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John Horton Conway
60 quotes
“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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“... 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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“We are planning a sequel... The Geometry of Low-Dimensional Groups and Lattices which will contain two earlier papers...”
John Horton Conway (1937–2020) first seen 1983 wikiquote, CC BY-SA 4.0 source
Recorded in: Ref: John H. Conway, Complex and integral laminated lattices, TAMS 280 (1983) 463-490 [2,6,22]; John H. Conway, The , The Mitchell group, and related sphere packings PCPS 93 (1983) 421-440 [2,4,7,8,22]
“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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“The classical... problem... asks: is this the greatest density..? an unsolved problem, one of the most famous...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[[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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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. ...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“The densest possible packings are known for n \leqslant 8. ...\Lambda_n... represents a laminated lattice... the densest lattices in up to 8 dimensions...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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?”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“He'd been working with... Jim Ax for two decades on a theory that is now contradicted by... [our] . ...I admire Simon tremendously.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“Neither of us could have done it without the other.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“I find it... fantastic that somebody 2,000 years ago could have suggested that atoms... have free will.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“Most scientists until fairly recently are determinists.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“Descartes... his type of determinism is only partial. I call it disconnected... that the physical world and... our bodies, but not our minds operate mechanically... We're robots, but the mind is different... that "The will is by its nature so free that it can never be constrained." This is normally called Descartes' dualism... mind, soul or spirit, and matter... [M]atter operates according to one set of laws, and mind or spirit... to another.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“Descartes' dualism... is explicitly contradicted by our theorem.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[P]articles making choices... seems to involve deliberation. ...We are not pretending particles are conscious...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[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...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“I believe, and most people believe until they are educated out of this belief, that we are... free.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“I'm going to present arguments... to strongly support... that we do... have free will, but not... to prove it at the deductive level.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[Leibniz] has this ... "that nothing happens without a reason why it should be so...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[S]omebody said, ..."We have to believe in free will. There's no choice." That's a nice ...sort of paradox ...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“Dennis Overbye ...says ~everything we know of science convinces me ...that the world is deterministic, nevertheless I cling to the illusion of free will.~ (I'm not quoting his exact words.) ..~I can't run my life without this illusion.~ Some people have described it as a necessary illusion. We don't think it's an illusion ...It's not ...We think we have free will.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[T]he arguments that suggest that we have free will can't be turned into logical deductive proofs. There's nothing wrong with that. If you argue in a court of law... we ...conclude, although we haven't got a proof, that probably the culprit was Mr. X.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“There isn't anything really, in the last resort that isn't inductive. Even the laws of deductive logic have been established only by induction. ...We can't use deductive logic ...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“The consistency of determinism doesn't mean that it's true. Mathematicians are accustomed to have consistency proofs. They say nothing about truth. They just say something about what might be true.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[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.”
John Horton Conway (1937–2020) first seen 1967 wikiquote, CC BY-SA 4.0 source
Recorded in: Ref: S. Kochen; E. P. Specker (1967) "The problem of hidden variables in quantum mechanics". Journal of Mathematics and Mechanics Vol. 17, No. 1, pp. 59–87.
“[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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[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 .”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[R]elativity is an important part of the game.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[T]he proof of the Free will theorem... It's ...plausible ...from the start. ...Let's see ...the axioms SPIN, FIN and TWIN.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“FIN [from finite speed] is the axiom... from relativity theory that information... can't travel faster than the speed of light.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“There is... strong evidence that these three axioms... are true.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“You should... believe... in the things that have been tested... strongly until... evidence against them...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“Don't think... I believe unreservedly in quantum mechanics, but I'm not going to change my mind before there's some reason...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[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.”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source
“[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...”
John Horton Conway (1937–2020) said by 2020 at the latest wikiquote, CC BY-SA 4.0 source