さいは投げられたのか : 確率論の応用の正当化と科学的経験の超越論性

Abstract

Probability theory is now applied in many fields of science as well as in various aspects of our everyday life. In the wake of its extraordinarily wide use, this paper raises the following questions: what it is that we apply it to reality, and how, if possible, we can justify its applications? In answering these questions, this paper identifies the distinctive features of justification for such applications which are not shared by justifications for the use of other mathematical theories. The discussion then draws some philosophical implications from the distinctive features that hold, in particular, for the use of the theory in statistical tests of significance. Although many mathematical formulations and non-mathematical interpretations of probability have been proposed in the past, nevertheless this paper confines its considerations exclusively to Kolmogorovian measure-theoretic theory of probability with limit-of-frequency interpretation, on the grounds that that theory has acquired the status of the de facto standard in scientific and ordinary uses of probability. The distinctive features mentioned above are disclosed through detailed analyses of strong law of large number and Neyman-Peasonian test of significance. And those can be summarised in the following three points. (1) Impossibility of empirical and pragmatic justification: One might argue that applications of probability to some actual events could be justified empirically by means of observations, if there are any, of stability of their relative frequency. Or it would be claimed that predictive successes of some probabilistic models could serve as pragmatic justification for their applications. But either of these sorts of justification is impossible in so far as empirical tests of its applicability or of predictions of the models should be made within the framework of one or another form of significance tests in classical statistics, as that of any scientific hypothesis is expected to be done. (2) Justification by means of randomisation: Applications of probability can only be justified by such human interventions on their objects as randomisation. Randomisation or its ilk is intended to make the objects random in the sense that they meet a sufficient condition for the applications set by the strong law of large number. (3) Unrevocability of justification and incorrigibility of possible misapplications: There still remains the possibility that randomisation would fail to make the objects random. However, once the application of probability has been justified by randomisation, no choice is left open to revoke the justification and to correct potential misapplications in response to some new empirical evidence. So once randomised, the die is cast. We are then destined to apply the probability theory to another new sphere. It is also claimed that any significance test in classical statistics is built on a stochastic trial hypothesis or ST hypothesis, as I call it, that measurement in the test is a stochastic trial or random process. The three distinctive features also hold for justification of the ST hypothesis. The ST hypothesis cannot be justified empirically and pragmatically but only by randomisation. However, once randomisation has been made, it can never be revoked on the basis of some empirical evidence even if it is false.

Wherever it is available, statistical hypothesis testing is regarded as the most reliable or the only appropriate method for an empirical test of any scientific hypothesis. Given the dominance of this testing, the distinctive features with the ST hypothesis seem to have some significant philosophical implications that can be outlined as follows. (1) A transcendental characteristic of scientific experience: The ST hypothesis is a priori in that it is not susceptible of any empirical test, although it can be justified as true by means of randomisation. It is also synthetic in that it is a factual claim, that is, a statement of measurements as physical processes rather than that of logic or mathematics. Being a priori and synthetic, it is transcendental in that it constitutes, as an indispensable assumption of statistical testing, a necessary condition for the possibility of an empirical test of any scientific hypothesis, or, in short, for that of scientific experience. In other words, experience in science should be made under a transcendental assumption. (2) Justification of the transcendental assumption by an act: Randomisation is an act taken by a scientist or scientists. The transcendental assumption or the ST hypothesis cannot be justified by any product of the act, namely, the stability of relative frequency of a particular value in repeated measurements, but rather only by the act itself, or the fact that she or they actually did it. The act is also the final justifier of any scientific hypothesis so far as statistics is concerned. (3) Incorrigibility of the uncertain assumption: The transcendental assumption becomes incorrigible by empirical evidence once justified by an act although we still recognise the possibility that it might be false. Its incorrigibility does not mean its certainty at all. Rather it is incorrigible even though it is uncertain.