As standardly presented in axiomatic form, set theory exhibits two fundamental features which together give rise to a number of foundational and philosophical problems. These features are so basic that they are seldom even isolated. The first is that the axioms are taken as categorical assertions (e.g. “There exists an empty set”, “There exists an infinite set”, “The power set of any set exists”, etc.). The second is that actualist quantifiers are employed in these axioms, so that they appear to be about actual objects; the domain of these quantifiers forms the fixed universe of sets (up to a choice of urelement basis), relata of a unique, fixed membership relation entering into the statement of the axioms.

Such presentations are associated with the philosophical positions of platonism and set-theoretic foundationalism.

In this paper, I want to present a family of results that may seem to add up to a new proof of the impossibility of hidden variables. In fact, I very much doubt that that’s really what really emerges, but I think the results are nonetheless interesting because they help to sharpen the discussion of Jon Jarrett’s very useful decompostion theorem, in particular, of the condition he calls locality. Jarrett (1984) and Ballentine and Jarrett (1987) have suggested that the so-called condition of locality is the one that provides the conceptual link between hidden variable theories and relativity: if locality is violated, so is relativity. On the other hand, a theory may violate the condition Jarrett calls completeness without running afoul of relativity. Now I agree with Jarrett and Ballentine about completeness, but I strongly suspect that we have quite a way to go before we really understand what would be involved in a violation of locality.

Cantor was the founder of not one but two theories: an earlier theory of point-sets and a later theory of sets. How are they related? How much more about points and sets of points can be proved with the assumption of sets of sets of points, sets of sets of sets of points, and so on, than without? The present paper is a semi-popular (nontechnical except for presupposing some familiarity with first-order logic) survey of partial answers obtained by logicians during the last decades.

Tarski (1959) has shown how classical geometry (of any number of dimensions, say for definiteness three) can be formalized in a first-order language, here to be called L0, as a first-order theory, here to be called G0. L0 has variables x,y,z, and so on for points, and predicates for a couple of geometric relations among points. G0 has a dozen axioms and one scheme.

Causal decision theory (CDT) is the best theory of rational choice now available.2 I intend to provide some support for that claim in part I of this paper by responding to two criticisms of CDT. The first criticism says that CDT is superfluous, since it does no better in the problems that matter than does evidential decision theory (EDT) at recommending correct choices. A second criticism says that CDT by itself is flawed: according to this view, there are problems in which CDT makes bad recommendations, unless it is supplemented with an additional deliberation mechanism, either involving metatickles or screening for ratifiable choices.3 I will argue in response to the first criticism that CDT is genuinely better than any EDT of the most sophisticated sort: there are problems where CDT gives better recommendations, and they are problems that do matter.

In this paper I contrast two causal explanations of the outcome of a set of laboratory experiments in population ecology conducted by Thomas Park in the 1940s and 1950s. These experiments shed light on the problem of adducing evidence for the operation of competition (see Lloyd 1987 for a recent philosophical discussion) and are central to the group selection controversy because they form the empirical base for experimental studies of group selection conducted over the last 12 years (see Griesemer and Wade 1988 for an analysis).

The experiments investigated competition in laboratory populations of two flour beetle species, Tribolium confusum and Tribolium castaneum, but the results were more complex and interesting than expected; in addition to demonstrating competition in a laboratory system, they exhibited what Park called “competitive indeterminacy.” This phenomenon was of great interest to statisticians, who devised stochastic models to describe its demographic basis (e.g., M.S. Bartlett, P.H. Leslie, and J. Neyman; see Mertz 1972 for references).

Government agencies and private risk assessors use (quasi) scientific “risk assessment” procedures to try to estimate or predict risks to human health or the environment that might result from exposure to toxic substances in order to take steps to prevent such risks from arising or to eliminate the risks if they already exist We might think of this as an aspect of “regulatory science”.

Using carcinogen risk assessment as a model I consider several aspects of risk assessment—use of epidemiology, animal bioassays, and extrapolation models in predicting risks—to show how scientific procedures and uncertainties may determine (in a way often unbeknownst to the practitioners) the public policy debates concerning the estimation of risks from carcinogens.

For one thing, actual and possible scientific uncertainties are large enough that two different researchers using exactly the same data could come to quite different conclusions; such uncertainties permit risk assessors in large measure to determine regulatory decisions even though this exceeds their authority.

The field of quantified risk assessment is a new field, only about 20 years old, and already it is considered to be in a crisis. As Funtowicz and J.R. Ravetz (1985) put it:

Risk assessments done by the Ford Foundation-Mitre Corporation and the Union of Concerned Scientists agree on the hazard probabilities and consequences associated with commercial nuclear fission, but make opposed recommendations regarding using atomic energy to generate electricity. The UCS risk analysis decided against use of the technology; the Ford-Mitre study advised in favor of it (Union of Concerned Scientists 1977; Nuclear Energy Policy Study Group 1977; Cooke 1982, p. 334).

The two studies agreed on the data but made different policy recommendations because they followed different methodological rules at the third, risk-evaluation, stage of assessment. (The first two stages are risk identification and risk estimation.) The Ford-Mitre research was based on the widely accepted Bayesian decision criterion that it is rational to choose the action with the best expected value or utility, where ‘expected value’ or ‘expected utility’ is defined as the weighted sum of all possible consequences of the action, and where the weights are given by the probability associated with each consequence.