My purpose in this talk is to give an overview of the rediscovery of Frege's theorem together with certain of the issues that this rediscovery has raised concerning the evaluation of Frege's logicism—the ‘old doctrine’ of my title.

The contextual definition of the cardinality operator, suggested in §63 of Grundlagen— what, after George Boolos, has come to be known as Hume's principle—asserts

where F ≈ G (the Fs and the Gs are in one-to-one correspondence) has its usual, second order, explicit definition. The importance of this principle for the derivation of Peano's second postulate (‘Every natural number has a successor’) was emphasized by Crispin Wright (1983, §xix) who presented an extended argument showing that, in the context of the system of second-order logic of Frege's Begriffsschrift, Peano's second postulate is derivable from Hume's principle.

In one version of the debate between scepticism and scientific realism, the sceptic denies and the realist affirms the ability of science to home in on the unknown truth. Germane to this debate is the familiar claim that probabilistic updating by Bayesian conditionalization will almost surely arrive at the truth so long as one does not “close the door” on a hypothesis by assigning it probability zero. In this paper, we observe, on behalf of the sceptic, that there exists a coherent agent who in fact fails on an uncountable set of possible worlds to find the truth about a simple, universal hypothesis H even though he “keeps the door open” concerning H until H is logically refuted and even though there is a trivial, logically driven method that converges to the truth about H no matter what; that nonetheless, this agent “almost surely” arrives at the truth in the usual, Bayesian sense because the uncountable set of circumstances in which he fails is assigned probability zero by the agent; and that the guarantee that the set of all circumstances in which the agent fails has probability zero hinges on the axiom of countable additivity.

In his first philosophy book, Science and Hypothesis, Poincaré gives us a picture which relates the different sciences to different kinds of hypotheses. In fact, as Michael Friedman has pointed out (Friedman 1995), Poincaré arranges this book—chapter by chapter—in terms of a hierarchy of sciences. Arithmetic is the most general of all the sciences because it is presupposed by all the others. Next comes mathematical magnitude, or the analysis of the continuum, which presupposes arithmetic; then geometry which presupposes magnitude; the principles of mechanics which presuppose geometry; and finally experimental physics which presupposes mechanics. Poincaré's basic view was that experiment in science depends on fixing other concepts first. In particular he believed at the time that our concept of space had to be fixed before we could discover truths about the objects in space.

Inferential statistical tests—such as analysis of variance, t-tests, chi-square and Wilcoxin signed ranks—now constitute a principal class of methods for the testing of scientific hypotheses. In particular, inferential statistics (when properly applied) are normally understood, for better or worse, as warranting a theoretical (typically causal) inference from an observed sample to an unobserved part of the population. These methods have been applied with great effect in domains such as population genetics, mechanics, cognitive, perceptual and social psychology, economics, and sociology, and it is not often that scientists in these fields eschew causal explanation in favor of the statement of brute correlations among properties. However, the appropriateness of a particular statistical test, and the reliability of the inference from sample to population, are subject to the application of certain statistical principles and concepts.

The realism-instrumentalism debate (RID) is often seen as the core debate in philosophy of science, yet increasing numbers of philosophers have followed Fine's (1984) lead in questioning the point of this debate. Fine's largely unchallenged rendition of RID makes it is easy to see why. RID appears doomed to stalemate. According to Fine, realists and instrumentalists are both trying to account for the string of progressive episodes in the history of science. The two sides are said to agree on what those episodes are and that they constitute progress. However, it seems that every realist story of one such episode can be matched by an instrumentalist one—and vice versa. As Fine sees it, any such story is merely an attempt to extract surplus philosophical value from the historical labors of scientists. It only adds a misleading air of inevitability to their original efforts.

Sometimes themes exist only in the eye of the beholder and emerge from the history of mathematics only with the benefit (or otherwise) of hindsight. There is however at least one issue (no doubt there are others) which was clearly present throughout the foundational debates of the early Twentieth century. It formed the core component of one of the major traditions that grew up as responses to the paradoxes, though the intuition behind it has never been fully articulated nor has it been possible to give a fully fledged mathematical articulation to that intuition. Essentially the central idea is that of ‘indefinite extensibility’ generated by an appropriate form of diagonalisation and it has its fundamental source in Richard's paradox and in particular in Poincaré's response to that paradox.

Of the four great epistemological concerns around which much of the work in the foundations of mathematics this century has concentrated viz the notions of demonstrability, definability, set or class and computability only the latter clearly falls outside the scope of the Richard type reasoning.

I shall discuss two recently suggested quantum experiments. These experiments lead to paradoxical situations. I will argue that in the framework of a particular Many Worlds Interpretation of quantum theory the paradoxes do not arise.

The first experiment is the “interaction-free measurement” (Elitzur and Vaidman 1993). The experimental group at Innsbruck headed by Prof. Zeilinger is working now on its realization. Before describing the experiment let me discuss a general question: “How do we know that there is an object in a region of space?”

The simplest case is when the object itself causes some physical changes outside the region:

• i) The object is charged, so there is a field outside. We can measure this field.

• ii) The object yields a potential outside. Aharonov and Bohm have taught us that potential even without field leads to a measurable effect.

• iii) The objects radiates photons or other particles.

Aristotle's De Partibus Animalium (PA) is, if there ever was one, a classic. It consists, at the broadest level, of four books. The first is devoted to articulating (I'm quoting its introduction) “…standards, by reference to which one will judge the manner of the demonstrations <of natural inquiry>, apart from the question of how the truth has it, whether thus or otherwise.” (639al2-15). Books II-IV, on the other hand, are introduced as attempts to provide causal explanations for the facts regarding the parts that belong to the various kinds of animals, facts systematically organized in the Historia Animalium. (646a8-12). This means that Aristotle's De Partibus consists of an introductory book on the philosophy of biological science, and three books of biological science.

Such an arrangement provides the student of this great work an opportunity to explore one of the perennial issues in the history and philosophy of science: the connection between a scientist's theory of science, and his actual scientific practice.

The aim of this paper is to suggest that Galileo's Dialogue may be fruitfully interpreted and evaluated from the viewpoints of methodological judgment and critical reasoning. This approach has significant historical precedents in the readings of Thomas Salusbury and the Port-Royal logicians. This focus on the book's philosophical dimension is not meant to exclude its scientific, historical, rhetorical, and aesthetic dimensions; indeed, such anti-reductionism is suggested by methodological judgment applied at the metalevel; the latter also suggests a criticism of the excessively rhetorical readings prevalent today, as well as viable ways for resolving the problems of hermeneutical pluralism, interpretation versus evaluation, and theory versus practice. Finally, the book's methodological judgment and critical reasoning can be shown to correspond to Galileo's intentions and self-reflections.

Philosophers of science have not paid much attention to recent developments in quantum cosmology. This fact is surprising, since quantum cosmology is replete with conceptual issues involving (e.g.) the fundamental nature of time and space, the interpretation of quantum mechanics, and the ultimate meaning of probability. One notable exception, Quentin Smith, has recently examined the Hartle-Hawking (1983) proposal. Trying to make sense of the view, he resorts to an instrumentalist picture, which treats the proposal as merely a heuristic device for the algorithm responsible for predictions. While we do not examine Smith's account here, we would like to contrast it with the model presented in this note, in which a fully realistic interpretation of quantum cosmology is developed.

Recently there has been a resurgence of interest in the de Broglie-Bohm causal interpretation of quantum mechanics.

Although the relativistic, hot big bang (RHBB) model is generally regarded as “spectacularly successful: In short, it provides a reliable accounting of the history of the Universe from about 0.01 sec after the [big] bang until today, some 15 billion years later” (Turner 1992, 1), it has recently faced serious attacks from a number of physicists who cite long lists of (they claim) critical anomalies (e.g. Arp, et.al. 1990, Burbidge, 1992, Arp and van Flandern, 1992, Lerner, 1993a, Narlikar 1993). In this paper, we shall not attempt to adjudicate on such disputes, nor do we wish to take a position on the normative criteria that should govern science. Rather, our primary concern is to highlight the tensions between the normative criteria accepted by proponents of the RHBB (or demanded of rival theories) and the actual progress of the big bang program.

The opening paragraphs of Thomas Kuhn's Structure of Scientific Revolutions noted that different kinds of scientific writings have peculiar roles (Kuhn 1962). A textbook written for pedagogical purposes in a well-established scientific field gives a view of the field quite different from the actual history—an actual history that for the most part has to be reconstructed from the writings that originated the field. The originating writings correspond to the period of founding, to revolutionary science, whereas conventionalized textbooks belong to normal science.

This is almost too well known to be rehearsed again. Certain works found a science, like Newton's Opticks or Watson and Crick's original papers on the double-helical structure of DNA; others come in their wake, not just textbooks used for beginners but all the future texts that contribute to the paradigm project embodied in the originals.

The term ‘classic’ runs a risk that all our terms of approbation face in these inflationary times, the risk, that is, of losing its ability to mark off singular works of human achievement. A “classic” today might be anything from a horse-race to a hairstyle. But it was not always so. There is some debate about the origins of the term (“of the highest class”? “books for class use”?), but it long ago came to designate those works of Greek and Latin literature that for centuries shaped the education of young Western Europeans. These books were held up not only as models of literary accomplishment but also as sources of moral and social wisdom. Though they were composed in ages long past, their resonance was still felt in the present. And so the broader sense developed of a significant literary work of the past that still in one way or another speaks to us today.

Bayesians assess the inductive support for theoretical hypotheses on the basis of two sorts of factors, one fairly objective, the other highly subjective. The objective factor consists of the likelihoods or direct inference probabilities that theoretical hypotheses specify for evidential events. This is the means by which evidence affects inductive support. The subjective factor consists of the prior probabilities assigned to the various competing hypotheses. For a Bayesian agent the prior probability of a hypothesis represents how plausible the agent considers the hypothesis to be before the impact of evidence is considered, and Bayesian agents may radically differ in their initial plausibility assessments. Bayes’ formula combines likelihoods with an agent's prior probabilities to produce the agent's posterior probability for each hypothesis. Posterior probabilities represent how plausible an agent considers the hypothesis to be after the evidence is taken into account.

Traditional philosophy of science concentrated on logic and methodology, dismissing psychology and sociology as irrelevant (except for the context of discovery). Recent movements to ‘naturalize’ philosophy of science have reinstated psychological and social factors repudiated by the tradition, but these two types of factors have mostly been addressed independently and by different authors. Most cognitive approaches (e.g., Thagard 1989a, 1992; Churchland 1989) pursue the microcognition of science in abstraction from the social, and most sociologists of science ignore, post pone, or reject cognitive explanations of science (Latour and Woolgar 1986, 280). More recently, however, several philosophers of science have urged a blending of the cognitive and the social (Thagard 1993, in press, Solomon 1992, 1994a, 1994b, Kitcher 1993, Giere 1988), and this paper has a similar thrust. I shall not endorse a reduction of the social to the psychological, but a substantial overlap or intermingling of the two.

We discuss two general issues concerning diverging sets of Bayesian (conditional) probabilities—divergence of “posteriors”—that can result with increasing evidence. Consider a set of probabilities typically, but not always, based on a set of Bayesian “priors.” Incorporating sets of probabilities, rather than relying on a single probability, is a useful way to provide a rigorous mathematical framework for studying sensitivity and robustness in Classical and Bayesian inference. See: Berger (1984, 1985, 1990); Lavine (1991); Huber and Strassen (1973); Walley (1991); and Wasserman and Kadane (1990). Also, sets of probabilities arise in group decision problems. See: Levi (1982); and Seidenfeld, Kadane, and Schervish (1989). Third, sets of probabilities are one consequence of weakening traditional axioms for uncertainty. See: Good (1952); Smith (1961); Kyburg (1961); Levi (1974); Fishburn (1986); Seidenfeld, Schervish, and Kadane (1990); and Walley (1991).

Philosophers are particularly susceptible to the temptation to produce what Donna Haraway calls a “TOE (=a Theory of Everything)” (Haraway 1994). Many of us like to integrate the methods and results of different disciplines, discern underlying unity in bifurcations and thereby—so we think—get closer to a single, true, representation of the universe. Proposals to integrate cognitive and social models of scientific change have come from philosophers—and not from historians, sociologists, feminist critics, anthropologists, rhetoricians or semioticians of science. The proposals are also responses to the growing influence of sociologists of science (e.g. Collins, Latour, Shapin, Woolgar) who have taken stands on epistemological matters, formerly the province of philosophers. It is common among sociologists of science to reject all proposals to integrate cognitive and social accounts, as well as to reject all cognitive accounts: Bruno Latour's famous moratorium on cognitive studies of science (Latour 1987, 247) is one example of this.

“Maximizing expected utility (MEU)” is one assumption of (strict) Bayesian decision theory [Savage 1972]. According to the principle of MEU, in a given decision situation, the decision maker should choose one of the alternatives with maximal expected utility [For an excellent discussion of decision theory, see Jeffrey 1990]. However, MEU as the foundation of Bayesian decision theory has been under attack. One counterexample that seems to dispute MEU as a rationality principle is offered by Daniel Ellsberg[Ellsberg 1961]. While discussing what has been dubbed as the Ellsberg paradox, I will consider briefly three decision principles, each of which is different from the MEU principle.

Following a period during which philosophers of science focused on theory to the near exclusion of experiment, a number of philosophers, historians and sociologists of science have, in one way or another, turned their attention to experimentation, instrumentation, and laboratory practices. Considerable work in philosophy of science of the last decade reflects this surge of interest in experiment, as promoted by Ackermann, Cartwright, Franklin, Galison, Giere, Hacking and others. Where has this movement taken us and where do we still have to go?

In asking this question, my focus is on that subset of the experimentalist movement whose members, following Ackermann (1989), I dub the “New Experimentalists”. Although their agendas differ, members of this group share the core thesis that aspects of experiment might offer an important, though largely untapped, resource for addressing key problems in philosophy of science.

In 1979, Dr. J. Robin Warren of the Royal Perth Hospital in Australia discovered bacteria in the biopsies of stomach tissue taken from patients with digestive complaints. His colleague Dr. Barry Marshall followed up on Warren's work and found the bacteria in many patients with stomach inflammation and peptic ulcers (Marshall and Warren 1984). When Marshall claimed at a medical conference that bacteria are the principal cause of peptic ulcers, his remarks were rejected as preposterous. It was widely believed that the human stomach's caustic gastric juices made it too antiseptic for bacteria to survive for long. Moreover, alternative explanations of the principal cause of ulcers were available, focusing on excess acidity and emotional stress. Stung by rejection of his theory and failure of animal experiments, Marshall resorted in 1984 to drinking the bacteria himself, and underwent endoscopy and biopsy to show that his stomach had indeed become inflamed (Monmaney 1993).