RANDOMNESS, INDEPENDENCE, AND HYPOTHESES
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The last decade has seen a renewed interest in the problem of randomness. Two widely different approaches have emerged-the statistical tests criterion, and the complexity criterion. It is evident that these fresh looks necessitate a revision in the way that randomness is viewed; away from an absolute idea of randomness, and towards relativised concepts. The reasons provided by these two new approaches will be discussed here, together with some separate arguments for the same point of view. In particular, the role played by independence, traditionally a keystone of randomness, will be diminished, and the independence situation seen to be a special case of random behaviour.
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There seem to be two principal reasons for having an interest in randomness. The first approach regards randomness as interesting because probability theory is presumed to be the theory of random phenomena. This might be termed the domain approach. Alternatively, one's principal interest can be in arriving at an adequate definition of disorder, unpredictability, complexity, and related notions. A solution to problems in this area would give a better understanding of what it means for a sequence of events to occur 'at random'. Any interest along these lines will be called the irregularity approach. One c~m pursue either of these interests with small regard for the other, or, as yon Mises did, regard the two areas as coextensive. I shall argue below that by keeping the domain approach as the primary one, a satisfactory solution to the irregularity problem may be arrived at.
The principal claims to be argued here are these. Firstly, that there is no single absolute concept of randomness, but that each assignment of randomness has to be made relative to a particular probabilistic hypothesis or reference class. The contrary belief is perhaps more implicit than explicit, and is a legacy of the early attempts to define a chance-like sequence by yon Mises, Popper, Church, and others. For an excellent survey of these efforts, see [15]. These earlier attempts might be characterised as 'hypothesis-flee ', in Synthese 36 (1977) 415-426. All Rights Reserved.