C.A. Hooker - Böcker

To mathematicians, mathematics is a happy game, to scientists a mere tool and to philosophers a Platonic mystery - or so the caricature runs. The caricature reflects the alleged 'cultural gap' between the disciplines­ a gap for which there too often has been, sadly, sound historical evidence. In many minds the lack of communication between philosophy and the exact disciplines is especially prominent. Yet in the past there was no separation - exact knowledge, covering both scientists and mathemati­ cians, was known as natural philosophy and the business of providing a critical view of the nature of reality and an accurate mathematical de­ scription of it constituted a single task from the glorious tradition begun by the early Greek philosophers even up until Newton's day (but I am thinking of Descartes and Leibniz I). The lack of communication between these professional groups has been particularly unfortunate, for the past half century has seen the most ex­ citing developments in mathematical physics since Newton. These devel­ opments hinged on the introduction of vast new reaches of mathematics into physics (non-Euclidean geometries, covariant formulations, non­ commutative algebras, functional analysis and so on) and conversely have challenged mathematicians to develop the appropriate mathematical fields. Equally, these developments have posed profound philosophical problems to do with the rejection of traditional conceptions concerning the nature of physical reality and physical theorising.

The twentieth century has witnessed a striking transformation in the un­ derstanding of the theories of mathematical physics. There has emerged clearly the idea that physical theories are significantly characterized by their abstract mathematical structure. This is in opposition to the tradi­ tional opinion that one should look to the specific applications of a theory in order to understand it. One might with reason now espouse the view that to understand the deeper character of a theory one must know its abstract structure and understand the significance of that struc­ ture, while to understand how a theory might be modified in light of its experimental inadequacies one must be intimately acquainted with how it is applied. Quantum theory itself has gone through a development this century which illustrates strikingly the shifting perspective. From a collection of intuitive physical maneuvers under Bohr, through a formative stage in which the mathematical framework was bifurcated (between Schrödinger and Heisenberg) to an elegant culmination in von Neumann's Hilbert space formulation the elementary theory moved, flanked even at the later stage by the ill-understood formalisms for the relativistic version and for the field-theoretic altemative; after that we have a gradual, but constant, elaboration of all these quantal theories as abstract mathematical struc­ tures (their point of departure being von Neumann's formalism) until at the present time theoretical work is heavily preoccupied with the manip­ ulation of purely abstract structures.

In May of 1973 we organized an international research colloquium on foundations of probability, statistics, and statistical theories of science at the University of Western Ontario. During the past four decades there have been striking formal advances in our understanding of logic, semantics and algebraic structure in probabilistic and statistical theories. These advances, which include the development of the relations between semantics and metamathematics, between logics and algebras and the algebraic-geometrical foundations of statistical theories (especially in the sciences), have led to striking new insights into the formal and conceptual structure of probability and statistical theory and their scientific applications in the form of scientific theory. The foundations of statistics are in a state of profound conflict. Fisher's objections to some aspects of Neyman-Pearson statistics have long been well known. More recently the emergence of Baysian statistics as a radical alternative to standard views has made the conflict especially acute. In recent years the response of many practising statisticians to the conflict has been an eclectic approach to statistical inference. Many good statisticians have developed a kind of wisdom which enables them to know which problems are most appropriately handled by each of the methods available. The search for principles which would explain why each of the methods works where it does and fails where it does offers a fruitful approach to the controversy over foundations.

In May of 1973 we organized an international research colloquium on foundations of probability, statistics, and statistical theories of science at the University of Western Ontario. During the past four decades there have been striking formal advances in our understanding of logic, semantics and algebraic structure in probabilistic and statistical theories. These advances, which include the development of the relations between semantics and metamathematics, between logics and algebras and the algebraic-geometrical foundations of statistical theories (especially in the sciences), have led to striking new insights into the formal and conceptual structure of probability and statistical theory and their scientific applications in the form of scientific theory. The foundations of statistics are in a state of profound conflict. Fisher's objections to some aspects of Neyman-Pearson statistics have long been well known. More recently the emergence of Baysian statistics as a radical alternative to standard views has made the conflict especially acute. In recent years the response of many practising statisticians to the conflict has been an eclectic approach to statistical inference. Many good statisticians have developed a kind of wisdom which enables them to know which problems are most appropriately handled by each of the methods available. The search for principles which would explain why each of the methods works where it does and fails where it does offers a fruitful approach to the controversy over foundations.

In May of 1973 we organized an international research colloquium on foundations of probability, statistics, and statistical theories of science at the University of Western Ontario. During the past four decades there have been striking formal advances in our understanding of logic, semantics and algebraic structure in probabilistic and statistical theories. These advances, which include the development of the relations between semantics and metamathematics, between logics and algebras and the algebraic-geometrical foundations of statistical theories (especially in the sciences), have led to striking new insights into the formal and conceptual structure of probability and statistical theory and their scientific applications in the form of scientific theory. The foundations of statistics are in a state of profound conflict. Fisher's objections to some aspects of Neyman-Pearson statistics have long been well known. More recently the emergence of Bayesian statistics as a radical alternative to standard views has made the conflict especially acute. In recent years the response of many practising statisticians to the conflict has been an eclectic approach to statistical inference. Many good statisticians have developed a kind of wisdom which enables them to know which problems are most appropriately handled by each of the methods available. The search for principles which would explain why each of the methods works where it does and fails where it does offers a fruitful approach to the controversy over foundations.

In May of 1973 we organized an international research colloquium on foundations of probability, statistics, and statistical theories of science at the University of Western Ontario. During the past four decades there have been striking formal advances in our understanding of logic, semantics and algebraic structure in probabilistic and statistical theories. These advances, which include the development of the relations between semantics and metamathematics, between logics and algebras and the algebraic-geometrical foundations of statistical theories (especially in the sciences), have led to striking new insights into the formal and conceptual structure of probability and statistical theory and their scientific applications in the form of scientific theory. The foundations of statistics are in a state of profound conflict. Fisher's objections to some aspects of Neyman-Pearson statistics have long been well known. More recently the emergence of Bayesian statistics as a radical alternative to standard views has made the conflict especially acute. In recent years the response of many practising statisticians to the conflict has been an eclectic approach to statistical inference. Many good statisticians have developed a kind of wisdom which enables them to know which problems are most appropriately handled by each of the methods available. The search for principles which would explain why each of the methods works where it does and fails where it does offers a fruitful approach to the controversy over foundations.

For this book, we have selected papers from symposia and contributed sessions at the fourth biennial meeting of the Philosophy of Science Association, held at the University of Notre Dame on November 1-3, 1974. The meeting was lively and well-attended, and we regret that there was no way to record here the many stimulating discussions after the papers and during the informal hours. We also regret that we had in­ sufficient space for all the contributed papers. Even more, some of the symposia were not available: those on systems and decision theory (c. W. Churchman, P. Suppes, I. Levi), and on the Marxist philosophy of science (M. W. Wartofsky, R. S. Cohen, E. N. Hiebert). Unhappily several individual contributions to other symposia were likewise not available: I. Velikovsky in the session on his own work and the politics of science, D. Finkelstein in the session on quantum logic. Memorial minutes were read for Alan Ross Anderson (prepared by Nuel Belnap) and for Imre Lakatos (prepared by Paul Feyerabend). They initiate this volume of philosophy of science in the mid-seventies.

The twentieth century has witnessed a striking transformation in the understanding of the theories of mathematical physics. There has emerged clearly the idea that physical theories are significantly characterized by their abstract mathematical structure. This is in opposition to the tradi­ tional opinion that one should look to the specific applications of a theory in orrter to understand it. One might with reason now espouse the view that to understand the deeper character of a theory one must know its abstract structure and understand the significance of that structure, while to understand how a theory might be modified in light of its experimental inadequacies one must be intimately acquainted with how it is applied. Quantum theory itself has gone through a development this century which illustrates strikingly the shifting perspective. From a collection of intuitive physical manoeuvers under Bohr, through a formative stage in which the mathematical framework was bifurcated (between Schrodinger and Heisenberg) to an elegant culmination in von Neumann's Hilbert space formulation, the elementary theory moved, flanked even at this later stage by the ill-understood formalisms for the relativistic version and for the field-theoretic alternative; after that we have a gradual, but constant, elaboration of all these quantal theories as abstract mathematical structures (their point of departure being von Neumann's formalism) until at the present time theoretical work is heavily preoccupied with the manipulation of purely abstract structures.

The twentieth century has witnessed a striking transformation in the understanding of the theories of mathematical physics. There has emerged clearly the idea that physical theories are significantly characterized by their abstract mathematical structure. This is in opposition to the tradi­ tional opinion that one should look to the specific applications of a theory in orrter to understand it. One might with reason now espouse the view that to understand the deeper character of a theory one must know its abstract structure and understand the significance of that structure, while to understand how a theory might be modified in light of its experimental inadequacies one must be intimately acquainted with how it is applied. Quantum theory itself has gone through a development this century which illustrates strikingly the shifting perspective. From a collection of intuitive physical manoeuvers under Bohr, through a formative stage in which the mathematical framework was bifurcated (between Schrodinger and Heisenberg) to an elegant culmination in von Neumann's Hilbert space formulation, the elementary theory moved, flanked even at this later stage by the ill-understood formalisms for the relativistic version and for the field-theoretic alternative; after that we have a gradual, but constant, elaboration of all these quantal theories as abstract mathematical structures (their point of departure being von Neumann's formalism) until at the present time theoretical work is heavily preoccupied with the manipulation of purely abstract structures.

1. INTRODUCTION In the Spring of 1975 we held an international workshop on the Foundations and Application of Decision Theory at the University of Western Ontario. To help structure the workshop into ordered and manageable sessions we distri­ buted the following statement of our goals to all invited participants. They in turn responded with useful revisions and suggested their own areas of interest. Since this procedure provided the eventual format of the sessions, we include it here as the most appropriate introduction to these collected papers result­ ing from the workshop. The reader can readily gauge the approximation to our mutual goals. 2. STATEMENT or OBJECTIVES AND RATIONALE (Attached to this statement is a bibliography; names of persons cited in the statement and writing in this century will be found referenced in the biblio­ graphy - certain 'classics' aside. ) 2. 1. Preamble We understand in the following the Theory of Decisions in a broader sense than is presently customary, construing it to embrace a general theory of deciSion-making, induding social, political and economic theory and applica­ tions. Thus, we subsume the Theory of Games under the head of Decision Theory, regarding it as a particularly clearly formulated version of part of the general theory of decision-making.

1. INTRODUCTION In the Spring of 1975 we held an international workshop on the Foundations and Application of Decision Theory at the University of Western Ontario. To help structure the workshop into ordered and manageable sessions we distri­ buted the following statement of our goals to all invited participants. They in turn responded with useful revisions and suggested their own areas of interest. Since this procedure provided the eventual format of the sessions, we include it here as the most appropriate introduction to these collected papers result­ ing from the workshop. The reader can readily gauge the approximation to our mutual goals. 2. STATEMENT OF OBJECTIVES AND RATIONALE (Attached to this statement is a bibliography; names of persons cited in the statement and writing in this century will be found referenced in the biblio­ graphy - certain 'classics' aSide. ) 2. 1. Preamble We understand in the following the Theory of Decisions in a broader sense than is presently customary, construing it to embrace a general theory of decision-making, including social, political and economic theory and applica­ tions. Thus, we subsume the Theory of Games under the head of Decision Theory, regarding it as a particularly clearly formulated version of part of the general theory of decision-making.

In two earlier volumes, entitled The Logico-Algebraic Approach to Quan­ tum Mechanics (hereafter LAA I, II), I have presented collections of research papers which trace out the historical development and contem­ porary flowering of a particular approach to physical theory. One might characterise this approach as the extraction of an abstract logico-algebraic skeleton from each physical theory and the reconstruction of the physical theory as construction of mathematical and interpretive 'flesh' (e. g. , measures, operators, mappings etc. ) on this skeleton. The idea is to show how the specific features of a theory that are easily seen in application (e. g. , 'interference' among observables in quantum mechanics) arise out of the character of its core abstract structure. In this fashion both the deeper nature of a theory (e. g. , in what precise sense quantum mechanics is strongly statistical) and the deeper differences between theories (e. g. clas­ sical mechanics, though also a 'mechanics', is not strongly statistical) are penetratingly illuminated. What I would describe as the 'mainstream' logico-algebraic tradition is captured in these two collections of papers (LAA I, II). The abstract, structural approach to the characterisation of physical theory has been the basis of a striking transformation, in this century, in the understanding of theories in mathematical physics. There has emerged clearly the idea that physical theories are most significantly characterised by their abstract structural components.

1. INTRODUCTION In the Spring of 1975 we held an international workshop on the Foundations and Application of Decision Theory at the University of Western Ontario. To help structure the workshop into ordered and manageable sessions we distri­ buted the following statement of our goals to all invited participants. They in turn responded with useful revisions and suggested their own areas of interest. Since this procedure provided the eventual format of the sessions, we include it here as the most appropriate introduction to these collected papers result­ ing from the workshop. The reader can readily gauge the approximation to our mutual goals. 2. STATEMENT or OBJECTIVES AND RATIONALE (Attached to this statement is a bibliography; names of persons cited in the statement and writing in this century will be found referenced in the biblio­ graphy - certain 'classics' aside. ) 2. 1. Preamble We understand in the following the Theory of Decisions in a broader sense than is presently customary, construing it to embrace a general theory of deciSion-making, induding social, political and economic theory and applica­ tions. Thus, we subsume the Theory of Games under the head of Decision Theory, regarding it as a particularly clearly formulated version of part of the general theory of decision-making.