from the entire mass of the universe. It doesn’t cancel itself out, it doesn’t have positive or negative value, it all adds up." (Michael F Atiyah, [interview] 2013)

"The most familiar manifold, however, is the space-time manifold, which has 4 dimensions. It is described by a time coordinate and three spatial coordinates. In addition to being a differentiable manifold, space-time has much more additional structure. It is at the level of this additional structure, which will be the subject of later chapters, that the space-time of Newtonian physics differs from the space-time of special relativity and from the space-times of Einstein’s theory of gravity (also called general relativity)." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"String theory today looks almost fractal. The more closely people explore any one corner, the more structure they find. Some dig deep into particular crevices; others zoom out to try to make sense of grander patterns. The upshot is that string theory today includes much that no longer seems stringy. Those tiny loops of string whose harmonics were thought to breathe form into every particle and force known to nature (including elusive gravity) hardly even appear anymore on chalkboards at conferences." (K C Cole, "The Strange Second Life of String Theory", Quanta Magazine", 2016) [source]

"Granularity is ubiquitous in nature: light is made of photons, the particles of light. The energy of electrons in atoms can acquire only certain values and not others. The purest air is granular, and so, too, is the densest matter. Once it is understood that Newton’s space and time are physical entities like all others, it is natural to suppose that they are also granular. Theory confirms this idea: loop quantum gravity predicts that elementary temporal leaps are small, but finite." (Carlo Rovelli, "The Order of Time", 2018)

"[...] one thing that's worth mentioning, though, it that apart from the dream of understanding physics at a deeper level involving gravity, work in string theory has been useful in shedding lights on more conventional problems in quantum field theory and even in condensed matter physics and as well with applications to mathematics. Apart from its intrinsic interest, those successes are one of the things that tend to give us confidence that we're on the right track. Because, speaking personally, I find it implausible that a completely wrong new physics theory would give rise to useful insights about so many different areas." (Edward Witten, [in "Dirac Conversation: Edward Witten". Interview at Int'l Center for Theoretical Physics], 2024) [source]