What I have to say falls under four headings: What is unity of science, unity and reductionism, the search for certainty, and the search for completeness.

To answer this initial question, I turned to the introductory essay by Otto Neurath for Volume 1, Part 1, of the International Encyclopedia of Unified Science. He begins this way:

This contribution gives an outline of a game theoretic foundation of the logical structure inherent to the language of a science. Game theoretic approaches to a language were considered and developed by several authors. In this volume Saarinen examines the game theoretic semantics due to Hintikka and the dialog-game semantics due to Lorenzen. In the following I shall not re-examine the approaches by Hintikka and Lorenzen. However, some remarks about Lorenzen’s semantics are necessary since the game theoretic approach considered here is essentially based on the idea of Lorenzen to use dialog-games for a foundation of logic.

A systematic game theoretic presentation of the rules of a dialog game was given at first by Lorenz. Starting with certain structural rules which constitute the general scheme of a dialog, a further dialog rule is necessary in order to guarantee a finite game.

In this paper I shall attempt to compare the dialogue approach originally advocated by Lorenz and Lorenzen and the game-theoretical approach of Hintikka with each other. I shall not try to present any survey of either one of the approaches and will assume that the reader is familiar with the basic ideas of these theories.

The original works of Lorenzen and Lorenz have been reprinted in their while Stegmüller contains a survey (in English) of the basic ideas and results. Works in this tradition which apply the approach to quantum logic include Denecke, Mittelstaedt Mittelstaedt and Stachow, Stachow. The reader will also find van Dunn, Krabbe, Lorenz and Stachow to be relevant.

The literature on Hintikka’s game-theoretical semantics as applied both to different formal languages (including infinitary languages and intensional logic) and to natural language is rapidly growing. As for the latter, the reader is referred to papers by Hintikka, Carlson and myself reprinted in Saarinen, to Carlson and ter Meulen, and to Saarinen.

Early modern science, according to a misleading and now widely accepted thesis, introduced a split or schism between a world of colorless, imperceptible particles, on the one hand, and the familiar world of perception, on the other. One of the most important dilemmas of modern philosophy, of course, seems to follow directly from this alleged rupture: For how are the two seemingly incongruous worlds to be “reconciled”? This way of formulating the problem, however, seems to be based on a misunderstanding of early modern science, on an interpretation that underestimates the very real difficulties that scientists faced in putting forward hypotheses about unseen minute bodies. If we start to pay attention to the way in which knowledge claims about the properties of atoms or corpuscles were advanced in the sixteenth and seventeenth centuries, in fact, it becomes increasingly obvious that such fundamental properties were first attributed to minute bodies on grounds of strict analogy, that is, on the grounds that all perceptible bodies exhibited precisely the properties of size, shape, motion, and so forth.

Kant's Metaphysical Foundations of Natural Science is undoubtedly the most neglected and misunderstood of all his major works. It should, I believe, be interpreted as an attempted rational reconstruction of the mechanics of Newton and Euler. What Kant attempted to reconstruct, however, were the conceptual foundations of science rather than the logical structure. The reasons for this as well as the particular way in which it was done can best be understood in the light of Kant's own philosophical development. In summarizing this I will focus on two interrelated issues: Kant's concept of matter, which changed somewhat in the course of his long development; and his interpretation of scientific explanation, which changed considerably.

Kant's earliest concerns with scientific explanation involved three aspects of interest here: Christian Wolff's (1679-1754) systematization of philosophy, which served as his point of departure; the ideas on scientific methodology, which he derived from Isaac Newton; and Kant's own work in science.

The concept of randomness as applied to number sequences is important to the study of the relationship between the foundations of mathematics and physics. A reason is that while randomness is often defined in mathematical-logical terms, the only way one has to generate random number sequences is by means of repetitive physical processes. This paper will examine the question: What definition of randomness is correct in the sense of being the weakest allowable? Why this question is so important will become clear during the course of the discussion.

The main body of this paper is divided into three sections. Section 1. discusses the use of probability theory to describe various statistical processes and some of the alternative definitions of randomness that have been proposed.

In Section 2. a criterion which a definition of randomness should satisfy is proposed.

In previous technical work ([1] and [2]) on which his present paper [3] draws, Benioff has presented results conforming with the following argument-scheme:

How interesting is this? One can approach the matter on two levels, the level of the specific technical results, and the level of more general implications of this pattern of argument for understanding connections between mathematics and physics. The second more general level should be our primary focus, but it will pay at the outset to look briefly at the technical level.

Any discussion of probability theory in early nineteenth-century Britain requires a specially vigorous exercise of the historical imagination. A century of subsequent thought about the foundations of the subject has made both historians and philosophers of science only too aware of how elusive and polyvalent a term ‘probability’ is; indeed, since Carnap, it has scarcely been possible to use the word without a qualifying adjective or subscript. Before the nineteenth century, on the other hand, during the ‘emergence’ of formalized probabilistic thinking, writers from Pascal and Fermat onwards were keenly conscious that the concept they used was often ill-defined and equivocally employed.

There is a sensation, therefore, of sailing unexpectedly into a patch of flat calm when one turns to such early Victorian writers (I use the term in its common, extended sense to refer to the period from around 1830 till the middle of the century) as Lubbock, Drinkwater-Bethune, Galloway, and, above all, De Morgan.

The conceptually puzzling features of quantum mechanics as a statistical theory all have their source in the impossibility of relating the probability assignments defined by the quantum state to distributions over determinate values of the physical magnitudes. While some interpretations of quantum mechanics, as well as “hidden variable” modifications of the theory, have proposed several constructions for assigning values to magnitudes (beyond the value assignments defined by the quantum state alone), a variety of theorems exist which impose severe restrictions on such theoretical constructions.

This note will deal with Bell's proof (in [2]), that any hidden variable theory satisfying a physically reasonable locality condition is characterized by an inequality which is inconsistent with the quantum statistics. Bell's result has initiated a series of experiments designed to refute the entire class of local hidden variable theories. We show that Bell's inequality actually characterizes a feature of hidden variable theories which is much weaker than locality in the sense considered physically motivated.

The topic of this session is “physical randomness”. It might be doubted whether such a subject exists, for definitions of randomness have hitherto almost all been mathematical in nature. The only exceptions of which I am aware are the preceding paper by Benioff and a paper by Wesley Salmon. These attempts to inject some empirical content into randomness are highly desirable. But anyone attempting to formulate a physically based definition of randomness should at some point make clear what the connection is (if any) with a more traditional notion of disorder - that of indeterminism. Repeated reference to quantum mechanical examples whenever physical randomness is discussed indicates that a primary motivation for considering physical randomness to be important is because of the current belief that data sequences associated with quantum mechanical experiments are irreducibly random. (As an indication of this, in any situation in which physical randomness is discussed, translate the remarks about quantum phenomena into remarks about coin tossing, and they will lose much of their interest.)

Position probabilities play a privileged role in the interpretation of quantum mechanics. The quantum wave function (usually designated “Ψ(r)’) cannot itself represent a physical quantity, because it is complex valued. Its square — |Ψ(r)|2 — however, is real; and it is |Ψ(r)|2 which is the fundamental interpreted quantity of the theory. Schrödinger, who originated wave mechanics, interpreted |Ψ(r)|2 as the density of a charged particle; but this suggestion was inadequate. A principal reason is that, for most physical systems, |Ψ(r)|2 spreads out in time, whereas particles themselves do not. Schrödinger's interpretation was replaced by Born who introduced probabilities into quantum mechanics. Born urged that |Ψ(r)|2 represents not the density of the system in space, but rather a probability density. This is almost universally how quantum mechanics is taught today: |Ψ(r)|2 gives the probability that a system in state Ψ(r) is located at r.

All biologists use classifications to one degree or another, and those of us who work on classifications use the results of all other biologists to one degree or another, so you might reasonably expect that biologists in general would share some common conception of how classifications should be constructed and how they can be used. Certainly one might expect that all taxonomists, at least, would share such a perspective. But this is not the case; in fact, the theory of taxonomy is at present a very controversial subject, with much of the controversy revolving around the question of how to go about constructing an ideal classification.

Surprisingly, there is relatively little controversy about what properties an ideal classification would have. Admittedly, different terms are used to describe these properties, various workers arguing that classifications should be maximally stable, or maximally useful, or maximally informative, or maximally testable, or maximally refutable, or maximally predictive, and so on.

By a kinematical theory we mean a theory of space, time, and light signals as measured by rigid rods and clocks. In this paper we shall construct an axiomatization of possible kinematical theories which has two especially interesting properties: it is universal in the sense that it presents a threefold continuum of kinematical theories ranging from Special Relativity (STR) on one extreme to Classical Aether Theory on the other; and it is synchronization independent, that is, it does not assume any particular synchronization for clocks. Reichenbach's axiomatization has the threefold continuum implicitly in it, but he does not explicitly present it as a universal axiomatization and he does not give it a synchrony free formulation. On the other hand, the synchrony free axiomatization of STR by Winnie is not universal and it masks an important distinction within the light principle. However, both of these axiomatizations were central to developing the present axiomatization.

In recent years two groups of taxonomists have attempted to influence the general goals and methods of biological classification. The first group, which emerged in the late 1950’s, has been called variously neo-Adansonian, numerical, computer and phenetic taxonomy. The founders of this school, Robert R. Sokal and P.H.A. Sneath, termed their unified approach to systematics “neo-Adansonian” because of the affinities which they saw between their views and those of the 18th century botanist, Michel Adanson (1727-1806). Today little mention is made of Adanson. His ideas turned out not to be as prescient as Sokal and Sneath had at first thought. Besides, he had served his purpose as a patron saint ([79], p. 23). Sokal and Sneath also termed their approach “numerical” because they believed that taxonomists should make greater use of available mathematical techniques. Not only should taxonomic characters be coded quantitatively but also estimates of affinity should be made on the basis of explicitly stated clustering procedures.

Accounts of ancient mathematical astronomy tend to err in either of two directions as regards the relationship between theory and observation. On the one hand, there are overly positivistic accounts according to which it is assumed that all theoretical progress made by the ancients was rendered possible by the increasing bulk and accuracy of observational data. On the other hand, there are those authors who, whenever they are confronted with a discrepancy between the observational report of an ancient and a modern result, are only too quick to conclude that the alleged observations were fudged (or even manufactured out of thin air) in order to support certain pre-existing theoretical commitments. It would appear from the case studies cited below that the truth about the relationship between observational data and theoretical presuppositions is somewhat more complicated than either of these two extreme accounts would allow.

Classical statistical mechanics has commanded a modest but steady amount of attention from philosophers of science. By contrast, there has been an almost total neglect of relativistic statistical mechanics, or more precisely, a neglect of the prospects and problems of producing a relativistic version of classical statistical mechanics. The neglect is undeserved, for this area offers a fascinating array of case studies for those concerned with the history and sociology of science, with the structure and dynamics of scientific theories, or with foundations problems in physics. This paper is dedicated to the goal of ending the neglect. Towards this end, I will survey some of the issues which arise in attempting to marry statistical-thermodynamics with relativity theory. The choice of the issues to be discussed and their treatment naturally reflect my own preferences and prejudices, and I cannot hope for the reader’s agreement on all points.