John Horton Conway

John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician, and Professor Emeritus of Mathematics at Princeton University in New Jersey. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton with Conway's Game of Life.

Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. He died of complications from COVID-19 at age 82.

• 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.
  • Mark Ronan (18 May 2006). Symmetry and the Monster: One of the greatest quests of mathematics. Oxford University Press, UK. pp. 163. ISBN 978-0-19-157938-7. 

• ... 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."
  • Life, Death and the Monster - Numberphile. YouTube (9 May 2014).

by J.H. Conway & N.J.A. Sloane (3rd edition, 1993)

• [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.
  • Preface to 1st edition

• We are planning a sequel... The Geometry of Low-Dimensional Groups and Lattices which will contain two earlier papers...
  • Ref: John H. Conway, Complex and integral laminated lattices, TAMS 280 (1983) 463-490 [2,6,22]; John H. Conway, The Coxeter–Todd lattice, 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 Euclidean space and of packing points on the surface of a sphere. The kissing number 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 ${\displaystyle \pi /{\sqrt {18}}=.7405...}$ of the total space.... the lattice packing has density ${\displaystyle .7405...}$ . [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...

• The general... problem... packing... in n-dimensional space. ...[T]here is nothing mysterious about n-dimensional space. A point in real n-dimensional space ${\displaystyle \mathbb {R} ^{n}}$ is... a string of real numbers${\displaystyle x=(x_{1},x_{2},x_{3},...,x_{n})}$.A sphere in ${\displaystyle \mathbb {R} ^{n}}$ with center ${\displaystyle u=(u_{1},u_{2},u_{3},...,u_{n})}$ and radius ${\displaystyle \rho }$ consists of all points ${\displaystyle x}$... satisfying ${\displaystyle (x_{1}-u_{1})^{2}+(x_{2}-u_{2})^{2}+...+(x_{n}-u_{n})^{2}=\rho ^{2}}$. We can describe a sphere packing in ${\displaystyle \mathbb {R} ^{n}}$... by specifying the centers ${\displaystyle u}$ and the radius.

• 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.

• [L]attice packing... has the properties that 0 is a center and... if there are spheres with centers ${\displaystyle u}$ and ${\displaystyle v}$ then there are spheres with centers ${\displaystyle u+v}$ and ${\displaystyle u-v}$... [i.e.,] the sets of centers forms an additive group. In crystallography these... are... called Bravais lattices... We can find... in general ${\displaystyle n}$ centers ${\displaystyle v_{1},v_{2},...,v_{n}}$ for an n-dimensional lattice... such that the set of all centers consists of the sums ${\displaystyle \sum k_{i}v_{i}}$ where ${\displaystyle k_{i}}$ are integers.

• 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. ...

• [T]he best packings in up to eight dimensions belong to families ${\displaystyle A_{n},D_{n}}$ and ${\displaystyle E_{n}}$, and the corresponding Coxeter–Dynkin diagrams turn up in apparently unrelated areas... [I]n 24 dimensions the Leech lattice ${\displaystyle \Lambda _{24}}$ has... connections with hyperbolic geometry, Lie algebras, 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 Diophantine equations, and to "the geometry of numbers"... 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 optical fibers... 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 ${\displaystyle \mathbb {R} ^{n}}$ or in its interior. ...A related application ...n-dimensional search or approximation problems ...[I]n physics... dual theory and superstring theory... have involved the ${\displaystyle E_{8}}$ and ${\displaystyle \Lambda _{24}}$ lattices and the related Lorentzian lattices in dimensions 10 and 26...

• The densest possible packings are known for ${\displaystyle n\leqslant 8}$. ...${\displaystyle \Lambda _{n}}$... represents a laminated lattice... the densest lattices in up to 8 dimensions...

John Conway (presenter) & Simon B. Kochen (March 23 - April 27, 2009) weekly Princeton University public lecture series. A 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...

• 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?"

• He'd been working with... Jim Ax for two decades on a theory that is now contradicted by... [our] Free will theorem. ...I admire Simon tremendously.

• Neither of us could have done it without the other.

• Lucretius... was an atomist, a follower of Epicurus. The original people who invented the atomic theory were Leucippus and Democritus. ...Lucretius is discussing ...atoms ...he says, "at quite indeterminate times and places they swerve" ...because it allows for human free will... and "if the atoms never swerve... what is the source of the free will possessed by living things throughout the earth?" He says, "Although many men are driven by an external source, and often constrained involuntarily to advance or rush headlong, yet there is in the human breast something that can fight against it and resist it... So also in the atoms you must recognize the same possibility. Besides weight and impact, there must be a third cause of movement, the source of this inborn power... due to the slight swerve of the atoms... since nothing can come out of nothing." And then he goes on to say, "the fact that the mind itself has no internal necessity to determine its every act, this is due to the slight swerve of the atoms at no determinate time and place."

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

• Most scientists until fairly recently are determinists.

• 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.

• There are many varieties of compatibilism. ...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...
  • Ref: The Questions Concerning Liberty, Necessity, and Chance Clearly Stated and Debated Between Dr. Bramhall, Bishop of Derry, and Thomas Hobbes of Malmesbury, The English works of Thomas Hobbes of Malmesbury Vol.5. p.5.

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

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

• [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...

• 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.

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

• One consequence of the free will theorem... plus the assumption we actually have free will, is that even in the... inanimate world of particles, it's not true, this principle of sufficient reason. 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.

• 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..!

• 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.

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

• 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.

• [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.

• 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 ...

• 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.

• [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 principle of sufficient reason. ...[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 Kochen–Specker theorem is not as strong as the new theorem.
  • 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.

• The Free Will Theorem [s]ays roughly, that "if... humans have free will, then so do the elementary particles outside us.
  Uses three axioms SPIN, FIN, (or MIN) and TWIN." I only mean this in a very restrictive sense. I don't suppose much free will. The supposition is only that the human can choose which one of 33 buttons he want to press. ...[T]his choice affects the future history of the world in a minor way. ...We know that button was pressed ...before it was pressed we didn't know which one it was and ...a human experimenter has that much free will. That's the only amount of free will I suppose. We suppose nothing about these deep questions of moral responsibility and so on.

• [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.

• 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 Kochen–Specker theorem says is that it can't exist before you measure it... because there's no consistent set of answers to every question.

• When I was a little boy, I had two sisters... and we used to play Twenty questions ...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... [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.

• [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 Free will theorem.

• [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.

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

• [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 Kochen–Specker theorem 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.

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

• 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.

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

• 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.

• 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...

• 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.

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

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

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

• [T]he last lecture is going to be about the consequence of our Free will theorem... Descartes'... disconnected determinism won't work. ...That's it. It's gone. Leibniz's principle of sufficient reason won't work. It's gone.

• 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...

• 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.

• [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...

• 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.
  • John Horton Conway: the world’s most charismatic mathematician. The Guardian (23 July 2015).

• Complex and Integral Laminated Lattices by J.H. Conway & N.J.A. Sloane
• Free Will and Determinism in Science and Philosophy John Conway (presenter) & Simon Kochen, Princeton University video series
• John Conway Solved Mathematical Problems With His Bare Hands @QuantaMagazine.org
• John H. Conway Bibliography, Articles & Papers
• John Horton Conway 1937-2020 Princeton University Mathematics Dept
• Mathematician John Horton Conway, a ‘magical genius’ known for inventing the ‘Game of Life,’ dies at age 82 by Catherine Zandonella, Princetion University
• YouTube videos
  • John Conway - The Game of Life and Set Theory @Istrail Laboratory channel
  • John Conway Distinguished Lecture - The Symmetries of Things @Istrail Laboratory channel
  • John Conway on Numberphile playlist.
  • LOVE/HATE Relationship with LIFE John Horton Conway (2015) @Talks at Google
  • The Princeton Brick - John Conway (2014) @Sam Jacob channel.
  • Tangles, Bangles and Knots with John Conway @Graduate Mathematics channel