Mathematicians From The United States

"If the “universe” contains everything that exists, what can be outside it? If the answer is “Things that don’t exist”, then “multiverse” becomes an idea in the domain of psychology, not physics."

"Intelligent creatures [that] evolved to live deep within the atmosphere of a gas giant planet could be deluded, for eons, into thinking that the Universe is an approximately homogeneous expanse of gas, filling a three-dimensional space, but featuring anisotropic laws of motion (which we would ascribe to the planet’s gravitational field). Are we human scientists comparably blinkered?"

"Knowing how to calculate something is not the same as understanding it. Having a computer to calculate the origin of mass for us may be convincing, but is not satisfying. Fortunately we can understand it too."

"E = mc2 really applies only to isolated bodies at rest. In general, when you have moving bodies, or interacting bodies, energy and mass aren't proportional. E = mc2 simply doesn't apply. ...For moving bodies, the correct mass-energy equation is E=\frac {mc^2} {\sqrt{1-\frac{v^2} {c^2}}} where v is the velocity. For a body at rest (v=0), this becomes E = mc2. ...we must consider the special case of particles with zero mass... examples include photons, color gluons, and gravitons. If we attempt to put m = 0 and v = c in our general mass-energy equation, both the numerator and denominator on the right-hand-side vanish, and we get the nonsensical relation E = 0/0. The correct result is that the energy of a photon can take any value. ...The energy E of a photon is proportional to the frequency f of the light it represents. ...they are related by the Planck-Einstein-Schrödinger equation E = hf, where h is Plank's constant."

"Different ways of writing the same equation can suggest very different things, even if they are logically equivalent. ...In Einstein's original 1905 paper, you do not find the equation E = mc2. What you find is m = E/c2. ...the title of the paper is a question: "Does the Inertia of a Body Depend on Its Energy Content?" …If we can explain mass in terms of energy, we'll be improving our description of the world. We'll need fewer ingrediants in our world-recipe."

"An ordinary mistake is one that leads to a dead end, while a profound mistake is one that leads to progress. Anyone can make an ordinary mistake, but it takes a genius to make a profound mistake."

"For many centuries before modern science, and for the first two and a half centuries of modern science, the division of reality into matter and light seemed self-evident. ...As long as the separation between the massive and the massless persisted, a unified description of the physical world could not be achieved."

"We evolved to be good at learning and using rules of thumb, not at searching for ultimate causes and making fine distinctions. Still less did we evolve to spin out long chains of calculation that connect fundamental laws to observable consequences. Computers are much better at it!"

"We have heard that nature can sing some strange and unfamiliar songs. In coming to appreciate these songs, we develop a heightened perception... leavened by an admixture of our own creation..."

"The whole idea of science is really to listen to nature, in her own language, as part of a continuing dialogue."

"The main problem with many nonscientific world models is the vigor with which they insist upon their rightness. Once a world model claims to be completely right, it is no longer open to any changes. ...Closed systems can be comforting, but they are limited. ...It's not the best we can do. Neither is extreme "open-mindednesss" that slides into "empty headedness"—the ideal that we can never really know anything."

"It is quite easy to include a weight for empty space in the equations of gravity. Einstein did so in 1917, introducing what came to be known as the cosmological constant into his equations. His motivation was to construct a static model of the universe. To achieve this, he had to introduce a negative mass density for empty space, which just canceled the average positive density due to matter. With zero total density, gravitational forces can be in static equilibrium. Hubble's subsequent discovery of the expansion of the universe, of course, made Einstein's static model universe obsolete. ...The fact is that to this day we do not understand in a deep way why the vacuum doesn't weigh, or (to say the same thing in another way) why the cosmological constant vanishes, or (to say it in yet another way) why Einstein's greatest blunder was a mistake."

"Thinking along these lines will help prepare us for the day when we—or more likely, our distant descendents—will develop the machinery and cleverness to begin to program [create] worlds ourselves..."

"When we view nature stripped to essentials, so to speak, in this Unification Table, what we see is... a five bit register. ...It is in just this form that data is stored and manipulated within a digital computer. ...Every particle, then, can be specified by a five-bit word and stored in a five-bit register. It's eerie that even the odd restriction on how many minus signs are allowed is reminiscent of a trick used in computers, to detect errors in transmission. You see, if allowed words must have an odd number of minus signs, any single error in transmission of a word can be detected. ...our world might be an intricate program working itself out on a gigantic computing machine."

"In the table—and in nature—we find (leaving aside the antineutrino) fifteen fundamental fermions, with diverse strong, weak, and electromagnetic charges. ...They are so closely related by symmetry transformations that they are, so to speak, no more than different faces of the same cube."

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

"There is a simple rule for composite objects, such as nuclei or atoms. The rule is that if such an object contains an odd number of fermions, the composite object is a fermion. Otherwise, it is a boson. ...this simple rule doesn't care at all about the number of bosons in the composite object."

"When two identical 3He atoms collide... the interference is destructive. Particles that behave like 3He atoms are called fermions, short for "particles obeying Fermi–Dirac statistics." ...while bosons imitate one another... the "identity force" between fermions acts like a repulsion, and the probability of finding a fermion at some point in space is reduced if some of its identical siblings are nearby. ...It is the repulsive identity forces between electrons that support white dwarf stars... against their own gravity."

"Particles that, like 4He, show constructive interference are said to be bosons—a shorthand term for "particles obeying Bose–Einstein statistics." …One way to recognize bosons is their tendency to imitate one each other. ...the presence of one boson increases the chance that another of its identical siblings will also appear in the same spot. There's an attraction between them. We will speak ...of an attractive identity force drawing together identical bosons. Lasers are a spectacular example..."

"In science... the ultimate judges are not experts but experiments."

"The result will be points of quiescence—technically known as nodes—where the air's density varies not at all, and no sound is heard. Note the paradox here: either sphere alone creates a sound wave at this point; two spheres together add up to no sound there at all. Two sources can add up to give less than one. This is the essence of destructive interference. (When two sources are giving the same instruction, the resulting vibration bears not twice but four times the energy. This phenomenon, oxymoronically known as constructive interference, may seem puzzling.)"

"In a sense, all of Earth glows in the dark. The energy release from natural radioactivity, the lingering fluorescence of stellar explosions, keeps Earth dynamic. It melts the core and keeps it flowing, and heats the crust and mantle, with consequences ranging from the generation of earth's magnetic field to earthquakes and the motion of continents. By contrast, Luna and Mars, because they are smaller, derive less energy from radioactive decays. Geologically, they are dead."

"It is delightful in itself when we are able to interpret features of the present as signs confirming our understanding of the past."

"Once helium burning has occurred... the next possible reaction—carbon burning—is not necessarily slow... This reaction involves ...a strong as opposed to a weak interaction. ...Carbon burning results in magnesium. ...Taking a cross section of a highly evolved star would reveal a system of many layers. The inner layers have been subjected to the largest pressures, thereby forced to the highest temperatures, and burned the furthest; the outermost layers, by contrast, have not burned at all. Thus, as we proceed from outside in, there will be an outermost layer with the initial mix of hydrogen and helium, a layer of mostly helium, a layer of carbon, a layer of magnesium, and so on. ...So we arrive at the picture of a star, in the latest stages of its evolution... now composed of mostly carbon nuclei and other explosive material."

"Why can stars do better than the big bang? ...During the big bang, there were only a few minutes when nuclei could form. Very rare processes, or slow ones, played little role. A case in point is the key process from which the sun derives its energy. In this reaction, two protons collide to produce a deuterium nucleus, a neutrino, and a positron. ...This reaction belongs to the family of weak interactions. ...It remains... a remarkable—and for humanity, remarkably fortunate—circumstance that the central reaction that drives the sun is so rare. It is only this extraordinary rarity that allows the average proton in the sun to last so long, billions of years, even though it is colliding with other protons millions of times a second. ...an entertaining example of Treiman's theorem."

"The most abstract conservation laws of physics come into their being in describing equilibrium in the most extreme conditions. They are the most rigorous conservation laws, the last to break down. The more extreme the conditions, the fewer the conserved structures... In a deep sense, we understand the interior of the sun better that the interior of the earth, and the early stages of the big bang best of all."

"What should be most significant to us are not physical artifacts, but the meaning they embody. ...whenever we create paintings, songs, poems, books, computer programs—or ideas in the minds of children—we do something of this sort."

"What is conserved, in modern physics, is not any particular substance or material but only much more abstract entities such as energy, momentum, and electric charge. The permanent aspects of reality are not particular materials or structures but rather the possible forms of structures and the rules for their transformation."

"As the idea of permanence of objects has faded, the idea of permanence of physical laws has become better established and more powerful."

"Sam Treiman... has quoted something he called Treiman's theorem... Impossible things usually don't happen. ...With the discovery of radioactivity... it suddenly became apparent that the "impossible" was happening all the time. Uranium, thorium, radium... fit all the requirements of chemical elements. They could not be broken down by any of the standard methods... But occasionally... atoms of these elements spontaneously changed into other kinds of atoms. ...So what is left of the doctrine of the elements? Is alchemy reinstated? Not at all. The point is that the doctrine fails only under rare or special conditions. ...We can isolate the conditions in which they do, and retain a more restricted but still useful concept of the "impossible.""

"To understand this radiation [ cosmic microwave background ], it is easier to begin thinking about the radiation from a very hot gas like that inside a neon light. The same neon... is, at room temperature, utterly transparent... The character of matter in general changes abruptly when it gets heated above 3,000 degrees or so. Below this temperature, matter is electrically neutral... At high temperatures in a neon light, the electrically charged pieces of atoms become unstuck. Frequent and violent collisions break down neutral atoms into electrons and unbalanced nuclei. Matter in this state is called plasma, and it radiates much of its collision energy in the form of light. ...a gas of neutral atoms (like air) is virtually transparent. The free [charged] nuclei and electrons of plasma, by contrast, couple to light's electromagnetic fields and absorb it very efficiently. ...You ...see light only from the borderline layer of neon between opaque plasma and transparent neutral atoms."

"If you pass the light from a sodium flash through a prism, you get a pattern very different from the familiar continuous rainbow that Newton elicited from natural sunlight. Instead of a continuous pattern, in which all gradations of pure color are apparently represented, the sodium flash generates a series of lines of light. ...in the musical analogy, sodium produces a chord where sunlight produced all possible tones—"white noise." Other elements produce other chords."

"So let us listen to the light—what music do we hear? For one thing, we can elicit from each chemical element its own, unique chord. You may sometimes have noticed that a bright yellow flash is produced if ordinary table salt is sprinkled on a flame...a first bare hint of the subject of flame spectra... The fact that different elements emit light with different color characteristics is exploited by the makers of fireworks."

"The bases of music are rhythm and harmony. Rhythm is ordered recurrence in time... As the planets move around the sun, they repeat their orbits periodically; thus there is already a primitive kind of rhythm in their motion. ...Harmony ...can be considered a special kind of rhythm. ...pure musical tones are produced when the vibrations are... periodic or... repeat themselves regularly in time. Two tones harmonize if their intervals of repetition are in rhythm—or, in mathematical language, if their periods are in proportion. Kepler... in the third book of Harmonice mundi... attempted to make other... related, connections between musical harmony and mathematical proportion."

"If you don’t have all the clues, you can still do parts of the puzzle. There may be huge parts of the puzzle that we don’t even know about...It means that future generations can have the same kind of fun that we’re having."

"Science teaches us to be very suspicious of grand generalizations...Aristotle had a set theory of the universe, and he didn’t get too far. Galileo started with simple things like pendulums and balls sliding down inclined planes, and he got much further. You never find surprises when you think in terms of broad generalities. Both quantum mechanics and relativity grew out of trying to really understand essentially simple things."

"Does Michael Jackson or Archie Bunker or the president of General Motors need to know about quantum mechanics? Of course not. You can live a full life without that. But if you don’t believe that the universe is understandable, then it leads to the notion that one idea is just as good as another. And that’s horrible."

"Physics is my religious belief. In the sense that in physics we discover a fantastically wonderful world out there that’s rich in potential, rich in realization, and that has ample scope for fantasy, because the laws are so strange and there’s so much stuff out there to understand. And when you understand it, you understand how that could be. I learn that I myself am very small. But I am also very large because I contain multitudes, as Walt Whitman said. I can process information. I can understand things. I can imagine. I can have fun. That’s the essence of my religion. I learn my religion from the study of what the world is and how it works."

"[S]ome scientists focus on ideal beauty, others on empirical truth. My own approach, following a great tradition going back to Copernicus, Galileo and Kepler, has been to use beauty as a guide to truth."

"17:48 of 40:44"

"To me, dark matter is matter. It looks like matter, it quacks like matter, it waddles like matter. It has many, many properties that indicate it's matter. And I think I know what matter it is!"

"How do we get from symmetric laws to asymmetric appearances? (arrow of time problem) ... Why are the fundamental laws symmetric? ("naturality" problem) ..."

"Exoplanet astronomy will systematically survey our galaxy, gathering information on the masses, orbits, geology, and atmospheres of millions of planets. As a byproduct, we will learn how rare life is and what conditions it requires. What we discover might support tests and refinements of anthropic reasoning."

"Quite undeservedly, the ether has acquired a bad name."

"The possibility and significance of fractional angular momentum is discussed, and some simple physical realizations of it are mentioned. This leads naturally to consideration of the possibility of fractional quantum statistics, which is seen to be a possibility inherent in the kinematics of 2+1 dimensional quantum mechanics. Both sorts of fractionalization are intimately related to theories, and the classic considerations of Aharonov and Bohm on the significance of the vector potential in quantum mechanics. The meaning and importance of discrete gauge invariance in continuum theories is pointed out. Fractional statistics is shown to have a simple dynamical realization in the dynamics of charge-flux tube composites. Fractional statistics is shown to occur very naturally in the most geometrical quantum field theories in 2+1 dimensions, that is in the nonlinear sigma model and in quantum electrodynamics."

"... the magnificent structure of Gibbsian statistical mechanics [cannot] be viewed as founded upon ideal classical gases, Boltzmannian kinetic theory, and the virial and cluster expansions for dilute fluids! True, this last route was still frequently retravelled in textbooks more than 50 years after Gibbs’ major works were published; but it deeply misrepresents the power and range of statistical mechanics... asking ‘‘What does statistical mechanics convey to a physicist?’’ and replying: ‘‘It means that one can compute the second-virial coefficient to correct the ideal gas laws!’’ Of course, historically, that is not a totally irrelevant remark; but it is extremely misleading and, in effect, insults one of America’s greatest theoretical physicists, Josiah Willard Gibbs."

"... a comparison of the American regard for inventive skill as opposed to skill in pure science. Our greatest inventive genius, Thomas A. Edison, was all but canonized by the American public, and a legend has been built around him. One cannot, I suppose, expect that achievements in pure science would receive the same public applause that came to inventions as spectacular and as directly influential on ordinary life as Edison’s. But one might have expected that our greatest genius in pure science, Josiah Willard Gibbs, who laid the theoretical foundations for modern physical chemistry, would have been a figure of some comparable acclaim among the educated public. Yet Gibbs, whose work was celebrated in Europe, lived out his life in public and even professional obscurity at Yale, where he taught for thirty-two years. Yale, which led American universities in its scientific achievements during the nineteenth century, was unable in those thirty-two years to provide him with more than a half dozen or so graduate students who could understand his work, and never took the trouble to award him an honorary degree."

"[Gibbs' 1878] paper stands today as one of the most profound contributions to the world of human thought and […] places him with the greatest of the world's geniuses."

"Many men have had intuitions well ahead of their time; and this is not least true in mathematical physics. Gibbs' introduction of probability into physics occurred well before there was an adequate theory of the sort of probability he needed. But for all these gaps it is, I am convinced, Gibbs rather than Einstein or Heisenberg or Planck to whom we must attribute the first great revolution of twentieth century physics."

"Maxwell, and then Boltzmann, and then... J. Willard Gibbs... expended enormous intellectual effort in devising... , or... . The uses... extend far beyond gases... describing electric and magnetic interactions, chemical reactions, phase transitions... and all other manner of exchanges of matter and energy. The success... has driven the belief among many physicists that it could be applied with similar success to society. ...[E]verything from the flow of funds in the stock market to the flow of traffic on interstate highways ..."