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)

This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their (...) ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method. (shrink)

The modern notion of the axiomatic method developed as a part of the conceptualization of mathematics starting in the nineteenth century. The basic idea of the method is the capture of a class of structures as the models of an axiomatic system. The mathematical study of such classes of structures is not exhausted by the derivation of theorems from the axioms but includes normally the metatheory of the axiom system. This conception of axiomatization satisfies the crucial (...) requirement that the derivation of theorems from axioms does not produce new information in the usual sense of the term called depth information. It can produce new information in a different sense of information called surface information. It is argued in this paper that the derivation should be based on a model-theoretical relation of logical consequence rather than derivability by means of mechanical (recursive) rules. Likewise completeness must be understood by reference to a model-theoretical consequence relation. A correctly understood notion of axiomatization does not apply to purely logical theories. In the latter the only relevant kind of axiomatization amounts to recursive enumeration of logical truths. First-order “axiomatic” set theories are not genuine axiomatizations. The main reason is that their models are structures of particulars, not of sets. Axiomatization cannot usually be motivated epistemologically, but it is related to the idea of explanation. (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)

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)

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)

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.

Feferman argues that category theory cannot stand on its own as a structuralist foundation for mathematics: he claims that, because the notions of operation and collection are both epistemically and logically prior, we require a background theory of operations and collections. Recently [2011], I have argued that in rationally reconstructing Hilbert’s organizational use of the axiomatic method, we can construct an algebraic version of category-theoretic structuralism. That is, in reply to Shapiro, we can be structuralists all the way (...) down ; we do not have to appeal to some background theory to guarantee the truth of our axioms. In this paper, I again turn to Hilbert; I borrow his distinction between the genetic method and the axiomatic method to argue that even if the genetic method requires the notions of operation and collection, the axiomatic method does not. Even if the genetic method is in some sense epistemically or logically prior, the axiomatic method stands alone. Thus, if the claim that category theory can act as a structuralist foundation for mathematics arises from the organizational use of the axiomatic method, then it does not depend on the prior notions of operation or collection, and so we can be structuralists all the way up. (shrink)

Although this excellent introductory and intermediate level text is intended for students of mathematics, it could serve well in any course for philosophers on that level. The first two chapters present the propositional and predicate calculi, along with an informal discussion of some of the set-theoretic concepts needed to study logic. The third chapter discusses what exactly an axiomatic system is, and examples of various mathematical systems cast in axiomatic form are provided; the discussion here, as elsewhere in (...) the book, is generally self-contained and presupposes only an ability to deal with abstract ideas. The study of the propositional calculus is resumed in chapter four, but its formulation is now more rigorous and thoroughly axiomatic; in a similar manner the predicate calculus is treated in the next chapter, and the usual metatheorems are proved about each, but the specific techniques differ from the usual. The last chapter concentrates on the completeness proof of quantification theory, relating it to some of the axiomatic algebraic systems considered earlier. An appendix sketches some sophisticated results of Vaught and Robinson on model theory and completeness. In sum, this is one of the very best of a plethora of logic texts on the market, and it should serve well for years to come.—P. J. M. (shrink)

By way of these investigations, we hope to understand better the rationale behind Kant's theory of intuition, as well as to grasp many facets of the relations ...

Dr Batóg is Lecturer in Logic at the University of Poznan. In Poland mathematical logic has been as characteristic an object of study and development as that of the atom elsewhere. The names of Fr Bochenski and Tarski are of as much importance in the logical field as are those of Bohr and Rutherford in the physical. The Polish approach seems to have been based chiefly on Whitehead & Russell’s Principia Mathematica, that Homeric work whose effect abroad was far greater (...) than at home. The bearing of this fact on the present book will be discussed in due course, for it has determined our author’s treatment of his subject. (shrink)

The present paper investigates the philosophical relationship between John von Neumann’s Nohidden-variable theorem and Bell’s inequalities. Bell erroneously takes the axiomatic method as implying a finality claim and thus ignores von Neumann’s strongly pragmatist stance towards mathematical physics. If one considers, however, Hilbert’s axiomatic method as a critical enterprise, Bell’s theorem improves von Neumann’s by defining a more appropriate notion of ‘ hidden variable’ that permits one to include Bohm’s interpretation which recovers the predictive content of (...) quantum mechanics. Contrary to Bell’s belief, accepting this model does not require adopting the metaphysically realist Böhm picture. If one takes the latter as a physical research programme one sees that it only partly disputes a common domain of facts with the mathematically oriented research programme of von Neumann. (shrink)

The criticism of Hilbert's axiomatic system of geometry by Mate Meršić (Merchich, 1850-1928), presented in his work "Organistik der Geometrie" (1914, also in "Modernes und Modriges", 1914), is analyzed and discussed. According to Meršić, geometry cannot be based on its own axioms, as a logical analysis of spatial intuition, but must be derived as a "spatial concretion" using "higher" axioms of arithmetic, logic, and "rational algorithmics." Geometry can only be one, because space is also only one. It cannot be (...) reduced to manipulation with symbols but has to deal with concepts and provide real knowledge. According to Meršić, aiming at some particular proof of the consistency of the whole axiomatic system is meaningless because the consistency follows from the truth of individual axioms based on spatial intuition. It is impossible, according to Meršić, for the geometric axioms to be mutually independent because they must be interconnected as essential parts of a whole. Although Meršić did not recognize all the advantages of Hilbert's formalization of geometry, he clearly pointed out some crucial limitations of the axiomatic method. (shrink)

There is, of course, one clear sense in which Spinoza’s axiomatic method is a method of presentation: this is the sense which contrasts a method of presentation with a method of discovery. In the Ethics, Spinoza is stating and explaining his views, not describing how he arrived at them or telling us how to make discoveries for ourselves. Nor does he elsewhere present the axiomatic method as a method of discovery. In the (...) seventeenth century, the distinction between a method of discovery and a method of presentation was often described as a distinction between "analytic" and "synthetic" method; Arnauld and Nicole write. (shrink)

Hilbert’s methodological reflection has certainly shaped a new image of the axiomatic method. However, the discussion on the procedural character of the method is still open, with commentators subs...

This lecture will deal with the heuristic power of the deductive method and its contributions to the scientific task of finding new knowledge. I will argue for a new reading of the term 'deductive method.' It will be presented as an architectural scheme for the reconstruction of the processes of gaining reliable scientific knowledge. This scheme combines the activities of doing science with the activities of presenting scientific results. It combines the heuristic and the deductive side of science. (...) The heuristic side is represented, e.g., by the creative methods to find the 'best' hypotheses, to design experimental systems for empirical research in order to formulate general laws, or to create axiomatic systems. The other side consists of the production of deductive knowledge. This combination leads to a clear hierarchy: the heuristic side provides the basic presuppositions from which the deductive side takes off. The former is used to make deductions possible. The deductive method can be presented as an analysis-synthesis scheme as it can be found, e.g., in the tradition of Kant, Jakob Friedrich Fries, and Leonard Nelson. Nelson's critical philosophy can be seen as a key for understanding the philosophy behind David Hilbert's early axiomatic method. This axiomatic method is usually restricted to a non-philosophical approach to pure mathematics. But Hilbert was not an exclusive formalist; he proposed a mathesis universalis in the Cartesian-Leibnizian sense according to which mathematics is the syntactical tool for a general philosophy of science, applicable to all scientific disciplines. In this function, mathematics takes its problems from the sciences. Hilbert did not deny that mathematics should play a role in explaining the world. The analysis-synthesis scheme helps to provide a consistent interpretation of these two sides of Hilbert's attitude towards his working field. (shrink)

In this paper I provide a brief reconstruction of Otto Hölder’s conception of proof. My reconstruction focuses on Hölder’s critical assessment of David Hilbert’s account of axiomatics in general, and of Hilbert’s conception of metamathematics in particular. I argue that Hölder’s analysis of Hilbert’s general methodological ideas and, more importantly, Hölder’s analysis of the logical structure of the proofs provided by Hilbert in his Grundlagen der Geometrie of 1899 are helpful in reaching a clearer understanding of van der Waerden’s claim (...) linking Hölder’s conception of proof to the tradition established by Kurt Gödel. (shrink)

In this paper I provide a brief reconstruction of Otto Hölder’s conception of proof. My reconstruction focuses on Hölder’s critical assessment of David Hilbert’s account of axiomatics in general, and of Hilbert’s conception of metamathematics in particular. I argue that Hölder’s analysis of Hilbert’s general methodological ideas and, more importantly, Hölder’s analysis of the logical structure of the proofs provided by Hilbert in his Grundlagen der Geometrie of 1899 are helpful in reaching a clearer understanding of van der Waerden’s claim (...) linking Hölder’s conception of proof to the tradition established by Kurt Gödel. (shrink)