On Chaos: Does God Play Dice?

"God's dice always have a lucky roll." (Sophocles, 5th century BC)

"A perfect equity adjusts its balance in all parts of life. The dice of God are always loaded. The world looks like a multiplication-table, or a mathematical equation, which, turn it how you will, balances itself." (Ralph W Emerson, "Compensation", 1841)

"It seems hard to sneak a look at God's cards. But that He plays dice and uses 'telepathic' methods [...] is something that I cannot believe for a single moment." (Albert Einstein, [Letter to Cornel Lanczos] 1942)

"You believe in the God who plays dice, and I in complete law and order in a world that objectively exists, and which I, in a wildly speculative way, am trying to capture. [...] Even the great initial success of the quantum theory does not make me believe in the fundamental dice-game, although I am well aware that our younger colleagues interpret this as a consequence of senility. No doubt the day will come when we will see whose instinctive attitude was the correct one." (Albert Einstein, [Letter to Max Born] 1944)

"If God has made the world a perfect mechanism, He has at least conceded so much to our imperfect intellect that in order to predict little parts of it, we need not solve innumerable differential equations, but can use dice with fair success." (Max Born, "Albert Einstein: Philosopher-Scientist", 1949)

"Consideration of particle emission from black holes would seem to suggest that God not only plays dice, but also sometimes throws them where they cannot be seen." (Stephen Hawking, "The Quantum Mechanics of Black Holes", Scientific American, 1977)

"Not only does God play dice with the world - He does not let us see what He has rolled." (Stanisław Lem, "Imaginary Magnitude", 1981)

"Specialists [...] are slowly coming to the realization that the universe is biased and leans to the left. [...] Many scientists have come to believe that this odd state of affairs has somethittg to do with the weak nuclear force. It seems that the weak force tends to impart a left-handed spin to electrons, and this effect may bias some kinds of molecular synthesis to the left. [...] But scientific speculation of this ilk leads to a deeper question. Was it purely a matter of chance that left-handedness became the preferred direction in our universe, or is there some reason behind it? Did the sinister bent of existence that scientists have observed stem from a roll of the dice, or is God a semiambidextrous southpaw?" (Malcolm W Browne, 1986)

"So Einstein was wrong when he said, 'God does not play dice'. Consideration of black holes suggests, not only that God does play dice, but that he sometimes confuses us by throwing them where they can't be seen." (Stephen Hawking, 1994)

"Yet, Einstein's theories are also not the last word: quantum theory and relativity are inconsistent, and Einstein himself, proclaiming that 'God does not play dice!', rejected the basic reliance of quantum theory on chance events, and looked forward to a theory which would be deterministic. Recent experiments suggest that this view of Einstein's conflicts with his other deeply held beliefs about the nature of the physical universe. Certain it is that somewhere, beyond physicists' current horizons, are even more powerful theories of how the world is." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Chaos teaches us that anybody, God or cat, can play dice deterministically, while the naïve onlooker imagines that something random is going on." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1997)

"Indeed a deterministic die behaves very much as if it has six attractors, the steady states corresponding to its six faces, all of whose basins are intertwined. For technical reasons that can't quite be true, but it is true that deterministic systems with intertwined basins are wonderful substitutes for dice; in fact they're super-dice, behaving even more ‘randomly’ - apparently - than ordinary dice. Super-dice are so chaotic that they are uncomputable. Even if you know the equations for the system perfectly, then given an initial state, you cannot calculate which attractor it will end up on. The tiniest error of approximation - and there will always be such an error - will change the answer completely." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1997)

"Perhaps God can play dice, and create a universe of complete law and order, in the same breath." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1997)

"Simple laws may not produce simple behaviour. Deterministic laws can produce behaviour that appears random. Order can breed its own kind of chaos. The question is not so much whether God plays dice, but how God plays dice.", 1997)

"The chance events due to deterministic chaos, on the other hand, occur even within a closed system determined by immutable laws. Our most cherished examples of chance - dice, roulette, coin-tossing - seem closer to chaos than to the whims of outside events. So, in this revised sense, dice are a good metaphor for chance after all. It's just that we've refined our concept of randomness. Indeed, the deterministic but possibly chaotic stripes of phase space may be the true source of probability." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1997)

"[…] we would like to observe that the butterfly effect lies at the root of many events which we call random. The final result of throwing a dice depends on the position of the hand throwing it, on the air resistance, on the base that the die falls on, and on many other factors. The result appears random because we are not able to take into account all of these factors with sufficient accuracy. Even the tiniest bump on the table and the most imperceptible move of the wrist affect the position in which the die finally lands. It would be reasonable to assume that chaos lies at the root of all random phenomena." (Iwo Bialynicki-Birula & Iwona Bialynicka-Birula, "Modeling Reality: How Computers Mirror Life", 2004)

"Freedom, then, is the ability to choose at will from the greatest number of options available to us. In other words, entropy is freedom; and the equal opportunity (rather than equality per se) that maximizes the number of options available is the second law of thermodynamics. When the number of options available to us is infinite, choice becomes random and the microstate in which we exist is our fate that is determined by God’s game of dice." (Oded Kafri & Hava Kafri, "Entropy: God's Dice Game", 2013)

"There is no such thing as randomness. No one who could detect every force operating on a pair of dice would ever play dice games, because there would never be any doubt about the outcome. The randomness, such as it is, applies to our ignorance of the possible outcomes. It doesn’t apply to the outcomes themselves. They are 100% determined and are not random in the slightest. Scientists have become so confused by this that they now imagine that things really do happen randomly, i.e. for no reason at all." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"God may not play dice with the universe, but something strange is going on with the prime numbers." (Paul Erdos)

"God plays dice with the universe, but they’re loaded dice. And the main objective of physics now is to find out by what rules were they loaded and how can we use them for our own ends." (Joseph Ford)

"I shall never believe that God plays dice with the world." (Albert Einstein)