Relativity The Special And The General Theory Quotes

If you've never done anything wrong it's probably because you have never tried anything new.

Dear Habicht, / Such a solemn air of silence has descended between us that I almost feel as if I am committing a sacrilege when I break it now with some inconsequential babble... / What are you up to, you frozen whale, you smoked, dried, canned piece of soul...?

Silence is the hardest scream

of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended.

result: every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for ‘distances’, the ‘distance’ being represented physically by means of the convention of two marks on a rigid body.

if, relative to K, K’ is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K’ according to exactly the same general laws as with respect to K. This statement is called the Principle of Relativity (in the restricted sense).

Despite the earnest belief of most of his fans, Einstein did not win his Nobel Prize for the theory of relativity, special or general. He won for explaining a strange effect in quantum mechanics, the photoelectric effect. His solution provided the first real evidence that quantum mechanics wasn’t a crude stopgap for justifying anomalous experiments, but actually corresponds to reality. And the fact that Einstein came up with it is ironic for two reasons. One, as he got older and crustier, Einstein came to distrust quantum mechanics. Its statistical and deeply probabilistic nature sounded too much like gambling to him, and it prompted him to object that “God does not play dice with the universe.” He was wrong, and it’s too bad that most people have never heard the rejoinder by Niels Bohr: “Einstein! Stop telling God what to do.

The purpose of mechanics is to describe how bodies change their position in space with ‘time’.

I have known a great many intelligent people in my life. I knew Max Planck, Max von Laue, and Wemer Heisenberg. Paul Dirac was my brother-in-Iaw; Leo Szilard and Edward Teller have been among my closest friends; and Albert Einstein was a good friend, too. And I have known many of the brightest younger scientists. But none of them had a mind as quick and acute as Jancsi von Neumann. I have often remarked this in the presence of those men, and no one ever disputed me. [...] But Einstein's understanding was deeper than even Jancsi von Neumann's. His mind was both more penetrating and more original than von Neumann's. And that is a very remarkable statement. Einstein took an extraordinary pleasure in invention. Two of his greatest inventions are the Special and General Theories of Relativity; and for all of Jancsi's brilliance, he never produced anything so original.

Many of us have no grasp of quantum theory, or Einstein’s theories of special and general relativity, but this does not in itself lead us to oppose these theories! Darwinism, unlike ‘Einsteinism’, seems to be regarded as fair game for critics with any degree of ignorance.

In gravitational fields there are no such thing as rigid bodies with Euclidean properties; thus the fictitious rigid body of reference is of no avail in the general theory of relativity. ... For this reason non-rigid reference-bodies are used, which are, as a whole, not only moving in any way whatsoever, but which also suffer alterations in form ad lib. during their motion... This non-rigid reference-body, ... might appropriately be termed a "reference mollusc,"...

We thus obtain the following result: every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for ‘distances’, the ‘distance’ being represented physically by means of the convention of two marks on a rigid body.

In gravitational fields there are no such things as rigid bodies with Euclidean properties; thus the fictitious rigid body of reference is of no avail in general relativity.

An observer who is sitting eccentrically on the disc K' is sensible of a force which acts outwards in a radial direction, and which would be interpreted as an effect of inertia (centrifugal force) by an observer who was at rest with respect to the original reference-body K. But the observer on the disc may regard his disc as a reference body which is “at rest”; on the basis of the general principle of relativity he is justified in doing this. The force acting on himself, and in fact on all other bodies which are at rest relative to the disc, he regards as the effect of a gravitational field.

On graduating Buck and my grandfather were given three days’ leave in Baltimore, where Buck got my grandfather so drunk that he was able to directly experience, if not to communicate, some of the unlikelier effects on time and space called for by Einstein’s Special and General Theories of Relativity.

The importance of experimental proof, on the other hand, does not mean that without new experimental data we cannot make advances. It is often said that science takes steps forward only when there is new experimental data. If this were true, we would have little hope of finding the theory of quantum gravity before measuring something new, but this is patently not the case. Which new data were available to Copernicus? None. He had the same data as Ptolemy. Which new data did Newton have? Almost none. His real ingredients were Kepler's laws and Galileo's results. What new data did Einstein have to discover general relativity? None. His ingredients were special relativity and Newton's theory. It simply isn't true that physics only advances when it is afforded new data.

Đối với những con người hành động, nhận thức một lần về chân lý là không đủ; ngược lại, nhận thức này phải được luôn luôn làm mới lại một cách không mệt mỏi nếu không muốn nó bị mai một. Nhận thức giống một bức tượng cẩm thạch đứng giữa sa mạc và luôn có nguy cơ bị gió cát chôn vùi. Những bàn tay siêng năng phải luôn luôn hoạt động để cho cẩm thạch có thể tiếp tục lấp lánh dưới ánh mặt trời.

At the highest level, there is the general Subject-Self metaphor, which conceptualizes a person as bifurcated. The exact nature of this bifurcation is specified more precisely one level down, where there are five specific instances of the metaphor. These five special cases of the basic Subject-Self metaphor are grounded in four types of everyday experience: (1) manipulating objects, (2) being located in space, (3) entering into social relations, and (4) empathic projection-conceptually projecting yourself onto someone else, as when a child imitates a parent. The fifth special case comes from the Folk Theory of Essences: Each person is seen as having an Essence that is part of the Subject. The person may have more than one Self, but only one of those Selves is compatible with that Essence. This is called the "real" or "true" Self.

was sitting in a chair in the patent office at Bern when all of a sudden a thought occurred to me,” he recalled. “If a person falls freely, he will not feel his own weight.” That realization, which “startled” him, launched him on an arduous eight-year effort to generalize his special theory of relativity and “impelled me toward a theory of gravitation.”16 Later, he would grandly call it “the happiest* thought in my life.

universe, Einstein felt, and the goal of science was to discover it. His quest began in 1895, when as a 16-year-old he imagined what it would be like to ride alongside a light beam. A decade later came his miracle year, described in the letter above, which laid the foundations for the two great advances of twentieth-century physics: relativity and quantum theory. A decade after that, in 1915, he wrested from nature his crowning glory, one of the most beautiful theories in all of science, the general theory of relativity. As with the special theory, his thinking had evolved through thought experiments. Imagine being in an enclosed elevator

Einstein’s “physical strategy” began with his mission to generalize the principle of relativity so that it applied to observers who were accelerating or moving in an arbitrary manner. Any gravitational field equation he devised would have to meet the following physical requirements: • It must revert to Newtonian theory in the special case of weak and static gravitational fields. In other words, under certain normal conditions, his theory would describe Newton’s familiar laws of gravitation and motion. • It should preserve the laws of classical physics, most notably the conservation of energy and momentum. • It should satisfy the principle of equivalence, which holds that observations made by an observer who is uniformly accelerating would be equivalent to those made by an observer standing in a comparable gravitational field.

Enigmas answered. Not only is QFT the answer to Einstein's search, it also answers or resolves his Enigmas, and in a way that can be understood by the man (or woman) on the street. In Appendix A you will see how the paradoxes of special relativity become natural and understandable consequences of the way fields behave. In Appendix B you will see that the problematic curvature of space-time in general relativity is gone; in QFT gravity is just another force field and space and time are the same space and time we intuitively believe in. Finally, in Appendix C you will see how the infamous wave-particle duality of QM is eliminated because there are no particles - only fields - and hence there is no duality. However abandoning the familiar picture of solid particles and replacing it with intangible fields is not easy. It will require a leap of imagination greater than did the atomic picture that Eddington struggled with.

Einstein’s approach to general relativity again showed how his mind tended to work: • He was disquieted when there were two seemingly unrelated theories for the same observable phenomenon. That had been the case with the moving coil or moving magnet producing the same observable electric current, which he resolved with the special theory of relativity. Now it was the case with the differing definitions of inertial mass and gravitational mass, which he began to resolve by building on the equivalence principle. • He was likewise uncomfortable when a theory made distinctions that could not be observed in nature. That had been the case with observers in uniform motion: there was no way of determining who was at rest and who was in motion. Now it was also, apparently, the case for observers in accelerated motion: there was no way of telling who was accelerating and who was in a gravitational field. • He was eager to generalize theories rather than settling for having them restricted to a special case. There should not, he felt, be one set of principles for the special case of constant-velocity motion and a different set for all other types of motion. His life was a constant quest for unifying theories.

Newton had bequeathed to Einstein a universe in which time had an absolute existence that tick-tocked along independent of objects and observers, and in which space likewise had an absolute existence. Gravity was thought to be a force that masses exerted on one another rather mysteriously across empty space. Within this framework, objects obeyed mechanical laws that had proved remarkably accurate—almost perfect—in explaining everything from the orbits of the planets, to the diffusion of gases, to the jiggling of molecules, to the propagation of sound (though not light) waves. With his special theory of relativity, Einstein had shown that space and time did not have independent existences, but instead formed a fabric of spacetime. Now, with his general version of the theory, this fabric of spacetime became not merely a container for objects and events. Instead, it had its own dynamics that were determined by, and in turn helped to determine, the motion of objects within it—just as the fabric of a trampoline will curve and ripple as a bowling ball and some billiard balls roll across it, and in turn the dynamic curving and rippling of the trampoline fabric will determine the path of the rolling balls and cause the billiard balls to move toward the bowling ball. The curving and rippling fabric of spacetime explained gravity, its equivalence to acceleration, and, Einstein asserted, the general relativity of all forms of motion.92 In the opinion of Paul Dirac, the Nobel laureate pioneer of quantum mechanics, it was “probably the greatest scientific discovery ever made.” Another of the great giants of twentieth-century physics, Max Born, called it “the greatest feat of human thinking about nature, the most amazing combination of philosophical penetration, physical intuition and mathematical skill.

a lecture in Leiden in May 1920, Einstein publicly proposed a reincarnation, though not a rebirth, of the ether. “More careful reflection teaches us, however, that the special theory of relativity does not compel us to deny ether,” he said. “We may assume the existence of an ether, only we must give up ascribing a definite state of motion to it.” This revised view was justified, he said, by the results of the general theory of relativity. He made clear that his new ether was different from the old one, which had been conceived as a medium that could ripple and thus explain how light waves moved through space. Instead, he was reintroducing the idea in order to explain rotation and inertia. Perhaps he could have saved some confusion if he had chosen a different term. But in his speech he made clear that he was reintroducing the word intentionally: To deny the ether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics do not harmonize with this view… Besides observable objects, another thing, which is not perceptible, must be looked upon as real, to enable acceleration or rotation to be looked upon as something real… The conception of the ether has again acquired an intelligible content, although this content differs widely from that of the ether of the mechanical wave theory of light… According to the general theory of relativity, space is endowed with physical qualities; in this sense, there exists an ether. Space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any spacetime intervals in the physical sense. But this ether may not be thought of as endowed with the qualities of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.

THEORY OF ALMOST EVERYTHING After the war, Einstein, the towering figure who had unlocked the cosmic relationship between matter and energy and discovered the secret of the stars, found himself lonely and isolated. Almost all recent progress in physics had been made in the quantum theory, not in the unified field theory. In fact, Einstein lamented that he was viewed as a relic by other physicists. His goal of finding a unified field theory was considered too difficult by most physicists, especially when the nuclear force remained a total mystery. Einstein commented, “I am generally regarded as a sort of petrified object, rendered blind and deaf by the years. I find this role not too distasteful, as it corresponds fairly well with my temperament.” In the past, there was a fundamental principle that guided Einstein’s work. In special relativity, his theory had to remain the same when interchanging X, Y, Z, and T. In general relativity, it was the equivalence principle, that gravity and acceleration could be equivalent. But in his quest for the theory of everything, Einstein failed to find a guiding principle. Even today, when I go through Einstein’s notebooks and calculations, I find plenty of ideas but no guiding principle. He himself realized that this would doom his ultimate quest. He once observed sadly, “I believe that in order to make real progress, one must again ferret out some general principle from nature.” He never found it. Einstein once bravely said that “God is subtle, but not malicious.” In his later years, he became frustrated and concluded, “I have second thoughts. Maybe God is malicious.” Although the quest for a unified field theory was ignored by most physicists, every now and then, someone would try their hand at creating one. Even Erwin Schrödinger tried. He modestly wrote to Einstein, “You are on a lion hunt, while I am speaking of rabbits.” Nevertheless, in 1947 Schrödinger held a press conference to announce his version of the unified field theory. Even Ireland’s prime minister, Éamon de Valera, showed up. Schrödinger said, “I believe I am right. I shall look an awful fool if I am wrong.” Einstein would later tell Schrödinger that he had also considered this theory and found it to be incorrect. In addition, his theory could not explain the nature of electrons and the atom. Werner Heisenberg and Wolfgang Pauli caught the bug too, and proposed their version of a unified field theory. Pauli was the biggest cynic in physics and a critic of Einstein’s program. He was famous for saying, “What God has torn asunder, let no man put together”—that is, if God had torn apart the forces in the universe, then who were we to try to put them back together?

Although the collection of known mathematical structures is large and exotic, and even more remain to be discovered, every single mathematical structure can be analyzed to determine its symmetry properties, and many have interesting symmetry. Intriguingly, one of the most important discoveries in physics has been that our physical reality also has symmetries built into it: for example, the laws of physics have rotational symmetry, which means that there's not special direction in our Universe that you can call "up." They also appear to have translation (sideways shifting) symmetry, meaning that there's no special place that we can call the center of space. Many of these spaces just mentioned have beautiful symmetries, some of which match the observed symmetries of our physical world. For example, Euclidean space has both rotational symmetry (meaning that you can't tell the difference if the space gets rotated) and translational symmetry (meaning that you can't tell the difference if the space gets shifted sideways). The four-dimensional Minkowski space has even more symmetry: you can't even tell the difference if you do a type of generalized rotation between the space and time dimensions-and Einstein showed that this explains why time appears to slow down if you travel near the speed of light, as mentioned in the last chapter. Many more subtle symmetries of nature have been discovered in the last century, and these symmetries form the foundations of Einstein's relativity theories, quantum mechanics, and the standard model of particle physics.

While this signifier can be difficult to pin down with precision, it can clearly be heard in the records of Duane Eddy and many other guitarists of the period. It usually involves a relatively nondistorted electric guitar timbre articulated with a strong attack and a melody played on the lower strings. Reverberation is ubiquitous, and almost equally common were echo, amplifier tremolo, and use of the guitar’s vibrato bar. This overall guitar sound is often called a Fender sound, but that is a bit misleading, since Gretsch guitars were equally specialized for the purpose, and many other brands were also used. What makes the twang guitar interesting in topical terms is that it not only signified the western topic but also was key to a linked set of genres that intersect one another in complex ways: western, spy, and surf. Because these were all signified by overlapping musical features and in turn resemble one another in some of their broader connotations, we could speak of a twang guitar continuum: a range of topics that coalesced only shortly before psychedelia and were cognate with it in a variety of ways. Philip Tagg and Bob Clarida point out that the twang guitar, often in a minor mode with a flat seventh, was a common factor between spaghetti western and Bond/spy scores in the late 1950s and early 1960s. I would add surf guitar to the list, with its sonic experimentation and general relationship to fun, escape, and exoticism: “[The twang guitar] probably owes some of its immediate success as a spy sound to its similarity with various pre-rock ‘Viennese intrigue’ sounds like Anton Karas’s Third Man zither licks (1949). But in the 1962–64 period that produced The Virginian (1962), Dr. No (1963) and Leone’s A Fistful of Dollars (1964), steely Fender guitar was well on its way to becoming an all-purpose excitement/adventure timbre” (Tagg and Clarida 2003, 367).

Unfortunately, treatment with modern medicine is still under the suppressive dictatorship of Newton’s science. We need a SHIFT!! In the next chapter we’ll review the new physics, namely quantum field theory, and a new paradigm to consider in the health and healing of your body, mind and spirit. The dark ages of allopathic medicine are OVER! It’s time to usher in a new science of energy medicine with PEMF therapy and natural and holistic healing at the forefront. It’s also time to take action and take responsibility for your OWN health. The transition to the new physics is Einstein’s theories of special and general relativity. These ideas radically changed the landscape of space/time, as well as matter being merely a form of energy. Keep in mind however that Einstein’s relativity is still a CLASSICAL model with some of the aforementioned flaws. It’s just a much more accurate one! Einstein’s ideas further paved the way for a paradigm shift in physics at the beginning of the 20th century; coupled with the advent of quantum mechanics, for which Einstein was an important contributor with his Nobel Prize winning paper on the photoelectric effect. Newton Under Fire - Special and General Relativity Theory In 1905, Albert Einstein changed the prevailing worldview of Newtonian physics for good with the introduction of his special relativity theory, followed in 1915 by general relativity. He proved Newtonian laws of physics are by no means static, but

The twentieth century was a tumultuous time for physics. Centuries-old theories were overturned, first by Einstein’s discovery of special and general relativity between 1905 and 1915; then by the development, during the mid-1920s, of quantum theory; and again during the 1940s, when these two new theories were combined into what is now known as quantum field theory.5 Little was left unscathed by these revolutions—even the physics of nothing.

Dr Einstein, how many feet are there in a mile? "I don't know.I make it a rule not to clutter my mind with simple information that I can find in a book in five minutes.

as is always the case with scientific geniuses, Einstein’s theories would exist even if he had not. Special relativity, general relativity, and the photon model of light might not have been developed by the same individual, but someone would have sussed them out. Henri Poincaré, Hendrik Lorentz and others worked out much of relativity before 1905, just as Gottfried Leibniz independently worked out the calculus in parallel with Newton, and Alfred Russel Wallace developed natural selection in isolation from Charles Darwin. Historians of science once subscribed to a ‘Great Man’ theory, but we now know that transformative ideas emerge from the work of many talented individuals, instead of emerging ex nihilo from one brilliant mind.

Royal Society and the Royal Astronomical Society held especially for that purpose recently in London, it was the general opinion that Einstein's prediction might be regarded as justified, and warm tributes to his genius were made on all sides. Nevertheless, I cannot refrain, while I am mentioning it, from expressing my surprise that, according to the report in The Times there should be so much complaint about the difficulty of understanding the new theory. It is evident that Einstein's little book “About the Special and the General Theory of Relativity in Plain Terms,” did not find its way into England during wartime. Any one reading it will, in my opinion, come to the conclusion that the basic ideas of the theory

There's another promising idea about what the dark matter is, which emerges from a different proposal for improving the equations of physics. As we've discussed, QCD is in a profound and literal sense constructed as the embodiment of symmetry. There is an almost perfect match between the observed properties of quarks and gluons and the most general properties allowed by local color symmetry, in the framework of special relativity and quantum mechanics. The only exception is that the established symmetries of QCD fail to forbid one sort of behavior that is not observed to occur. The established symmetries permit a sort of interaction among gluons that would spoil the symmetry of the equations of QCD under a change in the direction of time. Experiments provide severe limits on the possible strength of that interaction. The limits are much more severe than might be expected to arise accidentally. The Core theory does not explain this "coincidence." Roberto Peccei and Helen Quinn found a way to expand the equations that would explain it. Steven Weinberg and I, independently, showed that the expanded equations predict the existence of new, very light, very weakly interacting particles called axions. Axions are also serious candidates to provide the cosmological dark matter. In principle they might be observed in a variety of ways. Though none is easy, the hunt is on. It's also possible that both ideas are right, and both kinds of particles contribute to the total amount of dark matter. Wouldn't that be pretty?

While the ink was drying on special relativity, Einstein started looking for a way to include gravity in the new framework. It was the beginning of a ten-year search, of which Einstein later said, ...the years of searching in the dark for a truth that one feels, but cannot express; the intense desire and the alternations of confidence and misgiving, until one breaks through to clarity and understanding, are only known to him who has himself experienced them. In the end he produced a field-based theory of gravity, general relativity. We'll have much more to say about that theory later in this chapter. Sever other clever people, including notably Poincare, the great German mathematician Hermann Minkowski, and the Finnish physicist Gunnar Nordstrom, were also in the hunt, trying to construct theories of gravity consistent with the concepts of special relativity. All were led to field theories.

Over the twentieth century, as physics developed, Planck's construction took on ever greater significance. Physicists came to understand that each of the quantities c, G, and h plays the role of a conversion factor, one you need to express a profound physical concept: 1) Special relativity postulates symmetry operations (boosts, a.k.a. Lorentz transformations) that mix space and time. Space and time are measured in different units, however, so for this concept to make sense, there must be a conversion factor between them, and c does the job. Multiplying a time by c, one obtains a length. 2) Quantum theory postulates an inverse relation between wave-length and momentum, and a direct proportionality between frequency and energy, as aspects of wave-particle duality; but these pairs of quantities are measured in different units, and h must be brought in as a conversion factor. 3) General relativity postulates that energy-momentum density induces space-time curvature, but curvature and energy density are measured in different units, and G must be brought in as a conversion factor.

There's a good general reason to expect that physical theories consistent with special relativity will have to be field theories. Here it comes: A major result of the special theory of relativity is that there is a limiting velocity: the speed of light, usually denoted c. The influence of one particle on another cannot be transmitted faster than that. Newton's law for the gravitational force, according to which the force due to a distant body is proportional to the inverse square of its distance right now, does not obey that rule, so it is not consistent with special relativity. Indeed the concept "right now" itself is problematic. Events that appear simultaneous to a stationary observer will not appear simultaneous to an observer moving at constant velocity. Overthrowing the concept of a universal "now" was, according to Einstein himself, by far the most difficult step in arriving at special relativity: [A]ll attempts to clarify this paradox satisfactorily were condemned to failure as long as the axiom of the absolute character of times, viz., of simultaneity, unrecognizedly was anchored in the unconscious. Clearly to recognize this axiom and its arbitrary character really implies already the solution of the problem.

When an important change takes place in science, in which a new theory takes the stagw, the incoming theory is generally an extension of the old theory which has the property of becoming more and more like the old theory in some limiting situation. In effect, it reveals that the old theory was an approximation (usually a very good one) to the new one that holds under a particular range of conditions. Thus, Einstein's special theory of relativity becomes Newton's theory of motion when speeds are far less than that of light, Einstein's general theory of relativity becomes Newton's theory of gravity when gravitational fields are weak and bodies move at speeds less than that of light. In recent years we have even begun to map out what the successor to Einstein's theory may look like. It appears that Einstein's theory of general relativity is a limiting, low-energy case of a far deeper and wider theory, which has been dubbed M theory.

We know that the uncertainty relation implies not only the absence of empty space but also the nature of the minimal content of every space-fluctuating energy and virtual particle-antiparticle pairs. The fields of the general theory of relativity by themselves are permitted to have zero values, and thereby truly empty space. But a quantum theory of gravity must preclude that; its energy carriers will create fluctuations. These fluctuations may include the value zero but will never violate the uncertainty relation for energy. Since we do not yet have a theory that unifies general relativity and quantum mechanics, anything we might say about fluctuations of the fields of general relativity must remain mere speculation. We can say for sure that quantum theory, when joined to the special theory of relativity, mandates the permanent and ubiquitous existence of fluctuations that include the appearance and disappearance of virtual particle pairs. This mandate includes the emptiest of all imaginable space, a void we might surround with an impenetrable wall at temperature zero.

The predictions of general relativity have been uniformly confirmed. There is no longer any doubt that Einstein's description of gravity is not only compatible with special relativity, but yields predictions closer to experimental results than those of Newton's theory.

The challenge in debating pseudoscience, for a teacher, or with a friend, is not an intellectual one, it is one of politeness. No-one of good will would want to hurt an intellectually curious, although misguided, person's feelings, anymore than hurting the child's. (Canning 77-78)

Don’t galaxies receding faster than the speed of light violate relativity theory? Hubble’s law v = Hd implies that galaxies will move away from us faster than the speed of light c if their distance from us is greater than c/H ≈ 14 billion light-years, and we have no reason to doubt that such galaxies exist, so doesn’t this violate Einstein’s claim that nothing can go faster than light? The answer is yes and no: it violates Einstein’s special relativity theory from 1905 but not his general relativity theory from 1915, and the latter is Einstein’s final word on the subject, so we’re okay.

The ramifications of Pythagoras' theorem have revolutionized twentieth century theoretical physics in many ways. For example, Minkowski discovered that Einstein's special theory of relativity could be represented by four-dimensional pseudo-Euclidean geometry where time is represented as the fourth dimension and a minus sign is introduced into Pythagoras' law. When gravitation is present, Einstein proposed that Minkowski's geometry must be "curved", the pseudo-Euclidean structure holding only locally at each point. A complex vector space having a natural generalization of the Pythagorean structure (defined over functions in an abstract space rather than geometrical points in the familiar Euclidean space) is known as Hilbert space and forms the basis of quantum mechanics. It is remarkable to think that the two pillars of twentieth century physics, relativity and quantum theory, both have their basis in mathematical structures based on a theorem formulated by an eccentric mathematician over two and a half thousand years ago.

Seibel: Other than the possibility of implementing it at all, how do you decide whether your interfaces are good? Steele: I usually think about generality and orthogonality. Conformance to accepted ways of doing things. For example, you don't put the divisor before the dividend unless there's a really good reason for doing so because in mathematics we're used to doing it the other way around. So you think about conventional ways of doing things. I've done enough designs that I think about ways I've done it before and whether they were good or bad. I'm also designing relative to some related thing that I've already designed before. So, for example, while looking at the specifications for numeric functions in Java, I'd already done numeric functions for Common Lisp. And I'd documented numeric functions for C. I knew some of the implementation pitfalls and some of the specification pitfalls for those things. I spent a lot of time worrying about edge cases. That's something I learned from Trenchard More and his array theory for APL. His contention was that if you took care of the edge cases then the stuff in the middle usually took care of itself. Well, he didn't say it that way; I guess that's the conclusion I draw from him. To turn it around, you want to design the specification of what's in the middle in such a way that it naturally is also correct on the boundaries, rather than treating boundaries as special cases.

If different physics is ‘all’ you want, you can look (say) to Einstein’s theories of special and general relativity, in which motion and gravity slow time and bend space. That’s not easy to imagine, but I reckon you can do it. You just need to imagine time passing more slowly, distances contracting: distortions of your grid references. You can put those ideas into words. In quantum theory, words are blunt tools. We give names to things and processes, but those are just labels for concepts that cannot be properly, accurately expressed in any terms but their own.

A feature of the relativity theory which seems to have aroused special interest among philosophers is the absoluteness of the velocity of light. In general velocity is relative. If I speak of a velocity of 40 km/s, I must add “relative to the earth”, “relative to Arcturus”, or whatever reference body I have in mind. No one will understand anything from my statement unless this is added or implied. But it is a curious fact that if I speak of a velocity of 299.796 km/s, it is unnecessary to add the explanatory phrase. Relative to what? Relative to any and every star or particle of matter in the universe.

In a world of general relativity, special relativity, string and super-string theory among other scientific oddities, Mattie Bennings would not have been all too surprised to know that as Alex Wayne, her close friend, had been locked in a battle of minds with another person, his body lay unconscious until the end. The fact that it had happened during the whole time from school to hospital, still somehow, though, escaped her notice until they were in the very room Alex was wheeled into upon arrival. In the journey to the hospital and through the gurney being transported to a room Alex only conveyed muscle spasms and pain, but when they got to his newly acquired room all hell had broken loose.

Previously, the most popular philosophy of science was probably Karl Popper’s falsificationism—this is the old philosophy that the Bayesian revolution is currently dethroning. Karl Popper’s idea that theories can be definitely falsified, but never definitely confirmed, is yet another special case of the Bayesian rules; if P(X|A) ≈ 1—if the theory makes a definite prediction—then observing ¬X very strongly falsifies A. On the other hand, if P(X|A) ≈ 1, and we observe X, this doesn’t definitely confirm the theory; there might be some other condition B such that P(X|B) ≈ 1, in which case observing X doesn’t favor A over B. For observing X to definitely confirm A, we would have to know, not that P(X|A) ≈ 1, but that P(X|¬A) ≈ 0, which is something that we can’t know because we can’t range over all possible alternative explanations. For example, when Einstein’s theory of General Relativity toppled Newton’s incredibly well-confirmed theory of gravity, it turned out that all of Newton’s predictions were just a special case of Einstein’s predictions.