University Of Cambridge Faculty

"[Robert Nozick]: Philosophical Explanations "...it would be foolhardy ...to place ...significant weight upon the necessity or even truth of SR. ...Moreover theorems show that any theory that retains certain features of Quantum Mechanics also will not satisfye SR." SR is Leibniz's . ...[T]here's a reference to the Kochen-Specker paper ...in which Kochen, my co-author, and Specker ...both logicians, not physicists ...prove this ...From our point of view this is not enough. The is not as strong as the new theorem."

"The philosophical consequences of the General Theory of Relativity are perhaps more striking than the experimental tests. As Bishop Barnes has reminded us, "The astonishing thing about Einstein's equations is that they appear to have come out of nothing." We have assumed that the laws of nature must be capable of expression in a form which is invariant for all possible transformations of the space-time co-ordinates and also that the geometry of space-time is Riemannian. From this exiguous basis, formulae of gravitation more accurate than those of Newton have been derived. As Barnes points out..."

"[T]his Kochen-Specker paradox ...what it does ...[T]here's a problem in physics ...the measurement problem ...that's a wrong description. There's ...measuring the squared spin of a spin one particle. ...Let's say "measuring the spin" or measuring the [squared] component of spin ...of a spin one particle in a certain direction."

"I believe, and most people believe until they are educated out of this belief, that we are... free."

"I'm going to present arguments... to strongly support... that we do... have free will, but not... to prove it at the deductive level."

"The only Bishop (Ernest William Barnes...) in the Royal Society appears to have contracted the habit so prevalent among popular scientists of making stupid and unsupported statements... In his book, Scientific Theory and Religion, he tells us that the flatness and fixity of the earth were "taken over into the Christian Creeds." Which creeds? ...he describes as erroneous the "authorized teaching of the Roman Church" regarding the date of creation. What date did the Church authorize? He also refers to the "doctrine of the special creation of the species." There is no such doctrine. Incidentally one of the most eminent of English mathematicians, Professor Whittaker, F.R.S., severely criticizes the mathematical theories that are put forward in this book by Bishop Barnes."

"[Leibniz] has this ... "that nothing happens without a reason why it should be so...""

"[L]ots of philosophical discussions about free will are concerned with the question of assigning moral responsibility. ...If the judge is a determinist, I might argue it was all determined. I just had to do it. I'm not responsible. ...We're not concerned with assigning moral responsibility... I want you to think of free whim..."

"The rise of determinism, it's just a consequence of the fact, science advanced tremendously, as a result mostly of... Newton and people like Laplace. They could determine things, but it's only the large scale science that ...seems ...deterministic. This century it's not been so."

"Descartes' dualism... is explicitly contradicted by our theorem."

"[P]articles making choices... seems to involve deliberation. ...We are not pretending particles are conscious..."

"One consequence of the ... plus the assumption we actually have free will, is that even in the... inanimate world of particles, it's not true, this . A particle does one thing or another, and there's no reason... There's nothing in the previous history of the universe which tells it what to do."

"When I was a little boy, I had two sisters... and we used to play ...and I had no moral sense ...when playing ...[I]f I thought that my sisters were getting near to my object, I changed ...the object ...[Y]ou have to select a new object which answers the ...questions ...already ...asked ...That's what the particles do. ...If you ask them this ...spin question, they don't have an object in mind. Think of a cleverer boy... who never bothers to select an object... just gives... answers at random, and then starts thinking what the object is. ...[T]hat's what the particles do. They don't have an answer in mind for each direction... [[w:Simon B. Kochen|[K]ochen]] and Specker proved that a long time ago... and the reason... is that there's a little puzzle... that puzzle has no solution. ...[T]hey use 117 directions, but ...Asher Peres ...reduced this set of directions to 33... [T]here's no conceivable set of answers... that's consistent with ...the spin law."

"Here's a particle, and I... direct my finger at it... and ask... What's it's spin in that direction? ...This particle is quantized. ...[I]t can only give two answers ...1 and 0. If I hadn't put that word squared in it could give three answers, 1, -1 and 0 ...Initially, it was... obvious... to believe that this concept existed before you measured it, but that was found not to be so. ...[W]hat the says is that it can't exist before you measure it... because there's no consistent set of answers to every question."

"Laplace... wrote Mécanique céleste discussing the motion of the planets. ...[O]ne ...reason why ...determinacy got its ...impetus from the development of science, was that Newton's theory of gravitation ...was entirely deterministic. It left no room for freedom. ...Laplace, who did a lot of work on Newtonian ...theory says ...An intellect ...[or] intelligence, which knew ...where all the particles were at some moment and how fast they were moving, and so on, that every single thing could be known to that intelligence, provided a ...good calculator. ...It could reason out exactly what was going to happen in the future. ...It's the strongest ...assertion of determinism in the scientific literature. I don't believe it for one moment..!"

"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."

"[S]omebody said, ..."We have to believe in free will. There's no choice." That's a nice ...sort of paradox ..."

"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."

"The classical... problem is... how densely a large number of identical spheres ([e.g.,] ball bearings...) can be packed together. ...[C]onsider an aircraft hangar... [A]bout one quarter of the space will not be used... One... arrangement... the face-centered cubic (or fcc) lattice... spheres occupy \pi / \sqrt{18} = .7405... of the total space.... the lattice packing has density .7405... . [H]pwever, there are partial packings that are denser than the face-centered cubic... over larger regions..."

"The classical... problem... asks: is this the greatest density..? an unsolved problem, one of the most famous..."

"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."

"... 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.""

"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."

"[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."

"[T]he best packings in up to eight dimensions belong to families A_n, D_n and E_n, and the corresponding s turn up in apparently unrelated areas... [I]n 24 dimensions the \Lambda_{24} has... connections with , s, and the Monster simple group... [O]ne day someone will write an article on "The Ubiquity of the Leech lattice." ...There are applications of... packings to number theory... [e.g.,] solving s, and to "the "... There are... applications of sphere packings... in digital communications... a typical question from... spread-spectrum communications for mobile radio... how many spheres of radius 0.25 can be packed in a sphere of radius 1 in 100-dimensional space? ...Two and three-d... packings... circles in a two-d... packing may represent s... in... a cable. Three-d... packings have applications in chemistry and physics... biology... antenna design... choosing directions for X-ray tomography... and... statistical analysis on spheres... n-dimensional packings may be used in... numerical evaluation of integrals... on the surface of a sphere in \R^n or in its interior. ...A related application ...n-dimensional search or approximation problems ...[I]n physics... dual theory and superstring theory... have involved the E_8 and \Lambda_{24} lattices and the related Lorentzian lattices in dimensions 10 and 26..."

"There has been a great deal of nonsense written... about the mysterious fourth dimension. ...4-dimensional space just consists of points with four coordinates instead of three (...similarly for any number of dimensions). ...[I]magine a telegraph ...over which numbers are ...sent in sets of four. Each set... is a point in 4-d... space."

"The densest possible packings are known for n \leqslant 8. ...\Lambda_n... represents a laminated lattice... the densest lattices in up to 8 dimensions..."

"[[w:Sphere packing|[L]attice packing]]... has the properties that 0 is a center and... if there are spheres with centers u and v then there are spheres with centers u + v and u - v... [i.e.,] the sets of centers forms an . In crystallography these... are... called s... We can find... in general n centers v_1,v_2, ...,v_n for an n-dimensional lattice... such that the set of all centers consists of the sums \sum k_i v_i where k_i are s."

"Barnes was notorious for delivering what the press called his "guerilla sermons," in which he pointed out the need for the church to be honest in admitting how much of its traditional dogma would have to be abandoned if evolution theory was accepted. Even a progressionist, teleological evolutionism required a reinterpretation of the doctrine of original sin. To begin with, Barnes said little about the actual process of evolution, but he seems to have assumed that it was purposeful and aimed at the production of higher mental states. In 1930, though, he obtained a copy of R. A. Fisher's Genetical Theory of Natural Selection and began a correspondence with Fisher, who had studied under him while a student at Cambridge. Barnes was one of the few clergymen who could actually understand Fisher's mathematics (although even he admitted that it was hard going)... He did not concede that the selection theory offered a complete explanation, and he continued to believe that evolution was intended to produce beings with higher mental and spiritual qualities, but he was now aware that the more simpleminded forms of teleology were unacceptable."

"The Roman Catholics were already prominent in the debate on abortion in Britain in the 1930s. It is notable, for example, that only two religious groups were keen to give evidence before the Inter-Departmental Committee in Abortion between 1937 and 1939 or sent written statements to the Committee. One was the Modern Churchmen's Union, and in particular its most prominent supporter, though not a member, Ernest William Barnes, the Anglican Bishop of Birmingham, which was concerned to advance the cause of abortion on eugenic grounds, and the other was the Roman Catholics... No representatives of the Protestant Nonconformist Churches took part or made statements (Report of the Inter-Departmental Committee 1939)"

"Revelation can be supplemented by reason. Christ Himself gave reasons for His belief, and put in modern form, these reasons are, to my mind, conclusive. You remember the passage in the earliest Gospel: "But as touching the dead, that they are raised; have ye not read in the book of Moses, in the place concerning the bush, how God spake unto him, saying, I am the God of Abraham, and the God of Isaac, and the God of Jacob? He is not the God of the dead, but of the living: ye do greatly err" (Mark xii, 26 R.V.) Herein, in a form adapted to Jewish thought is "the one great argument which has made most sincere believers in God believers in Immortality also."

"Although he was trained in mathematics and was not a biblical scholar... Barnes, undertook to write a book about the origins of the Christian religion. Published in 1947, The Rise of Christianity caused a stir because it was so frankly dismissive of traditional Christian dogma, especially the miraculous. In this book, for example, Barnes calls the birth stories "edifying legend." He observes that the roots of the story of the Virgin Birth are "pagan." He questions the dogma of the Logos—the eternal word incarnate in this man, Jesus—set forth in the first chapter of John's Gospel. And he denies the bodily resurrection of Christ. Like Thomas Jefferson, he admires Jesus' character and teaching."

"Man is what he is, because a spiritual element has entered into, and taken possession of, animal consciousness. This spiritual element is not, according to Christian teaching, divine: but it is capable of entering into relations with God. It can perceive Him: in thought, it can reason as to His nature and actions: in will and feeling, it can serve and love Him, or disobey and fear Him. Such activity shows itself in what we call the working of conscience."

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

"In many respects, this is not in any way an unorthodox opinion. ...Physicists have lived with these paradoxes ...for 80 years now. They have been accustomed to the fact that quantum mechanics is not a totally predictive theory, and they've proved long ago that no extension of quantum mechanics can be. This is not a defect... It's not a temporary defect, anyway. No extension of quantum mechanics can recover... total predictivity. From our point of view that's... obvious. ...[N]o correct theory can predict what the particle is going to do before it's made its decision... while it's still free to do something else. ...[I]t's not to be seen as a defect in quantum mechanics that it doesn't predict. It's a merit. ...You shouldn't expect to be able to predict things."

"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."

"[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."

"[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 ."

"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."

"[W]hen... non-predictivity of quantum mechanics was discovered, it came as a great surprise. People tried to explain it away. ...Many ...invented larger theories. ...Nobody has succeeded in making one of these theories relativistically invariant, and physics appears to be relativistically invariant. ...Physicists have believed ...in the result that we're proving for a long time. It's no surprise. "I knew all that," they say. However, what they didn't know was that it can be deduced in this very precise, logical fashion, from so little information... that is not at all contentious. ...[T]hese 3 axioms ...they're routine, they're accepted ...They follow from quantum mechanics and relativity. There's nothing dubious about them ...and that's all we need... [T]he original deductions... [of] quantum mechanics... used more. They used all sorts... and some... were... pretty poorly understood..."

"Why do we care about finding dense packing in n-dimensional space? ...This is an interesting problem in pure geometry. Hilbert mentioned it in 1900 in his list open problems... [T[he best packings... have connections... with other branches of mathematics. ..."

"He is Archimedes, Mick Jagger, Salvador Dalí, and Richard Feynman, all rolled into one. He is one of the greatest living mathematicians, with a sly sense of humour, a polymath’s promiscuous curiosity, and a compulsion to explain everything about the world to everyone in it."

"We see in man three elements; the material body, the life principle and the element of human personality. The last has only slowly reached its present complexity and is still far from the power and perfection that we can imagine it will some day possess."

"The astonishing thing about Einstein's equations is that they appear to have come out of nothing."

"There are many varieties of . ...It says that there's nothing incompatible about determinism... and free will. ...This is from a dispute between Hobbes and.... Archbishop Bramhall... "A man was free, in those things... were in his power, to follow his will, but... he was not free to will..." The will was determinate, but that he was free. ...It's not contradictory, although it looks it at first sight..."

"My duty is, as a Judge, to be governed by fixed rules and former precedents."

"The Courts of law are not provided at the public expense, and were not intended by those who so provided them, for the settlement of any but differences which do arise in the ordinary course of business."

"An active imagination may find a bad tendency arising out of every transaction between imperfect morals."

"Human nature is imperfect."