indeterminacy   Indeterminacy    chance art   Chance Art    Radical Art     Radical Art     

Chance (What is it? Does it exist? Can we fake it?)

Is it possible to design conceptually definite processes with unpredictable outcomes? Can indeterminacy be implemented without invoking "nature", and without shifting artistic decisions to curators, performing artists, or the public itself? The obvious answer to this challenge is the use of chance procedures – a method that may be summarized as follows: (1) define a space of possibilities in explicit, mathematical terms; (2) define a probability distribution over this space; (3) draw random samples from the space, in accordance with the probability distribution.

This probabilistic art generation strategy highlights one artistic problem with relentless clarity: How to define the space of possible outcomes (and the concomitant probability distribution)? This problem is discussed in our page on chance art. The strategy also raises some slightly esoteric philosophical/physical questions: What is chance, and does it exist? For the practice of chance art, the answers to these questions are largely immaterial, but for an appreciation of its conceptual dimensions, they are indispensible.

What is chance?

The common-sense notion of chance refers to real-life unpredictability. (William Wollaston, 1722: "Chance seems to be only a term, by which we express our ignorance of the cause of any thing.") For predictions about an ongoing sequence of events that must be based on observations of an initial segment, a mathematical correlary of unpredictability can be developed: unpredictablity = the absence of regularity = the impossibility of a gambling strategy. This analysis was first proposed by Richard von Mises in 1919. It was perfected by Abraham Wald (1936/1937) and Alonzo Church (1940), criticized by Ville (1939), and saved by Per Martin-Löf (1966).

A different perspective on this matter, based on
Shannon's information theory, is due to Andrey Kolmogorov, who focussed directly on the absence of regularities in initial segments of a random sequence. Since any regularity in a sequence allows it to be encoded in a more efficient way, randomness may be equated with incompressibility. This idea was further developed by Gregory Chaitin. (Cf. Li & Vitanyi, 1993; Calude, 1994; Chaitin, 2001.)

Randomness implies various kinds of statistical uniformity – and for many practical purposes, that is all one needs from a "random" sequence. Effective criteria for statistical uniformity were first proposed by
Kermack & McKendrick (1936/1937) and Kendall & Babington Smith (1938). See Meyer (1956) for a bibliography of early work in this area. The current state of the art is the Diehard test-suite (cf. Marsaglia & Tsang, 2002).

Does it exist?

Unpredictability is often operationalized through uncontrolled physical processes, such as casting dice, tossing coins, and spinning the roulette wheel. For practical purposes, this works fine. We know, however, that events of this sort can in principle be predicted, by measuring initial conditions and applying the laws of classical mechanics. For roulette wheels this is even practically feasible (Bass, 1985). But prediction becomes increasingly difficult if we look at modern devices for random number generation, which generate fast bit streams from small-scale physical phenomena such as thermal noise (electric potential fluctuations in conducting materials) or atmospheric radio noise (cf.

Physical measurements at the quantum level are not predicted by any known theory; they are thus "random" in an unusually strong sense of that word. It is sometimes asserted that they are absolutely random, i.e., that we know that no conceivable deterministic theory could predict their outcomes. Von Neumann (1932) presented a formal proof to this effect, which was, however, based on an incorrect assumption (cf. Hermann, 1935; Bell, 1966). In the meantime, there is experimental evidence about the reality of quantum-entanglement, which implies that quantum-measurements cannot be accounted for by local hidden variables. HotBits is an online source of random numbers which uses quantum effects: radioactive decay.

Can we fake it?

An old challenge in computer science: can a deterministic computer be programmed to yield number sequences which are "random" in the mathematical sense of that word? In the strict sense demanded by Von Mises and Kolmogorov, this is obviously out of the question: the generating algorithm defines both a perfect gambling strategy and an extremely efficient compressed code. (John von Neumann, 1951: "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.") Mere statistical uniformity, on the other hand, is a difficult but not impossible challenge. Divisions between large incommensurable numbers often yield sequences with reasonable statistical properties (Knuth, 1969). Several other methods have been developed over the years; see Coddington (1996) for an overview. The current state of the art is the "Mersenne Twister" (Matsumoto & Nishimura, 1998).




Thomas A. Bass: The Eudaemonic Pie. Houghton-Mifflin, New York, 1985.

Jorge Luis Borges: The Lottery in Babylon. In: El Jard'n de senderos que se bifurcan, 1941.

J. Piaget & B. Inhelder: La Genèse de l'Idée de Hasard chez l'Enfant (Paris: PUF, 1951).

Luke Rhinehart: The Dice Man, 1971.

William Wollaston: The Religion of Nature Delineated. (1722), V. 83.

Characterizing Randomness: Complexity Theory

Cristian Calude: Information and Randomness. An Algorithmic Perspective. Berlin: Springer-Verlag, 1994.

Gregory J. Chaitin: Exploring Randomness. London: Springer-Verlag, 2001.

Alonzo Church: "On the concept of a random sequence." Bulletin of the American Mathematical Society 46 (1940), pp. 130-135.

Ming Li & Paul Vitányi: An Introduction into Kolmogorov Complexity and its Applications. New York: Springer-Verlag, 1993.

P. Martin-Löf: "The definition of random sequences." Information and Control 9 (1966), pp. 602-619

Richard von Mises: "Fundamentalsätze der Wahrscheinlichkeitsrechnung." Mathematische Zeitschrift 4 (1919), p. 1.

Richard von Mises: "Grundlagen der Wahrscheinlichkeitsrechnung." Mathematische Zeitschrift 5 (1919), p. 52.

J. Ville: Étude Critique de la Notion de Collectif. Gauthier-Villars, 1939.

Abraham Wald: "Sur la notion de collectif dans le calcul des probabilités." Comptes Rendus des Séances de l'Académie des Sciences, 202 (1936), pp. 1080-1083.

Abraham Wald: "Die Wiederspruchsfreiheit des Kollektivbegriffes der Wahrscheinlichkeitsrechnung." Ergebnisse eines mathematischen Kolloquiums 8 (1937), pp. 38-72.

Characterizing Randomness: Statistics

M.G. Kendall: "A Theory of Randomness." Biometrika, 1941.

M.G. Kendall & B. Babington Smith: "Randomness and Random Sampling Numbers." Journal of the Royal Statistical Society, Vol. 101, No. 1 (1938), pp. 147-166.

W.O. Kermack & A.G. McKendrick: "Tests for Randomness in a Series of Numerical Observations" Proceedings of the Royal Society, Edinburgh, Vol. 57 (1936-1937), Part 3, pp. 228-240.

G. Marsaglia, W.W. Tsang: "Some difficult-to-pass tests of randomness." Journal of Statistical Software, 2002

H.A. Meyer: "The generation and testing of random digits, and known sources of random digit tables. (An annotated bibliography.)" In H. A. Meyer (ed.): Symposium on Monte Carlo Methods, pp. 323-337. Wiley, 1956.

Hidden variables and the Einstein-Podolsky-Rosen effect

Cristian S. Calude: "Algorithmic Randomness, Quantum Physics, and Incompleteness."

J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 195 (1964)

J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys. 38, 447 (1966)

Arthur Fine (1982), Hidden Variables, Joint Probability, and the Bell Inequalities, Physical Review Letters 48, 5, pp. 291-294.

Grete Hermann (1935), Abhandlungen der Fries'schen Schule 6, 75.

Willem de Muynck: Foundations of quantum mechanics, an empiricist approach. Dordrecht: Kluwer Academic Publishers, 2002.

Willem de Muynck: Hidden-variables or subquantum theories

John von Neumann (1932), Mathematische Grundlagen der Quantenmechanic (Berlin: Springer)

Fergus Ray-Murray: Bibliography

[To add: TV program about this (VPRO). Victor Wentink has a tape.]

Pseudo-Random Numbers

Paul Coddington: Random Number Generators for Parallel Computers. NHSE Review, 1996.

R. Coveyou: "Random number generation is too important to be left to chance." Studies in Applied Mathematics 3 (1969), pp. 70-111.

Andrew Culver:
Software for John Cage.

Gardner, M. (1968). "On the meaning of randomness and some ways of achieving it." Scientific American, July 1968, 116-121.

Peter Hellekalek: Bibliography.

Donald Knuth: The Art of Computer Programming, Vol.2 (Seminumerical Algorithms), Ch. 3 (Random Numbers). Reading, MA: Addison-Wesley, 1969.

M. Matsumoto & T. Nishimura: "Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator" ACM Transactions on Modeling and Computer Simulation, 1998.

J. von Neumann: "Various techniques used in connection with random digits" National Bureau of Standards Applied Mathematics Series 12 (1951), 36-38. Bibliography until 1984.

   Random number tables

hardware      Hardware devices for generating random numbers

quotes      Quotes about chance

art      Chance Art