Random Numbers

“A random sequence is a vague notion embodying the idea of a sequence in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians and depending somewhat on the uses to which the sequence is to be put.” (Derrick H Lehmer, 1951)

“Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number - there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.” (John von Neumann, "Various techniques used in connection with random digits", 1951)

"[A] sequence is random if it has every property that is shared by all infinite sequences of independent samples of random variables from the uniform distribution." (J. N. Franklin (1962)

“[…] random numbers should not be generated with a method chosen at random. Some theory should be used.” (Donald E. Knuth, “The Art of Computer Programming” Vol. II, 1968)

"The generation of random numbers is too important to be left to chance." (Robert R. Coveyou, 1969)

“What will prove altogether remarkable is that some very simple schemes to produce erratic numbers behave identically to some of the erratic aspects of natural phenomena.” (Mitchell Figenbaum, “Universal Behavior in Nonlinear Systems”, 1980)

“[In statistics] you have the fact that the concepts are not very clean. The idea of probability, of randomness, is not a clean mathematical idea. You cannot produce random numbers mathematically. They can only be produced by things like tossing dice or spinning a roulette wheel. With a formula, any formula, the number you get would be predictable and therefore not random. So as a statistician you have to rely on some conception of a world where things happen in some way at random, a conception which mathematicians don’t have.” (Lucien LeCam, [interview] 1988)

"It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means.'' (Rob C Vaughan, 1990)

"According to the narrower definition of randomness, a random sequence of events is one in which anything that can ever happen can happen next. Usually it is also understood that the probability that a given event will happen next is the same as the probability that a like event will happen at any later time. [...] According to the broader definition of randomness, a random sequence is simply one in which any one of several things can happen next, even though not necessarily anything that can ever happen can happen next." (Edward N Lorenz, "The Essence of Chaos", 1993)

“Suppose that we think of the integers lined up like dominoes. The inductive step tells us that they are close enough for each domino to knock over the next one, the base case tells us that the first domino falls over; the conclusion is that they all fall over. The fault in this analogy is that it takes time for each domino to fall and so a domino which is a long way along the line won't fall over fora long time. Mathematical implication is outside time.” (Peter J Eccles, “An Introduction to Mathematical Reasoning”, 1997)

“Sequences of random numbers also inevitably display certain regularities. […] The trouble is, just as no real die, coin, or roulette wheel is ever likely to be perfectly fair, no numerical recipe produces truly random numbers. The mere existence of a formula suggests some sort of predictability or pattern.” (Ivars Peterson, “The Jungles of Randomness: A Mathematical Safari”, 1998)

“The practical definitions of randomness - a sequence is random by virtue of how many and which statistical tests it satisfies and a sequence is random by virtue of the length of the algorithm necessary to describe it [...].” (Deborah J. Bennett, “Randomness”, 1998)