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HILBERT, DUALITY, AND THE GEOMETRICAL ROOTS OF MODEL THEORY
Abstract
The article investigates one of the key contributions to modern structural mathematics, namely Hilbert’s Foundations of Geometry (1899) and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry (in particular, the development of non-Euclidean geometries), so far, little has been said about how exactly model-theoretic concepts grew out of methodological investigations within projective geometry. This article is supposed to fill this lacuna and investigates this geometrical prehistory of modern model theory, eventually leading up to Hilbert’s Foundations .
... That is, any concrete incidence relation between points and lines specified relative to one conic can be shown to correspond to a dualized relation between lines and points specified relative to the second conic. Accordingly, the theorems form an instance of the general principle of projective duality: one can deduce Brianchon's result from Pascal's result (and vice versa) by the previously mentioned technique of dualization, that is, by 8 The present subsection will closely follow Eder and Schiemer (2018) and Schiemer (2018) in the discussion of the principle of projective duality. 9 A corresponding principle of duality for solid geometry states that for any theorem of solid projective geometry we get another theorem by interchanging the words 'point' for 'plane' and 'plane' for 'point' (as well as of the primitive incidence relations). ...
... See againEder and Schiemer (2018) andSchiemer (2018) for closer discussions of Poncelet's transformation-based account of duality. 12 See, e.g.,Coxeter (1987) for a modern textbook presentation of polar theory. ...
The present article investigates Felix Klein’s mathematical structuralism underlying his Erlangen program . The aim here is twofold. The first aim is to survey the geometrical background of his 1872 article, in particular, work on the principle of duality and so-called transfer principles in projective geometry. The second aim is more philosophical in character and concerns Klein’s structuralist account of geometrical knowledge. The chapter will argue that his group-theoretic approach is best characterized as a kind of “methodological structuralism” regarding geometry. Moreover, one can identify at least two aspects of the Erlangen program that connect his approach with present philosophical debates, namely (i) the idea to specify structural properties and structural identity conditions in terms of transformation groups and (ii) an account of the structural equivalence of geometries in terms of transfer principles.
... Thirty years later, Veblen and Young, referring to this proof-oriented conception, note that the "method of formal inference from explicitly stated assumptions" makes duality "almost self-evident" (Veblen and Young 1910, p. 29). See also Eder and Schiemer (2018) for a discussion of duality in nineteenth century geometry. 16 See Gray (2007, ch. 2, 5) and Kline (1972, ch. ...
... It should also be mentioned that, in spite of passages like the one quoted earlier, there is still room for disagreement about how Hilbert understood his method of proving independence and consistency in detail at various times. See Eder and Schiemer (2018) for further discussion. 26 Frege held introductory courses in complex analysis and seminars on advanced topics such as elliptic functions and Abelian integrals, and in geometry he lectured on both synthetic and analytic geometry. ...
The aim of this article is to contribute to a better understanding of Frege’s views on semantics and metatheory by looking at his take on several themes in nineteenth century geometry that were significant for the development of modern model-theoretic semantics. I will focus on three issues in which a central semantic idea, the idea of reinterpreting non-logical terms, gradually came to play a substantial role: the introduction of elements at infinity in projective geometry; the study of transfer principles, especially the principle of duality; and the use of counterexamples in independence arguments. Based on a discussion of these issues and how nineteenth century geometers reflected about them, I will then look into Frege’s take on these matters. I conclude with a discussion of Frege’s views and what they entail for the debate about his stance towards semantics and metatheory more generally.
... Thus the segment MN/A consists of E points, and the segment MN/A of H points. 51 A non-self-conjugate line is an e-line or an h-line "according to the nature of the involution of conjugate points on it" [39, p. 82]. 52 In Sect. 4.2, we defined a polarity with respect to a conic Ω as an involutory correspondence between points and lines, such that each range corresponds to a projectively related pencil. ...
In the 1960s, H. S. M. Coxeter adopted a set of incidence axioms similar to one O. Veblen and J. W. Young proposed in 1910, to study projective spaces for which a field of elements of a commutative number system is an algebraic model. Coxeter introduced a theory called “accessibility,” which is applicable to the study of conics defined in 1847 by G. K. C. von Staudt, and the partitioning of projective planes in terms of them. In a conversation of 1989, Coxeter revealed that he used ideas from an 1898 paper of M. Pieri to define the relation of accessible points in a continuity-independent projective plane. In 1979, M.J. Greenberg reinterpreted that relation in such a plane using incidence and order axioms from another of Coxeter’s works. We compare the axiomatic bases on which Coxeter, Greenberg, and Pieri construct projective planes with and without continuity, using synthetic methods to introduce analytic ones. We show how several of their publications intersect in significant ways. Other paths emerge from these investigations, such as one that examines the origins of a statement postulated by Pieri, often credited to Pasch, which Coxeter attributed to Veblen.
... 9], [Detlefsen, 2014], [Peckhaus, 2003], and [Corry, 2004], as well as [Hallett, 1990] [Hallett, 1994] [Hallett, 2008], [Sieg, 2014], and [Wilson,202?]. For entry into interpreting Hilbert's Grundlagen der Geometrie, see [Giovannini, 2016] and [Eder and Schiemer, 2018]. For an account that emphasizes more of the foundationalist aspects of the axiomatic method, based on Hilbert's work related to general relativity, see [Brading and Ryckman, 2008] [Brading and Ryckman, 2018], as well as [Brading, 2014] and the editors' remarks in [Sauer and Majer, 2009]. ...
In this paper I provide a detailed history of von Neumann’s “No Hidden Variables” theorem, and I argue it is a demonstration that his axiomatization mathematically captures a salient feature of the statistical interpretation (namely, that hidden variables are incompatible). I show that this reading of von Neumann’s theorem is obvious once one recalls several contextual factors of his work. First, his axiomatization was what I call a Hilbert-style axiomatic completion; indeed, it developed from work initiated by Hilbert (and Nordheim). Second, it was responsive to specific mathematical and theoretical problems faced by Dirac and Jordan’s statistical transformation theory (then called ‘quantum mechanics’). Third, the axiomatization was essentially completed already in his 1927 papers, at least concerning the status of hidden variables, and this would have been obvious to the audience for those papers. Thus, the theorem’s statement and proof were only necessary when the material was presented for a general mathematical audience, i.e., in his 1932 Mathematical Foundations of Quantum Mechanics . With this reading in mind, his claim that quantum mechanics was in “compelling logical contradiction with causality” appears as a straightforward consequence of his theorem. I conclude by reassessing the theorem’s broader historical and scientific significance.
... The idea is that for each basic truth of projective geometry there exists another basic truth that 27 See Nagel (1939), Gray (2007, p. 53 ff.) and Lorenat (2015) for a discussion of the duality controversy. Some of the issues that are discussed in this section have been investigated in Eder (2019), which in turn is partly based on joint work with Georg Schiemer [see Eder and Schiemer (2018)]. 28 Chasles later clarified that really all that matters is that incidence is preserved. ...
In recent years, several scholars have been investigating Frege’s mathematical background, especially in geometry, in order to put his general views on mathematics and logic into proper perspective. In this article I want to continue this line of research and study Frege’s views on geometry in their own right by focussing on his views on a field which occupied center stage in nineteenth century geometry, namely, projective geometry.
This paper aims to show that Frege’s and Hilbert’s mutual disagreement results from different notions of Anschauung and their relation to axioms. In the first section of the paper, evidence is provided to support that Frege and Hilbert were influenced by the same developments of 19th-century geometry, in particular the work of Gauss, Plücker, and von Staudt. The second section of the paper shows that Frege and Hilbert take different approaches to deal with the problems that the developments in 19th-century geometry posed for the traditional Kantian philosophy of mathematics. Frege maintains that Anschauung is a source of knowledge by which we acknowledge geometrical axioms as true. For Hilbert, in contrast, axioms provide one of several correct “pictures” of reality. Hilbert’s position is thereby deeply influenced by epistemological ideas from Hertz and Helmholtz, and, in turn, influenced the neo-Kantian Cassirer.
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.
Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano’s understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that compose these theories. Second, I shall claim that, in the context of independence arguments, Peano developed a schematic understanding of the axioms which, despite diverging in some respects from Dedekind’s construction of arithmetic, should be considered structuralist. From this stance I shall argue that this schematic understanding of the axioms anticipates the basic components of a formal language.
In this paper I provide a detailed history of von Neumann’s “No Hidden Variables” theorem, and I argue it is a demonstration that his axiomatization mathematically captures a salient feature of the statistical transformation theory (namely, that hidden variables are incompatible). I show that this reading of von Neumann’s theorem is obvious once one recalls several factors of his work. First, his axiomatization was what I call a Hilbert-style axiomatic completion; indeed, it developed from work initiated by Hilbert (and Nordheim). Second, it was responsive to specific mathematical and theoretical problems faced by Dirac and Jordan’s statistical transformation theory (then called ‘quantum mechanics’). Third, the axiomatization was completed across his 1927 papers and 1932 book when he identified the basic assumptions underwriting quantum mechanics, showed that these suffice for deriving the trace rule, and showed that the trace rule is incompatible with hidden variables. With this reading in mind, his claim that quantum mechanics was in “compelling logical contradiction with causality” appears as a straightforward consequence of his theorem. I conclude by reassessing the theorem’s broader historical and scientific significance.
The symmetries between points and lines in planar projective geometry and between points and planes in solid projective geometry are striking features of these geometries that were extensively discussed during the nineteenth century under the labels “duality” or “reciprocity.” The aims of this article are, first, to provide a systematic analysis of duality from a modern point of view, and, second, based on this, to give a historical overview of how discussions about duality evolved during the nineteenth century. Specifically, we want to see in which ways geometers’ preoccupation with duality was shaped by developments that lead to modern logic towards the end of the nineteenth century, and how these developments in turn might have been influenced by reflections on duality.
This paper engages the question ‘Does the consistency of a set of axioms entail the existence of a model in which they are satisfied?’ within the frame of the Frege-Hilbert controversy. The question is related historically to the formulation, proof and reception of Gödel’s Completeness Theorem. Tools from mathematical logic are then used to argue that there are precise senses in which Frege was correct to maintain that demonstrating consistency is as difficult as it can be, but also in which Hilbert was correct to maintain that demonstrating existence given consistency is as easy as it can be.
This chapter presents Pasch’s structuralist methodology within his empiricist philosophy. Two criteria for a minimal version of methodological structuralism are proposed, and it is argued that they are met in Pasch’s work on projective geometry and the foundations of arithmetic, despite the fact that Pasch firmly held an empiricist standpoint according to which only empirical objects can ultimately serve as a foundation for mathematics. What drove Pasch toward his version of structuralism were his focus on rigorous deductions, the duality of projective geometry, and the independence of arithmetical results from the referents of numerals.
Those inquiring into the nature of mind have long been interested in the foundations of mathematics, and conversely this branch of knowledge is distinctive in that our access to it is purely through thought. A better understanding of mathematical thought should clarify the conceptual foundations of mathematics, and a deeper grasp of the latter should in turn illuminate the powers of mind through which mathematics is made available to us. The link between conceptions of mind and of mathematics has been a central theme running through the great competing philosophies of mathematics of the twentieth century, though each has refashioned the connection and its import in distinctive ways. The present collection will be of interest to students of both mathematics and of mind. Contents include: “Introduction” by Alexander George; “What is Mathematics About?” by Michael Dummett; “The Advantages of Honest Toil over Theft” by George Boolos; “The Law of Excluded Middle and the Axiom of Choice” by W.W. Tait; “Mechanical Procedures and Mathematical Experience” by Wilfried Sieg; “Mathematical Intuition and Objectivity” by Daniel Isaacson; “Intuition and Number” by Charles Parsons; and “Hilbert’s Axiomatic Method and the Laws of Thought” by Michael Hallett.
Our main purpose in this book is to present an English translation of Desargues' Rough Draft of an Essay on the results of taking plane sections of a cone (1639), the pamphlet with which the modem study of projective geometry began. Despite its acknowledged importance in the history of mathematics, the work has never been translated before in its entirety, although short extracts have appeared in several source books. The problems of making Desargues' work accessible to modem mathematicians and historians of mathematics have led us to provide a fairly elaborate introduction, and to include translations of other relevant works. The translation ofthe Rough Draft on Conics (as we shall call it) thus appears in Chapter VI, the five preceding chapters forming an introduction and the three following ones giving translations of other works by Desargues. Chapter I briefly reviews parts of ancient geometrical works available to Desargues which seem to be relevant to his own work, namely theorems in Euclid's Elements, the first four books of Apollonius' Conics and some remarks by Pappus in his Collection. These Hellenistic works belong to the 'high' mathematical tradition whose development has been the main theme of all histories of mathematics. It is from these works that Desargues took the theorems whose theory he was to reformulate in the Rough Draft on Conics.
Geometry has fascinated philosophers since the days of Thales and Pythagoras. In the 17th and 18th centuries it provided a paradigm of knowledge after which some thinkers tried to pattern their own metaphysical systems. But after the discovery of non-Euclidean geometries in the 19th century, the nature and scope of geometry became a bone of contention. Philosophical concern with geometry increased in the 1920's after Einstein used Riemannian geometry in his theory of gravitation. During the last fifteen or twenty years, renewed interest in the latter theory -prompted by advances in cosmology -has brought geometry once again to the forefront of philosophical discussion. The issues at stake in the current epistemological debate about geometry can only be understood in the light of history, and, in fact, most recent works on the subject include historical material. In this book, I try to give a selective critical survey of modern philosophy of geometry during its seminal period, which can be said to have begun shortly after 1850 with Riemann's generalized conception of space and to achieve some sort of completion at the turn of the century with Hilbert's axiomatics and Poincare's conventionalism. The philosophy of geometry of Einstein and his contemporaries will be the subject of another book. The book is divided into four chapters. Chapter 1 provides back ground information about the history of science and philosophy.
Between 1897 and 1902, there took place a brief correspondence between Frege and Hilbert, consisting of four letters from Frege, and two letters and three postcards from Hilbert. It centres on Frege's reactions to Hilbert's classic Grundlagen der Geometrie, first published in 1899, and Hilbert's restatements in his letters to Frege of the foundational positions which that work, sometimes only implicitly, embodies. Despite the obvious richness of common purpose between Frege and Hilbert, the correspondence is especially instructive because of the strong disagreements expressed. For example, the two disagreed on the form and function of definitions, the nature, purpose and formulation of axioms, the nature of (axiomatized) mathematical theories, the method of independence proofs in geometry, the role and form of consistency proofs and the nature of mathematical existence. Many of the articles of disagreement, especially those on axioms and independence proofs, also reveal or underline significant differences in their respective conceptions of logic. Frege followed the correspondence with two polemical, and wider-ranging, articles on similar or related themes, Hilbert himself having apparently declined Frege's suggestion that their exchange of views be published. These two papers help to fill out the picture on Frege's side, first by restating Frege's opposition, and then by presenting his insights into the formal structure of Hilbert's position. Especially important are Frege's attempts in his second article to render central results of Hilbert's project as read through his own system.
In the Conclusion to his Grundlagen der Geometrie of 1899, Hilbert stated that the concern with 'purity of method' is nothing more than a 'subjective interpretation' of the demand for a careful examination of central mathematical propositions, the search either for rigorous proofs from clearly specified axioms, or the proof of the impossibility of such a proof. This chapter examines Hilbert's treatment of purity in the lecture notes surrounding the Grundlagen. In particular, it presents three important case studies, concerning Desargues's Theorem, the Euclidean Isosceles Triangle Theorem, and the Three Chord Theorem. These examples show how important 'higher' mathematical knowledge is for Hilbert, and how this can often shape and instruct the intuitive level, which is often where a 'purity' question about geometry first arises; in the examples examined here, this forces a reassessment of what is 'appropriate' (or 'pure').
In a series of articles dating from 1903 to 1906, Frege criticizes Hilbert’s methodology of proving the independence and consistency
of various fragments of Euclidean geometry in his Foundations of Geometry. In the final part of the last article, Frege makes his own proposal as to how the independence of genuine axioms should
be proved. Frege contends that independence proofs require the development of a ‘new science’ with its own basic truths. This
paper aims to provide a reconstruction of this New Science that meets modern standards and to examine possible problems surrounding
Frege’s original proposal. The paper is organized as follows: the first two sections summarize the main points of the Frege–Hilbert
controversy and discuss some issues surrounding the problem of independence proofs. Section 3 contains an informal presentation
of Frege’s proposal. In section 4 a more detailed reconstruction of Frege’s New Science is set out while section 5 examines
what is left out. The concluding section is devoted to a discussion of Frege’s strategy and its significance from a broader
perspective.
This is an up-to-date and integrated introduction to model theory, designed to be used for graduate courses (for students who are familiar with first-order logic), and as a reference for more experienced logicians and mathematicians. Model theory is concerned with the notions of definition, interpretation and structure in a very general setting, and is applied to a wide variety of other areas such as set theory, geometry, algebra (in particular group theory), and computer science (e.g. logic programming and specification). Professor Hodges emphasises definability and methods of construction, and introduces the reader to advanced topics such as stability. He also provides the reader with much historical information and a full bibliography, enhancing the book's use as a reference.
The paper outlines an interpretation of one of the most important and original contributions of David Hilbert’s monograph Foundations of Geometry (1899), namely his internal arithmetization of geometry. It is claimed that Hilbert’s profound interest in the problem of the introduction of numbers into geometry responded to certain epistemological aims and methodological concerns that were fundamental to his early axiomatic investigations into the foundations of elementary geometry. In particular, it is shown that a central concern that motivated Hilbert’s axiomatic investigations from very early on was the aim of providing an independent basis for geometry. Accordingly, these concerns about an independent grounding for elementary geometry determined very clear methodological constraints in the process of embedding it into a formal axiomatic system. It is argued that Hilbert not only sought to show that geometry could be considered a pure mathematical theory, once it was presented as a formal axiomatic system; he also aimed at showing that in the construction of such an axiomatic system one could proceed purely geometrically, avoiding concept formations borrowed from other mathematical disciplines like arithmetic or analysis.
Frege's Coneption of Logic explores the relationship between Frege's understanding of conceptual analysis and his understanding of logic. It is argued that the fruitfulness of Frege's conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation. The first part of the book locates the role of conceptual analysis in Frege's logicist project. It is argued that, despite a number of difficulties, Frege's use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege's intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals. The second part of the book explores the resulting conception of logic itself, and some of the straightforward ways in which Frege's conception differs from its now-familiar descendants. In particular, it's argued that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege's position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.
It is a remarkable fact that Hilbert's programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic1 from the late 1890s is often understood from that vantage point. My essay pursues one main goal, namely, to contrast Hilbert's formal axiomatic method from the early 1920s with his existential axiomatic approach from the 1890s. Such a contrast illuminates the circuitous beginnings of the finitist consistency program and connects the complex emergence of existential axiomatics with transformations in mathematics and philosophy during the 19th century; the sheer complexity and methodological difficulties of the latter development are partially reflected in the well known, but not well understood correspondence between Frege and Hilbert. Taking seriously the goal of formalizing mathematics in an effective logical framework leads also to contemporary tasks, not just historical and systematic insights; those are briefly described as "one direction" for fascinating work.
In historical discussions of twentieth-century logic, it is typically assumed that model theory emerged within the tradition that adopted first-order logic as the standard framework. Work within the type-theoretic tradition, in the style of Principia Mathematica, tends to be downplayed or ignored in this connection. Indeed, the shift from type theory to first-order logic is sometimes seen as involving a radical break that first made possible the rise of modern model theory. While comparing several early attempts to develop the semantics of axiomatic theories in the 1930s, by two proponents of the type-theoretic tradition (Carnap and Tarski) and two proponents of the first-order tradition (Gödel and Hilbert), we argue that, instead, the move from type theory to first-order logic is better understood as a gradual transformation, and further, that the contributions to semantics made in the type-theoretic tradition should be seen as central to the evolution of model theory.
Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.
In this paper our major concern is with methodological issues of purity and thus we treat the connection to other areas of the planimetry/stereometry relation only to the extent necessary to articulate the problem area we are after.
Our strategy will be as follows. In the first part of the paper we will give a rough sketch of some key episodes in mathematical practice that relate to the interaction between plane and solid geometry. The sketch is given in broad strokes and only with the intent of acquainting the reader with some of the mathematical context against which the problem emerges. In the second part, we will look at a debate (on “fusionism”) in which for the first time methodological and foundational issues related to aspects of the mathematical practice covered in the first part of the paper came to the fore. We conclude this part of the paper by remarking that only through a foundational and philosophical effort could the issues raised by the debate on “fusionism” be made precise. The third part of the paper focuses on a specific case study which has been the subject of such an effort, namely the foundational analysis of the plane version of Desargues’ theorem on homological triangles and its implications for the relationship between plane and solid geometry. Finally, building on the foundational case study analyzed in the third section, we begin in the fourth section the analytic work necessary for exploring various important claims about “purity,” “content,” and other relevant notions.
Moritz Pasch (1843–1930) gave the first rigorous axiomatization of projective geometry in his Vorlesungen über neuere Geometrie (1882), in which he also clearly formulated the view that deductions must be independent from the meanings of the nonlogical terms involved. Pasch also presented in these lectures the main tenets of his philosophy of mathematics, which he continued to elaborate on throughout the rest of his life. This philosophy is quite unique in combining a deductivist methodology with a radically empiricist epistemology for mathematics. By taking into consideration publications from the entire span of Pasch’s career, the latter decades of which he devoted primarily to careful reflections on the nature of mathematics and of mathematical knowledge, Pasch’s highly original, but virtually unknown, philosophy of mathematics is presented.
The most important background factor in the development of twentieth-century logic has received insufficient attention in the literature. This factor is a largely tacit contrast between ways of looking at the relation of language and its logic to reality. I have called them the idea of language as the universal medium and the idea of language as calculus.1 I shall also refer the two traditions representing these two respective ideas as the universalist tradition and as the model-theoretical tradition.
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.
Worlds out of Nothing is the first book to provide a course on the history of geometry in the 19th Century, and it is based on the latest historical research. Emphasis is placed on understanding the historical significance of the new mathematics: why it was done, how, if at all, it was appreciated, what new questions did it generate? Topics covered in the first part of the course are projective geometry, especially the concept of duality (in the work of Gergonne, Poncelet and Chasles), and non-Euclidean geometry (the work of Gauss, Bolyai and Lobachevskii). The course then moves on to the study of the singular points of algebraic curves (Plücker’s equations) and their role in resolving a paradox in the theory of duality, to Riemann’s work on differential geometry, and to Beltrami’s role in successfully establishing non-Euclidean geometry as a rigorous mathematical subject. The final part of the course considers how projective geometry rose to a central position in geometry (exemplified by Klein’s Erlangen Program) and then looks at Poincaré’s ideas about non-Euclidean geometry and its possible physical and philosophical significance. It ends with a series of discussions about geometry: geometry and formalism (Italian work and Hilbert’s Foundations of Geometry), geometry and physics (a look at some ideas of Einstein’s), and geometry and truth. An Appendix describes how von Staudt gave an independent foundation for projective geometry and how his work it was received.
A miracle happens. In one hand we have a class of marvelously complex theories in predicate logic, theories with 'sufficient coding potential', like PA (Peano Arithmetic) or ZF (Zermelo Fraenkel Set Theory). In the other we have certain modal propositional theories of striking simplicity. We translate the modal operators of the modal theories to certain specic, fixed, defined predicates of the predicate logical theories. These special predicates generally contain an astronomical number of symbols. We interpret the propositional variables by arbitrary predicate logical sentences. And see: the modal theories are sound and complete for this interpretation. They codify precisely the schematic principles in their scope. Miracles do happen ....
The Modernist Transformation of Mathematics; citation_author=Gray
- Ghost
The Frege-Hilbert controversy
- P Blanchette
David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933
- D Hilbert
David Hilbert’s Lectures on the Foundations of Geometry 1891-1902
- D Hilbert
Traité des propriétés projectives des figures
- V Poncelet