Mathematicians From England

"The general... problem... packing... in n-dimensional space. ...[T]here is nothing mysterious about n-dimensional space. A point in real n-dimensional space \R^n is... a string of real numbersx = (x_1,x_2,x_3, ...,x_n).A sphere in \R^n with center u = (u_1,u_2,u_3, ...,u_n) and radius \rho consists of all points x... satisfying (x_1-u_1)^2 + (x_2-u_2)^2+ ... +(x_n-u_n)^2 = \rho^2. We can describe a sphere packing in \R^n... by specifying the centers u and the radius."

"[S]uppose I had a twin brother... [M]y sisters would have had a much better chance... [T]hey could... interrogate us separately... Then we can't change the object. ...Without transmitting information ...we couldn't win... [T]hat's what manages to happen in the particle case. ...[T]his ...Simon also thought of a long time ago.... but he didn't... deduce the ."

"... I have said for twenty-five or thirty years that the one thing I would really like to know before I die is why the monster group exists.""

"[I]n two dimensions the... [19 point] hexagonal lattice solves the packing, kissing, covering and quantizing problems. ...[T]his ...book is ...a search for similar nice patterns in higher dimensions."

"We are planning a sequel... The Geometry of Low-Dimensional Groups and Lattices which will contain two earlier papers..."

"When I was on the train from Liverpool to Cambridge to become a student, it occurred to me that no one at Cambridge knew I was painfully shy, so I could become an extrovert instead of an introvert."

"In this chapter we discuss the problem of packing spheres in and of packing points on the surface of a sphere. The problem is an important special case of the latter, and asks how many spheres can just touch another sphere of the same size."

"Let me phrase the free will theorem that Simon and I proved. ...[I]f we... have free will... then so do elementary particles have their... very small quantity of free will... to mean, our behavior is not a function of the past. ...[I]f some experimenters have free will ...then so do elementary particles... even the ones outside us..."

"It's one of the things I most admire about Simon Kochen, my co-author, that... in August 2006, we'd been talking about this... for years... Suddenly the scales fell away, that had been obscuring the thing, and I said... "We've proved if we have free will, so do the particles" and he... said "Yes... this means that my stuff with Ax is all nonsense, doesn't it?""

"[T]he last lecture is going to be about the consequence of our ... Descartes'... disconnected determinism won't work. ...That's it. It's gone. Leibniz's won't work. It's gone."

"With a view to recalling Clausen's identity, we begin by introducing the generalized Gaussian and Clausenian hypergeometric function defined, in the notations of Leo Pochhammer and Ernest William Barnes... as already pointed out by Barnes, the generalized hypergeometric function pFq originated with Clausen and was studied, among others, by Johannes Karl Thomae, Édouard Jean-Baptiste Goursat, and Pochhammer whose voluminous work on the subject provides a detailed development of the theory."

"SPIN... is a... curious axiom. If you take one of these particles and ask it what... it's squared component of spin is, in three... mutually perpendicular directions, it always happens that two of the answers are 1, and one of them is 0. That's most mysterious... and... it's not possible to solve this puzzle. ...[W]e have these 33 directions, and it's not possible to assign 0s and 1s to them, subject to that condition... the 1-0-1 rule. ...[T]he particle is acting somewhat like a little boy ...making up its mind as it goes along. It doesn't stop it from giving answers, but it does stop the answers from being determined ahead of time, and that's the guts of it."

"FIN [from finite speed] is the axiom... from relativity theory that information... can't travel faster than the speed of light."

"The conclusion seems to be irresistible that such laws of nature as the principle of conservation of energy, the principle of conservation of momentum and the law of gravitation are necessary consequences of our modes of measurement. They are, in fact, elaborately disguised identities which could have been predicted a priori by a being of sufficiently powerful analytical insight who fully understood all that is implied in the way we measure space-time intervals."

"TWIN... I just described... Even remotely separated particles... a condition in which, if asked the same questions, they will give the same answers. If they're not asked the same questions, all bets are off. ...We call that Twinning the particles ...an instance of ...entanglement."

"[T]he strangest contribution of quantum mechanics to this discussion is the EPR paradox. ...That's an essential contribution to our theorem too. ...Despite the fact that information can't be transmitted faster than the speed of light, ...remotely separated events can be correlated ...and this is the content of our TWIN axiom, you can put two particles into a... singleton state... the angular momentum of the pair of particles is zero... [B]y the conservation of angular momentum... if you measure the angular momentum of this in any direction, then for the angular momentum of the other you get the negative answer, but... we're going to square it, that means... the squared component of spin is the same... [T]hese particles have been sort of hypnotized. If you ask... they will give the same answer... like I and my twin brother... [T]he funny thing is, even though the proves that the answers do not exist ahead of time, the equality of the answers can exist..."

"If I ask this question of this particle, and... my colleague on Mars asks the same question of the other particle, then even though those questions aren't determined, ...they don't exist ahead of time, ...they'll give the same answer. ...It's meaningless to compare the times at which we do it, because time is not an invariant concept. ...[I]f my colleague on Mars has asked the same question, or ...will ask the same question... or if he's now asking the same question... he'll get the same answer. That is the EPR paradox, the fantastic thing that Einstein thought would disprove quantum mechanics. It is... perfectly consistent, but ever since it was discovered people have been trying to explain it away... because it's hard to believe."

"There has been the assumption that men are finite spirits. They are, that is to say, not only animals with a brief terrestrial existence, but in them is an element which comes from, and belongs to, the spiritual world. This world we postulate to be the world of eternal reality, of God; and we assume that in it whatever is of God, the things that are good, beautiful and true, will exist for ever with Him. We have then, to justify our belief that, because such God-like qualities exist in human personality, that personality will survive the destruction of the body."

"[R]elativity is an important part of the game."

"[Y]ou've probably heard of the theory of relativity. ...Most of us have heard the assertion that you can't transmit information faster than the speed of light. Most of us... hear it on authority only. We don't really understand why not. ...The reason is ...there's no absolute notion of time. Time depends on which coordinate system you're using... on your frame of reference... As seen from one frame of reference, event A can be before event B and as seen from another frame of reference, event B came first. The world [universe] hasn't got a standard definition of time."

"I have a very simple way of explaining relativity theory... and if you follow that you'll understand how... it's impossible to... transmit information faster than the speed of light. The reason is, if you could, then seen from another person's point of view, you'd transmit information backward in time... [W]e would know the result of somebody's experiment before they performed it, and if they have free will... you'd know the result of their choice before they've made it, and they're free to make another one."

"[T]he proof of the Free will theorem... It's ...plausible ...from the start. ...Let's see ...the axioms SPIN, FIN and TWIN."

"Continual miniaturisation allows resources to be conserved, efficiency to be increased, pollution to be reduced, and the remarkable flexibilities of the quantum world to be tapped. Very advanced civilizations elsewhere in the universe may have been forced to follow the same technological path. Their nano-scale space probes, their atomic-scale machines and nano-computers, would be imperceptible to our coarse-grained surveys of the universe. ...This may be the low-impact evolutionary path you need to follow in order to survive into the far, far future."

"In Theories of Everything... John Barrow argued that Gödel's incompleteness theorem undermines the very notion of a complete theory of nature. Gödel established that any moderately complex system of axioms inevitably raises questions that cannot be answered about the axioms. The implication is that any theory will always have loose ends. Barrow also pointed out that a unified theory of particle physics would not really be a theory of everything, but only a theory of all particles and forces. The theory would have little or nothing to say about phenomena that make our lives meaningful, such as love or beauty."

"Einstein had spent the previous thirty years showing how we could understand the behaviour of whole universes with simple maths. Gamow saw that those universes must have had a past that was unimaginably different to the present. What had stopped them both in their tracks was Gamow's suggestion that the laws of physics could describe something being created out of nothing."

"Just focusing on what exists now seems a bit exclusive. And if we include everything that has ever existed as part of the universe, why not include the future as well? This seems to leave us with the definition that the universe is everything that has ever existed, does exist, or will ever exist."

"Aristotle believed that the world did not come into being at some time in the past; it had always existed and it would always exist, unchanged in essence for ever. He placed a high premium on symmetry and believed that the sphere was the most perfect of all shapes. Hence the universe must be spherical. ...An important feature of the spherical shape... was the fact that when a sphere rotates it does not cut into empty space where there is no matter and it leaves no empty space behind. ...A vacuum was impossible. It could no more exist than an infinite physical quantity. ...Circular motion was the most perfect and natural movement of all."

"Einstein showed us how to find all the possible universes that were consistent with the laws of physics and the character of gravity, how to reconstruct their pasts and predict their futures. But actually finding them was no easy task."

"The Indian religious traditions... accepted the concept of non-being on an equal footing with that of being. Like many other Eastern religions, the Indian culture regarded Nothing as a state from which one might have come and to which one might return.. Where Western religious traditions sought to flee from nothingness... a state of non-being was something to be actively sought by Buddhist and Hindus in order to achieve Nirvana: oneness with the Cosmos."

"The Greek tradition was a complete contrast to that of the Far East. ...the Greeks placed logic at the pinnacle of human thinking. Their sceptical attitude towards the wielding of 'non-being' as some sort of 'something' that could be subject to logical development was exemplified by Parmenides' influential arguments against the concept of empty space. ...He maintained that you can only speak about what is: what is not cannot be thought of, and what cannot be thought of cannot be. ...more unexpected was the further conclusion that time, motion nor change could exist either."

"While we have no reason to expect that our position in the universe is special in every way, we would be equally misled were we to assume that it could not be special in any way."

"The Indian system of counting is probably the most successful intellectual innovation ever devised by human beings. It has been universally adopted. ...It is the nearest thing we have to a universal language."

"The physicist's concept of nothing—the vacuum... began as empty space—the void... turned into a stagnant ether through which all the motions of the Universe swam, vanished in Einstein's hands, then re-emerged in the twentieth-century quantum picture of how Nature works."

"The logic of the Greeks prevents them having the idea at all and it is to the Indian cultures that we must look to find thinkers who are comfortable with the idea that Nothing might be something."

"The quantum revolution showed us why the old picture of a vacuum as an empty box was untenable. ...Gradually, this exotic new picture of quantum nothingness succumbed to experimental exploration... in the form of vacuum tubes, light bulbs and X-rays. Now the 'empty' space itself started to be probed. ...There was always something left: a vacuum energy that permeated every fibre of the Universe."

"Einstein showed us that the Universe might contain a mysterious form of vacuum energy. ...Last year, two teams of astronomers used Earth's most powerful telescopes... to gather persuasive evidence for the reality of the cosmic vacuum energy. Its effects are dramatic. It is accelerating the expansion of the Universe."

"Location is not, as the estate agents say, everything. We must also consider our place in history."

"Scientific pictures are often not just about science. They may... have an undeniable aesthetic quality. They may even have been primarily works of art that possess a scientific message."

"The abstractions of Einstein's curved space and time gave rise to analogies and pictures that played a new explanatory role. Space and time gave way to space-time, visible light was augmented by images across the rest of the electromagnetic spectrum, and we realised that we could see back towards the apparent beginnings of time."

"Copernicus' picture did more than picture the solar system correctly: it painted a new world picture."

"Mathematics became an experimental subject. Individuals could follow previously intractable problems by simply watching what happened when they were programmed into a personal computer. ...The PC revolution has made science more visual and more immediate. ...by creating films of imaginary experiences of mathematical worlds. ...Words are no longer enough."

"We have witnessed a revolution in the history of science. Not the sort of revolution that philosophers of science once believed in—they don't happen any more—but a revolution brought about by new tools, different ways of seeing, and novel ways of understanding. Nothing old needed to be overthrown to make way for the new. The future of science will be increasingly dominated by artificial images and simulations."

"Images and pictures... have played a key role in shaping our scientific picture of the world. ...Carefully constructed families of pictures can act as a calculus all their own. Like any successful systems of symbols, with an appropriate grammar they enlarge the number of things that we can do without consciously thinking."

"In the spring of 1845, William Parsons, the third Earl of Rosse, began observing with his great six-foot telescope... The Earl was excited by what he was the first human to see: spiral patterns of stars, seemingly swirling in great 'spiral convolutions' about the centre of the galaxy. ...No one could ever have seen the spiral pattern of stars in a galaxy unless they had looked through Rosse's telescope or seen his drawings. ...I believe that Van Gogh would have seen those drawings in the press following the publicity attracted by them, or in Flammarion's book... and gained his astronomical inspiration from them."

"The advent of small, inexpensive computers with superb graphics has changed the way many sciences are practiced, and the way that all sciences present the results of experiments and calculations."

"The living world is not a marble palace. It is a higgledy-piggledy outcome of natural selection and the competition between many interacting factors. The outcome is often neither elegant nor symmetrical."

"Each of the most basic physical laws that we know corresponds to some invariance, which in turn is equivalent to a collection of changes which form a symmetry group. ...whilst leaving some underlying theme unchanged. ...for example, the conservation of energy is equivalent to the invariance of the laws of motion with respect to translations backwards or forwards in time... the conservation of linear momentum is equivalent to the invariance of the laws of motion with respect to the position of your laboratory in space, and the conservation of angular momentum to an invariance with respect to directional orientation... discovery of conservation laws indicated that Nature possessed built-in sustaining principles which prevented the world from just ceasing to be. There were fewer roles for the Deity to play..."

"Even today, there persists... a feeling that the creation of the Universe out of nothing must violate some basic conservation law that stops one from getting something for nothing. Nevertheless, there is actually no evidence that the Universe as a whole possesses a non-zero value of any such conserved quantity. The total mass-energy of all the constituents of a finite Universe appears to be always equal in magnitude but opposite in sign to the total gravitational potential energies of those particles. ...Similarly, there is no evidence that the Universe possesses any overall net rotation or electric charge."

"It is not hard to see why the Eastern holistic perspective made scientific progress so difficult. It denies the intuition that one can study the parts of the world in isolation from the rest—that one can analyze the world..."

"If one looks at the special problems that were the mainsprings of progress along the oldest and most persistent lines of human inquiry, then one finds Nothing, suitably disguised as something, never far from the centre of things."