The task of axiomatizing physical theories has attracted, in recent years, some interest among both empirical scientists and logicians. However, the axiomatizations produced by either one of these two groups seldom appear satisfactory to the members of the other. It is the purpose of this paper to develop an approach that will satisfy the criteria of both, hence permit us to construct axiomatizations that will meet simultaneously the standards and needs of logicians and of empirical scientists.

Although it has become a common place to refer to the ׳sixth problem׳ of Hilbert׳s (1900) Paris lecture as the starting point for modern axiomatized probability theory, his own views on probability have received comparatively little explicit attention. The central aim of this paper is to provide a detailed account of this topic in light of the central observation that the development of Hilbert׳s project of the axiomatization of physics went hand-in-hand with a redefinition of the status of (...) probability theory and the meaning of probability. Where Hilbert first regarded the theory as a mathematizable physical discipline and later approached it as a ׳vague׳ mathematical application in physics, he eventually understood probability, first, as a feature of human thought and, then, as an implicitly defined concept without a fixed physical interpretation. It thus becomes possible to suggest that Hilbert came to question, from the early 1920s on, the very possibility of achieving the goal of the axiomatization of probability as described in the ׳sixth problem׳ of 1900. (shrink)

This book is a wonderful resource for historians and philosophers of mathematics and physics alike, not just for Hilbert's own work in physics, but also because Corry sets Hilbert in context, bringing out the people with whom Hilbert had contact, describing their work and possible links with Hilbert's work, and describing the activities going on around Hilbert. The historical thesis of this book is that Hilbert worked on a wide range of issues in physics for a period (...) lasting more than two decades, employing and developing his axiomatic approach throughout. One conclusion that follows from this is that Hilbert's 1915–1917 work relating to Einstein's General Theory of Relativity was a natural continuation of Hilbert's pre-existing interests and activities, and not a one-off foray into foreign territory. 1Of especial interest to philosophers of mathematics are two further theses. Corry stresses that for Hilbert geometry is an empirical science, and related to this argues first, that Hilbert intends the axiomatic method to be used in enhancing our understanding of the content of a given theory via relating the results of the axiomatic investigation back to the intuitive content of the axioms; and, second, that to understand Hilbert's axiomatic approach in mathematics we must pay serious attention to his work in physics.Corry also hopes to show ‘the significant and unique contribution of Hilbert to certain important developments in twentieth-century physics’ . 2 In the end, this assessment of Hilbert's contribution to physics is far from clear cut: the two cases where Hilbert goes into the details of a physical theory show him lacking feel for what is important physically with respect to that theory. Nevertheless, philosophers and historians of physics will find a great deal to interest them in the story of Hilbert's involvement in physics, and in the details …. (shrink)

Sixteen years after his “Foundations of Geometry,” Hilbert published a communication that bears a similar and, by use of the definite article, even less mistakable title: “The Foundations of Physics.” In the opening paragraph of this article, Hilbert announced his intention self-confidently:In the following, I should like to set up — following the axiomatic method — a new system of fundamental equations of physics, constructed essentially from two simple axioms; equations that are of ideal beauty and in which, (...) as I believe, is contained the solution of both Einstein’s and Mie’s problems. (shrink)

Hilbert's axiomatization program of physical theories met an interesting challenge when it confronted the rise of quantum mechanics in the mid-twenties. The novelty of the mathematical apparatus of the then newly born theory was to be matched only by its substantial lack of any definite physical interpretation. The early attempts at axiomatization, which are described here, reflect all the difficulty of the task faced by Jordan, Hilbert, von Neumann and others. The role of von Neumann is examined in (...) considerable detail as he can be viewed here as the most outstanding of Hilbert's heirs. Von Neumann, especially in his work devoted to the proof of the impossibility of hidden variables, not only continued Hilbert's program but pushed it to its very limits, blending axiomatic rigor and interpretative commitment. (shrink)

Reichenbach wrote this book just after taking the first course Einstein ever taught on the theory of relativity. His important and influential work The Philosophy of Space and Time was written several years later and relied in part on the axiomatization of the special and general theories of relativity already worked out in this book. For special relativity Reichenbach divides his axioms into two sets, the light axioms which relate light signals to the topology and metric of time and (...) space, and the matter axioms which do the same for rigid rods and clocks. Thus the axioms focus on the conventions uniting theory and observation and give more insight into the physical foundations of the theory than do most axiomatizations. Reichenbach's approach has been criticized by Weyl and others, but at least some of the criticism seems to rest on a misunderstanding of what Reichenbach was attempting to do. Now that this early work has been translated into English, there is hope that it will be more widely read. There is no doubt that students of the theory or of recent philosophical discussion of space and time will profit from a careful reading of this book.--R. H. K. (shrink)

A well known conception of axiomatization has it that an axiomatized theory must be interpreted, or otherwise coordinated with reality, in order to acquire empirical content. An early version of this account is often ascribed to key figures in the logical empiricist movement, and to central figures in the early “formalist” tradition in mathematics as well. In this context, Reichenbach’s “coordinative definitions” are regarded as investing abstract propositions with empirical significance. We argue that over-emphasis on the abstract elements of (...) this approach fails to appreciate a rich tradition of empirical axiomatization in the late nineteenth and early twentieth centuries, evident in particular in the work of Moritz Pasch, Heinrich Hertz, David Hilbert, and Reichenbach himself. We claim that such over-emphasis leads to a misunderstanding of the role of empirical facts in Reichenbach’s approach to the axiomatization of a physical theory, and of the role of Reichenbach’s coordinative definitions in particular. (shrink)

Suppose that entities composed of two independent components are qualitatively ordered by a relation that satisfies the axioms of conjoint measurement. Suppose, in addition, that each component has a concatenation operation that, together either with the ordering induced on the component by the conjoint ordering or with its converse, satisfies the axioms of extensive measurement. Without further assumptions, nothing can be said about the relation between the numerical scales constructed from the two measurement theories except that they are strictly monotonic. (...) An axiom is stated that relates the two types of measurement theories, seems to cover all cases of interest in physics, and is sufficient to establish that the conjoint measurement scales are power functions of the extensive measurement scales. (shrink)

The aim of the paper is this: Instead of presenting a provisional and necessarily insufficient characterization of what mathematical physics is, I will ask the reader to take it just as that, what he or she thinks or believes it is, yet to be prepared to revise his opinion in the light of what I am going to tell. Because this is precisely, what I intend to do. I will challenge some of the received or standard views about mathematical (...) physics and replace them by a more sophisticated picture, which takes into account the methodological and philosophical roots of mathematical physics in Göttingen. (shrink)

On December 10th, 1947, John von Neumann wrote to the Spanish translator of his Mathematical Foundations of Quantum Mechanics: 1Your questions on the nature of mathematical physics and theoretical physics are interesting but a little difficult to answer with precision in my own mind. I have always drawn a somewhat vague line of demarcation between the two subjects, but it was really more a difference in distribution of emphases. I think that in theoretical physics the main emphasis (...) is on the connection with experimental physics and those methodological processes which lead to new theories and new formulations, whereas mathematical physics deals with the actual solution and mathematical execution of a theory which is assumed to be correct per se, or assumed to be correct for the sake of the discussion. In other words, I would say that theoretical physics deals rather with the formation and mathematical physics rather with the exploitation of physical theories. However, when a new theory has to be evaluated and compared with experience, both aspects mix. (shrink)

We have begun a theory of measurement in which an experimenter and his or her experimental procedure are modeled by algorithms that interact with physical equipment through a simple abstract interface. The theory is based upon using models of physical equipment as oracles to Turing machines. This allows us to investigate the computability and computational complexity of measurement processes. We examine eight different experiments that make measurements and, by introducing the idea of an observable indicator, we identify three distinct forms (...) of measurement process and three types of measurement algorithm. We give axiomatic specifications of three forms of interfaces that enable the three types of experiment to be used as oracles to Turing machines, and lemmas that help certify an experiment satisfies the axiomatic specifications. For experiments that satisfy our axiomatic specifications, we give lower bounds on the computational power of Turing machines in polynomial time using nonuniform complexity classes. These lower bounds break the barrier defined by the Church-Turing Thesis. (shrink)

Foundations research in physics, according to Bunge, has lagged behind its sister discipline, the foundations of mathematics. His book is an attempt to partially remedy this situation by analyzing the form and content of some basic ideas in physics and presenting some of the fundamental theories of physics in an axiomatic fashion. The heart of the book consists of axiomatizations of Classical Mechanics, Classical Field Theories, and Quantum Mechanics. Bunge does not claim to be working without predecessors. (...) While the idea of a separate discipline in the foundations of physics is rather new, the existence of material that clearly belongs to foundations research is not. This is attested to by the substantial bibliography at the end of the book. But he is ideally suited to the task of bringing this difficult material together and welding it in his own fashion into a coherent whole. His grasp of physical theory is wide-ranging and he writes with clarity and wit. In the first chapter he states his views on a number of issues in the philosophy of science: explanation, prediction, operationalism, laws, and other topics. The arguments are usually summarized very briefly, or reference is made to other books. Thus those who disagree with his philosophical presuppositions will not be convinced; but friends or foes will miss a great deal of informative material if they do not push on to the axiomatizations which follow and which are the main concern of the book. This book will undoubtedly be a source-book for future foundations research in the physical sciences.--R. H. K. (shrink)

Further arguments are offered in defence of the position that the variant of quantum field theory (QFT) that should be subject to interpretation and foundational analysis is axiomatic quantum field theory. I argue that the successful application of renormalization group (RG) methods within alternative formulations of QFT illuminates the empirical content of QFT, but not the theoretical content. RG methods corroborate the point of view that QFT is a case of the underdetermination of theory by empirical evidence. I also urge (...) caution in extrapolating interpretive conclusions about QFT from the application of RG methods in other contexts (e.g., condensed matter physics). This paper replies to criticisms advanced by David Wallace, but aims to be self-contained. (shrink)

Our main purpose here is to make some considerations about the definability of physical concepts like mass, force, time, space, spacetime, and so on. Our starting motivation is a collection of supposed definitions of closed system in the literature of physics and philosophy of physics. So, we discuss the problem of definitions in theoretical physics from the point of view of modern theories of definition. One of our main conclusions is that there are different kinds of definitions (...) in physics that demand different approaches. Within this context, we strongly advocate the use of the axiomatic method in order to discuss some issues concerning definitions. (shrink)

On the basis of the Woodhouse causal axiomatics, we show that conformal proper times and an extra variable in addition to those of space and time, together give a physical justification for the ‘chronometric hypothesis’ of general relativity. Indeed, we show that, with a lack of these latter two ingredients and of this hypothesis, clock paradoxes exist for which the unparadoxical asymmetry cannot be recovered when using the ‘clock and message functions’ only. These proper times originate from a given conformal (...) structure of the spacetime when ascribing different compatible projective structures to each Woodhouse particle, and then, each defines a specific Weylian ‘sheaf structure’. In addition, the proper time parameterizations are defined via path-dependent conformal scale factors, which act like sockets for any kind of physical interaction and also represent the values of the variable associated with the extra dimension. (shrink)

A basic aim of E. Beth's work in philosophy of science was to explore the use of formal semantic methods in the analysis of physical theories. We hope to show that a general framework for Beth's semantic analysis is provided by the theory of semi-interpreted languages, introduced in a previous paper. After developing Beth's analysis of nonrelativistic physical theories in a more general form, we turn to the notion of the 'logic' of a physical theory. Here we prove a result (...) concerning the conditions under which semantic entailment in such a theory is finitary. We argue, finally, that Beth's approach provides a characterization of physical theory which is more faithful to current practice in foundational research in the sciences than the familiar picture of a partly interpreted axiomatic theory. (shrink)

In recent years the works of Friedman, Howard and many others have made obvious what perhaps was always self-evident. Namely, that the philosophy of the logical empiricists was shaped primarily by Einstein and his invention of the theory of relativity, whereas Hilbert and his axiomatic approach to the exact sciences had comparatively little impact on the logical empiricists and their understanding of science — if they had any effect at all. This is in one respect quite astonishing, insofar as Einstein (...) himself confessed 1921 in his famous lecture before the Prussian Academy of Science that “without it [the axiomatic point of view] it would have been impossible for me to propound the theory of relativity”.2 Hence the simple question arises: why didn’t the logical empiricists pay more attention to Hilbert and his axiomatic point of view, than they in fact did? It is an aim of this paper to answer this question and in part to correct this one-sided view in the hypothetical or contra-factual sense that the logical empiricists would have done better, if they had paid somewhat more attention to Hilbert and his axiomatic approach to science. (shrink)

Although the introduction of the modern axiomatic method in physics is attributed to Hilbert, it is only recently that physicists and mathematicians have applied it significantly, i.e. on a basis extensive enough to promise fruitful results. Carnap, for one, stresses the importance of the axiomatic method, yet he considers its application in physics as a task for the future.

The present paper discusses Grosseteste’s reception of Proclus’ Elements of Physics (EP) in his Commentary on Aristotle’s Physics VI. In the first section I examine the method with which Grosseteste reconstructs Aristotelian texts. The second section initiates a study of the way Grosseteste evaluates Proclus’ EP on the basis of this method. Thus, the third section brings out Grosseteste’s moderate criticism of Proclus’ treatment of certain Aristotelian conclusiones and assumptions. The fourth section extends this study to the conceptual (...) relation between contiguity, continuity and succession. Finally, Grosseteste’s evaluation of Proclus’ tendency to omit, divide and merge Aristotelian conclusiones is studied in the fifth section. I conclude that Grosseteste is a careful and moderately critical reader of Proclus. He aptly grasps the dependence of the EP on Physics VI and conceives of Proclus’ EP as a forerunner of his own method of reconstructing Aristotelian texts. (shrink)

The dramatic development of physics in the twentieth century has thrown philosophy into a crisis regarding its self-image, from which today's philosophy has still not fully recovered.1 The crisis had two consequences or complementary manifestations: First, there was a gradual retreat of philosophy from the natural sciences. Because physics turned out to be an autonomous discipline, which aims at cognition of nature apparently totally independent of any kind of philosophy, "natural philosophy" as a particular branch of philosophy seemed (...) superfluous. Second, in order to prevent its own abdication, natural philosophy mutated into a kind of meta-science, called philosophy of science, that avoids any serious .. (shrink)

The debate between Fraser and Wallace over the foundations of quantum field theory has spawned increased focus on both the axiomatic and conventional formalisms. The debate has set the tone for future foundational analysis, and has forced philosophers to “pick a side”. The two are seen as competing research programs, and the major divide between the two manifests in how each handles renormalization. In this paper I argue that the terms set by the Fraser-Wallace debate are misleading. AQFT and CQFT (...) should be viewed as complementary formalisms that start from the same physical basis. Further, the focus on cutoffs as demarcating the two approaches is also highly misleading. Though their methods differ, both axiomatic and conventional QFT seek to use the same physical principles to explain the same domain of phenomena. (shrink)

The history of the publication of the gravitational field equations of general relativity in November 1915 by Einstein and Hilbert is briefly reviewed. An analysis of the internal structure and logic of Hilbert's theory as expounded in extant proofs and in the published version of his relevant paper is given with respect to the specific question what information would have been found on a missing piece of Hilbert's proofs. The existing texts suggest that the missing piece contained the explicit form (...) of the Riemann curvature scalar in terms of the Ricci tensor as a specification of the axiomatically underdetermined Lagrangian in Hilbert's action integral. An alternative reading that the missing piece of the proofs already may have contained the Einstein tensor, i.e. an explicit calculation of the gravitational part of Hilbert's Lagrangian is argued to be highly implausible. (shrink)

An ab-initio foundation for relativistic spacetime is given, which is a conservative extension of Zermelo’s set theory with urelemente. Primitive entities are worldlines rather than spacetime points. Spacetime points are sets of intersecting worldlines. By the proper axioms, they form a manifold. Entities known in differential geometry, up to a metric, are defined and have the usual properties. A set-realistic point of view is adopted. The intended ontology is a set-theoretical hierarchy with a broad base of the empty set and (...) urelemente. Sets generated from the empty set are mathematically interpreted, all other sets are physically interpreted. (shrink)

The following remarks are intended to show that some of Freudenthal's recent criticisms of Bunge'sFoundations of Physics are wide of the mark. Freudenthal sets his criticisms of detail in a framework of some general considerations of the role played by axiomatic theories in the foundations of physics. In particular, he considers the notion of the objects of an axiomatic theory, the relation of an axiomatic theory to reality, and the notion of the transformation group of a theory. These (...) topics are considered below. (shrink)

The paper justifies the following theses: The totality can found time if the latter is axiomatically represented by its “arrow” as a well-ordering. Time can found choice and thus information in turn. Quantum information and its units, the quantum bits, can be interpreted as their generalization as to infinity and underlying the physical world as well as the ultimate substance of the world both subjective and objective. Thus a pathway of interpretation between the totality via time, order, choice, and information (...) to the substance of the world is constructed. The article is based only on the well-known facts and definitions and is with no premises in this sense. Nevertheless it is naturally situated among works and ideas of Husserl and Heidegger, linked to the foundation of mathematics by the axiom of choice, to the philosophy of quantum mechanics and information. (shrink)

A realistic axiomatic formulation of Galilean Quantum Field Theories is presented, from which the most important theorems of the theory can be deduced. In comparison with others formulations, the formal aspect has been improved by the use of certain mathematical theories, such as group theory and the theory of rigged Hilbert spaces. Our approach regards the fields as real things with symmetry properties. The general structure is analyzed and contrasted with relativistic theories.

Reconstructions of quantum theory are a novel research program in theoretical physics which aims to uncover the unique physical features of quantum theory via axiomatization. I focus on Hardy’s “Quantum Theory from Five Reasonable Axioms” (2001), arguing that reconstructions represent a modern usage of axiomatization with significant points of continuity to von Neumann’s axiomatizations in quantum mechanics. In particular, I show that Hardy and von Neumann share similar methodological ordering, have a common operational framing, and insist on (...) the empirical basis of axioms. In the reconstruction programme, interesting points of discontinuity with historical axiomatizations include the stipulation of a generalized space of theories represented by a framework and the stipulation of analytic machinery at two levels of generality (first by establishing a generalized mathematical framework and then by positing specific formulations of axioms). In light of the reconstruction programme, I show that we should understand axiomatization attempts as being context–dependent, context which is contingent upon the goals of inquiry and the maturity of both mathematical formalism and theoretical underpinnings within the area of inquiry. Drawing on Mitsch (2022)’s account of axiomatization, I conclude that reconstructions should best be understood as provisional, practical, representations of quantum theory that are well suited for theory development and exploration. However, I propose my context–dependent re–framing of axiomatization as a means of enriching Mitsch’s account. (shrink)

A realistic axiomatic formulation of nonrelativistic quantum mechanics for a single microsystem with spin is presented, from which the most important theorems of the theory can be deduced. In comparison with previous formulations, the formal aspect has been improved by the use of certain mathematical theories, such as the theory of equipped spaces, and group theory. The standard formalism is naturally obtained from the latter, starting from a central primitive concept: the Galilei group.

We make a first attempt to axiomatically formulate the Montevideo interpretation of quantum mechanics. In this interpretation environmental decoherence is supplemented with loss of coherence due to the use of realistic clocks to measure time to solve the measurement problem. The resulting formulation is framed entirely in terms of quantum objects without having to invoke the existence of measurable classical quantities like the time in ordinary quantum mechanics. The formulation eliminates any privileged role to the measurement process giving an objective (...) definition of when an event occurs in a system. (shrink)

This article presents and compares various algebraic structures that arise in axiomatic unsharp quantum physics. We begin by stating some basic principles that such an algebraic structure should encompass. Following G. Mackey and G. Ludwig, we first consider a minimal state-effect-probability (minimal SEFP) structure. In order to include partial operations of sum and difference, an additional axiom is postulated and a SEFP structure is obtained. It is then shown that a SEFP structure is equivalent to an effect algebra with (...) an order determining set of states. We also consider σ-SEFP structures and show that these structures distinguish Hilbert space from incomplete inner product spaces. Various types of sharpness are discussed and under what conditions a Brouwer complementation can be defined to obtain a BZ-poset is investigated. In this case it is shown that every effect has a best lower and upper sharp approximation and that the set of all Brouwer sharp effects form an orthoalgebra. (shrink)

The standard coordinate-based formulation of the space-time theory of special relativity (Minkowski geometry) is philosophically unsatisfactory for various reasons. We here present an explicit axiomatic formulation of that theory in terms of primitives with a definitive physical interpretation, prove its equivalence to the standard coordinate formulation, and draw various philosophical conclusions concerning the physical content and assumptions of the space-time theory. The prevalent causal interpretation of physical Minkowski geometry deriving from Reichenbach is criticised on the basis of the present formulation.

In treatises or advanced textbooks on theoretical physics, it is apparent that the way mathematics is used is very different from what is to be found in books of mathematics. There is, for example, no close connection between books on analysis, on the one hand, and any classical textbook in quantum mechanics, for example, Schiff, [11], or quite recent books, for example Ryder, [10], on quantum field theory. The differences run a good deal deeper than the fact that the (...) books on theoretical physics are not written in the definition-theorem-proof style characteristic of pure mathematics. Although a good many propositions are proved in the books on physics, there are almost with exception no existential proofs, and consequently there is no really serious systematic use of quantifiers. Another important characteristic is the free use of infinitesimals. In fact, most results would not lose anything, from a physicist's point of view, by leaving them in approximate form, i.e., instead of strict equalities or inequalities, using equalities or inequalities only up to an infinitesimal.The discrepancy between the way mathematics is ordinarily done in theoretical physics and the way it is built up from a foundational standpoint in any of the standard modern views raises the question of whether it might be possible to construct quite directly a rigorous foundation that reflects a significant part of this standard practice in theoretical physics. Other parts of standard practice in physics, for example, the use of physically intuitive but nonrigorous arguments, are not present in our system. (shrink)

In 1900 David Hilbert announced his famous list of then-opened mathematical problems; the problem number 6 in this list is axiomatization of physical theories. Since then a lot of systematic efforts have been invested into solving this problem. However the results of these efforts turned to be less successful than the early enthusiasts of axiomatic method expected. The existing axiomatizations of physical and biological theories provide a valuable logical analysis of these theories but they do not constitute anything like (...) their standard presentation,which can be used for transmission, evaluation, and justification of physical and biological knowledge. This state of theart in the axiomatization of physics is strong evidence that the standard notion of axiomatic theory stemming from Hilbert and Tarski is not appropriate for thet ask. However in the recent years in mathematics there emerged a new axiomatic approach best represented by the Homotopy Type theory (HoTT). We argue that the constructive axiomatic architecture used in HoTT has better chances to be successfully applied in physics as well as in computer science and engineering. (shrink)

In this paper we examine Ellis and Bowman's argument, that simultaneity in inertial frames of reference is not conventional, from the axiomatic point of view. In Part I we examine the role of conventions in an axiomatic physical theory, and in Part II the relation of simultaneity in Reichenbach's axiomatization of the space-time theory of special relativity.