Theoretical physics

"[T]he program which Immanuel Kant proposed back in the 1760s... was this: our knowledge of the outside world depends on our modes of perception... Unfortunately, a great revolution took place in or about the year 1768, when he read a paper by Euler which intended to show that space was indeed absolute as Newton had suggested and not relative as Leibnitz suggested. (...in the eighteenth century the question of whether Newton's... or Leibnitz's view of the world was right profoundly affected all philosophy.) After reading Euler's argument... Kant... for the first time proposed that... we must be conscious of [absolute space] a priori. ...Kant died in 1804, long before new ideas about space... had been published... And since one of the things that happened in [our] lifetime has been the substitution of... a Leibnitz universe, the universe of relativity, for Newton's universe... we should think that out again."

"Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space."

"I hold in fact (1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them. (2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave. (3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial. (4) That in the physical world nothing else takes place but this variation, subject possibly to the law of continuity."

"No mathematician can give any meaning to the language about matter, force, inertia used in current text-books of mechanics."

"We may... be treating merely as physical variations effects which are really due to changes in the curvature of our space; whether, in fact, some or all of those causes which we term physical may not be due to the geometrical construction of our space. There are three kinds of variation in the curvature of our space which we ought to consider as within the range of possibility. (i) Our space is perhaps really possessed of a curvature varying from point to point, which we fail to appreciate because we are acquainted with only a small portion of space, or because we disguise its small variations under changes in our physical condition which we do not connect with our change of position. The mind that could recognise this varying curvature might be assumed to know the absolute position of a point. For such a mind the postulate of the relativity of position would cease to have a meaning. It does not seem so hard to conceive such a state of mind as the late Professor Clerk-Maxwell would have had us believe. It would be one capable of distinguishing those so-called physical changes which are really geometrical or due to a change of position in space. (ii) Our space may be really same (of equal curvature), but its degree of curvature may change as a whole with the time. In this way our geometry based on the sameness of space would still hold good for all parts of space, but the change of curvature might produce in space a succession of apparent physical changes. (iii) We may conceive our space to have everywhere a nearly uniform curvature, but that slight variations of the curvature may occur from point to point, and themselves vary with the time. These variations of the curvature with the time may produce effects which we not unnaturally attribute to physical causes independent of the geometry of our space. We might even go so far as to assign to this variation of the curvature of space 'what really happens in that phenomenon which we term the motion of matter.'"

"The modern theory of relativity, on its mathematical side, is merely an elaboration of Riemann's analysis."

"It is the reciprocity of these appearances—that each party should think the other has contracted—that is so difficult to realise. Here is a paradox beyond even the imagination of Dean Swift. Gulliver regarded the Lilliputians as a race of dwarfs; and the Lilliputians regarded Gulliver as a giant. That is natural. If the Lilliputians had appeared dwarfs to Gulliver, and Gulliver had appeared a dwarf to the Lilliputians—but no! that is too absurd for fiction, and is an idea only to be found in the sober pages of science. ...It is not only in space but in time that these strange variations occur. If we observed the aviator carefully we should infer that he was unusually slow in his movements; and events in the conveyance moving with him would be similarly retarded—as though time had forgotten to go on. His cigar lasts twice as long as one of ours. ...But here again reciprocity comes in, because in the aviator's opinion it is we who are travelling at 161,000 miles a second past him; and when he has made all allowances, he finds that it is we who are sluggish. Our cigar lasts twice as long as his."

"...The present revolution of scientific thought follows in natural sequence on the great revolutions at earlier epochs in the history of science. Einstein's special theory of relativity, which explains the indeterminateness of the frame of space and time, crowns the work of Copernicus who first led us to give up our insistence on a geocentric outlook on nature; Einstein's general theory of relativity, which reveals the curvature or non-Euclidean geometry of space and time, carries forward the rudimentary thought of those earlier astronomers who first contemplated the possibility that their existence lay on something which was not flat. These earlier revolutions are still a source of perplexity in childhood, which we soon outgrow; and a time will come when Einstein's amazing revelations have likewise sunk into the commonplaces of educated thought."

"If you don't take my words too seriously, I would say this: If we assume that all matter would disappear from the world, then, before relativity, one believed that space and time would continue existing in an empty world. But, according to the theory of relativity, if matter and its motion disappeared there would no longer be any space or time."

"Another topic deserving discussion is Einstein’s modification of Newton’s law of gravitation. In spite of all the excitement it created, Newton’s law of gravitation is not correct! It was modified by Einstein to take into account the theory of relativity. According to Newton, the gravitational effect is instantaneous, that is, if we were to move a mass, we would at once feel a new force because of the new position of that mass; by such means we could send signals at infinite speed. Einstein advanced arguments which suggest that we cannot send signals faster than the speed of light, so the law of gravitation must be wrong. By correcting it to take the delays into account, we have a new law, called Einstein’s law of gravitation. One feature of this new law which is quite easy to understand is this: In the Einstein relativity theory, anything which has energy has mass—mass in the sense that it is attracted gravitationally. Even light, which has an energy, has a “mass.” When a light beam, which has energy in it, comes past the sun there is an attraction on it by the sun. Thus the light does not go straight, but is deflected. During the eclipse of the sun, for example, the stars which are around the sun should appear displaced from where they would be if the sun were not there, and this has been observed."

"For over 200 years the equations of motion enunciated by Newton were believed to describe nature correctly, and the first time that an error in these laws was discovered, the way to correct it was also discovered. Both the error and its correction were discovered by Einstein in 1905.Newton’s Second Law, which we have expressed by the equation :F=\frac {d\left({mv}\right)}{dt} was stated with the tacit assumption that m is a constant, but we now know that this is not true, and that the mass of a body increases with velocity. In Einstein’s corrected formula m has the value :m=\frac {{m}_}{\sqrt} where the “rest mass” m0 represents the mass of a body that is not moving and c is the speed of light, which is about 3×105 km⋅sec−1 or about 186,000 mi⋅sec−1."

"Only around the end of the nineteenth century did scientists come across a few observations that did not fit well with Newton's laws, and these led to the net revolution in physics - the theory of relativity and quantum mechanics."

"One clock stayed on the ground; its double flew. / And it ran slow. So, then. The mad thing's true."

"The modern world began on 29 May 1919 when photographs of a solar eclipse, taken on the island of Principe off West Africa and at Sobral in Brazil, confirmed the truth about a new theory of the universe. It had been apparent for half a century that the Newtonian cosmology, based upon the straight lines of Euclidean geometry, and Galileo's notions of absolute time, was in need of serious modification. It had stood for more than two hundred years. It was the framework within which the European Enlightenment, the Industrial Revolution, the vast expansion of human knowledge, freedom, and prosperity which had characterized the nineteenth century, had taken place. But increasingly powerful telescopes were revealing anomalies. In particular, the motions of the planet Mercury deviated by forty-three seconds of an arc a century from its predictable behavior under Newtonian laws of physics. Why? In 1905, a twenty-six-year old German Jew, Albert Einstein, then working in the Swiss patent office in Berne, had published a paper, 'On the electrodynamics of moving bodies,' which became known as the Special Theory of Relativity. Einstein's observations on the way in which, in certain circumstances, lengths appeared to contract and clocks to slow down, are analogous to the effects of perspective in painting. In fact the discovery that space and time are relative rather than absolute terms of measurement is comparable, in its effect on our perception of the world, to the first use of perspective in art, which occurred in Greece in the two decades c.500-480 BC."

"The originality of Einstein, amounting to a form of genius, and the curious elegance of his lines of argument, which colleagues compared to a kind of art, aroused growing, world-wide interest. In 1907 he published a demonstration that all mass was energy, encapsulated in the equation E = mc2, which a latter age saw as the starting point in the race for the A-bomb. Not even the onset of the European war prevented scientists from following his quest for an all-embracing General Theory of Relativity which would cover gravitational fields and provide a comprehensive revision of Newtonian physics. In 1915 news reached London that he had done it. The following spring, as the British were preparing their vast and catastrophic offensive on the Somme, the key paper was smuggled through the Netherlands and reached Cambridge, where it was received by Arthur Eddington, Professor of Astronomy and Secretary of the Royal Astronomical Society"

"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."

"It was the space doctor who figured out the answer. He said that if our ideas about light were right, then our ideas about distance and seconds must be wrong. He said that time doesn’t pass the same for everyone. When you go fast, he said, the world around you changes shape, and time outside starts moving slower. The doctor came up with some numbers for how time and space must change to make the numbers for light work. With his idea, everyone would see light moving the right distance every second. This idea is what we call his special idea. The special idea is really, really strange, and understanding it can take a lot of work. Lots of people thought it must be wrong because it’s so strange, but it turned out to be right. We know because we’ve tried it out. If you go really fast, time goes slower. If you’re in a car, you see watches outside the car go slower. They only go a little slower, so you wouldn’t notice it in your normal life; it takes the best watches in the world to even tell that it’s happening. But it really does happen."

"After the doctor figured out the special idea, he started thinking about weight. Things with weight pull on each other. Earth pulls things down toward it, which is why you can’t jump to space. Earth also pulls on the moon, keeping it near us, and the sun pulls on Earth in the same way. It turns out that light gets pulled by weight, too. (People weren’t sure about this for a while, because it moves so fast that it only gets pulled a little.) Someone very careful might notice that this gives us a new problem: How can light turn? The numbers that explain how light moves also say that it can only go forward. It can’t change direction in empty space. That’s just what the numbers for light say—the same numbers that say it always moves a certain distance every second. If a light wave is pulled down, it has to turn to point down, since it can’t travel to the side. To turn, the bottom part of the wave has to go slower than the top part, since it’s going a shorter distance in the same time. But that can’t be right, because the numbers say that light can’t go faster or slower. We’re in trouble again. And, once again, the space doctor has an answer. The space doctor figured out that to explain how weight pulls things like light, we have to play around with time again. He showed that if time itself goes slower near heavy things, then the side of the light near the heavy thing won’t go as far every second. This lets the light turn toward the heavy thing. The doctor’s idea was that weight slows down time, and it explained how light could bend. But to figure out how much light bends, we need to look at the other part of the doctor’s big idea. To talk about that part, let’s forget about light and instead visit another world."

"There is a point of view according to which relativity theory is the end-point of "classical physics", which means physics in the style of Newton-Faraday-Maxwell, governed by the "deterministic" form of causality in space and time, while afterwards the new quantum-mechanical style of the laws of Nature came into play. This point of view seems to me only partly true, and does not sufficiently do justice to the great influence of Einstein, the creator of the theory of relativity, on the general way of thinking of the physicists of today. By its epistemological analysis of the consequences of the finiteness of the velocity of light (and with it, of all signal-velocities), the theory of special relativity was the first step away from naive visualization. The concept of the state of motion ,of the "luminiferous aether", as the hypothetical medium was called earlier, had to be given up, not only because it turned out to be unobservable, but because it became superfluous as an element of a mathematical formalism, the group-theoretical properties of which would only be disturbed by it."

"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."

"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."

"Einstein's famous theory of relativity states that while phenomena appear different to someone close to a black hole, traveling close to the speed of light, or in a falling elevator here on earth, scientists in profoundly different environments will nevertheless always discover the same underlying laws of nature."

"Relativity distorted classical expectations in a way that Clavain still did not find entirely intuitive. Slam two objects towards each other, each with individual velocities just below light-speed, and the classical result for their closing velocity would be the sum of their individual speeds: just under twice the speed of light. Yet the true result, confirmed with numbing precision, was that the objects saw each other approach with a combined speed that was still just below the speed of light. Similarly, the relativistic closing velocity for two objects moving towards each other with individual speeds of one-half of light-speed was not light-speed itself, but eight-tenths of it. It was the way the universe was put together, and yet it was not something the human mind had evolved to accept."

"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."

"This is the mathematical formulation of the theory of relativity. The metric properties of the four-dimensional continuum are described... by a certain number (ten, in fact) of quantities denoted by gαβ, and commonly called "potentials." The physical status of matter and energy, on the other hand, is described by ten other quantities, denoted by Tαβ, the set of which is called the "material tensor." This special tensor has been selected because it has the property which is mathematically expressed by saying that its divergence vanishes, which means that it represents something permanent. The fundamental fact of mechanics is the law of inertia, which can be expressed in its most simple form by saying that it requires the fundamental laws of nature to be differential equations of the second order. Thus the problem was to find a differential equation of the second order giving a relation between the metric tensor gαβ and the material tensor Tαβ. This is a purely mathematical problem, which can be solved without any reference to the physical meaning of the symbols. The simplest possible equation (or rather set of ten equations, because there are ten gs) of that kind that can be found was adopted by Einstein as the fundamental equation of his theory. It defines the space-time continuum, or the "field." The world-lines of material particles and light quanta are the geodesics in the four-dimensional continuum defined by the solutions gαβ of these field-equations. The equations of the geodesic thus are equivalent to the equations of motion of mechanics. When we come to solve the field-equations and substitute the solutions in the equations of motion, we find that in the first approximation, i.e. for small material velocities (small as compared with the velocity of light), these equations of motion are the same as those resulting from Newton's theory of gravitation. The distinction between gravitation and inertia has disappeared; the gravitational action between two bodies follows from the same equations, and is the same thing, as the inertia of one body. A body, when not subjected to an extraneous force (i.e. a force other than gravitation), describes a geodesic in the continuum, just as it described a geodesic, or straight line, in the absolute space of Newton under the influence of inertia alone. The field-equations and the equations of the geodesic together contain the whole science of mechanics, including gravitation."

"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. Second, 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."

"The best presentation of the general theory [of relativity] is still Eddington's book of 1923, The Mathematical Theory of Relativity. For the planetary motion and the motion of the moon, see: de Sitter, "On Einstein's theory of gravitation and and its astronomical consequences," Monthly Notices, R. Astr. Soc. London, 76:699; 77:155. The mathematical foundation, the calculus of tensors, is given very completely in Eddington's book. For an exhaustive treatment see: Levi-Cevita, The Absolute Differential Calculus, translated by Dr. E. Perisco (1927)."

"It did not last: the Devil howling 'Ho! Let Einstein be!' restored the status quo."

"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."

"With the new views advocated by Riemann... the texture, structure or geometry of space is defined by the metrical field, itself produced by the distribution of matter. Any non-homogeneous distribution of matter would then entail a variable structure of geometry for space from place to place. ... Riemann's exceedingly speculative ideas on the subject of the metrical field were practically ignored in his day, save by the English mathematician Clifford, who translated Riemann's works, prefacing them to his own discovery of the non-Euclidean Clifford space. Clifford realised the potential importance of the new ideas and suggested that matter itself might be accounted for in terms of these local variations of the non-Euclidean space, thus inverting in a certain sense Riemann's ideas. But in Clifford's day this belief was mathematically untenable. Furthermore, the physical exploration of space seemed to yield unvarying Euclideanism. ...it was reserved for the theoretical investigator Einstein, by a stupendous effort of rational thought, based on a few flimsy empirical clues, to unravel the mystery and to lead Riemann's ideas to victory. (In all fairness to Einstein... he does not appear to have been influenced directly by Riemann.) Nor were Clifford's hopes disappointed, for the varying non-Euclideanism of the continuum was to reveal the mysterious secret of gravitation, and perhaps also of matter, motion, and electricity. ... Einstein had been led to recognize that space of itself was not fundamental. The fundamental continuum whose non-Euclideanism was fundamental was... one of Space-Time... possessing a four-dimensional metrical field governed by the matter distribution. Einstein accordingly applied Riemann's ideas to space-time instead of to space... He discovered that the moment we substitute space-time for space (and not otherwise), and assume that free bodies and rays of light follow geodesics no longer in space but in space-time, the long-sought-for local variations in geometry become apparent. They are all around us, in our immediate vicinity... We had called their effects gravitational effects... never suspecting that they were the result of those very local variations in the geometry for which our search had been in vain....the theory of relativity is the theory of the space-time metrical field."

"Einstein's definition... does not differ in spirit from the definitions in classical science; its sole advantage is that it entails a minimum of assumptions, and is susceptible of being realised in a concrete way permitting a high degree of accuracy in our measurements. Einstein's definition, is then, as follows: If we consider a ray of light passing through a Galilean frame, its velocity in the frame will be the same regardless of the relative motion of the luminous source and frame, and regardless of the direction of the ray. ...when it was found that contrary to the anticipations of classical science not the slightest trace of anisotropy could be detected even by ultra-precise experiment, the objections which classical science may might have presented... lost all force. ...ether drift appeared to exert no influence one way or the other. ... Isotropy signifies that the velocity of light is the same in all directions. And how can we ascertain the equality of a velocity in all directions when we do not yet know how to measure time? Experimenters solved the difficulty by appealing to the observation of coincidences. ... Waves of light leaving the centre of a sphere simultaneously are found to return to the centre also in concidence, after having suffered a reflection against a highly polished inner surface of the sphere. ...the light waves have thus covered equal distances in the same time; whence we conclude that their speed is the same in all directions. Inasmuch as this experiment has been performed, yielding the results we have just described, even though the ether drift caused by the earth's motion should have varied in direction and intensity, the isotropy of space to luminous propogations was thus established. (The experiment described constitutes but a schematic form of Michelson's.) It is to be noted that in this experiment the observation of coincidences is alone appealed to (even spatial measurements can be eliminated). This is because in Michelson's experiment it is not necessary to consider a sphere. The two arms of the apparatus may be of different lengths; and all that is observed is the continued coincidence of the interference-bands with markings on the instrument. When it is realised that coincidences constitute the most exact form of observation, we understand why it is that Einstein's definition is justified"

"The most precise experiments have proved the correctness of the Einsteinian laws of mechanics and...Bucherer's experiment proving the increase in mass of an electron in rapid motion is a case in point. Very important differences distinguish the theory of Einstein from that of Lorentz. Lorentz also had deduced from his theory that the mass of the electron should increase and grow infinite when its speed neared that of light; but the speed in question was the speed of the electron through the stagnant ether; whereas in Einstein's theory it is merely the speed with respect to the observer. According to Lorentz, the increase in mass of the moving electron was due to its deformation or Fitzgerald contraction. The contraction modified the lay of the electromagnetic field round the electron; and it was from this modification that the increase in mass observed by Bucherer was assumed to arise. In Einstein's theory, however, the increase in mass is absolutely general and need not be ascribed to the electromagnetic field of the electron in motion. An ordinary unelectrified lump of matter like a grain of sand would have increased in mass in exactly the same proportion; and no knowledge of the microscopic constitution of matter is necessary in order to predict these effects, which result directly from the space and time transformations themselves. Furthermore, the fact that this increase in mass of matter in motion is now due to relative motion and not to motion through the stagnant ether, as in Lorentz's theory, changes the entire outlook considerably. According to Lorentz, the electron really increased in mass, since its motion through the ether remained a reality. According to Einstein, the electron increases in mass only in so far as it is in relative motion with respect to the observer. Were the observer to be attached to the flying electron no increase in mass would exist; it would be the electron left behind which would now appear to have suffered the increase. Thus mass follows distance, duration and electromagnetic field in being a relative and having no definite magnitude of itself and being essentially dependent on the conditions of observation. Owing to the general validity of the Lorentz-Einstein transformations, it becomes permissible to apply them to all manner of phenomena.. ...temperature, pressure and many other physical magnitudes turned out to be relatives. ...entropy, electric charge and the velocity of light in vacuo were absolutes transcending the observer's motion. ...a number of other entities are found to be absolutes, the most important of which is that abstract mathematical quantity called the Einsteinian interval, which plays so important a part in the fabric of the new objective world of science, the world of four-dimensional space-time."

"Consider two observers, one... moving uniformly along a straight line, the other [stationary] on the embankment. At the precise instant these two observers pass each other at a point P, a flash of light is produced at the point P. The light wave produced by the instantaneous flash will present the shape of an expanding sphere. Since the invariant velocity of light holds equally for either observer, we must assume that either observer will find himself at all times situated at the centre of the expanding sphere. Our first reaction might be to say: "What nonsense! How can different people, travelling apart, all be at the centre of the same sphere?" Our objection, however, would be unjustified."

"If our world-line and that of the body which is being observed are parallel, the body is said to be at rest. But if the two world-lines are not parallel, then, when interpreting things in terms of space and time, the body will be said to be in relative motion."

"Superstring theories demand our spacetime dimension to be 10, which means we should reduce them to an effectively 4-dimensional theory. The standard solution of string compactification, as a generalization of Kaluza-Klein compactification, renders the extra six dimensions Calabi-Yau (CY). Thus, the study of Calabi-Yau and algebraic geometry has entered the field of theoretical physics."

"Scientifically speaking, a butterfly is at least as mysterious as a superstring. When something ceases to be mysterious it ceases to be of absorbing interest to scientists. Almost all things scientists think and dream about are mysterious."

"Imagine, if you can, four things that have very different sizes. First, the entire universe. Second, the planet Earth. Third, the nucleus of an atom. Fourth, a superstring. The step in size from each of these things to the next is roughly the same... twenty powers of ten...."

"What philosophical conclusions should we draw from the abstract style of the superstring theory? We might conclude, as Sir James Jeans concluded long ago, that the Great Architect of the Universe now begins to appear as a Pure Mathematician, and that if we work hard enough at mathematics we shall be able to read his mind. Or we might conclude that our pursuit of abstractions is leading us far away from those parts of the creation which are most interesting from a human point of view. It is too early yet to come to conclusions."

"I have noticed when I was younger, that lots of old men in the field couldn't understand new ideas very well, and resisted them with one method or another, and that they were very foolish in saying these ideas were wrong — such as Einstein not being able to take quantum mechanics. I’m an old man now, and these are new ideas, and they look crazy to me, and they look like they’re on the wrong track. Now I know that other old men have been very foolish in saying things like this, and, therefore, I would be very foolish to say this is nonsense. I am going to be very foolish, because I do feel strongly that this is nonsense! I can’t help it, even though I know the danger in such a point of view. So perhaps I could entertain future historians by saying I think all this superstring stuff is crazy and is in the wrong direction."

"I don’t like that they’re not calculating anything. I don’t like that for anything that disagrees with an experiment, they cook up an explanation – a fix-up to say 'Well. it still might be true'. For example, the theory requires ten dimensions. Well, maybe there's a way of wrapping up six of the dimensions. Yes, that's possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these things get wrapped up, not the desire to agree with experiment. In other words, there's no reason whatsoever in superstring theory that it isn't eight of the ten dimensions that get wrapped up and that the result is only two dimensions, which would be completely in disagreement with experience. So the fact that it might disagree with experience is very tenuous, it doesn't produce anything; it has to be excused most of the time. It doesn't look right."

"Of all the fields in fundamental physical theory, the gravitational field is picked out as controlling, in Einsteinian fashion, the structure of space-time. This is true even in a unified description of all the fields and all the particles of nature. Today, in superstring theory, we have the first respectable candidate for such a theory, apparently finite in perturbation theory and describing, roughly speaking, an infinite set of local fields, one of which is the gravitational field linked to the metric of space-time. If all the other fields are dropped, the theory becomes an Einsteinian theory of gravitation."

"The notion of a smooth spatial geometry, the central principle of general relativity, is destroyed by the violent fluctuations of the quantum world on short distance scales. ...The equations of general relativity cannot handle the rolling frenzy of the quantum foam. ... Physicists have made numerous attempts at modifying either general relativity or quantum mechanics in some manner so as to avoid the conflict, but the attempts... have been met with failure after failure. That is, until the discovery of superstring theory."

"Superstring theory starts off by proposing a new answer to an old question: what are the smallest, indivisible constituents of matter? For many decades, the conventional answer has been that matter is composed of particles... that can be modeled as dots that are indivisible and that have no size and no internal structure. Conventional theory claims, and experiments confirm, that these particles combine in various ways to produce protons, neutrons, and a wide variety of atoms and molecules... Superstring theory tells a different story. ...it does claim that these particles are not dots. Instead... every particle is composed of a tiny filament of energy, some hundred billion billion times smaller than a single atomic nucleus, which is shaped like a string. And just as a violin string can vibrate in different patterns, each of which produces a different musical tone, the filaments of superstring theory can also vibrate in different patterns. But these vibrations... produce different particle properties. ...All species of particles are unified in superstring theory since each arises from a different vibrational pattern executed by the same underlying entity."

"Is string theory a futile exercise as physics, as I believe it to be? It is an interesting mathematical specialty and has produced and will produce mathematics useful in other contexts, but it seems no more vital as mathematics than other areas of very abstract or specialized math, and doesn't on that basis justify the incredible amount of effort expended on it."

"... within , string theory is a complex subject. There are few large particle theory groups within major universities that do not have at least one person doing string theory. In total, there are probably a couple of thousand people at universities around the world whose mortgage is paid either by doing string theory, using string theory, working with tools made within string theory, solving problems using methods developed in string theory, or simply having their mental map of the world at the set by string theory."

"The real world as we know it happens at energies well below the Planck scale, so it is very well described by effective field theory. There is a continuous infinity of consistent effective field theories. Remarkably, only a measure zero fraction of those seem to be obtainable from string theory. These effective field theories arise as low energy descriptions of certain "vacua" of string theory, which in some approximation schemes can be thought of as solutions to the equations of motion for the compactification space."

"... one of the main beefs with the string theory is that it is so flexible you can get almost anything out of it. ... String theorists themselves are not too happy about it."

"String theorists, of course, continue to do whatever it is that string theorists do."

"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. ... Physicists have found that a key signal that a quantum mechanical theory has gone haywire is that particular calculations yield "probabilities" that are not within... acceptable range. For instance... infinite probabilities. ...string theory cures these infinities. ...a residual ...problem remains. In the early days ...calculations yielded negative probabilities ...so string theory appeared to be awash in its own quantum-mechanical hot water. ... Physicists found that the troublesome calculations were highly sensitive to the number of independent directions to which a string can vibrate. ...if strings could vibrate in nine independent spatial directions, all of the negative probabilities would cancel out. ... Kaluza and Klein provide a loophole... in addition to our familiar three... there are six other curled-up... rather than just postulating the existence of extra dimensions, as had been done by Kaluza, Klein, and their followers, string theory requires them."

"Currently, string theorists are in a position analogous to an Einstein bereft of the equivalence principle. ...[A] central organizing principle that embraces ...all ...features of the theory within one overarching and systematic framework ...is still missing."

"Now, this change from point-particles to strings that are so small they look like points might not sound like it would accomplish much. But it does. Superstring theory successfully merges general relativity and quantum mechanics."

"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."

"Question 5. String Phenomenology Here there are many questions that can all be summarized by asking whether one can construct a totally realistic four-dimensional model which is consistent with string theory and agrees with observation?"

"The number 24 appearing in Ramanujan's function is also the origin of the miraculous cancellations occurring in string theory. ...each of the 24 modes in the Ramanujan function corresponds to a physical vibration of a string. Whenever the string executes its complex motions in space-time by splitting and recombining, a large number of highly sophisticated mathematical identities must be satisfied. These are precisely the mathematical identities discovered by Ramanujan. ...The string vibrates in ten dimensions because it requires... generalized Ramanujan functions in order to remain self-consistent."

"We actually have a candidate for the mind of God. The mind of God we believe is cosmic music, the music of strings resonating through 11 dimensional hyperspace. That is the mind of God."

"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."

"String theory is, in fact, a textbook case of a Deceitful Turkey, a beautiful set of ideas that will always remain just barely out of reach. Far from a wonderful technological hope for a greater tomorrow, it is instead the tragic consequence of an obsolete belief system—in which emergence plays no role and dark law does not exist."

"One can ask whether the situation today in string theory is really as favorable as it was for field theory in the early 60's. It is difficult to know. Then, of course we had many more experiments to tell us how quantum field theories actually behave. To offset that, we have today more experience and greater mathematical sophistication."

"Consistent, relativistic string theories had already been written down in two, ten or twenty-six dimensions (the last being relevant only to bosonic strings) in the 1970s. A closed string is a loop which replaces a spacetime point. Its quantum oscillations correspond to particles of higher spins and higher masses, which may be arranged in a linear trajectory in a spin-versus-mass ... (Regge) plot. If the slope parameter of this trajectory — the only parameter in the theory — is adjusted to equal the Newtonian constant, one can show, quite miraculously, that in the zeroth order of the closed bosonic string there emerges from the string theory Einstein's gravity in its fullness! (The higher orders give modifications to Einstein's theory, with corrections which have a range of length = 10–33 cm.)"

"The most recent chapter in our new understanding of nonperturbative effects in string theory has been the incorporation of unstable branes and open string tachyons into the overall framework of the theory. It has turned out that an understanding of unstable D-branes is necessary to properly describe all D-branes. This is natural from the point of view of K-theory, where brane configurations which are equivalent under the annihilation of unstable branes are identified ... The long-mysterious tachyon instability of open string theory has finally been given a physical interpretation: it is the instability of the D-brane that supports the existence of open strings. The instability disappears in the tachyon vacuum, in which the D-brane decays. Moreover, the belief that D-branes are solitonic solutions of string theory has been confirmed: starting with the appropriate tachyonic field theory of unstable space-filling branes, one can describe lower dimensional D-branes as solitonic solutions. Lower dimensional D-branes are thereby essentially obtained as solitons of the tachyon field theory, so, in some sense, lower-dimensional D-branes can be thought of as being made of tachyons! It has also been shown that the physics of unstable D-branes is captured by string field theory, thus making it a candidate for a non-perturbative formulation of string theory capable of describing changes of the string background."

"... all of these caveats really work only against the idea that the final theory of nature is a quantum field theory. They leave open the view, which is in fact the point of view of my book, that although you can not argue that relativity plus quantum mechanics plus cluster decomposition necessarily leads only to quantum field theory, it is very likely that any quantum theory that at sufficiently low energy and large distances looks Lorentz invariant and satisfies the cluster decomposition principle will also at sufficiently low energy look like a quantum field theory. Picking up a phrase from Arthur Wightman, I’ll call this a folk theorem. At any rate, this folk theorem is satisfied by string theory, and we don’t know of any counterexamples."

"From the beginning it was clear that, despite its successes, the Standard Model of elementary particles would have to be embedded in a broader theory that would incorporate gravitation as well as the strong and electroweak interactions. There is at present only one plausible candidate for such a theory: it is the theory of strings, which started in the 1960s as a not-very-successful model of hadrons, and only later emerged as a possible theory of all forces."

"It's possible that the true structures in the underlying theory are something quite different — and the obvious candidate is strings. That direction, I think, is certainly worth working on. We don't whether that's the correct direction."

"String theory at its finest is, or should be, a new branch of geometry. ...I, myself, believe rather strongly that the proper setting for string theory will prove to be a suitable elaboration of the geometrical ideas upon which Einstein based general relativity."

"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."

"It's been said that string theory is part of the physics of the twenty-first century that fell by chance into the twentieth century. That's a remark that was made by a leading physicist about fifteen years ago. ...String theory was invented essentially by accident in a long series of events, starting with the Veneziano model... No one invented it on purpose, it was invented in a lucky accident. ...By rights, string theory shouldn't have been invented until our knowledge of some of the areas that are prerequisite... had developed to the point that it was possible for us to have the right concept of what it is all about."

"Generally speaking, all the really great ideas of physics are really spin-offs of string theory... Some of them were discovered first, but I consider that a mere accident of the development on planet earth. On planet earth, they were discovered in this order [general relativity, quantum field theory, superstrings, and supersymmetry]... But I don't believe, if there are many civilizations in the universe, that those four ideas were discovered in that order in each civilization."

"I feel that we are so close with string theory that—in my moments of greatest optimism—I imagine that any day, the final form of the theory may drop out of the sky and land in someone's lap. But more realistically, I feel that we are now in the process of constructing a much deeper theory than anything we have had before and that well into the twenty-first century, when I am too old to have any useful thoughts on the subject, younger physicists will have to decide whether we have in fact found the final theory."

"String theory is not like anything else ever discovered. It is an incredible panoply of ideas about math and physics, so vast, so rich you could say almost anything about it."

"Unlike a Feynman graph, which is divided into different lines, which can represent particles of different types with different masses and spins, any part of a string world sheet is equivalent to any other so "there is only one string." Whatever particles there are going to be represent different states of vibration of one basic string. Also there are not any vertices in the string world sheet so we do not have the freedom to tell the string how to interact."

"Among the many significant ideas and developments that connect Mathematics with contemporary Physics one of the most intriguing is the role that Quantum Field Theory (QFT) plays in Geometry and Topology. We can argue back and forth on the relevance of such a role, but the perspective QFT offers is often surprising and far reaching. Examples abound, and a fine selection is provided by the revealing insights offered by Yang–Mills theory into the topology of 4-manifolds, by the relation between Knot Theory and topological QFT, and most recently by the interaction between Strings, Riemann moduli space, and enumerative geometry. Doubtless many of the most striking connections suggested by physicists failed to pass the censorship of the Department of Mathematics, and so do not appear in the above official list."

"The disillusionment with QFT as a basis for the theory of elementary particles was also premature. What was missing was many ingredients, including the identification of the underlying gauge symmetry of the weak interactions, the concept of spontaneous symmetry breaking that could explain how this symmetry was hidden, the identification of the fundamental constituents of the nucleons as colored quarks, the discovery of asymptotic freedom which explained how the elementary colored constituents of hadrons could be seen at short distances yet evade detection through confinement, and the identification of the underlying gauge symmetry of the strong interactions. Once these were discovered, it was but a short step to the construction of the standard model, a gauge theory modeled on QED, which opened the door to the understanding of mesons and nucleons."

"In parallel with the changes it brought in our attitude toward symmetries, the birth of the Standard Model marked changes also in our attitude toward quantum field theory. After 't Hooft's breakthrough in 1971, it became clear that the old problem of infinities in the weak interactions had been solved by the use of spontaneous symmetry breaking to give masses to the W and Z particles. Then the asymptotic freedom of quantum chromodynamics gave us a framework in which we could actually calculate something about the strong interactions - not everything, but at least something. But in scoring these victories, quantum field theory was preparing the way for a further change in our attitude, in which quantum field theory would lose its central position."

"Starting in the late 1960s and then accelerating in the 1970s, our optimism about quantum field theory once again returned with the advent of the Standard Model — first of electromagnetic and weak interactions and then chromodynamics for the strong interactions. And quantum field theory became the tool of choice for doing calculations in particle physics."

"Mathematical gauge theory studies connections on principal bundles, or, more precisely, the solution spaces of certain partial differential equations for such connections. Historically, these equations have come from mathematical physics. Gauge theory as a tool for studying topological properties of four-manifolds was pioneered by the fundamental work of Simon Donaldson in the early 1980's. Since the birth of the subject, it has retained its close connection with symplectic topology, a subject whose intricate structure was illuminated by Mikhail Gromov's introduction of pseudo-holomorphic curve techniques, also introduced in the early 1980's."

"Symmetry is one of the great unifying themes in physics. From cosmology to and from to , symmetries determine which shapes, interactions, and evolutions occur in nature. Perhaps the most important aspect of symmetry in theories of physics, is the idea that the states of a system do not need to have the same symmetries as the theory that describes them. Such spontaneous breakdown of symmetries governs the dynamics of , the emergence of new particles and s, the rigidity of collective states of matter, and is one of the main ways emerges in a quantum world. The basic idea of spontaneous symmetry breaking is well known, and repeated in different ways throughout all fields of physics."

"We investigate the possibility that radiative corrections may produce spontaneous symmetry breakdown in theories for which the semiclassical (tree) approximation does not indicate such breakdown. The simplest model in which this phenomenon occurs is the electrodynamics of massless s. We find (for small coupling constants) that this theory more closely resembles the theory with an imaginary mass (the Abelian Higgs model) than one with a positive mass; spontaneous symmetry breaking occurs, and the theory becomes a theory of a massive vector meson and a massive scalar meson."

"The secret of nature is symmetry, but much of the texture of the world is due to mechanisms of symmetry breaking. The are a variety of mechanisms wherein the symmetry of nature can be hidden or broken. The first is explicit symmetry breaking where the is only approximately symmetric, but the magnitude of the symmetry breaking forces is small, so that one can treat the symmetry violation as a small correction. Such approximate symmetries lead to approximate conservation laws. Many of the symmetries observed in nature are of this sort, not really symmetries of the laws of physics at all, but—for what appears sometimes to be accidental reasons—approximate symmetries for a certain class of phenomena. The isotopic symmetry of the is an example of an approximate symmetry; good due to the small values of the and masses and the weakness of the electromagnetic force. A more profound way of hiding symmetry is the phenomenon of spontaneous symmetry breaking. Here the laws of physics are symmetric but the state of the system is not. This situation is common in . The is an example of a solution of Newton’s equations that is not rotationally invariant, although the equations are. Consequently, for an observer of the solar system, the rotational invariance of the law of gravitation is not manifest."

"The observed is assumed to be due to the spontaneous symmetry-breaking mechanism; the Lagrangian is CP invariant but its particular solution is not. The general classification of such theories when coupled with different unified gauge models of the weak and electromagnetic interactions is given. All such theories lead naturally to a basically milliweak CP noninvariant solution. The possibility that for most weak transitions the result may resemble a superweak theory is analysed, and possible experiments to distinguish these two different types of theories are discussed. Detailed calculations for various CP violating amplitudes are carried out for a generalized Georgi-Glashow model."

"Quantum electrodynamics is studied analytically in a quenched, planar approximation in four dimensions. At sufficiently strong coupling, chiral symmetry is broken spontaneously and the corresponding pseudoscalar Goldstone boson is observed. This phase of the theory is governed by a novel ultraviolet fixed point which requires the mixing of four-fermion interactions with the electrodynamic interactions."

"The for scalar electrodynamics in Fermi gauges are shown to imply a homogeneous first-order partial differential equation for the effective potential involving only the gauge parameter and the external scalar field. Spontaneous symmetry breaking is consequently a gauge-invariant phenomenon. Also observable quantities, including masses, physical coupling constants, and elements, of a theory with spontaneous symmetry breaking are found to be invariant, if a change in the gauge parameter is accompanied by a suitable change in the ground-state expectation value of the scalar field. The generalization to a non-Abelian gauge theory is briefly indicated."

"There is nothing mysterious about spontaneous symmetry breaking. There are many examples in physics. Hold a drinking straw between the palms of your hands and you have a physical system that can be described by equations possessing rotational symmetry. Press your palms together and the straw bends. The symmetry is broken. You cannot necessarily predict how the straw will bend; it could bend "up," "down," "sideways," or in any other direction. This unsymmetrical situation, however, is the stable solution of perfectly symmetrical equations."

"Will we ever derive the necessary mathematical tools to analytically demonstrate from first principles that confinement is indeed a mathematical property of quantum chromodynamics? This is the million-dollar question, literally. The Clay Mathematics Institute has announced a million-dollar prize for a rigorous mathematical proof that quantum chromodynamics does not allow free quarks or gluons to be produced. While no claimants to the prize have yet come forward, we nevertheless have strong indirect support of this idea, coming not only from experimental observations, but also from numerical simulations that closely approximate the complicated interactions in quantum chromodynamics. This is heartening, if not definitive. We still have to confirm that it is some property of the theory and not of the computer simulation. However, for physicists, if not mathematicians, this seems pretty convincing."

"If all you have is a hammer, everything looks like a nail."

"[T]he single equation of nature, aimed at by Lagrange and Hamilton, by Weber and Maxwell in their several ways, has... reached a more profound significance and now even holds dynamics, awkwardly it is true but none the less inexorably, in its grasp. That it is not complete, that it never can be complete, is admitted (for the absolute truth poured into the vessel of the human mind would probably dissolve it); but that it is immeasurably more complete to-day than it was yesterday is as incontrovertably true as it is inspiring."

"Concerning such dimensionless constants] I would like to state a theorem which at present cannot be based upon anything more than upon faith in the simplicity, i.e., intelligibility, of nature: there are no arbitrary constants of this kind; that is to say, nature is so constituted that it is possible logically to lay down such strongly determined laws that within these laws only rationally completely determined constants occur (not constants, therefore, whose numerical value could be changed without destroying the theory)."

"All science has one aim, namely, to find a theory of nature. We have theories of races and of functions, but scarcely yet a remote approach to an idea of creation. We are now so far from the road to truth, that religious teachers dispute and hate each other, and speculative men are esteemed unsound and frivolous. But to a sound judgment, the most abstract truth is the most practical. Whenever a true theory appears, it will be its own evidence. Its test is, that it will explain all phenomena. Now many are thought not only unexplained but inexplicable; as language, sleep, madness, dreams, beasts, sex."

"[A]greement with observed facts" never singles out one individual theory. There is never only one theory that is in complete agreement with all observed facts, but several theories that are in partial agreement. We have to select the final theory by a compromise. The final theory has to be in fair agreement with observed facts and must also be fairly simple. If we consider this point, it is obvious that such a "final" theory cannot be "The Truth."

"[F]inding the T.O.E. would in no way mean that psychology, biology, geology, chemistry, or even physics had been solved or in some sense subsumed. The universe is such a wonderfully rich and complex place that the discovery of the final theory... would not spell the end of science. Quite the contrary: The discovery of the T.O.E.—the ultimate explanation of the universe at its most microscopic level, a theory that does not rely on any deeper explanation—would provide the firmest foundation on which to build our understanding of the world. Its discovery would mark the beginning, not the end. The ultimate theory would provide an unshakable pillar of coherence forever assuring us that the universe is a comprehensible place."

"We are all, in our own way, seekers of the truth... each generation stands firmly on the shoulders of the previous... Whether any of our descendants will ever take in the view from the summit and gaze out on the vast and elegant universe with a perspective of infinite clarity, we cannot predict. But... we are fulfilling our part, contributing our rung to the human ladder reaching for the stars."

"The clearest expression of this modern "religious" mission can be recognized wherever one encounters the ancient Pythagorean search for nature's mathematical symmetry and harmony. This Pythagorean religion was transformed by early mechanists into a search for the mind of the Christian God. That quest has been tempered since the seventeenth century by a concern to find more practical mathematical relationships in nature, but it has not disappeared. ...wherever the religious mission has been retained in its pure form, as, for example, in the quest for a Theory of Everything... fewer women scientists will be found."

"Ever since Newton, and especially since Einstein, the goal of physics has been to find simple mathematical principles of the kind Kepler envisioned, and with them to create a unified theory of everything that would account for every detail of the matter and forces we observe in nature. ...The goal was to find not just a single theory that explains all forces but also one that explains the fundamental numbers... such as the strength of the forces and the masses and charges of the elementary particles. ...A unique theory would be unlikely to have the fine-tuning that allows us to exist. But if... we interpret Einstein's dream to be that of a unique theory that explains this and other universes, with their whole spectrum of different laws, then M-theory could be that theory."

"Any concepts or words which have been formed in the past through the interplay between the world and ourselves are not really sharply defined with respect to their meaning: that is to say, we do not know exactly how far they will help us in finding our way in the world. We often know that they can be applied to a wide range of inner or outer experience, but we practically never know precisely the limits of their applicability. This is true even of the simplest and most general concepts like "existence" and "space and time". Therefore, it will never be possible by pure reason to arrive at some absolute truth. The concepts may, however, be sharply defined with regard to their connections. This is actually the fact when the concepts become part of a system of axioms and definitions which can be expressed consistently by a mathematical scheme. Such a group of connected concepts may be applicable to a wide field of experience and will help us to find our way in this field. But the limits of the applicability will in general not be known, at least not completely."

"The ideal theory of everything, in the minds of physicists... is a mathematical system of uncommon tidiness and rigor, which may, if all works out correctly, have the ability to accommodate the physical facts... Perhaps physicists will one day find a theory of such compelling beauty that its truth cannot be denied; truth will be beauty... because, in the absence of any means to make practical tests, what is beautiful is declared ipso facto to be the truth. This theory of everything will be... a myth... a story that makes sense within its own terms, offers explanations for everything... but can be neither tested nor disproved... an explanation that everyone agrees on because it is convenient to agree on it, not because its truth can be demonstrated. This... will indeed spell the end of physics... not because physics has at last been able to explain everything... but because physics has reached the end of all things it has the power to explain."

"Do quarks and galaxies play by the same rules? Physicists believe they should, even though they don't quite know why. For decades, physicists have been searching for a "theory of everything"—a comprehensive description of the laws of nature. In particular, they want to bridge the gap between the large and the small with a quantum theory of gravity—a reconciliation of general relativity with quantum mechanics. String theory appears to be the current best bet..."

"It can rightly be said that symmetry, gauge theories, and spontaneous symmetry breaking have been the three pegs upon which modern particle physics rests."

"Back in the 1970s there was a simple dream about how physics would end. A unified theory would be found that incorporated quantum theory, general relativity, and the various particles and forces known to us. This would not only be a theory of everything, it would be unique. We would discover that there was only one mathematically consistent quantum theory that unified elementary particle physics with gravity. ...Because it was unique, this theory would have no free parameters—there would be no adjustable masses or charges. ...There would be only one scale, against which everything would be measured... the Planck scale. The theory would allow us to calculate the results of any experiment to whatever accuracy we desired. ...Looking back, it is clear that the assumption that a unified theory would be unique was no more than that—an assumption. ....we know that there can be no such theory."

"Our particular laws are not at all unique. ...they could change from place to place and from time to time. The Laws of Physics are much like the weather... controlled by invisible influences in space almost the same way as that temperature, humidity, air pressure, and wind velocity control how rain and snow and hail form. ...The Landscape... is the space of possibilities... all the possible environments permitted by the theory. ...[T]heoretical physicists ...have always believed that the laws of nature are the unique, inevitable consequence of some elegant mathematical principle. ...the empirical evidence points much more convincingly to the opposite conclusion. The universe has more in common with a Rube Goldberg machine than with a unique consequence of mathematical symmetry. ...Two key discoveries are driving the paradigm shift—the success of inflationary cosmology and the existence of a small cosmological constant."

"I will... argue that... our universe is not just described by mathematics—it is mathematics. ...this hypothesis... should... be useful in narrowing down what an ultimate theory of everything can look like. The foundation of my argument is the assumption that there exists an external physical reality independent of us humans. ...Our most successful theories, such as general relativity and quantum mechanics, describe only parts of this reality... In contrast, the holy grail of theoretical physics is a theory of everything—a complete description of reality."

"If we assume that reality exists independently of humans, then for a description to be complete, it must also be well-defined according to non-human entities—aliens or supercomputers, say—that lack any understanding of human concepts. Put differently, such a description must be expressible in a form that is devoid of any human baggage like “particle”, “observation” or other English words."

"The true mathematical structure isomorphic to our world, if it exists, has not been found."

"We need to distinguish between two different ways of viewing the external physical reality: the outside view or bird perspective of a mathematician studying the mathematical structure and the inside view or frog perspective of an observer living in it. ...a mathematical structure is an abstract, immutable entity existing outside of space and time. If history were a movie, the structure would therefore correspond not to a single frame of it but to the entire videotape. ... If a future physics textbook contains the TOE, then its equations are the complete description of the mathematical structure that is the external physical reality. ...is rather than corresponds to...If our external physical reality is isomorphic to a mathematical structure, it therefore fits the definition of being a mathematical structure."

"We are at the end of our extensive journey into the heart of matter. It has been marked as a relentless march toward Unity. As physics progressed, intimate connections were discovered between phenomena once thought to be completely distinct. A unified description of Nature, sometimes referred to a bit pompously as "the theory of everything," has become the holy grail of modern physics. Symmetry has constantly and reliably guided the physicists' first hesitant steps in this quest for Unity. ...becoming increasingly abstract. ...Now scientists routinely deal with "matter-light symmetry.""

"The theory of superstrings... described the world as a vast symphony of vibrations of infinitesimal strings in a 10-dimensional spade-time. The world would then be governed by a single superforce, melding all four currently known forces, and whose rule would extend over the entire universe. Are we about to reach the goal? ...I am not convinced. It is, for the time being... to be verified experimentally, since it would require phenomenal energies. ...protons have displayed a longevity surpassing initial predictions. ...the unification of everything is predicted at energies that defy human imagination... the theory is shrouded in such a thick mathematical veil that it no longer has any connection with reality. ...as long as physics is not rooted in reality, it is no more than metaphysics."

"It is possible that when we finally understand how particles and forces behave at energies up to 1018 GeV, we will just find new mysteries, with a final unification as far away as ever. But I doubt it. There are no hints of any fundamental energy scale beyond 1018 GeV, and string theory even suggests that higher energies have no meaning."

"I feel that we are so close with string theory that—in my moments of greatest optimism—I imagine that any day, the final form of the theory may drop out of the sky and land in someone's lap. But more realistically, I feel that we are now in the process of constructing a much deeper theory than anything we have had before and that well into the twenty-first century, when I am too old to have any useful thoughts on the subject, younger physicists will have to decide whether we have in fact found the final theory."

"[I]t was upon... inequality of motions in point of velocity that Galileo built his theory of flux and reflux of the sea; supposing that the earth revolved faster than the water could follow; and that the water was therefore first gathered in a heap and then fell down, as we see in a basin of water moved quickly. But this he devised upon an assumption which cannot be allowed, viz. that the earth moves; and also without being well informed as to the sexhorary motion of the tide."

"When Gilbert of Colchester, in his “New Philosophy,” founded on his researches in magnetism, was dealing with tides, he did not suggest that the moon attracted the water, but that “subterranean spirits and humors, rising in sympathy with the moon, cause the sea also to rise and flow to the shores and up rivers”. It appears that an idea, presented in some such way as this, was more readily received than a plain statement. This so-called philosophical method was, in fact, very generally applied, and Kepler, who shared Galileo’s admiration for Gilbert’s work, adopted it in his own attempt to extend the idea of magnetic attraction to the planets."

"A law explains a set of observations; a theory explains a set of laws. The quintessential illustration of this jump in level is the way in which Newton’s theory of mechanics explained Kepler’s law of planetary motion. Basically, a law applies to observed phenomena in one domain (e.g., planetary bodies and their movements), while a theory is intended to unify phenomena in many domains. Thus, Newton’s theory of mechanics explained not only Kepler’s laws, but also Galileo’s findings about the motion of balls rolling down an inclined plane, as well as the pattern of oceanic tides. Unlike laws, theories often postulate unobservable objects as part of their explanatory mechanism. So, for instance, Freud’s theory of mind relies upon the unobservable ego, superego, and id, and in modern physics we have theories of elementary particles that postulate various types of quarks, all of which have yet to be observed."

"Among all the great men who have philosophized about this remarkable effect, I am more astonished at Kepler than at any other. Despite his open and acute mind, and though he has at his fingertips the motions attributed to the earth, he nevertheless lent his ear and his assent to the moon's dominion over the waters, to occult properties, and to such puerilities."

"J. Kepler was the first (that I know of) that discover'd the true cause of the Tide, and he explains it largely in his Introduction to the Physics of the Heavens, given in his Commentaries to the Motion of the Planet Mars, where after he has shewn the Gravity or Gravitation of all Bodies towards another, he thus writes: "The Orb of the attracting Power, which is in the Moon is extended as far as the Earth, and draws the Waters under the Torrid Zone, acting upon places where it is vertical, insensibly on included Seas, but sensibly on the Ocean, whose Beds are large, and the Waters have the liberty of reciprocation, that is, of rising and falling"; and in the 70th Page of his Lunar Astronomy,—"But the cause of the Tides of the Sea appear to be the Bodies of the Sun and Moon drawing the Waters of the Sea.""

"Afterwards that incomparable Philosopher Sir Isaac Newton, improv'd the hint, and wrote so amply upon this Subject as to make the Theory of the Tides his own, by shewing that the Waters of the Sea rise under the Moon and the Place opposite to it: For Kepler believ'd "that the Impetus occasion'd by the presence of the Moon, by the absence of the Moon, occasions another Impetus; till the Moon returning, stops and moderates the Force of that Impetus, and carries it round with its motion." Therefore this Spheroidical Figure which stands out above the Sphere (like two Mountains, the one under the Moon and the other in the place opposite to it) together with the Moon (which it follows) is carried by the Diurnal Motion, (or rather, according to the truth of the matter, as the Earth turns towards the East it leaves those Eminencies of Water, which being carried by their own motion slowly towards the East, are as it were unmov'd) in its journey makes the Water swell twice and sink twice in the space of 25 Hours, in which time the Moon being gone from the Meridian of any Place, returns to it again."

"Astronomy teaches the correct use of the sun and the planets. These may be put on a frame of little sticks and turned round. This causes the tides."

"But to return to Kepler, his great sagacity, and continual meditation on the planetary motions, suggested to him some views of the true principles from which these motions flow. In his preface to the commentaries concerning the planet Mars, he speaks of gravity as of a power that was mutual betwixt bodies, and tells us that the earth and moon tend towards each other, and would meet in a point so many times nearer to the earth than to the moon, as the earth is greater than the moon, if their motions did not hinder it. He adds that the tides arise from the gravity of the waters towards the moon. But not having just enough notions of the laws of motion, he does not seem to have been able to make the best use of these thoughts; nor does he appear to have adhered to them steadily, since in his epitome of astronomy, published eleven years after, he proposes a physical account of the planetary motions, derived from different principles."

"The allusion to the "puzzling" problem of [the orbit of] Mars shows that Galileo ought not to have been unaware of the great work of Kepler published in 1609: Astronomia nova... in which the first two of Kepler's laws were formulated. Yet he does not mention here at all Kepler's success in solving the problem, nor his laws, nor his name even, which is brought up... only to criticize his belief in the Moon's attraction [effect upon tides], which is quite reasonably presented in the Astronomia nova and founded on astronomical reasons and not on mystical speculations."

"This world was once a fluid haze of light, Till toward the centre set the starry tides, And eddied into suns, that wheeling cast The planets: then the monster, then the man."

"The Academy of Sciences at Paris proposed The Tides as the subject for a prize essay in 1740. Four essays were published in consequence at Paris. One essay was by a Jesuit named Cavallieri; this adopted the Cartesian system of vortices. The other essays were by Daniel Bernoulli, Maclaurin, and Euler; these are reprinted in the Jesuits' edition of the Principia, and it is stated that many errors in the original impression have been corrected. ...The second chapter of Daniel Bernoulli's essay contains some lemmas relating to the Attraction of Bodies. ...he determines the attraction at any superficial or internal point of an ellipsoid of revolution which is nearly spherical, neglecting powers of the ellipticity beyond the first. The method used consists in finding accurately the attraction of a sphere, and then approximately the attraction of the difference between the sphere and the ellipsoid on a particle at the pole or at the equator... this method had been previously used by Clairaut. But Daniel Bernoulli seems to claim the method as his own... Although Daniel Bernoulli employed attraction for the purpose of his essay, yet he seems to have had but a weak faith in the principle... Daniel Bernoulli added nothing to our subject; all his results respecting Attraction are included in the formulæ given by Clairaut in 1737. But his theory of the Tides is very important in the history of that subject..."

"A Frenchman who arrives in London, finds a great alteration in philosophy, as in other things. He left the world full, he finds it empty. At Paris you see the universe composed of vortices of subtile matter, at London we see nothing of the kind. With you it is the pressure of the moon which causes the tides of the sea, in England it is the sea which gravitates towards the moon; so that when you think the moon ought to give us high water, these gentlemen believe that you ought to have low water; which unfortunately we cannot test by experience; for in order to do that, we should have examined the moon and the tides at the moment of the creation. You will observe also that the sun, which in France has nothing to do with the business, here comes in for a quarter of it. Among you Cartesians, all is done by an impulsion which one does not well understand; with the Newtonians, it is done by an attraction of which we know the cause no better. At Paris you fancy the earth shaped like a melon, at London it is flattened on the two sides."

"The theory of the tides has been reduced [in this work] into an extremely simple form, which appears to agree better with all the phenomena, than the more intricate calculations which they have commonly been supposed to require."

"Here lies Isaac Newton, Knight, Who, by a Vigour of Mind almost supernatural, First demonstrated The Motions and Figures of the Planets, The Paths of the Comets, and the Tides of the Ocean. He diligently investigated The different Refrangibilities of the Rays of Light, And the Properties of the Colours to which they give rise. An assiduous, sagacious, and faithful Interpreter Of Nature, Antiquity, and the Holy Scriptures, He asserted his Philosophy of the Majesty of God, And exhibited in his conduct the Simplicity of the Gospel. Let mortals rejoice That there has existed such and so great An Ornament of Human Nature."

"The application of the general doctrines of mechanics to fluids was a natural and inevitable step, when the principles of the science had been generalised. It was easily seen that a fluid is, for this purpose, nothing more than a body of which the parts are moveable amongst each other with entire facility; and that the mathematician must trace the consequences of this condition upon his equations. This accordingly was done, by the founders of mechanics, both for the cases of the equilibrium and of motion. ... The explanation of the Tides, in the way in which Newton attempted it in the third book of the Principia, is another example of a hydrostatical investigation: for he considered only the form that the ocean would have if it were at rest. The memoirs of Maclaurin, Daniel Bernoulli, and Euler, on the question of the tides, which shared among them the prize of the Academy of Sciences in 1740, went upon the same views. The Treatise of the Figure of the Earth by Clairaut, in 1743, extended Newton's solution of the same problem, by supposing a solid nucleus covered with a fluid of different density. No peculiar novelty has been introduced into this subject, except a method employed by Laplace for determining the attractions of s of small eccentricity, which is, as Professor Airy has said, "a calculus the most singular in its nature, and the most powerful in its effects, of any which has yet appeared.""

"Laplace... took up the subject of waves propagated along the surface of water; and deduced a very celebrated theory of the tides, in which he considered the ocean to be, not in equilibrium, as preceding writers had supposed, but agitated by a constant series of undulations, produced by the solar and lunar forces. The difficulty of such an investigation may be judged of from this, that Laplace, in order to carry it on, is obliged to assume a mechanical proposition, unproved, and only conjectured to be true; namely, that "in a system of bodies acted upon by forces which are periodical, the state of the system is periodical like the forces." Even with this assumption, various other arbitrary processes are requisite; and it appears still very doubtful whether Laplace's theory is either a better mechanical solution of the problem, or a nearer approximation to the laws of the phenomena, than that obtained by D. Bernoulli, following the views of Newton."

"In most cases, the solutions of problems of hydrodynamics are not satisfactorily confirmed by the results of observation. Poisson and Cauchy have prosecuted the subject of waves, and have deduced very curious conclusions by a very recondite and profound analysis. The assumptions of the mathematician here do not represent the conditions of nature; the rules of theory, therefore, are not a good standard to which we may refer the aberrations of particular cases; and the laws which we obtain from experiment are very imperfectly illustrated by à priori calculation. The case of this department of knowledge, hydrodynamics, is very peculiar... we want, in addition to what we have, true and useful principles, intermediate between the highest and the lowest;—between the extreme and almost barren generality of the laws of motion, and the endless varieties and inextricable complexity of fluid motions in special cases. The reason of this peculiarity in the science of hydrodynamics appears to be, that its general principles were not discovered with reference to the science itself, but by extension from the sister science of the mechanics of solids...by a perception that the parts of fluids are included in that range of generality which we are entitled to give to the supreme laws of motion of solids. ...[S]olid and fluid dynamics resemble two edifices which have their highest apartment in common, and though we can explore every part of the former building, we have not yet succeeded in traversing the staircase of the latter, either from the top or from the bottom. If we had lived in a world in which there were no solid bodies, we should probably not yet have discovered the laws of motion; if we had lived in a world in which there were no fluids, we should have no idea how insufficient a complete possession of the laws of motion may be, to give us a true knowledge of particular results."

"That all the parts of the universe are drawn and held together by love, or harmony, or some affection to which, among other names, that of attraction may have been given, is an assertion which may very possibly have been made at various times, by speculators writing at random, and taking their chance of meaning and truth. The authors of such casual dogmas have generally nothing accurate or substantial, either in their conception of the general proposition, or in their reference to examples of it... But among those who were really the first to think of the mutual attraction of matter, we cannot help noticing Francis Bacon; for his notions were so far from being chargeable with the looseness and indistinctness to which we have alluded, that he proposed an experiment which was to decide whether the facts were so or not;—whether the gravity of bodies to the earth arose from an attraction of the parts of matter towards each other, or was a tendency towards the centre of the earth. And this experiment is, even to this day, one of the best which can be devised, in order to exhibit the universal gravitation of matter: it consists in the comparison of the rate of going of a clock in a deep mine, and on a high place. Huyghens, in his book "De Causâ Gravitatis," published in 1690, showed that the earth would have an oblate form, in consequence of the action of the centrifugal force; but his reasoning does not suppose gravity to arise from the mutual attraction of the parts of the earth. The influence of the moon upon the tides had long been remarked; but no one had made any progress in truly explaining the mechanism of this influence; and all the analogies to which reference had been made, on this and similar subjects, as magnetic and other attractions, were rather delusive than illustrative, since they represented the attraction as something peculiar in particular bodies, depending upon the nature of each body. That all such forces, cosmical and terrestrial, were the same single force, and that this was nothing more than the insensible attraction which subsists between one stone and another, was a conception equally bold and grand; and would have been an incomprehensible thought, if the views which we have already explained had not prepared the mind for it."

"Newton, in the Principia, had inserted a series of propositions, the object of which was to prove, that the machinery of vortices could not be accommodated to one part of the celestial phenomena, without contradicting another part. A more obvious difficulty was the case of gravity of the earth; if this force arose, as Descartes asserted, from the rotation of the earth's vortex about its axis, it ought to tend directly to the axis, and not to the centre. The asserters of vortices often tried their skill in remedying this vice in the hypothesis, but never with much success. ...The mathematical prize-questions proposed by the French Academy, naturally brought the two sets of opinions into conflict. The Cartesian Memoir of John Bernoulli... was the one which gained the prize in 1730. ...The last act of homage of this kind to the Cartesian system was performed in 1740, when the prize on the question of the tides was distributed between Daniel Bernoulli, Euler, Maclaurin, and Cavallieri; the last of whom had tried to amend and patch up the Cartesian hypothesis on this subject."

"We propose... to enter at some length into the mathematical theories, and the experimental observations, applying to the two subjects of Tides and Waves of water. But we do not intend to treat them with the same extension. We shall give the various theories of Tides in detail sufficient to enable the reader to understand the present state of the science... and we shall advert to the principal observations which throw light either on the ordinary phænomena of tides, or on the extraordinary deviations that occur in peculiar circumstances. In thus treating the Tides, it will be necessary for us to enter largely into the theory of Waves. We shall take advantage of this circumstance for the introduction several propositions, not applying to the theory Tides, but elucidating some of the ordinary observations upon small Waves. But these investigations will be limited to that class which is most closely connected with tides, namely, that in which similar waves follow each other in a continuous series, or in which the same mathematical process may be used as when similar waves follow each other. In this class will be included nearly all the phænomena of waves produced by natural causes, and therefore possessing general interest. But it will not include the waves of discontinuous nature produced by the sudden action of arbitrary causes, which have been the subject of several remarkable mathematical memoirs, but which possess no interest for the general reader."

"We shall describe cursorily the ordinary phænomena of tides."

"We shall explain the Equilibrium-Theory of Tides, including the first tidal theory given by Newton, and the more detailed theory of his successors, especially Daniel Bernoulli."

"We shall give a sketch of Laplace's investigations, (founded essentially on the theory of the motion of water,) in the general form in which he first attempted the theory, as well as with the arbitrary limitations which he found it necessary to use for practical application."

"We shall give an extended Theory of Waves on water, applying principally to the motion of water in canals of small breadth, but with some indications of the process to be followed for the investigation of the motion of Waves in extended surfaces of water."

"The results of a few Experiments on Waves will be given, in comparison with the preceding theory."

"We shall investigate the mathematical expressions for the Disturbing Forces of the Sun and Moon which produce the Tides, and shall use them in combination with the theory of Waves to predict some of the laws of Tides."

"We shall advert to the methods which been used, or which may advantageously be used, for Observation of Tides, and for the Reduction of the Observations."

"We shall give the results of extensive observations of the Tides, as well with regard to the change of the phænomena of tides at different times in the same place, as with respect to the relation which the time and height of tide at one place bear to the time and height at other places, and shall compare these with the results of the preceding theories, as far as possible."

"And as Conclusion, we shall point out what we consider to be the present Desiderata in the Theory and Observations of Tides."

"Caesar, in his account of the invasion of Britain, (De Bella Gallico, lib. iv.) alludes to the nature of spring tides as perfectly well understood in connection with the moon’s age. Some of the peculiarities of river tides, however, were not published in scientific works till the beginning of the last century; and some of the properties of the tides in the English and other channels were not known till the end of that century. ...In the present century, the elaborate discussions of immense collections of accurate tide-observations by M. Laplace, Sir John W. Lubbock, and Professor Whewell, have brought to light and reduced to law many irregularities which were before that time unknown."

"Before entering upon either of the theories explaining the Tides, we must allude to their inadequacy, perhaps not to the explanation of the facts already observed, but certainly to the prediction of new ones. This inadequacy does not appear to arise from any defect in the principles upon which the theory is based,... but from the extreme difficulty of investigating mathematically the motions of fluids under all the various circumstances in which the waters of the sea and of rivers are found. For the problem of the Tides, it is evident, is essentially one of the motion of fluids. Yet so difficult are the investigations of motion that, till the time of Laplace, no good attempt was made to determine, by theory, the laws of the Tides, except on the supposition that the water was at -rest. Since that time theories of motion have been applied..."

"Indeed, throughout the whole of this subject, the selection of the proper theoretical ground of explanation is a matter of judgment. In some cases we may conceive that we are justified in using the Equilibrium theory; in others the Wave-theory will apply, completely or partially... as a last resource, in almost every case, we shall be driven to the same arbitrary suppositions which Laplace introduced. ...In the instances which it does not master completely, it will show that there are ample grounds for the arbitrary alterations of constants introduced by Laplace in his suppositions..."

"[W]e are precluded from further advance, partly by our almost necessary ignorance of the forms of the bottom in deep seas, and partly by the imperfection of our mathematics. ...the first principles of our explanation are correct."

"The popular explanation of the Equilibrium-theory is very simple. If we conceive the earth to be wholly or in a great degree with water, and consider that the attraction of the moon upon different particles (according to the law of gravitation) is inversely as the square of their distance, and is therefore greatest for those particles which are nearest to it; then it will be obvious that the moon attracts the water on that side which is next to her, more than she attracts the great mass of the earth, and therefore tends to raise the water from the earth on the side next to her; but she also attracts the great mass of the earth more than she attracts the water upon the side most distant from her, and therefore tends to draw the earth from the water on the side most distant from her; which will produce exactly the same effect as if a force tended to draw the water away from the earth on that side. Thus the moon’s action tends to raise the water on two opposite sides of the earth; and similarly the sun’s action tends to raise the water on two opposite sides. The close relation, however, which the times of high water bear to the times of the moon’s passage, shows that the moon’s influence in raising the tides must be much greater than the sun's. If the sun and moon are together, as seen from the earth, the elevations produced by these two bodies will coincide in place, and will therefore be added together. Thus Spring Tides will be produced. In other relative positions of the sun and moon, it may happen that the elevation produced by the sun will occur at a place where the moon causes depression: the action of the sun there tends to counteract that of the moon, and Neap Tides will be produced."

"Newton pointed out and assigned generally, not only the nature and the magnitude of the periodical forces which are concerned in producing the tides, but likewise indicated their true character as undulations, in one very remarkable proposition, as well as in a special explanation of... the tides of the Port of Batsha. The equilibrium theory of Daniel Bernoulli adopted the first part of Newton's views but altogether neglected the second."

"It had been shown that if the earth was a spherical body covered with water, and if both the earth and moon were at rest, the water would assume the form of a spheroid of equilibrium, of extremely small eccentricity, such as would be due to the disturbing action of the moon's forces. A similar but less eccentric spheroid would be formed beneath the sun. Under such circumstances the joint effect of the elevations or depressions of the two spheroids would produce the elevation or depression of the water, or the tide. The theory further assumes that the same effects would follow if the earth revolved round her axis and the earth and moon in their orbits, and that no effect was produced by the spontaneous oscillations of the sea. Totally false as are the principal assumptions upon which this theory is founded, it is extremely remarkable that it not only sufficiently separates from each other the principal movements of the tides, but represents generally the law and order of succession of the periodical phenomena which they present. "The greatest mathematicians and the most laborious observers of the present day," says Professor Airy, "including Sir John Lubbock and Dr. Whewell... have agreed equally in rejecting the foundation of this theory, and comparing all their observations with its results.""

"The same eminent authority [Professor Airy] has pronounced the theory proposed by La Place in the Mécanique Céleste,—if viewed with reference to the boldness and comprehensive character of its design rather than to the success of its execution—"as one of the most splendid works of the greatest mathematician of the past age." The problem, however, was not considered by him [La Place] in the most general form which it is capable of receiving. He assumed the earth to be entirely covered by water, and its depth to be uniform, at least throughout the same parallel of latitude, and he neglected the resistance both of the particles of the fluid amongst each other, and of that which arises from the irregular surfaces in the channels over which the tide is transmitted. He was consequently obliged to omit the consideration of the tides in canals, rivers, and narrow seas, which constitute some of the most interesting, and by no means the most unmanageable, of the problems which later, and even in some respects more simple, investigations of the oscillations of the sea have brought within the control of analysis. Imperfect, however, as the results of this theory were as it came from the hand of its author, their importance cannot easily be estimated too highly. Dr. Young adopted the general principles which they involved, though he has subjected them to a totally different treatment; and Professor Airy, who has materially simplified the investigations which it contains, by rejecting some conditions which they included, such as the density of the sea, by which they were made needlessly difficult and complicated, has not only verified the more remarkable of the conclusions at which La Place arrived, but has also made important use of his methods in his own theory of waves and tides, which is by far the most complete and comprehensive that has ever yet appeared."

"There is one result of a very unexpected kind, which La Place regarded as one of the happiest of his discoveries,—it is the entire evanescence, if the sea be of uniform depth, of the diurnal tide in elevation, but not in horizontal motion. At the equator, under such circumstances, the water moves north and south, resting for a moment at the change of motion. At the poles the motion is transverse to the meridian passing through the luminary. At all other points on the earth's surface it is perpetually changing. Few persons have attempted to follow the mazes of the difficult analysis by which this great mathematician has arrived at this conclusion, which has been verified by the Astronomer Royal. Its correctness, however, has been disputed by Dr. Young, who contends that the diurnal tide will not disappear, unless the depth of the sea be not merely uniform, but evanescent."

"Though Dr. Young was not disposed to give his assent to the results of an extremely difficult analysis,—which few persons of his age could venture to follow, and which might appear to those who could not trace them through the long train of consequences... to be either paradoxical or contradictory to the first principles of mechanics—he was sufficiently prepared to seize the general purport of other parts of this comprehensive theory; and by divesting it of the unnecessary generalizations by which it was encumbered, not only to bring its principles to bear immediately upon the ordinary phenomena of the tides, but to apply it to cases which it was otherwise incompetent to reach. Such were the tides of narrow seas and rivers, and the modifications which those tides undergo from the effects of the resistance of the particles of water upon each other, or upon the channels through which they are propagated. The same questions have been made the principal subject of the investigations of the Astronomer Royal, in his Article on Tides and Waves, in the Encyclopædia Metropolitana, where they have been treated with that rare combination of mathematical skill and clearness and completeness of exposition for which all his writings are so remarkable. It will be found, however, that there are not many of his results which Young had not already attained, though in a much less definite form, by methods which are, it is true, much less regular and systematic, but which are not less distinguished for the sagacity and philosophical power which they display."

"... the discovery of supersymmetry ... would be the first extension of our notions of spacetime since Einstein."

"Without SUSY, there is nothing like a chiral symmetry to protect scalar masses from heavy mass scales. But with SUSY, the chiral symmetry in the fermionic sector protects the scalars too."

"The observed Higgs mass is compatible with supersymmetry only if the superpartners are quite heavy (tens of TeV) or under special circumstances."

"Some theorists are led to supersymmetry because it emerges as part of the low energy theory from a superstring theory of everything. Others are particularly confident that nature will be supersymmetric because the Higgs mechanism is not an extra mechanism added ad hoc to the rest of the gauge theory, but emerges as a derived result (see the chapter by Ibáñez and Ross) if Mtop ≳ MW (which seems to be true). That is, supersymmetry can explain the ratio of the weak scale to the unification scale."

"If the Standard Model describes the world successfully, how can there be physics beyond it, such as supersymmetry? There are two reasons. First, the Standard Model does not explain aspects of the study of the large-scale universe, cosmology. For example, the Standard Model cannot explain why the universe is made of matter and not antimatter, nor can it explain what constitutes the dark matter of the universe. Supersymmetry suggests explanations for both of these mysteries. Second, the boundaries of physics have been changing. Now scientists ask not only how the world works (which the Standard Model answers) but why it works that way (which the Standard Model cannot answer). Einstein asked "why" earlier in the twentieth century, but only in the past decade or so have the "why" questions become normal scientific research in particle physics rather than philosophical afterthoughts."

"It would not be an exaggeration to say that today supersymmetry dominates theoretical high energy physics. Many believe it will play the same revolutionary role in the physics of the 21st as special and general relativity did in the physics of the 20th century. This belief is based on aesthetical appeal, on indirect evidence, and the fact that no theoretical alternative is in sight."

"Natural SUSY expects s and s to be light. This motivated and to carry out extensive in various final states using full Run-2 data."

"The mathematical consistency of string theory depends crucially on supersymmetry, and it is very hard to find consistent solutions (quantum vacua) that do not preserve at least a portion of this supersymmetry. This prediction of string theory differs from the other two (general relativity and gauge theories) in that it really is a prediction. It is a generic feature of string theory that has not yet been discovered experimentally."

"If dark matter is truly made of the lightest SUSY particle, then experiments designed to see it such as CDMS, XENON, Edelweiss and more should have detected it. Furthermore, SUSY dark matter should annihilate in a very particular way which hasn't been seen. Constraints on WIMP dark matter are quite severe, experimentally. The lowest curve rules out WIMP (weakly interacting massive particle) cross-sections and dark matter masses for anything located above it. This means that most models for SUSY dark matter are no longer viable."

"Of the proposed extensions to the Standard Model, supersymmetry (SUSY) has remained among the most popular for decades. It provides exactly the needed compensation to stabilize the Higgs mass, while additionally providing an ideal candidate for dark matter with a stable weakly interacting lightest supersymmetric particle (LSP)."

"The concept of naturalness is usually cited as the underlying motivation for supersymmetry. We will challenge that concept, and in any case need to point out that there is nothing natural about the development of the theory itself. Its main success is its agility in dodging the facts. The dubious explanation of the convergence of the three scaling coupling constants into a single point can not be taken seriously. It is just another fit, using some of the many free parameters."

"Shortly after the development of four-dimensional globally supersymmetric field theories, Zumino (1975) pointed out that supersymmetry in these theories would, if unbroken, imply a vanishing vacuum energy."

"There is an infinite number of Lie groups that can be used to combine particles of the same spin in ordinary symmetry multiplets, but there are only eight kinds of supersymmetry in four spacetime dimensions, of which only one, the simplest, could be directly relevant to observed particles."

"Steven Weinberg,"

"Arkani-Hamed and Dimopoulos ... have even shown how it is possible to keep the good features of supersymmetry, such as a more accurate convergence of the SU(3) × SU(2) × U(1) couplings to a single value, and the presence of candidates for dark matter WIMPs. The idea of this “split supersymmetry” is that, although supersymmetry is broken at some very high energy, the gauginos and higgsinos are kept light by a chiral symmetry. [An additional discrete symmetry is needed to prevent lepton-number violation in higgsino-lepton mixing, and to keep the lightest supersymmetric particle stable.]"

"Supersymmetry is a subject of considerable interest among physicists and mathematicians. Not only is it fascinating in its own right, but there is a growing belief that it may play a fundamental role in particle physicis. This belief is based on an important result of Haag, Sohnius, and Lopuszanski, who proved that the supersymmetry algebra is the only graded Lie algebra of symmetries of the S-matrix consistent with relativistic quantum field theory."

"Julius Wess and Jonathan Bagger: (p. 3)"

"In the absence of a canonical model for why and how supersymmetry breaking occurs, the predicted consequences of supersymmetry are not sharply defined."

"Frank Wilczek,"

"The unification of forces, even if it were perfected, would leave us with two great kingdoms of particles, still not unified. Technically, these are the fermion and boson kingdoms. More poetically, we may call them the kingdoms of substance (fermions) and force (bosons). By postulating that the fundamental equations enjoy the property of supersymmetry, we heal the division of particles into separate kingdoms. Supersymmetry can be approached from several different angles, but perhaps the most appealing is to consider it as an expansion of space-time, to include quantum dimensions. The defining characteristic of quantum dimensions is that they are represented by coordinates that are Grassmann numbers (i.e., anticommuting numbers) rather than real numbers. Supersymmetry posits that the fundamental laws of physics remain invariant transformations that correspond to uniform motion in the quantum dimensions. Thus supersymmetry extends Galileo/Lorentz invariance."

"Supersymmetry is an updating of special relativity to include fermionic as well as bosonic symmetries of spacetime. In developing relativity, Einstein assumed that the spacetime coordinates were bosonic; fermions had not yet been discovered! In supersymmetry the structure of spacetime is enriched by the presence of fermionic as well as bosonic coordinates."

"... as a very rough analogy, supersymmetric quantum theory is to ordinary quantum theory as differential forms on a manifold are to functions on a manifold. A very large fraction of geometrical applications of quantum field theory found in the eighties and nineties depend on supersymmetry. (Examples include the supersymmetric proofs of the positive energy theorem, the Atiyah-Singer index theorem, and the Morse inequalities, and the quantum field approaches to elliptic cohomology and to Donaldson theory.) ... Surely, if supersymmetry is confirmed in accelerators, mathematical attention will be focussed on this fruitful branch of quantum field theory roughly as the discovery of general relativity focussed attention on Riemannian geometry."

"… Supersymmetry … the virtues: * SUSY can make a “small” Higgs mass natural; • SUSY is part of a larger vision of physics, not just a technical solution; • the measured value of favors SUSY ’s; • SUSY survives tests; and • the mass has turned out to be heavy, as needed for electroweak symmetry breaking in the context of SUSY."

"SUSY is a unique new symmetry that relates s to s, in a sense explaining why fermions exist. Relating bosons to fermions also makes it possible to explain the smallness of the Higgs mass, since we do know why the smallness of fermion masses can be natural. So that is at least the germ of how SUSY solves the fine-tuning problem."

"... I knew quantum field theory well enough to know that saying that the potential energy for the scalar field is zero is not a meaningful statement quantum mechanically. If it were, we would not have a gauge hierarchy problem in particle physics. ... with supersymmetry the mass renormalization (and even the full effective potential) of a scalar can be zero."

"Edward Witten as quoted by Hirosi Ooguri in (p. 483)"

"Fundamental to the idea of MOND is that it is an 'effective' theory, playing a role similar to Kepler's laws (as stressed by Felten 1984). The proponents of MOND have yet to develop the analogue of Newtonian mechanics to explain the effective theory. The absence of a full theory seriously limits the predictive power of MOND, and leads various authors to disagree as to what the observational consequences of this revision will be."

"... Milgrom and those few who work on it, are quite aware of the pressing need to have a fully consistent theory that goes beyond the Newtonian non-relativistic limit to a theory that can be applied to cosmology. They don't have one. They fully admit it and they agree that this is a big gap, big lack in the theory. There it is. They do insist that on the scales of galaxies and smaller where it is intended to apply it works remarkably well, and they're right. There are just a few people working on this theory. The most active of the young people is Stacy McGaugh at the University of Maryland. If you ever get a chance you might be amused to talk to him."

"MOND is so successful that, as a minimum, it is telling us the exact functional form of the force in galaxies. Any theory of galaxy and structure formation must therefore be able to reproduce the MOND phenomenology."

"I've had conversations about MOND with several of the most imaginative theorists I know. Often it went like this: We would be talking about some sober mainstream problem and one of us would mention galaxies. We would look at each other with a glint of recognition and one of would say, "So you worry about MOND, too," as if admitting a secret vice. Then we would share our crazy ideas — because all ideas about MOND that are not immediately wrong turn out to be crazy."

"In matter of fact, whether MOND is a fundamental theory or not, the very special and central role the constant a0 plays in galaxy dynamics is well established and is here to stay. For instance, you will find it everywhere in the data itself. All round systems, from giant molecular clouds, through globular clusters and elliptical galaxies, to clusters of galaxies lie, in the mass-radius plane, near the line with constant M/R2. The value of this ratio when multiplied by G gives a0. Another example: the baryonic Tully-Fisher relation agrees well (over many orders of magnitude in mass) with a relation of the form α M = V–4. The proportionality constant α has dimensions of G times acceleration and when divided by G gives a0 (this is independent of the previous appearance as it refers to asymptotic regions in the galaxies)."

"Today we can probe regions of physics where space-time curvature is extremely small finding that, again, Newtonian mechanics fails. This may mean that the Theory of General Relativity needs an extension or that we do not yet understand what “space-time” and “mass” are nor how they are fundamentally related. Perhaps it just boils down to the problem of us not understanding the vacuum."

"Mild failures aside, it is clear that there is a broad range of masses, 106 — 1011 M☉, in which systems adhere to MOND in their systematics. This must be telling us something; logically there are the following possibilities. a) MOND is merely an efficient summary of the way DM is distributed in the said systems? b) MOND reveals the dependence of inertia on acceleration for small accelerations? c) MOND betrays hitherto unknown forces particularly effective at astronomical scales? d) MOND encapsulates departures from standard Newtonian-Einsteinian gravity theory at the mentioned scales?"

"Ten years ago, it was perfectly respectable to speculate that there was no such thing as dark matter, just a modification of gravity. (It couldn’t have been MOND alone, which was ruled out by clusters, but it could have been some more elaborate modification.) That’s no longer true. The Bullet Cluster and the CMB both provide straightforward evidence that there is gravity pointing in the direction of something other than the ordinary matter. The source for that gravity is “dark matter.” It could be simple, like an axion or a thermal relic, or it could be quite baroque, like TeVeS + sprinkles of other dark matter as required, but it’s definitely there."

"Viewed simply, MOND is an algorithm that, with one additional fundamental parameter having units of acceleration, allows calculation of the distribution of the effective gravitational force in astronomical objects from the observed distribution of baryonic dark matter — and it works remarkably well. This is evidenced primarily by the use of the MOND algorithm in the determination of rotation curves of disk galaxies where the agreement with observed rotation curves is often precise, even in details. The existence of such an algorithm is problematic for CDM because this is not something that dissipationless dark matter on the scale of galaxies can naturally do; it would seem to require a coupling between dark matter and baryonic matter which is totally at odds with the perceived properties of CDM."

"We do not know to what extent and how MOND affects nongravitational phenomena such as electromagnetism (EM). For example, if there is a consistent way to extend and apply the basic tenets to nongravitational physics."

"Another crucial point is that MOND as we know it now is arguably only an approximate 'effective field theory' that approximates some more fundamental scheme at a deeper stratum — some 'FUNDAMOND' — conceptually, in a similar way to thermodynamics being an approximation of the statistical-mechanics, microscopic description."

"... one really has to stand on one's head to reconcile MOND with what is well-established about relativistic physics, and the results are pretty obscure and far-fetched looking."

"We have in MOND a formula that has had repeated predictive successes. Many of these have been true a priori predictions, like the absolute nature of the Tully-Fisher relation, the large mass discrepancies evinced by low surface brightness galaxies, and the velocity dispersions of many individual dwarf spheroidal galaxies like Cluster 2. I don't see how this can be an accident. But what we lack is an underlying theoretical basis for the observed MONDian phenomenology: Why does this happen?"

"I think the existence of (something like) dark matter is incontrovertible. It would be nice to understand why Modified Newtonian dynamics (MOND) works so well."

"The key appeal of MOND is that we only need ordinary matter, the matter we can actually see, to explain the universe. But opponents are not happy with the ad-hocness of aspects of the theory, the messiness of the underlying mathematics, and the tweaking of parameters required to make MOND work. Most cosmologists would bet on dark matter, but MOND advocates show little sign of slowing down."

"Milgromian theorists have understood for a long time that there is just no way that a formless entity such as dark matter can spontaneously rearrange itself – and keep rearranging itself – so as to produce the striking regularities that we observe in the kinematics of nearby galaxies."

"... the interpretation of MOND as modified inertia (MI) ... ... Who is afraid of modified inertia? The interpretation of MOND as MI was on the table from the very inception of MOND ..."

"... when non-perturbative phenomena are included, there is no problem from the string theory point of view in effecting continuous transitions between Calabi-Yau spaces of different topology. This shows that stringy ideas about geometry are really more general than those found in classical Riemannian geometry. The moduli space of Calabi-Yau manifolds should thus be regarded as a continuously connected whole, rather than a series of different ones individually associated with different topological objects ... Thus, questions about the topology of Calabi-Yau spaces must be treated on the same footing as questions about the metric on the spaces. That is, the issue of topology is another aspect of the the moduli fields. These considerations are relevant to understanding the ground state of the universe."

"Calabi-Yau manifolds admit Kähler metrics with vanishing Ricci curvatures. They are solutions of the Einstein field equation with no matter. The theory of motions of circles inside of a Calabi-Yau manifold provide a model of a conformal field theory. (It is called a σ-model in physics.) Because of this, Calabi-Yau manifolds are pivotal in superstring theory. ... It has long been argued that, in order to solve certain classic problems of unified gauge theories such as the gauge hierarchy problem, the 4-dimensional effective theory should admit an N = 1 supersymmetry. In a fundamental paper, Candelas-Horowitz-Strominger-Witten ... analyzed what the constraint of that N = 1 supersymmetry would mean for the geometry of the internal space X. They found that, for the most basic product models with N = 1 supersymmetry, the space X must be a Calabi-Yau manifold of complex dimension 3. Shortly afterwards, Strominger ... considered slightly more general models, allowing warped products. For these models, the N = 1 supersymmetry constraint results in a modification of the Ricci-flat equation of the earlier model."

"Extra spatial dimensions could be a good thing. ... In a nongravitational theory, the spacetime geometry is a rigid background on which the dynamics takes place. In that setup, the fact that we observe four-dimensional Minkowski spacetime is a compelling argument to formulate the theory in that background geometry. As you know very well, this is part of the story of the Standard Model. However, in a gravitational theory that abides by the general principles laid out by Einstein, the spacetime geometry is determined by the dynamical equations. In such a setup extra dimensions can make sense provided that the equations of the theory have a solution for which the geometry is the product of four-dimensional Minkowski spacetime and a compact manifold that is sufficiently small to have eluded detection. It turns out that there are many such solutions. Moreover, the details of the compact manifold play a crucial role in determining the symmetries and particle content of the effective low-energy theory in four dimensions, even when the compact dimensions are much too small to observe directly."

"Why try to unite the four forces in a single theory? Why not simply use Einstein’s theory of general relativity to govern big things and quantum mechanics for little ones? Some concepts, such as the Big Bang or how black holes form, live in both domains. When we combine equations of the four forces to describe these ideas, our answers usually end up being either zero or infinity. … Here’s where string theory comes to the rescue. By adding seven hidden dimensions to the familiar three and another for time, plus antiparticles and a mirror set of particles called superparticles, the math starts to make sense. The force of gravity is diluted because it permeates into one or more of the hidden dimensions. Dark matter and dark energy also may invisibly shape our universe from these phantom dimensions."

"There seems to be a vast landscape of possible universes. ... We live in one in which life is possible, but if the universe were only slightly different, beings like us could not exist. What are we to make of this fine-tuning? Is it evident that the universe, after all, was designed by a benevolent creator? Or does science offer a different explanation?"

"The old cosmological constant problem is to understand why the is so small; the new problem is to understand why it is comparable to the present mass density. ... does not help with either; anthropic considerations offer a possibility of solving both. In theories with a that takes random initial values, the anthropic principle may apply to the cosmological constant, but probably to nothing else."

"Once one starts to admit anthropic interpretations of fine-tuning problems like the cosmological constant, it is clear that such a proposal might be made for other fine-tuning problems, such as the problem of the Higgs boson mass. Certainly, we would not be here if the Higgs boson mass, and hence also the and and and masses, were greatly bigger. If they were near the , for example, any collection of more than a few elementary particles would collapse into a Black Hole. More generally, if the elementary particle masses were scaled up by a factor N, the number of elementary particles in a star or planet would scale down like N–3, and for very modest N the stars would stop shining."

"... the theory of relativity makes it appear probable that Mach was on the right road in his thought that inertia depends upon a mutual action of matter. For we shall show in the following that, according to our equations, inert masses do act upon each other in the sense of the relativity of inertia, even if only very feebly. What is to be expected along the line of Mach's thought? 1. The inertia of a body must increase when ponderable masses are piled up in the neighborhood. 2. A body must experience an accelerating force when neighbouring masses are accelerated, and, in fact, the force must be in the same direction as that acceleration. 3. A rotating hollow body must generate inside itself a "Coriolis field," which deflects moving bodies in the sense of the rotation, and a radial centrifugal field as well. We shall now show that these three effects, which are to be expected in accordance with Mach's ideas, are actually present according to our theory, although their magnitude is so small that confirmation of them by laboratory experiments is not to be thought of."

"Mach's profound critique of the foundations of Newtonian mechanics played a key role in Einstein's development of . Mach's principle has also guided other developments in gravitation theory such as the ... It has inspired interesting experiments, such as the ... and continues to be of current interest ... Mach identified the essential epistemological shortcoming of the Newtonian foundations of physics, namely, that the intrinsic state of a particle in Newtonian mechanics, i.e. its mass, has no immediate connection with its extrinsic state in space and time, i.e. its position and velocity. Mach's observation can be re-stated in terms of the a priori independence of position (x) and momentum (p) of a particle in Newtonian mechanics."

"Most terrestrial motions are of such brief duration and extent, that it is wholly unnecessary to take into account the earth's rotation and the changes of its progressive velocity with respect to the celestial bodies. This consideration is found necessary only in the case of projectiles cast great distances, or in the case of the vibrations of , and in similar instances. When now Newton sought to apply the mechanical principles discovered since Galileo's time to the planetary system, he found that, so far as it is possible to form any estimate at all thereof, the planets, irrespectively of dynamic effects, appear to preserve their direction and velocity with respect to bodies of the universe that are very remote and as regards each other apparently fixed, the same bodies moving on the earth do with respect to the fixed bodies of the earth. The comportment of terrestrial bodies with respect to the earth is reducible to the comportment of the earth with respect to the remote heavenly bodies. If we were to assert that we knew more of moving objects than this their last-mentioned, experimentally-given comportment with respect to the celestial bodies, we should render ourselves culpable of a falsity. When, accordingly, we say, a body preserves unchanged its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe. The use of such an abbreviated expression is permitted the original author of the principle, because he knows, that as things are no difficulties stand in the way of carrying out its implied directions. But no remedy lies in his power, if difficulties of the kind mentioned present themselves; if, for example, the requisite, relatively fixed bodies are wanting."

"The so-called Mach's Principle is surely one of the most elusive concepts in physics. On one hand, Machian aspects have been present either explicitly or implicitly in theoretical astronomy, general physics, and dynamics from their Greek infancy up the present day ... On the other hand, most of practical physics is done, and successfully done, without ever thinking of the 'deep questions' connected with Mach's Principle. (The situation is similar in quantum theory, which functions extremely well using established prescriptions notwithstanding deep and unresolved questions about its interpretation, its measuring process, and its classical limit.)"

"This circumstance of an expanding universe is irritating. ...To admit such possibilities seems senseless to me."

"If a distant galaxy is moving relative to us, its entire is Doppler-shifted in frequency. Its s are displaced relative to those of stationary light sources. Thanks to this effect, we know that distant galaxies recede from the solar system at speeds proportional to their distances from us. That's the effect that told us of the expanding universe, and of its birth, long ago, in the Big Bang."

"All kinds of questions remain. Many have to do with cosmology. How did the universe originate? How did the galaxies become distributed in space like the suds in the kitchen sink..? Why is the cosmological constant apparently very tiny but non-zero and has a peculiar value that leads the universe to expand more rapidly?"

"All of this picture of the expansion is exciting, pleasant, coherent, well in order. But what if the s are not to be interpreted by the Doppler-Fizeau law in the classical mechanical view, or general relativistically, by the fact that the ratio of the of a photon (as measured by a co-moving observer) to the space radius of curvature is independent of ? Not speaking of quasars, the first indications for non-Doppler redshifts for a galaxy have been provided... What if not all galaxies were formed at the dawn of the Big Bang; what if some are being formed now? Then, at least, the can be anything larger than the age of our own Galaxy..."

"Red-shifts are produced either in the nebulae, where the light originates, or in the intervening space through which the light travels. If the source is in the nebulae, then red-shifts are probably velocity-shifts and the nebulae are receding. If the source lies in the intervening space, the explanation of red-shifts is unknown, but the nebulae are sensibly stationary."

"A book, too, can be a star, explosive material, capable of stirring up fresh life endlessly, a living fire to lighten the darkness, leading out into the expanding universe."

"The definition of inflation is extraordinarily simple: it is any period of the Universe's evolution during which the scale factor, describing the size of the Universe, is accelerating. This leads to a very rapid expansion of the Universe, though perhaps a better way of thinking of this is that the characteristic scale of the Universe, given by the Hubble length, is shrinking relative to any fixed scale caught up in the rapid expansion. In that sense, inflation is actually akin to zooming in on a small part of the initial Universe."

"One of the few authors to have explicitly connected the physical issue of the expansion of the universe with the philosophical topic of the metaphysical status of space is Gerald James Whitrow."

"In 1917 de Sitter showed that Einstein's field equations could be solved by a model that was completely empty apart from the cosmological constant—i.e. a model with no matter whatsoever, just . This was the first model of an expanding universe. although this was unclear at the time. The whole principle of general relativity was to write equations for physics that were valid for all observers, independently of the coordinates used. But this means that the same solution can be written in various different ways... Thus de Sitter viewed his solution as static, but with a tendency for the rate of ticking clocks to depend on position. This phenomenon was already familiar in the form of gravitational ... so it is understandable that the de Sitter effect was viewed in the same way. It took a while before it was proved (by Weyl, in 1923) that the prediction was of a redshifting of spectral lines that increased linearly with distance (i.e. ). ..."

"This model of the expanding universe I shall call the substratum. It achieves in the private Euclidean space of each fundamental observer the objects for which Einstein developed his closed spherical space. Although it is finite in volume, in the measures of any chosen observer, it has all the properties of an infinite space in that its boundary is forever inaccessible and its contents comprise an infinity of members. It is also homogeneous in the sense that each member stands in the same relation to the rest. This description of the substratum holds good in the scale of time in which the galaxies or fundamental particles are receding from one another with uniform velocities. This choice of the scale of time, together with the theory of equivalent time-keepers... makes possible the application of the Lorentz formulae to the private Euclidean spaces of the various observers. It thus brings the theory of the expanding universe into line with other branches of physics, which use the Lorentz formulæ and adopt Euclidean private spaces. ...[T]here is no more need to require a curvature for space itself in the field of cosmology than in any other department of physics. The observer at the origin is fully entitled to select a private Euclidean space in which to describe phenomena, and when he concedes a similar right to every other equivalent observer and imposes the condition of the same world-view of each observer, he is inevitably led to the model of the substratum which we have discussed."

"The ideas that prove to be of lasting interest are likely to build on the framework of the now standard world picture, the hot big bang model of the expanding universe. The full extent and richness of this picture is not as well understood as I think it ought to be, even among those making some of the most stimulating contributions to the flow of ideas."

"We should, of course, expect that any universe which expands without limit will approach the empty de Sitter case, and that its ultimate fate is a state in which each physical unit—perhaps each nebula or intimate group of nebulae—is the only thing which exists within its own observable universe."

"If the general picture, however, of a Big Bang followed by an expanding Universe is correct, what happened before that? Was the Universe devoid of all matter and then the matter suddenly somehow created? How did that happen? In many cultures, the customary answer is that a God or Gods created the Universe out of nothing. But if we wish to pursue this question courageously, we must of course ask the next question: where did God come from? If we decide that this is an unanswerable question, why not save a step and conclude that the origin of the Universe is an unanswerable question? Or, if we say that God always existed, why not save a step, and conclude that the Universe always existed? That there's no need for a creation, it was always here. These are not easy questions. Cosmology brings us face to face with the deepest mysteries, questions that were once treated only in religion and myth."

"The most far-reaching implication of general relativity... is that the universe is not static, as in the orthodox view, but is dynamic, either contracting or expanding. Einstein, as visionary as he was, balked at the idea... One reason... was that, if the universe is currently expanding, then... it must have started from a single point. All space and time would have to be bound up in that "point," an infinitely dense, infinitely small "singularity." ...this struck Einstein as absurd. He therefore tried to sidestep the logic of his equations, and modified them by adding... a "cosmological constant." The term represented a force, of unknown nature, that would counteract the gravitational attraction of the mass of the universe. That is, the two forces would cancel... it is the kind of rabbit-out-of-the-hat idea that most scientists would label ad-hoc. ...Ironically, Einstein's approach contained a foolishly simple mistake: His universe would not be stable... like a pencil balanced on its point."

"The cosmological constant['s]... most important consequence: the repulsive force, acting at cosmological distances, causes space to expand exponentially. There is nothing new about the universe expanding, but without a cosmological constant, the rate of expansion would gradually slow down. Indeed, it could even reverse itself and begin to contract, eventually imploding in a giant cosmic crunch. Instead, as a consequence of the cosmological constant, the universe appears to be doubling in size about every fifteen billion years, and all indications are that it will do so indefinitely."

"De Sitter proposed three types of nonstatic universes: the oscillating universes and the expanding universes of the first or second kiind. The main characteristic of the expanding "family" of the first kiind is that the radius is continually increasing from a definite initial time when it had the value zero. The universe becomes infinitely large after an infinite time. In the second kind... the radius possesses at the initial time a definite minimum value... in the Einstein model... the cosmological constant is supposed to be equal to the reciprocal of R2, whereas de Sitter computed for his interpretation the constant to be equal to 3/R2. Whitrow correctly points out the significant fact that in special relativity the cosmological constant is omitted..."

"[W]e stress... the wide range of validity exhibited by s in theoretical physics. ...[I]t has ...been demonstrated how they can be employed to derive equations of optics, dynamics of particles and rigid bodies, and electromagnetism. In addition, physicists have succeeded in formulating the laws of elasticity and hydrodynamics as variational principles, and even Einstein's law of gravitation was included in this category by Hilbert, who found a scaler function... for which \partial\int\mathfrak{h}\,dx_0\,dx_1\,dx_2\,dx_3=0 is equivalent to Einstein's law. This function has been called the "curvature," an identification which induced Whittaker to describe Hilbert's principle in the laconic words, "gravitation simply represents a continual effort of the universe to straighten itself out.""

"The general theory of relativity considers physical space-time as a four-dimensional manifold whose line element coefficients g_{\mu \nu} satisfy the differential equationsG_{\mu \nu} = \lambda g_{\mu \nu} \qquad .\;.\;.\;.\;.\;.\; (1)in all regions free from matter and electromagnetic field, where G_{\mu \nu} is the contracted Riemann-Christoffel tensor associated with the fundamental tensor g_{\mu \nu}, and \lambda is the ."

"An "empty world," i.e., a homogeneous manifold at all points at which equations (1) are satisfied, has, according to the theory, a constant Riemann curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy."

"In considerations involving the nature of the world as a whole the irregularities caused by the aggregation of matter into stars and stellar systems may be ignored; and if we further assume that the total matter in the world has but little effect on its macroscopic properties, we may consider them as being determined by the solution of an empty world."

"The solution of (1), which represents a homogeneous manifold, may be written in the form:ds^2 = \frac{d\rho^2}{1 - \kappa^2\rho^2} - \rho^2 (d\theta^2 + sin^2 \theta \; d\phi^2) + (1 - \kappa^2 \rho^2)\; c^2 d\tau^2, \qquad (2)where \kappa = \sqrt \frac{\lambda}{3}. If we consider \rho as determining distance from the origin... and \tau as measuring the proper-time of a clock at the origin, we are led to the de Sitter spherical world..."

"O. Heckmann has pointed out that the non-static solutions of the field equations of the general theory of relativity with constant density do not necessarily imply a positive curvature of three-dimensional space, but that this curvature may also be negative or zero. There is no direct observational evidence for the curvature, the only directly observed data being the mean density and the expansion, which latter proves that the actual universe corresponds to the non-statical case. It is therefore clear that from the direct data of observation we can derive neither the sign nor that value of the curvature, and the question arises whether it is possible to represent the observed facts without introducing the curvature at all. Historically the term containing the "cosmological constant" λ was introduced into the field equations in order to enable us to account theoretically for the existence of a finite mean density in a static universe. It now appears that in the dynamical case this end can be reached without the introduction of λ."

"The determination of the coefficient of expansion h depends on the measured red-shifts, which do not introduce any appreciable uncertainty, and the distances of the extra-galactic nebulae, which are still very uncertain. The density depends on the assumed masses of these nebulae and on the scale of distance, and involves, moreover, the assumption that all the material mass in the universe is concentrated in the nebulae. It does not seem probable that this latter assumption will introduce any appreciable factor of uncertainty."

"Although... the density... corresponding to the assumption of zero curvature and to the coefficient of expansion... may perhaps be on the high side, it... is of the correct order of magnitude, and we must conclude that... it is possible to represent the facts without assuming a curvature of three-dimensional space. The curvature is, however, essentially determinable, and an increase in the precision of the data derived from observations will enable us in the future to fix its sign and to determine its value."

"Why should not the space be there already, and the material system expand into it..? ...[I]f the speed of recession continues to increase outwards, it will ere long approach the speed of light, so that something must break down. The result is that the system becomes a ... such a system cannot expand without the space also expanding. ...[E]xpansion of space has often been given too much prominence ...and readers have been led to think that it is more directly concerned in the explanation of the motions of the nebule than is... the case. ...If we adopt open space we encounter certain difficulties (not necessarily insuperable) which closed space entirely avoids; and we do not want... speculation as to the solution of difficulties which need never arise. If we wish to be noncommittal, we shall naturally work in terms of a closed universe of finite radius R, since we can at any time revert to an infinite universe by making R infinite."

"The immediate results of introducing the cosmical term into the law of gravitation was the appearance... of two universes—the Einstein universe and the de Sitter universe. Both were closed spherical universes; so that a traveller going on and on in the same direction would at last find himself back at the starting-point... Both claimed to be static universes... thus they provided a permanent framework within which the small-scale systems—galaxies and stars—could change and evolve. ...[H]owever ...in de Sitter's universe there would be an apparent recession of remote objects ...At that time only three radial velocities were known, and these ...lamely supported de Sitter ...2 to 1. ...But in 1922 ...V. M. Sipher furnished me ...measures of 40 spiral nebulæ for ...my book Mathematical Theory of Relativity. ...[T]he majority had become 36 to 4 ..."

"The situation has been summed up in the statement that Einstein’s universe contains matter but no motion and de Sitter’s contains motion but no matter. ...[T]he actual universe containing both matter and motion does not correspond exactly to either... Which is the better choice for a first approximation? Shall we put a little motion into Einstein’s world of inert matter, or... a little matter into de Sitter’s ?"

"The choice between Einstein’s and de Sitter’s models... [W]e are not now restricted to these... extremes; we have... the whole chain of intermediate solutions between motionless matter and matterless motion... [W]e can pick... the right proportion of matter and motion to correspond with what we observe. ...[E]arlier... it was the preconceived idea that a static solution was a necessity... an unchanging background of space. ...[T]his ...should strictly have barred... de Sitter’s solution, but ...it was the precursor of the other non-static solutions..."

"[I]nvestigation of non-static solutions was carried out by A. Friedmann in 1922. His solutions were rediscovered in 1927 by Abbé G. Lemaître, who brilliantly developed the astronomical theory... and... remained unknown until 1930... In the meantime the solutions had been discovered... by H. P. Robertson, and through him... interest was... realised. The astronomical application, stimulated by Hubble and Humason’s observational work on the spiral nebule, was also being rediscovered, but it had not been carried so far as in Lemaître’s paper."

"The intermediate solutions of Friedmann and Lemaitre are "expanding universes." Both the material system and the closed space, in which it exists, are expanding. At one end we have Einstein’s universe with no motion and therefore in equilibrium. Then... we have model universes showing more and more rapid expansion until we reach de Sitter’s... The rate of expansion increases all the way along the series and the density diminishes; de Sitter’s universe is the limit when the average density of celestial matter approaches zero. The series of expanding universes then stops... but because there is nothing left to expand."

"[T]he most satisfying theory would be one which made the beginning not too unæsthetically abrupt. This... can only be satisfied by an Einstein universe with all... major forces balanced. Accordingly, the primordial state of things... is an even distribution of s and electrons, extremely diffuse and filling all (spherical) space, remaining nearly balanced for an exceedingly long time until its inherent instability prevails. ...[T]he density of this distribution can be calculated ...[at] about one proton and electron per litre. ...[S]mall irregular tendencies accumulate, and evolution gets under way. ...[T]he formation of condensations ultimately ...become the galaxies; this ...started off an expansion, which ...automatically increased in speed until ...now manifested ...in the recession of the spiral nebulae. As the matter drew closer... in the condensations... evolutionary processes followed—evolution of stars... of... more complex elements... of planets and life."

"Within the galaxy the average world-curvature is... thousands of times greater than Lamaître's average for the universe... his formulæ are inapplicable. The result... only the intergalactic distances expand. The galaxies... are unaffected... —s, stars, human observers and their apparatus, atoms—are entirely free from expansion. Although the cosmical repulsion or expansive tendency is present in all of these... it is checked by much larger forces... [T]he demarcation between permanent and dispersing systems is... abrupt. It corresponds to the distinction between periodic and aperiodic phenomena."

"If you think... the shattering of the bubble universe is... tragic... [W]hen the worst has happened our galaxy... will be left intact. ...not so bad a prospect."

"All change is relative. The universe is expanding relatively to our common material standards; our material standards are shrinking relatively to the size of the universe. The theory of the "expanding universe" might also be called the theory of the "shrinking atom". ...[T]ake the... universe as our standard of constancy... he sees us shrinking... only the intergalactic spaces remain the same. The earth spirals round the sun in an ever‑decreasing orbit. ...Our years will ...decrease in geometrical progression in the cosmic scale of time. ... Owing to the property of geometrical progressions an infinite number of our years will add up to a finite cosmic time; so that what we should call the end of eternity is an ordinary finite date in the cosmic calendar. But on that date the universe has expanded to infinity in our reckoning, and we have shrunk to nothing in the reckoning of the cosmic being. ...When the last act opens the curtain rises on midget actors rushing through their parts at frantic speed. Smaller and smaller. Faster and faster. One last microscopic blurr of intense agitation. And then nothing."

"If the astronomers are right, it is a straightforward conclusion from the observational measurements that the system of galaxies is expanding—or, since the system of the galaxies is all we know—that the universe is expanding. There is no subtlety or metaphysics about it ...But are we sure of the observational facts? Scientific men are rather fond of saying pontifically that one ought to be quite sure of one's observational facts before embarking on theory. Fortunately those who give this advice do not practice what they preach. Observation and theory get on best when they are mixed together, both helping one another in the pursuit of truth. It is a good rule not to put overmuch confidence in a theory until it has been confirmed by observation. I hope I shall not shock the experimental physicists too much if I add that it is also a good rule not to put overmuch confidence in the observational results that are put forward until they have been confirmed by theory. So in starting to theorise about the expanding universe I am not taking it for granted that the observational evidence which we have been considering is entirely certain."

"It is scarcely true... that we observe these velocities of recession. We observe a shift of the spectrum to the red; and although such... is usually due to recession... it is not inconceivable that it should arise from another cause."

"[I]t was theory that first suggested a systematic recession of the spiral nebulae and so led to a search for this effect. The theoretical possibility was first discovered by W. de Sitter in 1917. Only three radial velocities were known at that time, and they... lamely supported his theory by... 2 to 1. Since then... support is far more unanimous... mainly due to V. M. Slipher... and M. L. Humason... The linear law of proportionality between speed and distance was found by E. H. Hubble. Meanwhile the theory has also developed, and... taken the form... associated with... A. Friedman and G. Lemaître."

"The theory of relativity predicts... a... force... we call the cosmical repulsion... directly proportional to the distance... It is so weak... we can leave it out of... motions of the planets... or any motion within... our... galaxy. ...[S]ince it increases... to the distance we... if we go far enough, find it significant."

"I have said the repulsion is proportional to the distance... Distance from what? From anywhere you like. ...Cosmical repulsion is a dispersing force tending to make a system expand uniformly—not diverging from any centre in particular, but such that all internal distances increase at the same rate. That corresponds precisely to the kind of expansion we observe in the system of the galaxies."

"I have said that relativity theory predicts a force of cosmical repulsion. ...[R]elativity theory does not talk of anything so crude as force; it describes... curvature of space-time. But for practical purposes... nearly equivalent to the Newtonian force of gravitation... [T]he actual relativity effect is represented with sufficient accuracy by a force of cosmical repulsion... up to the greatest distances... we... observe."

"The galaxies exert on one another their ordinary gravitational attraction approximately according to Newton's law. This makes them tend to cling together. So we... have a contest of two forces, Newtonian attraction... and cosmical repulsion... If our theory is right cosmical repulsion must have got the upper hand... Having got the advantage, cosmical repulsion will keep it; because, as the nebulae become further apart, their mutual attraction will become weaker..."

"is a congruence geometry, or equivalently the space comprising its elements is homogeneous and isotropic; the intrinsic relations between... elements of a configuration are unaffected by the position or orientation of the configuration. ...[M]otions of are the familiar translations and rotations... made in proving the theorems of Euclid."

"[O]nly in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained."

"That the existence of these motions (the "axiom of free mobility") is a desideratum, if not... a necessity, for a geometry applicable to physical space, has been forcefully argued on a priori grounds by von Helmholtz, Whitehead, Russell and others; for only in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained."

"Euclidean geometry is only one of several congruence geometries... Each of these geometries is characterized by a real number K, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces... K may be interpreted as the ' of the surface into the third dimension—whence it derives its name..."

"[W]e propose... to deal exclusively with properties intrinsic to the space... measured within the space itself... in terms of... inner properties."

"Measurements which may be made on the surface of the earth... is an example of a 2-dimensional congruence space of positive curvature K = \frac{1}{R^2}... [C]onsider... a "small circle" of radius r (measured on the surface!)... its perimeter L and area A... are clearly less than the corresponding measures 2\pi r and \pi r^2... in the Euclidean plane. ...for sufficiently small r (i.e., small compared with R) these quantities on the sphere are given by 1):L = 2 \pi r (1 - \frac{Kr^2}{6} + ...), A = \pi r^2 (1 - \frac{Kr^2}{12} + ...)"

"In the sum \sigma of the three angles of a triangle (whose sides are arcs of s) is greater than two right angles [180°]; it can... be shown that this "spherical excess" is given by 2)\sigma - \pi = K \deltawhere \delta is the area of the spherical triangle and the angles are measured in s (in which 180° = \pi [radians]). Further, each full line (great circle) is of finite length 2 \pi R, and any two full lines meet in two points—there are no parallels!"

"[T]he space constant K... "" may in principle at least be determined by measurement on the surface, without recourse to its embodiment in a higher dimensional space."

"These formulae [in (1) and (2) above] may be shown to be valid for a circle or a triangle in the hyperbolic plane... for which K < 0. Accordingly here the perimeter and area of a circle are greater, and the sum of the three angles of a triangle are less, than the corresponding quantities in the Euclidean plane. It can also be shown that each full line is of infinite length, that through a given point outside a given line an infinity of full lines may be drawn which do not meet the given line (the two lines bounding the family are said to be "parallel" to the given line), and that two full lines which meet do so in but one point."

"The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces... The intrinsic geometry of such a space of curvature K provides formulae for the surface area S and the volume V of a "small sphere" of radius r, whose leading terms are 3)S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + ...), V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + ...)."

"In all these congruence geometries, except the Euclidean, there is at hand a natural unit of length R = \frac{1}{K^\frac{1}{2}}; this length we shall, without prejudice, call the "radius of curvature" of the space."

"We have merely (!) to measure the volume V of a sphere of radius r or the sum \sigma of the angles of a triangle of measured are \delta, and from the results to compute the value of K."

"What is needed is a homely experiment which could be carried out in the basement with parts from an old sewing machine and an Ingersoll watch, with an old file of Popular Mechanics standing by for reference! This I am, alas, afraid we have not achieved, but I do believe that the following example... is adequate to expose the principles..."

"Let a thin, flat metal plate be heated... so that the temperature T is not uniform... clamp or otherwise constrain the plate to keep it from buckling... [and] remain [reasonably] flat... Make simple geometric measurements... with a short metal rule, which has a certain coefficient of expansion c... What is the geometry of the plate as revealed by the results of those measurements? ...[T]he geometry will not turn out to be Euclidean, for the rule will expand more in the hotter regions... [T]he plate will seem to have a negative curvature K... the kind of structure exhibited... in the neighborhood of a ".""

"What is the true geometry of the plate? ...Anyone examining the situation will prefer Poincaré's common-sense solution... to attribute it Euclidean geometry, and to consider the measured deviations... as due to the actions of a force (thermal stresses in the rule). ...On employing a brass rule in place of one of steel we would find that the local curvature is trebled—and an ideal rule (c = 0) would... lead to Euclidean geometry."

"In what respect... does the general theory of relativity differ...? The answer is: in its universality; the force of gravitation in the geometrical structure acts equally on all matter. There is here a close analogy between the gravitational mass M...(Sun) and the inertial mass m... (Earth) on the one hand, and the heat conduction k of the field (plate)... and the coefficient of expansion c... on the other. ...The success of the general relativity theory... is attributable to the fact that the gravitational and inertial masses of any body are... rigorously proportional for all matter."

"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."

"Now it is the practice of astronomers to assume that brightness falls off inversely with the square of the "distance" of an object—as it would do in Euclidean space, if there were no absorption... We must therefore examine the relation between this astronomer's "distance" d... and the distance r which appears as an element of the geometry."

"All the light which is radiated... will, after it has traveled a distance r, lie on the surface of a sphere whose area S is given by the first of the formulae (3). And since the practical procedure... in determining d is equivalent to assuming that all this light lies on the surface of a Euclidean sphere of radius d, it follows...4 \pi d^2 = S = 4 \pi r^2 (1 - \frac{K r^2}{3} + ...);whence, to our approximation 4)d = r (1- \frac{K r^2}{6} + ...), or r = d (1 + \frac{K d^2}{6} + ...)."

"[T]he astronomical data give the number N of nebulae counted out to a given inferred "distance" d, and in order to determine the curvature... we must express N, or equivalently V, to which it is assumed proportional, in terms of d. ...from the second of formulae (3) and... (4)... to the approximation here adopted, 5)V = \frac{4}{3} \pi d^2 (1 + \frac{3}{10} K d^2 + ...);...plotting N against... d and comparing... with the formula (5), it should be possible operationally to determine the "curvature" K."

"This... is an outrageously over-simplified account of the assumptions and procedures..."

"The search for the curvature K indicates that, after making all known corrections, the number N seems to increase faster with d than the third power, which would be expected in a Euclidean space, hence K is positive. The space implied thereby is therefore bounded, of finite total volume, and of a present "radius of curvature" R = \frac{1}{K^\frac{1}{2}} which is found to be of the order of 500 million light years. Other observations, on the "red shift" of light from these distant objects, enable us to conclude with perhaps more assurance that this radius is increasing..."

"Hubble was inclined, from about 1936, to reject the Doppler-effect interpretation of the red shifts and to regard the nebulae as stationary; but theoretical cosmologists, notably McVittie... and Heckmann... severely criticized Hubble’s method...and disputed his conclusions. Although these criticisms... came to be generally accepted, it still seemed that the available data were open to rival interpretations, depending on the method of analysis..."

"At last, in 1949, the... ... was ready... Humason... succeeded in photographing the spectra of two remote galaxies in the . These exhibited red-shifts which, on the Doppler interpretation, indicated... one-fifth of the velocity of light. [I]n 1956, with... photoelectric equipment attached... [W. A.] Baum obtained a red-shift... recessional velocity of about two-fifths of the velocity of light."

"[I]n... 1952, Baade... announced that Hubble’s entire distance scale was in error... According to Baade, the distances formerly assigned to all extragalactic objects must be multiplied by a factor of about two. Later it was generally accepted that this... was probably nearer three. ...[I]t followed that the sizes of all such objects had been underestimated. ...Therefore ...this nebula must be... twice as far away... [T]he average absolute magnitude at maximum brightness of novae... in the Milky Way attain on the average... 7.4, whereas those... in the Andromeda... 5.7... [T]he apparent anomaly could be removed by placing... Andromeda... rather more than twice as far as previously. ...[[w:Distance measure|[E]xtragalactic distances]] had ...been underestimated because of an error in converting... relative distances of s into an absolute scale. ...Baade's revision ...applied only to extragalactic objects... [and] had momentous consequences concerning the size and , for the scale of both was correspondingly increased."

"An important new survey of the law relating red-shifts and magnitudes published in 1956 by Humason, Mayall and Sandage suggested... that the expansion of the universe may have been faster in the past... so that its age may be somewhat less than that estimated on the hypothesis of uniform expansion. But... caution, for a recent review (1958) by Sandage of Hubble's criteria for constructing the extragalactic distance-scale has revealed that, not only must his Cepheid criterion be corrected but also... the brightest star criterion..."

"As for Hubble’s brightest star criterion, Sandage... has shown that objects in the of galaxies which Hubble believed to be highly luminous stars are... regions of glowing of intrinsic luminosity... two magnitudes brighter... If Sandage’s result is accepted, then the distances of all galaxies beyond those in which Cepheids can be detected... must be augmented by a factor... between 5 and 10... with the result that the rate of increase of velocity with distance will be reduced to between 5O and 100 kilometres per second per megaparsec. Consequently, taking 80 as a rough average... the , if it has expanded uniformly, will have to be increased to about 13-5 thousand million years. If... it was expanding more rapidly in the past... this... might be reduced to about 9 thousand million years."

"There is still something in the system which gravels me. I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties. ... If with your alchemy you can make three pounds of gold, why should you stop there?"

"It is possible to form an analogous theory with seven imaginary roots of (-1)."

"The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative."

"Besides their possible role in physics, the octonions are important because they tie together some algebraic structures that otherwise appear as isolated and inexplicable exceptions."

"The octonions are much stranger. Not only are they noncommutative, they are also break another familiar law of arithmetic: the associative law (xy)z = x(yz)."

"But mathematicians know that the number system we study in school is but one of many possibilities. And indeed, other kinds of numbers are important for understanding geometry and physics. Among the strangest alternatives is the octonions. Largely neglected since their discovery in 1843, in the last few decades they have assumed a curious importance in string theory. And indeed, if string theory is a correct representation of the universe, they may explain why the universe has the number of dimensions it does."

"But perhaps most important, it wasn’t clear in Hamilton’s time just what the octonions would be good for. They are closely related to the geometry of 7 and 8 dimensions, and we can describe rotations in those dimensions using the multiplication of octonions. But for over a century that was a purely intellectual exercise. It would take the development of modern particle physics—and string theory in particular—to see how the octonions might be useful in the real world."

"Despite its counter-culture status, the octonions have long drawn the curiosity of generations of physicists. The algebra is known to appear without warning in apparently disparate areas of mathematics, within algebra, geometry, and topology. However, despite its ubiquity, its practical uses in physics have remained elusive, due to the algebra's non-associativity, which must be handled with care."

"It is nearly irresistible to ask if the octonions, \mathbb O, the last of the set of four normed division algebras over \mathbb R, have a calling in nature. Certainly several have thought so, but for the most part, the octonions have remained as a well kept secret from mainstream physics. More often than not, the octonions are passed by in haste because they are non-associative, and hence at times temperamental. As we will show, this property is in fact a gift, which will offer a way to streamline some of the standard model’s complex structure."

"Octonions are to physics what the Sirens were to ."

"That the pion-nucleon (πN) coupling constant is fundamental in our understanding of the Cosmos has been adequately emphasised in numerous works. In meson-exchange models of the , a significantly weaker coupling between the pions and the nucleons would have prevented the s from combining fast with s in the ; they would have decayed before they had any chance to be enmeshed first in s, then in other light nuclei. According to the , within half an hour of the Big Bang, all existing matter had assumed the form of free electrons, protons, and helium nuclei (as well as traces of other nuclei up to 7). On the contrary, a significantly stronger coupling would have resulted in the rapid creation of bound diprotons and would have led to a helium-dominated Universe. It is hard to imagine how life could emerge in such a Universe: typical stars burn hydrogen to helium for about 90 % of their lives."

"... Roughly speaking, in renormalizable theories no coupling constants can have the dimensions of negative powers of . But every time we add a field or a space-time derivative to an interaction, we reduce the dimensionality of the associated coupling constant. So only a few simple types of interaction can be renormalizable. ..."