People From Liverpool

"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?""

"I find it... fantastic that somebody 2,000 years ago could have suggested that atoms... have free will."

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

"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 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."

"[[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."

"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..."

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

"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. ..."

"... 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.""

"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."

"[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]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."

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

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

"[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..."

"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."

"[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..."

"[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."

"Don't think... I believe unreservedly in quantum mechanics, but I'm not going to change my mind before there's some reason..."

"Physicists... have seen lots of instances of people... without... qualifications in physics, and presenting some... loony theory... and they don't read it. ...[W]e had this thing in mathematics once. People... thought they'd proved Fermat's Last Theorem... Eventually somebody did, but he was a... distinguished... high-powered mathematician. I'm... prepared for the fact... that physicists, especially ones that don't read our paper, don't believe it. They think it's just another... of those strange things. However, it's... better than that. ...I hope that the physicists stay and... learn something..."

"There is... strong evidence that these three axioms... are true."

"FIN, the axiom that information can't travel faster than a finite speed, (that's where it gets its name from)... does not have that nice property. ...I can't disprove... that somewhere there isn't an as yet undiscovered way of transmitting information faster than... light. ...FIN ...follows from a symmetry principle that the laws of physics are independent of the coordinate frame ...If you're traveling ...at half the speed of light, you still have the same physics ...That symmetry principle ...that's been tested in countless ways, and that's ...why we believe FIN."

"You should... believe... in the things that have been tested... strongly until... evidence against them..."

"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."

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

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

"[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..."

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

"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."

"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."

"[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 ."

"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."

"[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."

"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."

"SPIN and TWIN are operationally definable... Do the operation that's called, measuring... as many times as you like, and see that they always give the same answers. That's what is meant by saying that those things are operationally definable..."

"Suffer fools gladly; they may be right."

"There are idle students and cavillers, who have advertised Burton as the creator of a peculiar anthologic maze, an amusing literary chaos, a farrago of quotations, a mere olla podrida of quaintness, a pot pourri of pleasant delites, a florilegium of elegant extracts, a tangled fardel of old-world flowers of thought, a faggot of odd fancies, quips, facetiæ, loosely tied, the creator, in short, of a book for a rainy day and a cosy corner."

"How we started the game and how we're going to finish the game, are just extraordinary. Polar opposites."

"Men against boys, Ian. Isn't it?"

"Ian, I think? This will definitely be the best World Cup we'll have ever seen. We're lucky, we're privileged."

"Yeah. This game will be referred to many, many, many, many, years to come. Yes, we're talking about 1950 games now. This game, is going to like that. We will refer to this in fifty years time."

"But, their lack of effort today has just astounded me. It really has; you could just get ten people sitting around just to work harder than a lot of these players have worked today. Oh, yeah."

"Have just thrown in the towel; this towel was thrown in. Forty, fifty minutes ago. They're not even getting close to the ball."

"Please, do not adjust your television set."

"It's hard to decipher this game. It's been a very good German performance, but Ian? You cannot underestimate how bad Brazil have been today, it's like? It's like amateur hour watching them. But, their attitude has just been appalling. I know, I keep saying appalling. Because, I'm running out words. They've just given in. They've just completely given in, actually trying to run around and play football."

"That's a Sunday morning goal; it's certainly Sunday morning defending. Simple as you'd like, thank you very much! What a present!"