The work by J.H. Conway (1976, ch. 4), Philip Ehrlich (1987, with further references) and others on absolute continua derives much of its philosophical and mathematical interest from its instantiating a more general approach to the foundations of mathematics. This approach focuses on the role of extremality (rnaximality and minimality) assumptions as a part of the basis of mathematical theories. The idea of extremality is potentially much more important than has recently been pointed out in the literature. It is illustrated by the principle of mathematical induction, which can be thought of as an attempt to enforce the requirement of minimality on the models of elementary number theory as well as by Hilbert's Axiom of Completeness (1971), which was calculated to enforce a kind of maximality on the models of axiomatic geometry. (These models were of course intended to be a species of continua.)

Perhaps one of the chief items of pride of mathematical philosophy in the last century and a half is the insight that mathematics is the science of formal structures; as opposed to the traditional view, that ‘the proper and exclusive subject matter of mathematics is…quantity.’ Closely associated with this insight is the distinction between pure mathematics, the beneficiary of the freedom conferred by the new status, and applied mathematics (in the philosopher's rather than the mathematician's sense of the word), which has been sent into philosophical limbo, supposedly under the care of philosophy of empirical science.

These two insights allow us to avoid many embarrassments. Our postulated freedom to adore gods other than Quantity allows us to break the seemingly endless circle of pointless debate, whether or not negative and complex numbers are legitimate quantities; our assumption that mathematical truth is independent of physical truth relieves us of the worry, whether Euclidean or non-Euclidean geometries are legitimate.

What is the use of probability theory in the sciences? A common answer to this question is: probability is the conceptual tool for statistics, i.e. for the evaluation of collections of numerical data. This reply, that covers a wide range of applications may be rendered by saying that the calculus of probability is the theory for statistical practice. Under this interpretation, probability is part of scientific method; it belongs to the ‘logic’ of science, i.e. it regulates scientific inference. At most, it says something about the state of justified belief of a scientist, but nothing about the nature of his objects.

If I speak of probability as a theoretical concept, I intend to refer to a different use of probability in the sciences; an explanatory use, or - more cautiously put - a use within the framework of explanatory theories, e.g. in microphysics or evolutionary biology. Compared with the standard application to statistics, this use is exceptional and problematical.

Any plausible treatment of human knowledge must acknowledge two shapers of opinion: the psychological and the social. The formation of almost any belief, even the mere perception of familiar objects or recollection of recent events, involves the execution of a complex array of mental operations or computations. So the psychological component in belief-formation is ubiquitous. The social component, while not quite ubiquitous, is almost as crucial. A large proportion of the mature cognizer's beliefs are acquired from others; and typically the vehicles of acquisition — public language, and internal techniques for assessing the claims of others — are themselves socially or culturally acquired. Scientists, of course, are no different in these fundamental respects. They too are cognizers with human brains, and they work in professional communities. Their beliefs result from use of their psychological equipment, and from the scientific lore and methodology that is largely indebted to interpersonal training and communication.

Goldman has offered a novel conception of epistemology, or, to use his terminology, epistemics. Although the project is complex and terminologically intricate, the central point that sets it apart from traditional epistemology is his conception of the objects of epistemic evaluation. Goldman claims that the objects of epistemic evaluation include not merely the beliefs a person holds and the relation of these beliefs to the evidence, but the processes and methods that are used in arriving at the beliefs. Processes and methods are epistemically commendable, on Goldman's view, to the extent that they reliably produce true beliefs, where “truth” is defined independently of epistemic norms. It is the role of the cognitive and social sciences to identify the processes and methods that are to be evaluated; the cognitive and social sciences thus have an essential role to play in epistemology, though they can by no means replace it.

When attempting to introduce new ideas it is usually best to provide an exemplar which demonstrates, in a particular case, the usefulness of those ideas. It is also occasionally helpful to talk about why the new ideas might be useful. Given limited space, one cannot do both. I have chosen here to talk about the use of cognitive models in the philosophy of science, leaving the exhibition of their usefulness for other occasions.

Now if one is just going to talk about something rather than exhibit its potential, it is best not to be too general or abstract. After all, the objective is to convince one's audience that one has something to offer. Few people, including even philosophers, are easily convinced by generalities or abstractions. So I will begin by sketching an account of how I personally came to the conclusion that the cognitive sciences provide models that are useful to the philosopher of science.

This paper is a summary of my oral presentation at PSA 86. Since that presentation was in turn a summary of views that will appear in great detail elsewhere (Thagard forthcoming), I shall be very brief here. My aim is to outline how computational models can provide the same benefits to the philosophy of science that they currently provide to cognitive psychology. I shall begin by situating an enterprise that I call computational philosophy of science in relation to the more familiar fields of historical philosophy of science, artificial intelligence, and cognitive psychology. After some general remarks about the use of computational models, I shall describe a particular model of problem solving and learning that has been used to simulate the discovery and justification of the wave theory of sound.

Michael Faraday (1791-1867) was a productive, historically significant, figure who has served as the subject of several recent attempts to construct a cognitive model of the nature of scientific thinking (Gooding, 1985; Nersessian, 1984; Tweney, 1985a; in press). Of particular interest in the present context is the relation of such an analysis to the broader issue of the role of cognitive accounts in the philosophy of science. In accord with Giere's call for a ‘cognitive realism,’ I believe there are substantial philosophical implications that stem from one overriding characteristic of Faraday's work, namely his reliance on procedural ways of thinking. By procedural, I mean thinking which is rooted in specific activities whether concrete or abstract. As will become clear, my use of this term places me at odds with those psychologists, like Piaget, who regard concrete thought as different in kind and less powerful than formal thought. While it is indeed different in kind, it is not less powerful.

Ever since Aristotle, the philosophy of science has had two clearly delineated components: (a) conceptual foundations and (b) methodology. These rather closely correspond to what one might call the metaphysical and the epistemic bases of scientific knowledge. Both branches of philosophy of science have co-existed for a long time. If anything, methodology has usually tended to predominate over foundations.

Recently, however, methodology has come onto hard times and, since the 1960s especially, it has come in for considerable abuse. Some of that abuse is doubtless well-deserved. Much of traditional methodology made assumptions about the nature of scientific inference which simply cannot be sustained.

The problem of scientific rationality is one of the most controversial in the contemporary philosophy of science. If I am so foolhardy as to raise it again, it is because I do not feel comfortable with most present-day solutions and because I hope I may suggest a more promising, although still tentative, line of research.

In order to illustrate the particular angle from which I intend to deal with this problem, I shall make use of the words of a great cosmologist, Fred Hoyle, and I shall also refer to the current debate in cosmology. What I have to say, however, is intended to apply to science in general and not only to a single discipline or to a specific episode.

Risk assessment is one part of the multistage process of determining the acceptability of risks. Risk assessment, as I understand it, and as I think Dale Hattis is talking about it (Hattis, 1986), is the estimation of the hazards likely to be incurred by proceeding with a given course of action—adopting a new pesticide, using a sugar substitute in processing foods, employing some method of generating energy, installing a toxic or radioactive waste disposal system in a particular location. The hazards in question are generally health hazards as distinct from economic costs, although many critics of current technology assessment (or non-assessment) programs might well argue that this is too limited a conception of hazard—that there may be social and political effects of a technology's adoption that are not encompassed in calculations of biological risk or economic cost. Risk evaluation generally refers to the procedures for deciding whether to employ a particular technology, chemical, etc. i.e. whether to accept the risks involved or not.