The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) survey of the major, traditional philosophies of mathematics indicating how each is prepared to deal with the present problem. It is shown that (the standard formulations of) some views seem to deny outright that there is a relationship between mathematics and any non-mathematical reality; such philosophies are clearly unacceptable. Other views leave the relationship rather mysterious and, thus, are incomplete at best. The final, more speculative section provides the direction of a positive account. A structuralist philosophy of mathematics is outlined and it is proposed that mathematics applies to reality though the discovery of mathematical structures underlying the non-mathematical universe.

It is not easy to assess the extent to which decision theory constitutes a contribution to philosophy. On the one hand, the enthusiasts seem tirelessly to churn out highly technical Bayesian reconstructions of scientific practice. On the other hand, their opponents dismiss the result as a body of theory that is overidealized and falsely precise, bearing no important relation to the actual conduct of inquiry. And most philosophers seem happy to ignore the debate altogether and pursue a theory of inquiry as best they can. I take the opponents’ criticism seriously, but I am still an enthusiast. The aim of this paper is to explain why. The paper consists of an accessible argument for a Bayesian decision theory designed to enable any interested philosopher to see how the insights of decision theory provide, without idealization or false precision, a fundamental (and somewhat revolutionary) contribution to our understanding of the enterprise of rational inquiry.

One of the most interesting programs in the foundations of quantum mechanics is the realist quantum logic approach associated with Putnam, Bub, Demopoulos and Friedman (and which is the focus of my own research.) I believe that realist quantum logic is our best hope for making sense of quantum mechanics, but I have come to suspect that the usual version may not be the correct one. In this paper, I would like to say why and to propose an alternative.

Criteria are given to characterize mature theories in contradistinction to developing theories. We lean heavily on the physical sciences. An established theory is defined as a mature one with known validity limits. The approximate truth of such theories is thereby given a quantitative character. Superseding theories do not falsify established theories because the latter are protected by their validity limits. This view of scientific realism leads to ontological levels and cumulativity of knowledge. It is applied to a defense of realism against recent attacks by Laudan.

Ronald Giere's analysis of causal effectiveness in populations involves the comparison of two hypothetical populations, one in which every individual has the suspected causal factor, and the other in which none do. Elliott Sober has argued that in cases where causal effectiveness depends on relative population sizes, Giere's analysis breaks down. I take issue with this claim, and argue to the contrary that Giere's analysis can help provide insight into these cases.

Scientific arguments often attribute the credit or blame for particular experimental or observational results to fragments of theories rather than to the whole of them. The thesis of Theory and Evidence (Glymour 1980) is that there are structural criteria that account for at least part of the distribution of credit and blame; the book also attempts a tentative formal characterization of one structural criterion. David Christensen (1983) has produced a series of counterexamples which show that the formal characterization is untenable. The question remains whether the failure is due to an oversight in the formal theory or to the falsity of the very idea that there are structural criteria for evidential relevance. Since the counterexamples are uniformly eliminated by a single formal principle, I incline to the first answer.

In (1981), Levi has laid out the issues involving chances, frequencies, and direct inference with admirable precision. Nevertheless, puzzles remain. The chief puzzle to which I wish to draw attention is this: Under certain circumstances, we can combine knowledge of chances and knowledge of frequencies to yield new knowledge of chances. If Petersen is “drawn at random” from among Swedes, and we also know that the proportion of Protestants among Swedes is 0.9, then we can say that the chance that Petersen is a Protestant is 0.9. But if we apply this principle generally, we are led to generally trivial results: direct inference yields probabilities constrained only to lie in a broad interval like [0, 1]. On the other hand, if we can't always combine knowledge of chances and knowledge of frequencies to get new knowledge of chances, or if this knowledge can be overridden by other considerations, how do we know when we can usefully apply direct inference?

To ground judgments of credal probability on knowledge of chance via direct inference, one should appeal to the information about chances available relative to the most specific description known to be true of the trial event.

Thus, to obtain a judgment of credal probability concerning the hypothesis that coin a landed heads at t given that it is known that at t it is known that a was tossed by Levi in 728 Philosophy Hall, the pertinent knowledge of chances concerns the chances of coin a landing heads on a toss of coin a which is (i) a toss at t, (ii) a toss in 728 Philosophy Hall and (iii) a toss by Levi. In many contexts, appeal is made to knowledge of the chance of coin a landing heads on a toss. This appeal, when legitimate, takes for granted that the additional information about the specific tossing listed in (i), (ii) and (iii) is stochastically irrelevant to landing heads and, hence, may be ignored. Such an assumption is a substantive presupposition about chances and not an a priori, self evident or pragmatic principle of methodology.

Philosophy of science may indeed perform a valuable normative function through historical analyses of characteristic errors, as Bechtel (1982) argues. But considerations of the two types of errors elected for this role, and harder looks at some of the historical examples prosecuted for these errors, disclose few helpful hints beyond the wisdom provided in Chapter I of freshman science texts coupled to a clear view of the future. To avoid churlishly spurning at first glance this offer of assistance, however, let us look in turn at the two candidates with their supporting examples.

Rosenberg (1983), in his comments on our article (Sober and Lewontin 1982) concerning the units of selection controversy, has matters precisely backwards. He chides us for conflating a causal theory with a mathematical model, whereas this is precisely the error we argued is at the core of the position defended by Williams (1966) and Dawkins (1976). Their idea—genic selectionism—confuses the computational adequacy of a mathematical model with a substantive causal theory. The main point of our paper was simply this: the fact that evolution by natural selection can be “represented” in terms of selection coefficients that attach to single genes does not imply that selection is always selection for or against the possession of those single genes.