¹ Popper remarks of the problem raised by Compton there that it is the only form of the problem of determinism worth discussing seriously. He characterizes it as ‘the problem which arises from a physical theory which describes the world as physically complete or a. physically closed system’. He goes on, ‘By a physically closed system I mean a set or system of physical entities, such as atoms or elementary particles or physical forces or fields of forces, which interact with each other—and only with each other—in accordance with definite laws of interaction that do not leave any room for interaction with, or interference by, anything outside that closed set of physical entities. It is this closure of the system that creates the deterministic nightmare.’ (OK, p. 219. The footnotes to this passage are omitted; the italics are Popper's).Google Scholar

² ‘Epistemology Without a Knowing Subject’, OK, pp. 107–152 and ‘On the Theory of the Objective Mind’, OK, pp. 153–190. Popper undoubtedly accepted the reality of abstract objects, like numbers and sets for example but his account of them and of how we come to know them was certainly neither Fregean nor constructivist. World Three contains theories which are presumably to be thought of as free creations of the human intellect. But it also contains abstract objects like number which are not as Frege thought to be conceived of as logical objects nor are they to be conceived of as entirely free creations.Google Scholar

³ These arguments have been effectively criticized by Urbach, Peter. See his ‘Is Any of Popper's Arguments Against Historicism Valid?, British Journal for the Philosophy of Science, 29 (1978), pp. 117–130.CrossRefGoogle Scholar

⁴ The quotations from Laplace occur in the famous passage from his A Philosophical Essay on Probabilities (1820), p. 4; Dover reprint, 1951.Google Scholar

⁵ Indeed there are some unequivocal examples of the failure of recursiveness in classical physics, see particularly Pour-El, Marian and Richards, Ian, ‘Non-Computability in Analysis and Physics’, Advances in Mathematics, 48 (1983), pp. 44–74.CrossRefGoogle Scholar

⁶ Earman's excellent A Primer on Determinism (Dordrecht: D. Reidel, 1986) is by far the best treatment of determinism and its relation to physical theory in the literature. Pollard's example mentioned above is noted by Earman, Ibid, p. 54. It is given as an exercise in Pollard, 's Mathematical Introduction to Celestial Mechanics (Carus Mathematical Monographs, Vol. 18, 1976), p. 59, Exercise 1.3.Google Scholar

⁷ It is important to recognize that long before chaos theory was fashionable in philosophical circles Popper drew attention to a very important property of some deterministic systems, namely, how their large scale macroscopic states were so finely dependent on initial conditions. It is an open question of considerable difficulty as to what extent, if any, this instability can be used to mimic indeterministic, or even much more strongly, random behaviour. Popper's ideas on the subject, based on classic papers by Hadamard and Hopf, are summarized in OU, pp. 39–40 and QTSP, pp. 104–116.Google Scholar

⁸ Popper is surely right in his contention (OK, p. 220) that it is not much use to appeal to well-worn maxims such as ‘Every event has a cause’ and ‘Like causes produce like effects’, when trying to formulate the intuitive notion of determinism more formally. This is so first and foremost because the notion of cause plays no role in either the formulation or the content of physical theory, and secondly because the notion of cause is so notoriously vague and context-dependent that it is quite unsuitable to employ it to capture the general notion of determinism. Neither would it be sensible to characterize deterministic theories as those which make no reference (or merely trivial reference) to probability, since a physical theory may well involve a parameter defined using a probability measure which itself evolves deterministically with time. The problem with formulating the thesis of determinism is the obvious difficulty of trying to steer a course between triviality on the one hand and far too strong a claim on the other. Thus Russell and Hempel have with great clarity established the charge of triviality against some syntactic formulations of the condition employing merely the existence of functional dependencies between the time and evolving states of a unique Universe (Russell ‘On the Notion of Cause’, Mysticism and Logic (London: Allen & Unwin, 1918), pp. 132–151.), and involving conditions sufficient for an event to occur (Hempel, ‘Some Reflections on the “The Case for Determinism”’, in Hook, S. (ed.), Determinism and Freedom, (London: Macmillan, 1958), pp. 170–175.)Google Scholar

⁹ Montague, Richard, ‘Deterministic Theories’, in Thomason, R. H. (ed.), Formal Philosophy (New Haven: Yale University Press, 1974).Google Scholar See also van Fraassen, B., ‘A Formal Approach to the Philosophy of Science’, in Colodny, R. G. (ed.), Paradigms and Paradoxes, (Pittsburgh: University of Pittsburgh Press, 1972), pp. 306–366.Google Scholar Montague's definition is not itself without some formal difficulties, see especially Hellman, G., ‘Randomness and Reality’, PSA, 2 (1978), pp. 79–97.Google Scholar

¹⁰ Earman, Primer on Determinism, chapter IV.Google Scholar

¹¹ An extensive account of the propensity theory can be found in Popper's RAS, Part II.Google Scholar

¹² ‘A note on Popper, Propensities and the Two Slit Experiment’, British Journal for the Philosophy of Science, 36 (1987), pp. 66–70.Google Scholar