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.

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.

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.)

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.

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.

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.

It is without doubt that the earth sciences were rocked to the core by a revolution in the late sixties with the initial acceptance of Harry Hess’s hypothesis of seafloor spreading and subsequent development and acceptance of plate tectonics. The major aim of this essay is to show precisely why this revolution is not Kuhnian. However, my reasons for claiming that Kuhn’s model fails to apply are at variance with Ruse [27] but in much agreement with R. Laudan [18]. I shall highlight our differences as space permits.

There are two general accounts of the rise of continental drift and plate tectonics which satisfy Kuhn’s model of scientific growth and change. (1) The geosciences prior to the acceptance of seafloor spreading and plate tectonics were in a pre-paradigm state. Earth scientists solved no problems within geological domains, but only argued among themselves about which of the various schools of thought provided the best framework for attacking geological problems.

Our view of the first half of the 20th century has been influenced by what we suppose to have occurred in the middle of that century. It is by now part of the conventional wisdom of the geological community that during the 1950’s and 1960’s a revolution occurred. It is further supposed by many, that before the revolution there was among geologists an uneasiness resulting from the lack of an organizing principle in terms of which accumulating facts could be understood. It is difficult to see our own times in the kind of perspective that historians consider so important an ingredient of ‘good’ history. Geologists now only in their middle years began their careers before the ‘new tectonics’ came to pervade their discipline. They do not look back upon prerevolutionary days as a time waiting to be saved from crisis nor do they see themselves as having collected facts in anticipation of being able to weave them into some pattern on the basis of some as yet undiscovered truth.

The 1960s witnessed a striking change in geology. Since at least the seventeenth century, one of the central problems of the subject had been the origin of the major irregularities of the surface of the globe—continents and oceans, mountain chains and ocean islands—irregularities that were not anticipated by most physical theories. Traditionally these features had usually been explained either as residual traces of events occurring during the very early history of the globe, or as the result of vertical movements of the earth’s crust, caused, for example, by changes in the heat budget. The last two decades have seen an end to all this. The vast majority of geologists now believe that these irregularities largely result from the lateral movement of thin rigid plates covering the earth, a theory now known as “plate tectonics”, but a theory which also has obvious parallels with the hypothesis of continental drift, in which it was postulated that continents can move laterally.

The one thing upon which we can all agree is that just over ten years ago a major revolution occurred in the science of geology. Geologists switched from accepting a static earth-picture, to endorsing a vision of an earth with its surface constantly in motion. (Cox [4]; Hallam [12]; Marvin [28]; Wilson [56]). It is true that early in this century the German geologist Alfred Wegener argued that the continents as we today find them have “drifted” to their positions from other positions widely different. However, other than amongst a number of scientists drawn almost exclusively from the Southern Hemisphere, his ideas fell on deaf — or more precisely, contemptuous — ears. Then in the mid 1960’s, almost literally overnight, the geological community swung around and embraced the hypothesis of continental drift, or what we shall see is perhaps more accurately called “plate tectonics”.

I became acquainted with Lakatos’s work in 1965 when I started studying at London School of Economics—where Lakatos taught. As his work was developed over the succeeding years until his death in 1974, one thing always puzzled me: his work seemed to contain such conflicting tendencies. He would continue developing his ideas along a progressive line, and suddenly would insert an element which appeared to me quite reactionary. By ‘reactionary’, I should hasten to add, I mean imbued with the spirit of Positivism—a person of different bias might reverse the labels!

When I was given this opportunity to reflect again on Lakatos’s work I did not try to resolve the puzzle by looking into Lakatos’s intellectual history1; rather I attempted to separate clearly those aspects of Lakatos’s work which seemed to me driving in the right and the wrong directions. Both aspects concern the research program which has dominated both philosophy of science and philosophy of mathematics in the twentieth century.

Mathematics offers us a puzzling contrast. On the one hand it is supposed to be the paradigm of certain and final knowledge: not fixed to be sure, but a steadily accumulating coherent body of truths obtained by successive deduction from the most evident truths. By the intricate combination and recombination of elementary steps one is led incontrovertibly from what is trivial and unremarkable to what can be non-trivial and surprising.

On the other hand, the actual development of mathematics reveals a history full of controversy, confusion and even error, marked by periodic reassessments and occasional upheavals. The mathematician at work relies on surprisingly vague intuitions and proceeds by fumbling fits and starts with all too frequent reversals. In this picture the actual historical and individual processes of mathematical discovery appear haphazard and illogical.

The first view is of course the currently conventional one which descends from the classic work of Euclid.

I once wrote to Imré Lakatos that if he had not already established himself as an accomplished philosopher and provocateur, he could have made a successful career as a Hollywood script writer. I had in mind, at that time, a beautifully written paper he had constructed by pasting together the cuttings from other papers. I had forgotten that his reputation as a dramatist was already established, with the production of his Proofs and Refutations [9]. For that work, it seems to me, is simply a play within a play. The main character, whose part is written in the footnotes, delivers a monologue which is a commentary on the drama, between teacher and students, that occupies center stage. This main character is not named in the play, so for ease of reference I shall call him “Imré”. What Imré points out is that the drama on center stage imitates life, in showing us the essential features of the methodology by virtue of which mathematical science grows.

Theories of chemistry have come to depend inescapably upon the framework provided by atomic physics. This is a dependency which is anchored upon the ability within quantum theory — in particular since the wave mechanics of Schrödinger — to account for chemical bonding by the pairing of electrons, the stability of their resulting structures, and thereby to provide a basis from which to deal with chemical behavior generally. The optimism generated fifty years ago by the initial attempts to apply this framework1 has bolstered the supposition, within the sciences, that chemistry is exclusively an experimental science, it has no theory except what it borrows from atomic physics. It seems to have established, within philosophy, so secure an example of intertheoretic reduction that however often we hear it cited, it remains a case rarely, if ever, examined.