Theorems

"I’ve had experts in quantum field theory – people who’ve spent years calculating path integrals of mind-boggling complexity – ask me to explain the Bell inequality to them, or other simple conceptual things like Grover’s algorithm. I felt as if Andrew Wiles had asked me to explain the Pythagorean Theorem."

"The purpose of the first part is to convince the reader that the formalism leading to Bell's inequalities is very general and reasonable. What is surprising is that such a reasonable formalism conflicts with quantum mechanics. In fact, situations exhibiting a conflict are very rare, and quantum optics is the domain where the most significant tests of this conflict have been carried out"

"One of these articles, written by N. David Mermin, gave me a tremendous shock. Mermin described the results of experiments that had been carried out as recently as 1982 to test something called Bell's theorem using two-photon 'cascade' emission from excited calcium atoms. Put simply, Bell's theorem says that my idea of naive realism is in conflict with the predictions of quantum theory in a way that can be tested in the laboratory in special experiments on pairs of quantum particles. These experiments had been done: quantum theory had been proved right and naive realism wrong! There in a montage was a pictorial history of the debate about reality and the experiments that had been done to test it (reproduced opposite). This work struck me as desperately important to my understanding of physical reality, something that as a scientist I felt I ought to know about. This discovery also made me feel rather embarrassed. Here I was, proud of my scientific qualifications and with almost 10 years' experience in chemical physics research at various prestigious institutions around the world, and I had been going around with a conception of physical reality that was completely wrong! Why hadn't somebody told me about this before?"

"Again, part of that psychohistorical study I would like to see is why it did not impress the Copenhagen people, especially Bohr. But in the end it turns out that these other people were, in a way, right, because what I am notorious for, the so-called Bell's theorem, is just for showing that Einstein's explanation doesn't work. Einstein's explanation works so long as you have perfect correlations, which means measuring the same component of spin on the two sides [spin is a measure of a property similar but not identical to the rotation of a particle on its axis]. But as soon as you are measuring in a nonparallel direction, you get results that cannot be explained by Einstein's idea that the answers existed before the experiment."

"The theorem tells you that maybe there must be something happening faster than light, although it pains me even to say that much. The theorem certainly implies that Einstein's concept of space and time, neatly divided up into separate regions by light velocity, is not tenable. But then, to say that there's something going faster than light is to say more than I know."

"The main impact of Bell's theorem is from a philosophical-historical perspective: it reinforces, outside the physics of quantum entanglement, the incompatibility of hidden variable theories with quantum mechanics."

"That's all. That's the difficulty. That's why quantum mechanics can't seem to be imitable by a local classical computer. I've entertained myself always by squeezing the difficulty of quantum mechanics into a smaller and smaller place, so as to get more and more worried about this particular item. It seems to be almost ridiculous that you can squeeze it to a numerical question that one thing is bigger than another. But there you are—it is bigger than any logical argument can produce, if you have this kind of logic."

"Bell’s theorem is the most profound discovery of science."

"The gist of Bell's theorem is this: no local model of reality can explain the results of a particular experiment."

"Bell himself managed to devise such a proof which rejects all models of reality possessing the property of "locality". This proof has since become known as Bells theorem. It asserts that no local model of reality can underlie the quantum facts. Bell's theorem says that reality must be non-local."

"Physicists continue to debate whether Bell's theorem is airtight or not. However, the real question is not whether Bell can prove beyond doubt that reality is non-local, but whether the world is in fact non-local."

"There's an interesting scientific principle that a wrong answer can be much more stimulating to the field than just sort of finding the answer that's in the back of the book. A wrong result gets people excited. Worried. Obviously, you don't really want that to be happening—it's OK for a theorist to come up with a speculative new theory that gets shot down, but experimentalists are supposed to be very careful and their error limits are supposed to be realistic. Unfortunately, with this experiment, whenever you're looking for a stronger correlation, any kind of systematic error you can imagine typically weakens it and moves it toward the hidden-variable range. It was a hard experiment. In those days, at any rate, with the kind of equipment I had, and … well, what can I say? I screwed up."

"The experimental verification of violations of Bell’s inequality for randomly set measurements at space-like separation is the most astonishing result in the history of physics. Theoretical physics has yet to come to terms with what these results mean for our fundamental account of the world. Experimentalists, from Freedman and Clauser and Aspect forward, deserve their share of the credit for producing the necessary experimental conditions and for steadily closing the experimental loopholes available to the persistent skeptic. But the great achievement was Bell’s. It was he who understood the profound significance of these phenomena, the prediction of which can be derived easily even by a freshman physics student. Unfortunately, many physicists have not properly appreciated what Bell proved: they take the target of his theorem— what the theorem rules out as impossible—to be much narrower and more parochial than it is. Early on, Bell’s result was often reported as ruling out determinism, or hidden variables. Nowadays, it is sometimes reported as ruling out, or at least calling in question, realism. But these are all mistakes. What Bell’s theorem, together with the experimental results, proves to be impossible (subject to a few caveats we will attend to) is not determinism or hidden variables or realism but locality, in a perfectly clear sense. What Bell proved, and what theoretical physics has not yet properly absorbed, is that the physical world itself is non-local."

"No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics."

"Bell's theorem, for which he is most famous, was more a triumph of character than of intellect. The difficult thing about it was the realization of what was understood and what was not understood in the discussion of hidden variables. Bell's honesty about his own understanding provided the impetus for his formulation and proof of the theorem."

"In the same paper, Bell also discussed two rather unwelcome properties of hidden-variables theories. The first was contextuality. This tells us that, except in trivial cases, any hidden-variable theory must be such that the result of measuring a particular observable will depend on which other observable) are measured simultaneously. The second was nonlocality. All me hidden-variable models that Bell examined, including Bohm's, had the unpleasant feature that the behaviour of a particular particle depended on the properties of all others, however far away they were. In the EPR case, the measurement result obtained on one particle would depend on what measurement is performed on the second. As Bell said, this was the resolution of the EPR problem that Einstein would have liked least, and it is in this sense that it may be said that Bell proved Einstein wrong."

"At the very least, Bell's Theorem prevents us from interpreting quantum amplitudes as probability in the obvious way. You cannot point at a single configuration, with probability proportional to the squared modulus, and say, "This is what the universe looked like all along.""

"When I was a student at the University of Chicago, where I was a direct student of in the classroom, and an indirect student of Frank Knight, I could not learn why price should equal marginal cost. Even when I got to Harvard in 1935, I went around asking everybody: “What is the proof that this is so?” Of course, I did not know the 1892–1893 work of Vilfredo Pareto in which he essentially shows that a perfectly competitive equation system gives you the necessary and sufficient condition, not for ethical optimality—he was always a little slippery on that problem—but for what came to be called Pareto optimality so that there is no avoidable deadweight loss. I think I had most to learn from Abba Lerner, although I, of course, worked it out for myself."

"If I had had perfect teachers, they would have known the Pareto work; they would have known Enrico Barone and what you might call the fundamental theorems of welfare economics that the conditions for Pareto optimality would be exactly realized by competitive arbitrage. Before Bergson, Lerner–Hicks–Hotelling–Kaldor–Scitovsky insufficiently understood that the full set of Pareto optimality conditions constituted an incomplete set of conditions for ethical maximization. You must ask the right questions and make the right distinctions. All of my teachers believed there was something to Adam Smith’s invisible hand—that each person pursuing their self interest would, by some miraculous action of the invisible hand, be led to contrive in some vague sense the best interest of all. However, none of them could explain properly what the truth and falsity was in that position. I would say that if I had been a bright student in 1894 and read Pareto’s Italian journal article, I would have understood what I now understand to be the germ of truth in the invisible hand argument. All it refers to is the avoidance of deadweight loss. Here is where my association with Abram Bergson becomes relevant."

"There is little doubt that the observation that quality may depend on price (productivity on wages; default probability on interest rates) has provided a rich mine for economic theorists: A simple modification of the basic assumptions results in a profound alteration of many of the basic conclusions of the standard paradigm. The Law of Supply and Demand has been repealed. The Law of the Single Price has been repealed. The Fundamental Theorem of Welfare Economics has been shown not to be valid. More than that, the theories that we describe here provide the basis of progress toward a unification of macroeconomics and microeconomics. They pro vide an explanation of unemployment and credit rationing, derived from basic microeconomic principles. It is a theory in which the extensive idleness that periodically confronts society's resources, human and capital, is seen as but the most obvious example of market failures that prevasively and persistently distort the allocation of resources."

"It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible arguments, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, choosing rather to acknowledge their ignorance, than affirm anything rashly. They affirm nothing among their arguments or assertions which is not most manifestly known and examined with utmost rigour, rejecting all probable conjectures and little witticisms. They submit nothing to authority, indulge no affection, detest subterfuges of words, and declare their sentiments, as in a court of justice, without passion, without apology; knowing that their reasons, as Seneca testifies of them, are not brought to persuade, but to compel."

"There is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master."

"Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet."

"The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around."

"A “good theorem,” as Tate puts it, lasts forever. Once proved, it will always stay proved, and other mathematicians are free to use it and build on it as they please, sometimes to great effect."

"Bell's theorem is the most profound discovery of science."

"To put it another way, if we assume everybody should count equally, then there is no method of voting which will avoid some possibility of violating the rules."

"Contrary to the rationalist followers of the American economist Kenneth Arrow, for whom the instability of majority rule was a problem, Dahl’s Madisonian insight was that instability is actually an advantage. It keeps majorities fluid in ways that stop politics from becoming winner-take-all contests in which losers might as well reach for their guns."

"Arrow’s Impossibility Theorem is quite surprising. It shows that three very plausible and desirable features of a social decision mechanism are inconsistent with democracy: there is no “perfect” way to make social decisions. There is no perfect way to “aggregate” individual preferences to make one social preference. If we want to find a way to aggregate individual preferences to form social preferences, we will have to give up one of the properties of a social decision mechanism described in Arrow’s theorem."

"The Einstein–Podolsky–Rosen (EPR) argument has been enormously influential in the debate on the foundations of quantum mechanics. While EPR argue for the incompleteness of quantum mechanics, Bell's 'no-go' theorem, which is in a sense an extension of the EPR argument, appears to support the opposite conclusion."

"The Kochen–Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind of hidden states models."

"One possibility that comes to mind is that the spin-two graviton might arise as a composite of two spin-one gauge bosons. This interesting idea would seem to be rigorously excluded by a no-go theorem of Weinberg & Witten ... The Weinberg–Witten theorem appears to assume nothing more than the existence of a Lorentz-covariant energy momentum tensor, which indeed holds in gauge theory. The theorem does forbid a wide range of possibilities, but (as with several other beautiful and powerful no-go theorems) it has at least one hidden assumption that seems so trivial as to escape notice, but which later developments show to be unnecessary. The crucial assumption here is that the graviton moves in the same spacetime as the gauge bosons of which it is made!"

"The question of the possibility for a completion of quantum mechanics received its most famous (partial) answer in 1964 by, again, Bell ... He proved what today is known simply as Bell's theorem, to wit, that is such a more complete description exists, it cannot be local, i.e. dependent only on the events in a system's past lightcone, and agree with quantum mechanics in all instances. To this day, this result forms the paradigm example of a 'no-go' theorem."

"... If you start off with switches and gears, or whatever, you can never construct a universe in which you see quantum mechanical phenomena, according to Bell. We call such a thing a 'no-go theorem'. You may already suspect that I still believe in the hidden variables hypothesis. Surely our world must be constructed in such an ingenious way that some of the assumptions that Einstein, Bell and others found quite natural will turn out to be wrong. But how this will come about, I do not know. Anyway, for me, the hidden variables hypothesis is still the best way to ease my conscience about quantum mechanics. And as for 'no-go theorems', we will encounter several of these and discuss their fate."

"The cord stretched in the diagonal of an oblong produces both [areas] which the cords forming the longer and shorter sides of an oblong produce separately."

"The diagonal cord of a rectangle makes both (the squares) that the vertical side and the horizontal side make separately."

"Some of the things, which belong particularly to the history of mathematics, we will not refrain from attributing to Pythagoras himself. Among them is the Pythagorean theorem, which we want to preserve under all circumstances. (Cantor 1880: 129)"

"In the Shulba Sutra appended to Baudhayana’s Shrauta Sutra, mathematical instructions are given for the construction of Vedic altars. One of its remarkable contributions is the theorem usually ascribed to Pythagoras, first for the special case of a square (the form in which it was discovered), then for the general case of the rectangle: “The diagonal of the rectangle produces the combined surface which the length and the breadth produce separately.”"

"The first writer to attribute this proposition to Pythagoras is Vitruvius, hardly a reliable witness. From then on, this account became more widespread, but always in connection with the famous hecatomb that Pythagoras is said to have offered in celebration of discovering the proposition—an anecdote that severely undermines the credibility of the entire story. This sacrifice is incompatible with the strict prohibition of all bloody sacrifices, which writers of the same period, indeed often the very same ones who elsewhere recount the hecatomb, have handed down to us from the Pythagorean ritual laws. Cicero himself took offense at this anecdote, and in the latest Neopythagorean tradition, the bloody sacrifice is replaced by that of an ox formed from flour. For this reason, the hecatomb is not only incongruous with the Pythagoreans, but also with the Pythagoreans themselves. Proclus, an insightful writer, expresses himself remarkably vaguely: ‘When we listen to those who want to tell old stories, we find that they trace this theorem back to Pythagoras.’ As this shows, he too was unaware of any reliable source."

"Though this is the proposition universally associated by tradition with the name of Pythagoras, no really trustworthy evidence exists that it was actually discovered by him."

"‘These operations are all founded on a very distinct conception of what happens in the case of an eclipse, and on the knowledge of this theorem, that, in a right-angled triangle, the square on the hypotenuse is equal to the squares of the other two sides. It is curious to find the theorem of PYTHAGORAS in India, where, for aught we know, it may have been discovered, and from whence that philosopher may have derived some of the solid, as well as the visionary speculations, with which he delighted to instruct or amuse his disciples.’"

"If we listen to those who like to record antiquities, we shall find them attributing this theorem to Pythagoras and saying that he sacrificed an ox on its discovery. For my part, though I marvel at those who first noted the truth of this theorem, I admire more the author of the Elements for the very lucid proof by which he made it fast."

"It is more likely that Pythagoras was influenced by India than by Egypt. Almost all the theories, religions, philosophical and mathematical taught by the Pythagoreans, were known in India in the sixth century B.C., and the Pythagoreans, like the Jains and the Buddhists, refrained from the destruction of life and eating meat and regarded certain vegetables such as beans as taboo" "It seems that the so-called Pythagorean theorem of the quadrature of the hypotenuse was already known to the Indians in the older Vedic times, and thus before Pythagoras."

"If we consider the results obtained together, we will not be able to doubt the conclusion to be drawn from them. The ancient priestly geometry of the Indians not only knew the Pythagorean theorem, but it even played the main role in their calculations; with its help, they constructed elements that the Greeks found in a completely different way; with its help, they also found the irrational quantities. And it was precisely these two things that Pythagoras introduced into the Greek-Italian world; these two things, according to the Greeks, he invented. Indeed, even more! The way in which Pythagoras proved his theorem was also, in all likelihood, the same as that which we find in the Vedic Shulba Sutras. After examining the Shulba Sutras, we could have said: If Pythagoras really was in India, as we previously suggested, and initiated himself into the priestly wisdom of the Brahmins, then he could have brought precisely these theorems of geometric science to Greece; — and history has been telling us for several millennia now that this was indeed the case!"