Theories: On Relativity Theory (1925-1949)

"The meaning of relativity [...] has been widely misunderstood. Philosophers play with the word, like a child with a doll. Relativity, as I see it, merely denotes that certain physical and mechanical facts, which have been regarded as positive and permanent, are relative with regard to certain other facts in the sphere of physics and mechanics. It does not mean that everything in life is mischievously topsy-turvy." (Albert Einstein [in George Sy Viereck, "What Life Means to Einstein", The Saturday Evening Post, 1929]) 

"[...] the most outstanding achievement of twentieth-century physics is not the theory of relativity with its welding together of space and time, or the theory of quanta with its present apparent negation of the laws of causation, or the dissection of the atom with the resultant discovery that things are not what they seem; it is the general recognition that we are not yet in contact with ultimate reality." (James H Jeans, "The Mysterious Universe", 1930) 

"There is another side to the theory of relativity. [...]×the development of science is in the direction to make it less subjective, to separate more and more in the observed facts that which belongs to the reality behind the phenomena, the absolute, from the subjective element, which is introduced by the observer, the relative. Einstein's theory is a great step in that direction. We can say that the theory of relativity is intended to remove entirely the relative and exhibit the pure absolute." (Willem de Sitter, "Relativity and Modern Theories of the Universe", 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 fundamental presuppositions of classical physics, which led to the scientific picture of the 19th century, had been challenged for the first time by Einstein’s special relativity." (Werner K Heisenberg, 1934)

"We know, since the theory of relativity at least, that empirical sciences are to some degree free in defining dynamical concepts or even in assuming laws, and that only a system as a whole which includes concepts, coordinating definitions, and laws can be said to be either true or false, to be adequate or inadequate to empirical facts. This 'freedom', however, is a somewhat doubtful gift. The manifold of possibilities implies uncertainty, and such uncertainty can become rather painful in a science as young as psychology, where nearly all concepts are open and unsettled. As psychology approaches the state of a logically sound science, definitions cease to be an arbitrary matter. They become far-reaching decisions which presuppose the mastering of the conceptual problems but which have to be guided entirely by the objective facts." (Kurt Lewin, "Principles of topological psychology", 1936)

"The modern theory of relativity, on its mathematical side, is merely an elaboration of Riemann's analysis." (Julian L Coolidge, "A History of Geometrical Methods", 1940)

"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)

"But Einstein came along and took space and time out of the realm of stationary things and put them in the realm of relativity - giving the onlooker dominion over time and space, because time and space are modes by which we think and not conditions in which we live." (Dimitri Marianoff & Palma Wayne, "Einstein: An Intimate Study of a Great Man", 1944)

"Not the state of rest, but the states of uniform translation form an objectively distinguished class of motions, and this puts an end to the substantial ether. Finally, and fourthly, the general relativity theory re-endows this metric world structure with the capacity of reacting to the forces of matter. Thus, in a sense, the circle is closed." (Hermann Weyl, "Philosophy of Mathematics and Natural Science" II, 1949

"The field equation may [...] be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium - in complete analogy with […] the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish." (Howard P Robertson, "Geometry as a Branch of Physics", 1949)

"We may sum up as follows: According to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, an ether exists. Space without an ether is inconceivable." (Albert Einstein, "The World as I See It", 1949) 

Theories: On Relativity Theory (1975-1999)

"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)

"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)

"The theory of relativity does, however, force us to change fundamentally our ideas of space and time. We must accept that time is not completely separate from and independent of space, but is combined with it to form an object called space-time." (Stephen Hawking, "A Brief History of Time: From the Big Bang to Black Holes", 1988)

"Theoretical physicists accept the need for mathematical beauty as an act of faith.… For example, the main reason why the theory of relativity is so universally accepted is its mathematical beauty." (Paul A M Dirac,"Methods in Theoretical Physics", 1989)

"All the implications of special relativity [...] have been con firmed by direct experiments. There are still people who believe it is “just a theory.” But they are wrong." (Paul C W Davies & John Gribbin, "The Matter Myth: Dramatic Discoveries That Challenge Our Understanding of Physical Reality", 1992)

"Our theories are very esoteric - necessarily so, because we are forced to develop these theories using a language, the language of mathematics, that has not become part of the general equipment of the educated public. Physicists generally do not like the fact that our theories arc so eso teric. On the other hand, I have occasionally heard artists talk proudly about their work being accessible only to a band of cognoscenti and justify this attitude by quoting the example of physical theories like general relativity that also can be understood only by initiates. Artists like physicists may not always be able to make themselves understood by the general public, but esotericism for its own sake is just silly." (Steven Weinberg, "Dreams of a Final Theory: The Scientist’s Search for the Ultimate Laws of Nature", 1992)

"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)

"Quantum mechanics - the theory that explains phenomena on the size of atoms - is right. It is also so conceptually weird that physicists to this day feel uncomfortable with it." (Tony Rothman, "Instant Physics: From Aristotle to Einstein, and Beyond", 1995)

"Relativity theory, of course, does not find that truth depends on the point of view of the observer but, on the contrary, reformulates the laws of physics so that they hold good for all observers, no matter how they move or where they stand. Its central meaning is that the most valued truths in science are independent of the point of view. [...] Einstein did not prove the work of Newton wrong; he provided a larger setting within which some limitations, contradictions, and asymmetries in the earlier physics disappeared." (Gerald Holton, Einstein, History, and Other Passions: The Rebellion Against Science at the End of the Twentieth Century, 1995)

"Use of the term 'model' makes it easier to keep in mind this distinction between theory and reality. By its very nature a model cannot include all the details of the reality it seeks to represent, for then it would be just as hard to comprehend and describe as the reality we want to model. At best, our model should give a reasonable picture of some small part of reality. It has to be a simple (even crude) description; and we must always be ready to scrap or improve a model if it fails in this task of accurate depiction. That having been said, old models are often still useful. The theory of relativity supersedes the Newtonian model, but all engineers use Newtonian mechanics when building bridges or motor cars, or probing the solar system." (David Stirzaker, "Probability and Random Variables: A Beginner’s Guide", 1999)

"What makes writing relativity so tricky is this: Built into ordinary language — in its use of tenses, for example - are many implicit assumptions about the nature of temporal relations that we now know to be false. Most importantly, we have known since 1905 that when you say that two events in different places happen at the same time you are not referring to anything inherent in the events themselves. You are merely adopting a conventional way of locating them that can differ from other equally valid conventional assignments of temporal order which do not have the events happening at the same time." (N David Mermin, "Writing Physics", 1999)

Theories: On Relativity Theory (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)

"The most important result of a general character to which the special theory has led is concerned with the conception of mass. Before the advent of relativity, physics recognized two conservation laws of fundamental importance, namely, the law of conservation of energy and the law of the conservation of mass; these two fundamental laws appeared to be quite independent of each other. By means of the theory of relativity they have been united into one law." (Albert Einstein, 1920)

"The discovery of Minkowski […] is to be found […] in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, √-1*ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry." (Albert Einstein,"Relativity: The Special and General Theory", 1920)

"The relativity theory of physics reduces everything to relations; that is to say, it is structure, not material, which counts. The structure cannot be built up without material; but the nature of the material is of no importance." (Arthur S Eddington, "Space, Time and Gravitation: An Outline of the General Relativity Theory", 1920)

"There is something attractive in presenting the evolution of a sequence of ideas in as brief a form as possible, and yet with a completeness suffi cient to preserve throughout the continuity of development. We shall endeavor to do this for the Theory of Relativity, and to show that the whole ascent is composed of small, almost self-evident steps of thought." (Albert Einstein, "A Brief Outline of the Development of the Theory of Relativity Nature Vol. 106 (2677) 1921) 

"Einstein's theory of relativity has advanced our ideas of the structure of the cosmos a step further. It is as if a wall which separated us from Truth has collapsed. Wider expanses and greater depths are now exposed to the searching eye of knowledge, regions of which we had not even a presentiment. It has brought us much nearer to grasping the plan that underlies all physical happening." (Hermann Weyl, "Space - Time - Matter (1922)

"In the realm of physics it is perhaps only the theory of relativity which has made it quite clear that the two essences, space and time, entering into our intuition, have no place in the world constructed by mathematical physics. Colours are thus 'really' not even æther-vibrations, but merely a series of values of mathematical functions in which occur four independent parameters corresponding to the three dimensions of space, and the one of time." (Hermann Weyl, "Space, Time, Matter", 1922)

"The numerical side of the theory of relativity is derived from the failure of all attempts to detect the relative motion of matter and ether." (Herbert Dingle, "Relativity for All", 1922) 

"Results of measurements are the subject-matter of physics; and the moral of the theory of relativity is that we can only comprehend what the physical quantities stand for if we first comprehend what they are." (Arthur S Eddington, "The Mathematical Theory of Relativity", 1923)

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

Theories: On Relativity Theory (1950-1974)

 "In the realm of physics it is perhaps only the theory of relativity which has made it quite clear that the two essences, space and time, entering into our intuition, have no place in the world constructed by mathematical physics. Colours are thus 'really' not even æther-vibrations, but merely a series of values of mathematical functions in which occur four independent parameters corresponding to the three dimensions of space, and the one of time." (Hermann Weyl, "Space, Time, Matter", 1952)

"The theory of relativity is a fine example of the fundamental character of the modern development of theoretical science. The initial hypotheses become steadily more abstract and remote from experience. On the other hand, it gets nearer to the grand aim of all science, which is to cover the greatest possible number of empirical facts by logical deduction from the smallest possible number of hypotheses or axioms." (Albert Einstein, 1954)

"[...] 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)

"Within the field of modern physics the theory of relativity has played a very important role. It was in this theory that the necessity for a change in the fundamental principles of physics was recognized for the first time." (Werner K Heisenberg, [Gifford Lecture, delivered at the University of St Andrews, 1955/56]) 

"By the widening of the transformation group in general relativity the idea of distinguished inertial coordinate systems could also be eliminated by Einstein as inconsistent with the group-theoretical properties of the theory. Without this general critical attitude, which abandoned naive visualizations in favour of a conceptual analysis of the correspondence between observational data and the mathematical quantities in a theoretical formalism, the establishment of the modern form of quantum theory would not have been possible." (Wolfgang Pauli, 1956)

"I consider the theory of relativity to be an example showing how a fundamental scientific discovery, sometimes even against the resistance of its creator, gives birth to further fruitful developments, following its own autonomous course." (Wolfgang Pauli, 1956)

"Understanding mathematical logic, or the theory of relativity, is not an indispensable attribute of the cultured mind. But if one wishes to learn anything about these subjects, one must learn something. It is necessary to master the rudiments of the language, to practice a technique, to follow step by step a characteristic sequence of reasoning and to see a problem through from beginning to end." (James R Newman, "The World of Mathematics” Vol. I, 1956)

"I believe the coordinate-free approach fosters the cultivation of intuition, a scarce commodity in relativity because the phenomena this theory is intended to describe are as yet rather remote from our daily experience." (Walter Noll, "Euclidean Geometry and Minkowskian Chronometry", The American Mathematical Monthly, 1964)

"Anyone who studies relativity without understanding how to use simple space-time diagrams is as much inhibited as a student of functions of a complex variable who does not understand the Argads diagram." (John L Synge, "Relativity: The Special Theory", 1965) 

"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)

"In science [...] it is impossible to open up new territory unless one is prepared to leave the safe anchorage of established doctrine and run the risk of a hazardous leap forward. With his relativity theory, Einstein had abandoned the concept of simultaneity, which was part of the solid ground of tra ditional physics, and, in so doing, outraged many leading physicists and philosophers and turned them into bitter opponents. In general, scientific progress calls for no more than the absorption and elaboration of new ideas - and this is a call most scientists are happy to heed." (Werner K Heisenberg, "Physics and Beyond: Encounters and Conversations", 1969) 

"Many cumbersome developments in the standard treatments of mechanics can be simplified and better understood when formulated with modern conceptual tools, as in the well-known case of the use of the 'universal' definition of tensor products of vector spaces to simplify some of the notational excesses of tensor analysis as traditionally used in relativity theory" (Saunders Mac Lane, "Hamiltonian Mechanics and Geometry", The American Mathematical Monthly Vol. 77 (6), 1970)

Herbert Dingle - Collected Quotes

"A great idea invariably creates as many problems as it solves: that is a sign of its greatness." (Herbert Dingle, "Relativity for All", 1922)

"The aim of the scientist is to express, in as simple a statement as possible, the principles underlying the order and arrangement of phenomena." (Herbert Dingle, "Relativity for All", 1922) 

"The numerical side of the theory of relativity is derived from the failure of all attempts to detect the relative motion of matter and ether." (Herbert Dingle, "Relativity for All", 1922) 

"A science in its infancy is the least satisfactory, and, at the same time, the most profitable theme for a general description. It is the leas satisfactory because its conclusions - if we can call them conclusions are, at the best, little more than tentative summaries of observed facts, liable at any moment to be superseded by wider generalisations: the inconsequential playfulness of childhood has not given place to the graver consistency of mature age. It is the most profitable theme because it has not yet lost the quickening inspiration that alone can produce great things. It is in touch with the poetry and romance that go side by side with all true science. In its eyes still shines 'the light that never was on sea or land'." (Herbert Dingle, "Modern Astrophysics", 1924) 

"It is as though a star throws the whole secret history of its being into its spectrum, and we have only to learn how to read it aright in order to solve the most abstruse problems of the physical Universe." (Herbert Dingle, "Modern Astrophysics", 1924)

"Modem physics is, indeed, not unlike a ship, drawing nearer to a goal not yet in sight, but so tossed about by the buffetings of experiment and working hypothesis that the passenger scarcely knows whether he is progressing or drifting." (Herbert Dingle, "Through Science to Philosophy", 1937)

"Success in scientific theory is won, not by rigid adherence to the rules of logic, but by bold speculation which dares even to break those rules if by that means new regions of interest may be opened up." (Herbert Dingle, "Through Science to Philosophy", 1937)

"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)

"Mathematics in itself, as I say, is independent of experience. It begins with the free choice of symbols, to which are freely assigned properties, and it then proceeds to deduce the necessary rational implications of those properties." (Herbert Dingle, "Science at the Crossroads", 1972)

On Tensors I

"The conception of tensors is possible owing to the circumstance that the transition from one co-ordinate system to another expresses itself as a linear transformation in the differentials. One here uses the exceedingly fruitful mathematical device of making a problem 'linear' by reverting to infinitely small quantities." (Hermann Weyl, "Space - Time - Matter", 1922)

"The field equation may [...] be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium - in complete analogy with […] the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish." (Howard P Robertson, "Geometry as a Branch of Physics", 1949)

"The physicist who states a law of nature with the aid of a mathematical formula is abstracting a real feature of a real material world, even if he has to speak of numbers, vectors, tensors, state-functions, or whatever to make the abstraction." (Hilary Putnam, "Mathematics, matter, and method", 1975)

"Maxwell's equations […] originally consisted of eight equations. These equations are not 'beautiful'. They do not possess much symmetry. In their original form, they are ugly. […] However, when rewritten using time as the fourth dimension, this rather awkward set of eight equations collapses into a single tensor equation. This is what a physicist calls 'beauty', because both criteria are now satisfied.  (Michio Kaku, "Hyperspace", 1995)

 "(…) the bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Nothing else has a baggage-free description. In other words, we all live in a gigantic mathematical object - one that’s more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names such as Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced physics theories. Everything in our world is purely mathematical - including you." (Max Tegmark, "Our Mathematical Universe: My Quest for the Ultimate Nature of Reality", 2014)

"Curvature is a central concept in differential geometry. There are conceptually different ways to define it, associated with different mathematical objects, the metric tensor, and the affine connection. In our case, however, the affine connection may be derived from the metric. The 'affine curvature' is associated with the notion of parallel transport of vectors as introduced by Levi-Civita. This is most simply illustrated in the case of a two- dimensional surface embedded in three- dimensional space. Let us take a closed curve on that surface and attach to a point on that curve a vector tangent to the surface. Let us now transport that vector along the curve, keeping it parallel to itself. When it comes back to its original position, it will coincide with the original vector if the surface is flat or deviate from it by a certain angle if the surface is curved. If one takes a small curve around a point on the surface, then the ratio of the angle between the original and the final vector and the area enclosed by the curve is the curvature at that point. The curvature at a point on a two-dimensional surface is a pure number." (Hanoch Gutfreund, "The Road to Relativity", 2015) 

"In geometric and physical applications, it always turns out that a quantity is characterized not only by its tensor order, but also by symmetry." (Hermann Weyl, 1925)

"Ultra-modern physicists [are tempted to believe] that Nature in all her infinite variety needs nothing but mathematical clothing [and are] strangely reluctant to contemplate Nature unclad. Clothing she must have. At the least she must wear a matrix, with here and there a tensor to hold the queer garment together." (Sydney Evershed)

String Theory II

"To build matter itself from geometry - that in a sense is what string theory does. It can be thought of that way, especially in a theory like the heterotic string which is inherently a theory of gravity in which the particles of matter as well as the other forces of nature emerge in the same way that gravity emerges from geometry. Einstein would have been pleased with this, at least with the goal, if not the realization. [...] He would have liked the fact that there is an underlying geometrical principle - which, unfortunately, we don’t really yet understand." (David Gross, [interview] 1988)

"I have no idea whether the properties of the universe as we know it are fundamental or emergent, but I believe that the mere possibility of the latter should give the string theorists pause, for it would imply that more than one set of microscopic equations is consistent with experiment - so that we are blind to these equations until better experiments are designed - and also that the true nature of the microscopic equations is irrelevant to our world." (Robert B Laughlin, "Fractional quantization", Reviews of Modern Physics vol. 71 (4), [Nobel lecture] 1999)

"String theory has the potential to show that all of the wondrous happenings in the universe - from the frantic dance of subatomic quarks to the stately waltz of orbiting binary stars; from the primordial fireball of the big bang to the majestic swirl of heavenly galaxies - are reflections of one, grand physical principle, one master equation."  (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 2000)

"String theory [...] resolves the central dilemma confronting contemporary physics - the incompatibility between quantum mechanics and general relativity - and that unifies our understanding of all of nature's fundamental material constituents and forces. But to accomplish these feats, [...] string theory requires that the universe have extra space dimensions."  (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 2000)

"In string theory, to understand the nature of the Big Bang, or the quantum fate of a black hole, or the nature of the vacuum state that determines the properties of the elementary particles, requires information beyond perturbation theory [...] Perturbation theory is not everything. It is just the way the [string] theory was discovered." (Edward Witten, "The Past and Future of String Theory", 2003)

"Replacing particles by strings is a naive-sounding step, from which many other things follow. In fact, replacing Feynman graphs by Riemann surfaces has numerous consequences: 1. It eliminates the infinities from the theory. [...] 2. It greatly reduces the number of possible theories. [...] 3. It gives the first hint that string theory will change our notions of spacetime." (Edward Witten, "The Past and Future of String Theory", 2003)

"String theory seems to be incompatible with a world in which a cosmological constant has a positive sign, which is what the observations indicate." (Lee Smolin, "The New Humanists: Science at the Edge", 2003)

"Some string theorists prefer to believe that string theory is too arcane to be understood by human beings, rather than consider the possibility that it might just be wrong." (Lee Smolin, "The Trouble with Physics: The Rise of String Theory, The Fall of a Science and What Comes Next", 2006)

"Even though it is, properly speaking, a postprediction, in the sense that the experiment was made before the theory, the fact that gravity is a consequence of string theory, to me, is one of the greatest theoretical insights ever." (Edward Witten)

"I would expect that a proper elucidation of what string theory really is all about would involve a revolution in our concepts of the basic laws of physics - similar in scope to any that occurred in the past. (Edward Witten [interview])

On Spacetime (2000-2019)

"Neither space nor time has any existence outside the system of evolving relationships that comprises the universe. Physicists refer to this feature of general relativity as background independence." (Lee Smolin, "Three Roads to Quantum Gravity", 2000)

"The relational picture of space and time has implications that are as radical as those of natural selection, not only for science but for our perspective on who we are and how we came to exist in this evolving universe of relations." (Lee Smolin, "Three Roads to Quantum Gravity", 2000)

"Time is described only in terms of change in the network of relationships that describes space." (Lee Smolin, "Three Roads to Quantum Gravity", 2000)

"In string theory one studies strings moving in a fixed classical spacetime. […] what we call a background-dependent approach. […] One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves - at the rate predicted by general relativity to the […] accuracy of eleven decimal place - tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up." (Lee Smolin, "Loop Quantum Gravity", The New Humanists: Science at the Edge, 2003)

"Spacetime […] turns out to be discrete, described by a structure called spin foam." (Lee Smolin, “The New Humanists: Science at the Edge”, 2003)

"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Space and time capture the imagination like no other scientific subject. For good reason. They form the arena of reality, the very fabric of the cosmos." (Brian Greene, "The Fabric of the Cosmos", 2004)

"The space and time of the universe that we humans inhabit contain symmetries. These are almost obvious yet subtle, even mysterious. Space and time form the stage upon which the dynamics - that is, the motion and interactions of the physical systems, atoms, atomic nuclei, protozoa, and people - are played out. The symmetries of space and time control the dynamics of the physical interactions of matter." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Minkowski calls a spatial point existing at a temporal point a world point. These coordinates are now called 'space-time coordinates'. The collection of all imaginable value systems or the set of space-time coordinates Minkowski called the world. This is now called the manifold. The manifold is four-dimensional and each of its space-time points represents an event." (Friedel Weinert," The Scientist as Philosopher: Philosophical Consequences of Great Scientific Discoveries", 2005) 

"Mathematicians call the infinite curvature limit of spacetime a singularity. In this picture, then, the big bang emerges from a singularity. The best way to think about singularities is as boundaries or edges of spacetime. In this respect they are not, technically, part of spacetime itself." (Paul Davies," Cosmic Jackpot: Why Our Universe Is Just Right for Life", 2007) 

"We can describe general relativity using either of two mathematically equivalent ideas: curved space-time or metric field. Mathematicians, mystics and specialists in general relativity tend to like the geometric view because of its elegance. Physicists trained in the more empirical tradition of high-energy physics and quantum field theory tend to prefer the field view, because it corresponds better to how we (or our computers) do concrete calculations." (Frank Wilczek, "The Lightness of Being: Mass, Ether, and the Unification of Forces", 2008)

"The hypothesis underlying all approaches to the landscape is that there is a cosmological setting in which different regions or epochs of the universe can have different effective laws. This implies the existence of spacetime regions not directly observable […] These regions must either be in the past of our big bang, or far enough away from us to be causally unrelated." (Lee Smolin," A perspective on the landscape problem", 2012)

"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)

"For a moving object, time contracts. Not only is there no single time for different places - there is not even a single time for any particular place. A duration can be associated only with the movement of something, with a given trajectory." (Carlo Rovelli, "The Order of Time", 2018)

"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

"Spacetime is a physical object like an electron. It, too, fluctuates. It, too, can be in a 'superposition' of different configurations." (Carlo Rovelli, "The Order of Time", 2018)

"The basic units in terms of which we comprehend the world are not located in some specific point in space. They are - if they are at all - in a where but also in a when. They are spatially but also temporally delimited: they are events." (Carlo Rovelli, "The Order of Time", 2018)

On Spacetime (1940-1949)

"Of all the fantastic ideas that belong to science fiction, the most remarkable - and, perhaps, the most fascinating - is that of time travel [...] Indeed, so fantastic a notion does it seem, and so many apparently obvious absurdities and bewildering paradoxes does it present, that some of the most imaginative students of science refuse to consider it as a practical proposition." (Idrisyn O Evans, "Can We Conquer Time?", Tales of Wonder, 1940) 

"The revolution in scientific ideas just mentioned is primarily logical. It is due to recognition that the very method of physical science, with its primary standard units of mass, space, and time, is concerned with measurements of relations of change, not with individuals as such." (John Dewey, "Time and Individuality", 1940)

"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)

"Yet a review of receipt physics has shown that all attempts at mechanical models or pictures have failed and must fail. For a mechanical model or picture must represent things as happening in space and time, while it has recently become clear that the ultimate processes of nature neither occur in, nor admit of representation in, space and time. Thus an understanding of the ultimate processes of nature is for ever beyond our reach: we shall never be able - even in imagination - to open the case of our watch and see how the wheels go round. The true object of scientific study can never be the realities of nature, but only our own observations on nature." (James H Jeans, "Physics and Philosophy", 1942)

"There is a reality outside the world, that is to say, outside space and time, outside man's mental universe, outside any sphere whatsoever that is accessible to human faculties." (Simone Weil, "Draft for a Statement of Human Obligation", 1943)

"A model, like a novel, may resonate with nature, but it is not a ‘real’ thing. Like a novel, a model may be convincing – it may ‘ring true’ if it is consistent with our experience of the natural world. But just as we may wonder how much the characters in a novel are drawn from real life and how much is artifice, we might ask the same of a model: How much is based on observation and measurement of accessible phenomena, how much is convenience? Fundamentally, the reason for modeling is a lack of full access, either in time or space, to the phenomena of interest." (Kenneth Belitz, Science, Vol. 263, 1944)

"But Einstein came along and took space and time out of the realm of stationary things and put them in the realm of relativity - giving the onlooker dominion over time and space, because time and space are modes by which we think and not conditions in which we live." (Dimitri Marianoff & Palma Wayne, "Einstein: An Intimate Study of a Great Man", 1944)

"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) 

"I cannot seriously believe in [the quantum theory] because it cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance." (Albert Einstein, [Letter to Max Born] 1948)

"It seems significant that according to quantum physics the indestructibility of energy on one hand - which expresses its timeless existence - and the appearance of energy in space and time on the other hand correspond to two contradictory (complementary) aspects of reality. In fact, both are always present, but in individual cases the one or the other may be more pronounced. (Wolfgang Pauli, "Moderne Beispiele zur Hintergrundsphysik" ["Modern Examples of Background Physics", 1948)

"[…] the universe is not a rigid and inimitable edifice where independent matter is housed in independent space and time; it is an amorphous continuum, without any fixed architecture, plastic and variable, constantly subject to change and distortion. Wherever there is matter and motion, the continuum is disturbed. Just as a fi sh swimming in the sea agitates the water around it, so a star, a comet, or a galaxy distorts the geometry of the spacetime through which it moves." (Lincoln Barnett, "The Universe and Dr. Einstein", 1948)

"Space-time is curved in the neighborhood of material masses, but it is not clear whether the presence of matter causes the curvature of space-time or whether this curvature is itself responsible for the existence of matter." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949) 

On Spacetime (1975-1999)

"Our look is as bound by time-space as our brain. We never look, we never see beyond this limitation; we do not know how to look through and beyond these fragmentary frontiers. But the eyes have to see beyond them, penetrating deeply and widely, without choosing, without shelter; they have to wander beyond man-made frontiers of ideas and values and to feel beyond love. Then there is a benediction which no god can give." (Jiddu Krishnamurti," Krishnamurti's Notebook", 1976)

"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) 

"The structure of space-time, taken as a whole, is the subject matter of the science called cosmology. Since you are asking about all space and all time in cosmology, you are interested in the entire universe, everywhere and everywhen, viewed as a static geometrical object." (Rudolf B Rucker," Geometry, Relativity and the Fourth Dimension", 1977)

"Instead of thinking of space and time as a stage, on which the drama of matter unfolds, we have to imagine some ultra- modern theater, in which the stage itself becomes one of the actors." (John Stachel, "Einstein's Odyssey: His Journey from Special to General Relativity", 1979)

"It is unscientific to say that within the many billions of galactic systems, ours is the only planet that supports life in advanced form. Nature shuns one of a kind as much as it abhors a vacuum. Given infinite time and space, anything that occurs at one place or time in the universe will occur elsewhere or ‘elsewhen’." (Norman Cousins, "Rendezvous with Infinity", Cosmic Search Magazine Vol. 1 (1), 1979)

"No matter how finely you subdivide time and space, each tiny division contains infinity." (Frank Herbert, "Dune Genesis", 1980)

"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)

"The quantum is that embarrassing little piece of thread that always hangs from the sweater of space-time. Pull it and the whole thing unravels." (Fred A Wolf, "Star Wave: Mind Consciousness of Quantum Physics", 1984)

"The nothingness ‘before’ the creation of the universe is the most complete void that we can imagine - no space, time, or matter existed. It is a world without place, without duration or eternity, without number - it is what mathematicians call ‘the empty set’. Yet this unthinkable void converts itself into the plenum of existence - a necessary consequence of physical laws. Where are these laws written into that void? What ‘tells’ the void that is pregnant with a possible universe? It would seem that, even the void is subject to law, a logic that exists prior to space and time." (Heinz R Pagels, "Perfect Symmetry: The Search for the Beginning of Time", 1985)

"The theory of relativity does, however, force us to change fundamentally our ideas of space and time. We must accept that time is not completely separate from and independent of space, but is combined with it to form an object called space-time." (Stephen Hawking, "A Brief History of Time: From the Big Bang to Black Holes", 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)

“Symmetry is bound up in many of the deepest patterns of Nature, and nowadays it is fundamental to our scientific understanding of the universe. Conservation principles, such as those for energy or momentum, express a symmetry that (we believe) is possessed by the entire space-time continuum: the laws of physics are the same everywhere.” (Ian Stewart & Martin Golubitsky, “Fearful Symmetry: Is God a Geometer?”, 1992)

"Time and space are finite in extent, but they don't have any boundary or edge. They would be like the surface of the earth, but with two more dimensions." (Stephen Hawking, "Black Holes and Baby Universes and Other Essays", 1993)

"Minkowski, building on Einstein's work, had now discovered that the Universe is made of a four-dimensional ‘spacetime’ fabric that is absolute, not relative." (Kip S Thorne, "Black Holes and Time Warps: Einstein's Outrageous Legacy" , 1994)

"Arm chair reflections on the concept of causation [are] not going to yield new insights. The grandfather paradox is simply a way of pointing to the fact that if the usual laws of physics are supposed to hold true in a chronology violating spacetime, then consistency constraints emerge. [To understand these constraints] involves solving problems in physics, not armchair philosophical reflections." (John Earman, “Recent Work on Time Travel", 1995) 

"Yet everything has a beginning, everything comes to an end, and if the universe actually began in some dense explosion, thus creating time and space, so time and space are themselves destined to disappear, the measure vanishing with the measured, until with another ripple running through the primordial quantum field, something new arises from nothingness once again." (David Berlinski, "A Tour of the Calculus", 1995)

"And of course the space the wave function live in, and (therefore) the space we live in, the space in which any realistic understanding of quantum mechanics is necessarily going to depict the history of the world as playing itself out […] is configuration-space. And whatever impression we have to the contrary (whatever impression we have, say, of living in a three-dimensional space, or in a four dimensional spacetime) is somehow flatly illusory." (David Albert, "Elementary Quantum Metaphysics", 1996)

"In an infinite universe, every point in space-time is the center." (David Zindell, "War in Heaven", 1998) 

"Spacetime tells matter how to move; matter tells spacetime how to curve." (John A Wheeler, "Geons, Black Holes and Quantum Foam: A Life in Physics" , 1998) 

Max Born - Collected Quotes

"The difficulty involved in the proper and adequate means of describing changes in continuous deformable bodies is the method of differential equations. […] They express mathematically the physical concept of contiguous action." (Max Born, "Einstein's Theory of Relativity", 1920)

"It is natural that a man should consider the work of his hands or his brain to be useful and important. Therefore nobody will object to an ardent experimentalist boasting of his measurements and rather looking down on the 'paper and ink' physics of his theoretical friend, who on his part is proud of his lofty ideas and despises the dirty fingers of the other." (Max Born, " Experiment and Theory in Physics", 1943)

"The conception of chance enters in the very first steps of scientific activity in virtue of the fact that no observation is absolutely correct. I think chance is a more fundamental conception that causality; for whether in a concrete case, a cause-effect relation holds or not can only be judged by applying the laws of chance to the observation." (Max Born, 1949)

"When a scientific theory is firmly established and confirmed, it changes its character and becomes a part of the metaphysical background of the age: a doctrine is transformed into a dogma." (Max Born, "Natural Philosophy of Cause and Chance", 1949)

"All great discoveries in experimental physics have been due to the intuition of men who made free use of models, which were for them not products of the imagination, but representatives of real things." (Max Born, "Physical Reality", Philosophical Quarterly Vol. 3 (11),1953)

"Every object that we perceive appears in innumerable aspects. The concept of the object is the invariant of all these aspects. From this point of view, the present universally used system of concepts in which particles and waves appear simultaneously, can be completely justified. The latest research on nuclei and elementary particles has led us, however, to limits beyond which this system of concepts itself does not appear to suffice. The lesson to be learned from what I have told of the origin of quantum mechanics is that probable refinements of mathematical methods will not suffice to produce a satisfactory theory, but that somewhere in our doctrine is hidden a concept, unjustified by experience, which we must eliminate to open up the road." (Max Born, "The Statistical Interpretations of Quantum Mechanics", [Nobel lecture] 1954)

"[...] if we can never actually determine more than one of the two properties (possession of a definite position and of a definite momentum), and if when one is determined we-can make no assertion at all about the other property for the same moment, so far as our experiment goes, then we are not justified in concluding that the 'thing' under examination can actually be described as a particle in the usual sense of the term." (Max Born, "Atomic Physics", 1957)

"Physics is in the nature of the case indeterminate, and therefore the affair of statistics." (Max Born, "Atomic Physics", 1957)

"The ultimate origin of the difficulty lies in the fact (or philosophical principle) that we are compelled to use words of common language when we wish to describe a phenomenon, not by logical or mathematical analysis, but by a picture appealing to the imagination. Common language has grown by everyday experience and can never surpass these limits. Classical physics has restricted itself to the use of concepts of this kind by analyzing visible motions it has developed two ways of representing them by elementary processes moving particles and waves. There is no other wav of giving a pictorial description of motions - we have to apply it even in the region of atomic process, where classical physics break down." (Max Born, "Atomic Physics", 1957)

"[...] the whole course of events is determined by the laws of probability; to a state in space there corresponds a definite probability, which is given by the de Brogile wave associated with the state." (Max Born, "Atomic Physics", 1957)

"The belief that there is only one truth and that oneself is in possession of it, seems to me the deepest root of all that is evil in the world." (Max Born, "Natural Philosophy of Cause and Chance", 1964)

"There are metaphysical problems, which cannot be disposed of by declaring them meaningless. For, as I have repeatedly said, they are ‘beyond physics’ indeed and demand an act of faith. We have to accept this fact to be honest. There are two objectionable types of believers: those who believe the incredible and those who believe that ‘belief’ must be discarded and replaced by "the scientific method." (Max Born, "Natural Philosophy of Cause and Chance", 1964)

"Science is not formal logic - it needs the free play of the mind in as great a degree as any other creative art. It is true that this is a gift which can hardly be taught, but its growth can be encouraged in those who already possess it." (Max Born)

Albert Einstein - Collected Quotes

"As soon as science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a theory. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the 'truth' of the theory lies.  " (Albert Einstein, "Relativity: The Special and General Theory ", 1916)

"No fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case. " (Albert Einstein, "Relativity, The Special and General Theory ", 1916)

"Since the introduction of the special principle of relativity has been justified, every intellect which strives after generalization must feel the temptation to venture the step towards the general principle of relativity." (Albert Einstein, 1917)

"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)

"Most teachers waste their time by asking questions which are intended to discover what a pupil does not know whereas the true art of questioning has for its purpose to discover what the pupil knows or is capable of knowing." (Albert Einstein, 1920)

"The discovery of Minkowski […] is to be found […] in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, √-1*ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry." (Albert Einstein,"Relativity: The Special and General Theory", 1920)

"A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To ‘visualise’ a theory, or bring it home to one's mind, therefore means to give a representation to that abundance of experiences for which the theory supplies the schematic arrangement " (Albert Einstein,  "Geometry and Experience ", 1921)

"Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience, but not on logical inferences only. We call this 'practical geometry'. [...] The question whether the practical geometry of the universe is Euclidean or not has a clear meaning, and its answer can only be furnished by experience." (Albert Einstein, [lecture] 1921)

"It seems that the human mind has first to construct forms independently, before we can find them in things. Kepler’s marvelous achievement is a particularly fine example of the truth that knowledge cannot spring from experience alone, but only from the comparison of the inventions of the intellect with observed fact." (Albert Einstein, 1930)

"Physics is the attempt at the conceptual construction of a model of the real world and its lawful structure. " (Albert Einstein, [letter to Moritz Schlick] 1931)

"It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience. " (Albert Einstein, [lecture] 1933)

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships.  In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature. " (Albert Einstein, [Obituary for Emmy Noether], 1935)

"Creating a new theory is not like destroying an old barn and erecting a skyscraper in its place. It is rather like climbing a mountain, gaining new and wider views, discovering unexpected connections between our starting point and its rich environment. But the point from which we started out still exists and can be seen, although it appears smaller and forms a tiny part of our broad view gained by the mastery of the obstacles on our adventurous way up. " (Albert Einstein & Leopold Infeld,  "The Evolution of Physics ", 1938)

"Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone."  (Albert Einstein & Leopold Infeld,  "The Evolution of Physics", 1938)

"Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world In our endeavor to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears its ticking, but he has no way of opening the case. If he is ingenious he may form some picture of a mechanism which could be responsible for all the things he observes, but he may never be quite sure his picture is the only one which could explain his observations. He will never be able to compare his picture with the real mechanism and he cannot even imagine the possibility of the meaning of such a comparison." (Albert Einstein & Leopold Infeld,  "The Evolution of Physics ", 1938)

"The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advances in science." (Albert Einstein & Leopold Infeld, "The Evolution of Physics", 1938)

"With the help of physical theories we try to find our way through the maze of observed facts, to order and understand the world of our sense impressions. " (Albert Einstein & Leopold Infeld,  "The Evolution of Physics ", 1938)

"The development of science and of creative activities of the spirit in general requires still another kind of freedom, which may be characterized as inward freedom. It is this freedom of the spirit which consists in the independence of thought from the restrictions of authoritarian and social prejudices as well as from unphilosophical routinizing and habit in general." (Albert Einstein, "On Freedom", 1940)

"When the number of factors coming into play in a phenomenological complex is too large, scientific method in most cases fails us. One need only think of the weather, in which case prediction even for a few days ahead is impossible. Nevertheless no one doubts that we are confronted with a causal connection whose causal components are in the main known to us.

Occurrences in this domain are beyond the reach of exact prediction because of the variety of factors in operation, not because of any lack of order in nature." (Albert Einstein, "Science and Religion", 1941)

"The words of the language, as they are written or spoken, do not seem to play any role in any mechanism of thought. The physical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced or combined. […]  But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought - before there is any connection with logical construction in words or other kinds of signs which can be communicated to others. The above-mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will. " (Albert Einstein, [letter to Hadamard, in (Jacques Hadamard,  "The Psychology of Invention in the Mathematical Field,1945)])

"A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended its area of applicability." (Albert Einstein, "Autobiographical Notes", 1949)

"In speaking here of ‘comprehensibility’, the expression is used in its most modest sense. It implies: the production being produced by the creation of general concepts, relations between these concepts and sense experience. It is in this sense that the world of our sense experiences is comprehensible. The fact that it is comprehensible is a miracle." (Albert Einstein, "Out of My Later Years", 1950)

"Physics too deals with mathematical concepts; however, these concepts attain physical content only by the clear determination of their relation to the objects of experience." (Albert Einstein, "Out of My Later Years", 1950)

"Science is the attempt to make the chaotic diversity of our sense-experience correspond to a logically uniform system of thought."  (Albert Einstein, "Out of My Later Years", 1950)

"Space-time does not claim existence on its own, but only as a structural quality of the field." (Albert Einstein, 1954)

"The important point for us to observe is that all these constructions and the laws connecting them can be arrived at by the principle of looking for the mathematically simplest concepts and the link between them. In the limited number of mathematically existent simple field types, and the simple equations possible between them, lies the theorist’s hope of grasping the real in all its depth." (Albert Einstein, "Ideas and Opinions", 1954)

"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,"Ideas and Opinions", 1954) 

"The theory of relativity is a fine example of the fundamental character of the modern development of theoretical science. The initial hypotheses become steadily more abstract and remote from experience. On the other hand, it gets nearer to the grand aim of all science, which is to cover the greatest possible number of empirical facts by logical deduction from the smallest possible number of hypotheses or axioms." (Albert Einstein, 1954)

"We have thus assigned to pure reason and experience their places in a theoretical system of physics. The structure of the system is the work of reason: the empirical contents and their mutual relations must find their representation in the conclusions of the theory. In the possibility of such a representation lie the sole value and justification of the whole system, and especially of the concepts and fundamental principles which underlie it. Apart from that, these latter are free inventions of human intellect, which cannot be justified either by the nature of that intellect or in any other fashion a priori." (Albert Einstein, "Ideas and Opinions", 1954)

"What distinguishes the language of science from language as we ordinarily understand the word? […] What science strives for is an utmost acuteness and clarity of concepts as regards their mutual relation and their correspondence to sensory data." (Albert Einstein, "Ideas and Opinions", 1954)

"When a man after long years of searching chances upon a thought which discloses something of the beauty of this mysterious universe, he should not therefore be personally celebrated. He is already sufficiently paid by his experience of seeking and finding." (Albert Einstein, [The New York Times] 1978)

 "All great achievements in science start from intuitive knowledge, namely, in axioms, from which deductions are then made. […] Intuition is the necessary condition for the discovery of such axioms. " (Albert Einstein) 

 "Equations are more important to me, because politics is for the present, but an equation is something for eternity. " (Albert Einstein)

"How can it be that mathematics, a product of human thought independent of experience, is so admirably adapted to the objects of reality." (Albert Einstein)

 "It is a miracle that curiosity survives formal education. " (Albert Einstein)

 "It is an outcome of faith that nature - as she is perceptible to our five senses - takes the character of such a well formulated puzzle. " (Albert Einstein)

 "It stands to the everlasting credit of science that by acting on the human mind it has overcome man's insecurity before himself and before nature. " (Albert Einstein)

"It’s not that I’m so smart, it’s just that I stay with problems longer." (Albert Einstein)

"Look deep, deep, deep into nature, and then you will understand everything." (Albert Einstein)

"Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas. " (Albert Einstein)

 "That deep emotional conviction of the presence of a superior reasoning power, which is revealed in the incomprehensible universe, forms my idea of God. " (Albert Einstein)

"The aim [of education] must be the training of independently acting and thinking individuals who, however, see in the service to the community their highest life achievement." (Albert Einstein)

"The supreme task 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 lead to them." (Albert Einstein)

"The truth of a theory is in your mind, not in your eyes." (Albert Einstein)

 "Truth is what stands the test of experience. " (Albert Einstein)

"To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advance in science." (Albert Einstein)

 "We can not solve our problems with the same level of thinking that created them." (Albert Einstein)

 "What I’m really interested in is whether God could have made the world in a different way; that is, whether the necessity of logical simplicity leaves any freedom at all. " (Albert Einstein)

Bridging the Gap: Science vs Divinity

“That deep emotional conviction of the presence of a superior reasoning power, which is revealed in the incomprehensible universe, forms my idea of God.” (Albert Einstein)

“Nothing in the universe is contingent, but all things are conditioned to exist and operate in a particular manner by the necessity of the divine nature.” (Baruch Spinoza)

“What you can show using physics, forces this universe to continue to exist. As long as you're using general relativity and quantum mechanics you are forced to conclude that God exists.” (Frank Tipler)

"The equations of physics have in them incredible simplicity, elegance and beauty. That in itself is sufficient to prove to me that there must be a God who is responsible for these laws and responsible for the universe" (Paul Davies, 1984)

“In our study of natural objects we are approaching the thoughts of the Creator, reading his conceptions, interpreting a system that is His and not ours.”​ (Louis Agassiz)

“What one man calls God, another calls the laws of physics.” (Nikola Tesla)

“Nature does not consist entirely, or even largely, of problems designed by a Grand Examiner to come out neatly in finite terms, and whatever subject we tackle the first need is to overcome timidity about approximating.” (Sir Harold Jeffreys & Bertha S Jeffreys, “Methods of Mathematical Physics”, 1946)

"The idea of a universal mind or Logos would be, I think, a fairly plausible inference from the present state of scientific theory." (Arthur Eddington)

"Such properties seem to run through the fabric of the natural world like a thread of happy coincidences. But there are so many odd coincidences essential to life that some explanation seems required to account for them." (Sir Fred Hoyle)

"I find it as difficult to understand a scientist who does not acknowledge the presence of a superior rationality behind the existence of the universe as it is to comprehend a theologian who would deny the advances of science." (Wernher von Braun)