Search

This is a survey of axiom systems for fragments of naturally encountered geometries which are just barely strong enough to imply that there are infinitely many objects in the universe of any of its models.

Summary In the article, we continue [7] the formalization of the work devoted to Tarski’s geometry – the book “Metamathematische Methoden in der Geometrie” (SST for short) by W. Schwabhäuser, W. Szmielew, and A. Tarski [14], [9], [10]. We use the Mizar system to systematically formalize Chapter 8 of the SST book. We define the notion of right angle and prove some of its basic properties, a theory of intersecting lines (including orthogonality). Using the notion of perpendicular foot, we prove the existence of the midpoint (Satz 8.22), which will be used in the form of the Mizar functor (as the uniqueness can be easily shown) in Chapter 10. In the last section we give some lemmas proven by means of Otter during Tarski Formalization Project by M. Beeson (the so-called Section 8A of SST).

El artículo examina una de las contribuciones más importantes a los fundamentos de la geometría euclídea elemental lograda por David Hilbert en su obraFundamentos de la geometría(1899), a saber: la reconstrucción de la teoría euclídea de las proporciones y de los triángulos semejantes. Se argumenta que dicha reconstrucción no sólo estuvo motivada por la identificación de Hilbert de suposiciones implícitas en la teoría de Euclides, sino que además estuvo esencialmente ligada a la preocupación por la 'pureza del método'. Más aún, se afirma que, en este caso específico, el requerimiento de Hilbert por la pureza del método posee un carácter general o fundacional, esto es, no se refiere a la demostración de un teorema en particular, sino más bien es planteado respecto de la construcción axiomática de la teoría misma. The paper examines one the most important contributions to the foundations of elementary Euclidean geometry achieved by David Hilbert in his bookFoundations of Geometry(1899), namely the reconstruction of the Euclidean theory of proportion and similar triangles. It is argued that this reconstruction was not only motived by Hilbert's identification of implicit assumptions in Euclid's theory, but also it was essentially linked to the concern for the 'purity of methods'. Moreover, it is claimed that in this specific case, Hilbert's requirement of the purity of method has a general or foundational dimension, that is, it is not established with regard to the proof of a particular theorem, but rather it is referred to the very axiomatic construction of the theory.

In the 1930s Otto Selz developed a novel approach to the psychology of perception which he called “synthetic psychology of wholes”. This “synthetic psychology” is based on a phenomenological description of the structural relationships between elementary items (tones, colors, smells, etc.) building up integral wholes. The present article deals with Selz’s account of spatial cognition within this general framework. Selz Zeitschrift für Psychologie, 114, 351–362 (1930a) argues that his approach to spatial cognition delivers answers to the long-discussed question of the epistemological status of the laws of geometry. More specifically he tries to derive (a subset of) the Euclidean axioms from the structural laws valid for phenomenal space. After a brief description of the discussion of the status of geometry in the 1920s/1930 (section 2), the present article explains Selz’s understanding of “phenomenology” (section 3). Section 4 then deals with Selz’s attempt to derive the Euclidean laws from the structural phenomenological laws of space. Selz’s attempted derivation suffers from some formal shortcomings, which however can be repaired. The question arises, though, whether the necessary improvements do not rely upon more intricate geometric intuitions and thus render Selz’s attempt to base geometry upon the phenomenology of spatial cognition circular.

This study investigates students’ development of metamathematical understanding of axioms. Based on four semesters of experiments teaching neutral, axiomatic geometry, often through guided reinvent...

In this paper, I critically discuss Frege’s philosophy of geometry with special emphasis on his position in The Foundations of Arithmetic of 1884. In Sect. 2, I argue that that what Frege calls facultyof intuition in his dissertation (1873) is probably meant to refer to a capacity of visualizing geometrical configurations structurally in a way which is essentially the same for most Western educated human beings. I further suggest that according to his Habilitationsschrift (1874) it is through spatial intuition that we come to know the axioms of Euclidean geometry. In Sect. 3, I argue that Frege seems right in claiming in The Foundations, §14 that the synthetic nature of the Euclidean axioms follows from the fact that they are independent of one another and of the primitive laws of logic. If the former were dependent on (provable from) the latter, they would be analytic in Frege’s sense of analyticity. But then they would not be independent of one another and due to their mutual provability would lose their status as axioms of Euclidean geometry, since according to Frege an axiom of a theory T is per definitionen unprovable in T. I further argue that only by invoking pure spatial intuition can Frege “explain” the (alleged) epistemological status of the axioms of Euclidean geometry completely: their synthetic a priori nature. Finally, I argue that his view about independence in The Foundations, §14 seems to clash with his conception of independence in his mature period. In Sect. 4, I scrutinize Frege’s somewhat vague, but unduly neglected remarks in The Foundations, §26 on space, spatial intuition and the axioms of Euclidean geometry. I argue that for the sake of coherence Frege should have said unambiguously that space is objective, that it is independedent not only of our spatial intuition, but of our mental life altogether including our judgements about space, instead of encouraging the possible conjecture that in his view it contains an objective and a subjective component. I further argue that for Frege the objectivity of both space and the axioms of Euclidean geometry manifests itself in our universal and compulsory acknowledgement of the Euclidean axioms as true. I conclude Sect. 4 by arguing that there is a conflict between the subjectivity of our spatial intuitions as stressed in The Foundations, §26 and Frege’s thesis in his dissertation that the axioms of Euclidean geometry derive their validity from the nature of our faculty of intuition. To resolve this conflict, I propose that in the light of his avowed realism in The Foundations Frege could have replaced his early thesis by saying that although we come to know the axioms of Euclidean geometry through spatial intuition and are justified in acknowledging them as true on the basis of geometrical intuition, their truth is independent not only of the nature of our faculty of intuition and singular acts of intuition, but of our mental processes and activities in general, including the inner mental act of judging. In Sect. 5, I argue that Frege most likely did not adopt Kant’s method of acquiring geometrical knowledge via the ostensive construction of concepts in spatial intuition. In contrast to Kant, Frege holds that the axioms of three-dimensional Euclidean geometry express state of affairs about space obtaining independently of our spatial intuition. In Sect. 6, I conclude with a summarized assessment of Frege’s philosophy of geometry.

online first; International audience; In this paper we focus on the formalization of the proofs of equivalence between different versions of Euclid's 5 th postulate. Our study is performed in the context of Tarski's neutral geometry, or equivalently in Hilbert's geometry defined by the first three groups of axioms, and uses an intuitionistic logic, assuming excluded-middle only for point equality. Our formalization provides a clarification of the conditions under which different versions of the postulates are equivalent. Following Beeson, we study which versions of the postulate are equivalent , constructively or not. We distinguish four groups of parallel postulates. In each group, the proof of their equivalence is mechanized using intuitionistic logic without continuity assumptions. For the equivalence between the groups additional assumptions are required. The equivalence between the 34 postulates is formalized in Archimedean planar neutral geometry. We also formalize a theorem due to Szmiliew. This theorem states that, assuming Archimedes' axiom, any statement which hold in the Euclidean plane and does not hold in the Hyperbolic plane is equivalent to Euclid's 5 th postulate. To obtain all these results, we have developed a large library in planar neutral geometry, including the formalization of the concept of sum of angles and the proof of the Saccheri-Legendre theorem, which states that assuming Archimedes' axiom, the sum of the angles in a triangle is at most two right angles.

It might be folly to insist on one overall official characterization of the way scientific and mathematical theories ought to be presented. They are usually presented of course in various ways so that the authors and scholars in a field can find optimal means for communication with others, whether within or outside that field. Nevertheless, if theories were thought of as representations of our knowledge, then one way of looking at the representation of theories is to ask, as Hermann Weyl did (2009) “What is the ultimate purpose of forming theories?”, and he then cited the familiar proposal of Heinrich Hertz (1894):This powerful and compact statement of Hertz, resonated with David Hilbert’s revolutionary proposal for the axiomatic representation of theories. The kind of axiomatization Hilbert advocated was exemplified in his Foundations of Geometry, [1899], and was to become the model for his inquiries into the physical sciences as well as mathematical ones.

Summary In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation), of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1), congruence equivalence relation (A2), congruence identity (A3), segment construction (A4), SAS (A5), betweenness identity (A6), Pasch (A7). Also a simple model, which satisfies these axioms, was previously constructed, and described in [6]. In this paper, we deal with four remaining axioms, namely: the lower dimension axiom (A8), the upper dimension axiom (A9), the Euclid axiom (A10), the continuity axiom (A11). They were introduced in the form of Mizar attributes. Additionally, the relation of congruence of triangles cong is introduced via congruence of sides (SSS). In order to show that the structure which satisfies all eleven Tarski’s axioms really exists, we provided a proof of the registration of a cluster that the Euclidean plane, or rather a natural [5] extension of ordinary metric structure Euclid 2 satisfies all these attributes. Although the tradition of the mechanization of Tarski’s geometry in Mizar is not as long as in Coq [11], first approaches to this topic were done in Mizar in 1990 [16] (even if this article started formal Hilbert axiomatization of geometry, and parallel development was rather unlikely at that time [8]). Connection with another proof assistant should be mentioned – we had some doubts about the proof of the Euclid’s axiom and inspection of the proof taken from Archive of Formal Proofs of Isabelle [10] clarified things a bit. Our development allows for the future faithful mechanization of [13] and opens the possibility of automatically generated Prover9 proofs which was useful in the case of lattice theory [7].

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.

In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution.

*Doutor em Lógica e Filosofia da Ciência pela Unicamp. Professor no Instituto de Filosofia da Universidade Federal de Uberlândia (UFU).David Hilbert e o axioma de Arquimedes: entre a geometria e a físicaResumo: A relação entre geometria e física na obra de Hilbert é analisada através do caso do Axioma de Arquimedes. Começando com as questões geométricas e as formais, em particular a definição de modelos não arquimedeanos para provar a independência, passa-se logo à concepção de Hilbert da geometria como uma ciência empírica, para depois estudar a afirmação de Hilbert de que o Axioma de Arquimedes deve ser testado empiricamente. Nesse sentido, esse autor enuncia uma formulação empírica do axioma, a qual, segundo afirma, deveria ser submetida à experimentação. Tal enunciado coloca três tipos de questões. Primeiro, se é realmente um enunciado empírico ou se é um princípio metodológico que não pode ser testado. Segundo, se é uma interpretação adequada desse axioma. Por último, como poderiam ser ideali­zados testes a partir desse enunciado. Sendo as primeiras questões problemáticas, pior é o caso da terceira, pois resulta difícil conceber experimentos bem definidos nos quais a formulação empírica possa ser testada, diferente, por exemplo, do caso da medição dos ângulos de um triângulo entre três picos encomendada por Gauss. Na estudo da formulação empírica também são analisados os comentários de Leo Corry e de Michael Stöltzer sobre o assunto, resultando em questionamentos sobre sua adequação e verificabilidade. Além disso, é salientada a importância de diferenciar os conceitos de mensuração, próprio do Axioma de Arquimedes, e de continuidade no sentido definido por Dedekind, baseado fundamentalmente na crítica que Sommer faz a Hilbert.David Hilbert e o axioma de Arquimedes: entre a geometria e a físicaAbstract: The relationship between geometry and physics in the work of Hilbert is analyzed through the case of the axiom of Archimedes. Starting with geometrical and formal issues (in particular, the definition of non-Archimedean models used for proving its independence), following with Hilbert conception that the physics is an empirical science, finally it is studied the Hilbertian statement that the axiom must be empirically confirmed. In this sense, he conceives an empirical statement of the axiom which must be confirmed by experiment. This statement arises three kind of questions. First, whether it actually is an empirical statement or a methodological rule which is not able to test. Second, whether it is a suitable interpretation of the axiom. Third, how can be created tests in relation to the empirical statement. Besides the fact that the two initial questions are difficult, the case of the third is worst, because, as far as I know, nobody proposed such a test, i.e. how can be designed well defined experiments to confirm the empirical statement (different, for example, of the case of the measurement of the sum of the angles of a triangle, performed by Gauss). The criticisms of Leo Corry and Michael Stöltzer are analyzed too, in particular the questions about adequacy and verification of the empirical statement. Furthermore, it is emphasized the relevance of the distinction between the concepts of measurement, inherent to the Archimedean axiom, and the one of the continuity (in Dedekind's sense), based upon the criticism of Sommer on the Foundations of Geometry of Hilbert.Keywords: Hilbert. Axiom of Archimedes. Geometry. Physics.Data de registro: 11/10/2013Data de aceite: 23/04/2014Referências:ARNOLD, V. I. Mathematics and physics. In: BONIOLO, G.; BUDINICH, P.; TROBOK, M. (Ed.). The role of mathematics in physical sciences, Dordrecht: Springer, p. 225-233, 2005. https://doi.org/10.1007/s00220-005-1300-2BERKELEY, G. The analyst: a discourse addressed to an infidel mathematician. Whitefish: Kessinger, 2004.BIRKHOFF, G.; MACLANE, S.A survey of modern algebra. New York: MacMillan, 1965.BOYER, C. B.; MERZBACH, U. C. A history of mathematics. 2. ed., New York: John Wiley & Son, 1991.CANTOR, G. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts . Berlin: Springer, 1932.CARRIER, M. Geometric facts and geometrie theory: Helmholtz and 20th-century philosophy of physical geometry. Berlin: Akademie Verlag, p.276-291, 1994.CORRY, L. David Hilbert and the axiomatization of physics (1898-1918): from Grundlagen der Geometrie to Grundlagen der Physik. Dordrecht: Kluwer, 2004.https://doi.org/10.1007/978-1-4020-2778-9______. Axiomatics, empiricism, and anschauung in Hilbert's conception of geometry: Between arithmetic and general relativity. In: FERREIRÓS, J.; GRAY, J. J. (Ed.). The architecture of modern mathematics. Essays in History and Philosophy, p. 133-156, Oxford: Oxford University Press, 2006.______. The origin of Hilberts axiomatic method. In: RENN, J.; SCHEMMEL, M. (Ed.). Theories of gravitation in the twilight of classical physics: the promise of mathematics and the dream of a unified theory, n.4, p. 759-855, Dordrecht: Springer, 2007.DAUBEN, J. W. Georg Cantor: his mathematics and philosophy of the infinite. Princeton: Princeton University Press, 1990.DEDEKIND, R. Stetigkeit und irrationale Zahlen, v. 3, Braunschweig: Friedr. Vieweg & Sohn, p. 315-334, 1932.______;WEBER, H. (Ed.). Gesammelte mathematische Werke und wissenschaftlicher Nachlass. Leipzig: B. G. Teubners, 1876.ENRIQUES, F. Prinzipien der Geometrie. Leipzig: Teubner, 1907.EUCLID. Elements. Lulu.Com. Texto grego com tradução inglesa, 2007.EWALD, W. From Kant to Hilbert: a source book in the foundations of mathematics, v. 2. New York: Oxford University Press, 1996.GÖDEL, K. Remark on non-standard analysis. In: FEFERMAN, S. (Ed.). Collected works II, n. 2, New York: Oxford University Press. p. 759-855, 1990.GRAY, J. Gauss and non-Euclidean geometry. New York: Springer, p. 61-80, 2006. https://doi.org/10.1007/0-387-29555-0_2HEATH, S.; HEIBERG, J. The thirteen books of Euclid's elements. Cambridge: Cambridge University Press, 1908.HENSEL, K. Ãœber eine neue Begründung der Theorie der algebraischen Zahlen. Jahresbericht der Deutschen Mathematiker-Vereinigung, v. 6, n. 3, p. 83-88, Jul. 1897.HILBERT, D. (1899). Grundlagen der Geometrie. In Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen. I. Theil. B. G. Teubner, Leipzig, 1 edition.______.Grundlagen der Geometrie. 2. ed., Leipzig: B. G. Teubner, 1903.______. Axiomatische Denken, v.3, Berlin: Springer, p. 146-156, 1935a. https://doi.org/10.1007/978-3-662-38452-7_10______. Gesammelte Abhandlungen III. Berlin: Springer, 1935b.______.David Hilbert's Lectures on the Foundations of Physics, 1915-1927: Relativity, Quantum Theory and Epistemology. Dordrecht: Springer-Verlag, 2006.HRBACEK, K.; JECH, T. Introduction to set theory. 3. ed., New York: Marcel Dekker, 1999.MAJER, U. Hilbert's axiomatic approach to the foundations of science - a failed research program? In: HENDRICKS, V. F.; JØRGENSEN, K. F.; LÃœTZEN, J.;PEDERSEN, S. A. (Ed.). Interactions — Mathematics, Physics and Philosophy, 1860-1930, v. 251, p. 155-183. Dordrecht: Springer, 2006. https://doi.org/10.1007/978-1-4020-5195-1_5MOISE, E. E. Elementary geometry from an advanced standpoint. 3. ed., Massachusetts: Addison-Wesley, 1990.POINCARÉ, H. Science and hypothesis. New York: Walter Scott, 1905.RAMAL, R.; TOULOUSE, G.; VIRASORO, M. Ultrametricity for physicists. Reviews of Modern Physics, v. 58, n. 3, p. 765-788, Jul. 1986. https://doi.org/10.1103/RevModPhys.58.765.ROBINSON, A. Non-standard analysis. Princeton; Princeton University Press, 1996.ROWE, D. The calm before the storm: Hilbert's early views on foundations. In: HENDRICKS, V. F.; LÃœTZEN, J.; PEDERSEN, S. A.; JØRGENSEN, K. F. (Ed.). Proof theory: history and philosophical significance, of Synthese Library. Studies in Epistemology, Logic, Methodology and Philosophy of Science, v. 292, p. 55-93. Dordrecht: Kluwer, 2000. https://doi.org/10.1007/978-94-017-2796-9_4 RUDIN, W. Principles of mathematical analysis. 3. ed., New York: McGraw-Hill, 1976.SCHEINERMAN, E. Matemática discreta- uma introdução. São Paulo: Thomson Learning, 2006.SOMMER, J. Hilbert's foundations of geometry. Bulletin of the American Mathematical Society (New Series), v. 6, n. 7, p. 287-299, Abr. 1900. https://doi.org/10.1090/S0002-9904-1900-00719-1STÖLTZNER, M. How metaphysical is 'deepening the foundations' — Hahn and Frank on Hilbert's axiomatic method. History of Philosophy of Science. New Trends and Perspectives, v. 55, n. 3, p. 245-262, 2002.TOEPELL, M. The origins and the further development of Hilbert ́s "Grundlagen der Geometrie". Le Matematiche, v. 55, n. 3, p. 207-226, 2000.TORRETTI, R. La geometria del universo. Mérida: Universidad de los Andes - Consejo de Publicaciones, 1994.VERONESE, G. Grundzüge der Geometrie von mehreren Dimensionen und mehreren Arten gradliniger Einheiten in elementarer Form entwickelt. Leipzig: Teubner, 1894.

Realistic Mathematics Education supports students’ formalization of their mathematical activity through guided reinvention. To operationalize “formalization” in a proof-oriented instructional context, I adapt Sjogren's (2010) claim that formal proof explicates (Carnap, 1950) informal proof. Explication means replacing unscientific or informal concepts with scientific ones. I use Carnap's criteria for successful explication – similarity, exactness, and fruitfulness – to demonstrate how the elements of mathematical theory – definitions, axioms, theorems, proofs – can each explicate their less formal correlates. This lens supports an express goal of the instructional project, which is to help students coordinate semantic (informal) and syntactic (formal) mathematical activity. I demonstrate the analytical value of the explication lens by applying it to examples of students’ mathematical activity drawn from a design experiment in undergraduate, neutral axiomatic geometry. I analyze the chains of meanings (Thompson, 2013) that emerged when formal elements were presented readymade alongside those emerging from guided reinvention.

This paper is about Poincare’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincare, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincare’s core commitment in the philosophy of geometry: the view that geometry is the study of groups of operations. In place of the established view I offer a revised view, according to which Poincare held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincare. Poincare’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them.

Primordially a geometry was a science on properties of geometrical objects and their mutual disposition. A use of the proper Euclidean geometry generated the axiomatic conception of geometry, where the geometry is considered as a logical construction. There is the metrical conception of a geometry, where the geometry is considered as a science on properties of geometric objects. In the framework of metrical conception the space-time geometries form a more powerful set of geometries, than those do in the framework of the axiomatic conception. It is important at the construction of the general relativity.

Summary This is the translation of the Mizar article containing readable Mizar proofs of some axiomatic geometry theorems formulated by the great Polish mathematician Alfred Tarski [8], and we hope to continue this work. The article is an extension and upgrading of the source code written by the first author with the help of miz3 tool; his primary goal was to use proof checkers to help teach rigorous axiomatic geometry in high school using Hilbert’s axioms. This is largely a Mizar port of Julien Narboux’s Coq pseudo-code [6]. We partially prove the theorem of [7] that Tarski’s (extremely weak!) plane geometry axioms imply Hilbert’s axioms. Specifically, we obtain Gupta’s amazing proof which implies Hilbert’s axiom I1 that two points determine a line. The primary Mizar coding was heavily influenced by [9] on axioms of incidence geometry. The original development was much improved using Mizar adjectives instead of predicates only, and to use this machinery in full extent, we have to construct some models of Tarski geometry. These are listed in the second section, together with appropriate registrations of clusters. Also models of Tarski’s geometry related to real planes were constructed.

International audience; This paper describes the formalization of the arithmetization of Euclidean geometry in the Coq proof assistant. As a basis for this work, Tarski's system of geometry was chosen for its well-known metamathematical properties. This work completes our formalization of the two-dimensional results contained in part one of [SST83]. We defined the arithmetic operations geometrically and proved that they verify the properties of an ordered field. Then, we introduced Cartesian coordinates, and provided characterizations of the main geometric predicates. In order to prove the characterization of the segment congruence relation, we provided a synthetic formal proof of two crucial theorems in geometry, namely the intercept and Pythagoras' theorems. To obtain the characterizations of the geometric predicates, we adopted an original approach based on bootstrapping: we used an algebraic prover to obtain new characterizations of the predicates based on already proven ones. The arithmetization of geometry paves the way for the use of algebraic automated deduction methods in synthetic geometry. Indeed, without a " back-translation " from algebra to geometry, algebraic methods only prove theorems about polynomials and not geometric statements. However, thanks to the arithmetization of geometry, the proven statements correspond to theorems of any model of Tarski's Euclidean geometry axioms. To illustrate the concrete use of this formalization, we derived from Tarski's system of geometry a formal proof of the nine-point circle theorem using the Gröbner basis method.

One of the issues at stake in the discussion about the origin and meaning of geometrical axioms was to establish the preconditions for the possibility of spatial measurement. A related issue was to analyze the concept of number to gain insights into its relation to that of magnitude. Despite the traditional definition of arithmetic as the theory of quantities, numbers cannot be identified as magnitudes. Numbers can only represent magnitudes in measurement situations. In order to justify the use of numbers in modeling measurement situations, some conditions are required. The study of these conditions is now known as measurement theory. Helmholtz has been acknowledged as one of the forefathers of measurement theory. However, the connection between Helmholtz’s analysis of measurement and his inquiry into the foundations of geometry has not received much attention.

Klein’s classification of geometries by the use of group theory inaugurated a new phase in the debate on the geometry of space. On the one hand, the conclusion of Riemann’s and Helmholtz’s inquiries into the foundations of geometry appeared to be confirmed: Euclidean geometry does not provide us with the necessary presuppositions for empirical measurement, because both Euclidean and non-Euclidean assumptions can be obtained as special cases of a more general system of hypotheses. On the other hand, Helmholtz had believed that he had shown that the free mobility of rigid bodies implied and was implied by a metric of constant curvature, which includes spherical and elliptic geometries. The group-theoretical approach enabled Sophus Lie to disprove Helmholtz’s argument and provide a mathematically sound solution to the same problem. The most challenging argument against Helmholtz’s empiricism, however, was formulated by Henri Poincare: observation and experiment cannot contradict geometrical assumptions, because the application of geometrical concepts to empirical objects, including the characterization of solid bodies as “rigid,” already presupposes these kinds of assumptions. The present chapter is devoted to the reception of Poincare’s argument in neo-Kantianism. In particular, I contrast Poincare’s conclusion that geometrical axioms are conventions with Cassirer’s view that the interpretation of measurements depends on conceptual rules and ultimately on rational rather than conventional criteria. Cassirer relied on the group-theoretical analysis of space to infer such criteria from the relations of geometrical systems to one another.

The development of non-Euclidean geometry in the nineteenth century led mathematicians, scientists, and philosophers to reconsider the foundations of geometry. One of the issues at stake was to redefine the notion of geometrical axiom and to establish criteria of choice among different axiomatic systems in case of equivalent geometries. The possibility of considering a variety of hypotheses concerning physical space appeared to contradict Kant’s conception of geometrical axioms as a priori synthetic judgments. Therefore, Riemann called geometrical axioms hypotheses, and maintained that the geometry of physical space is a matter for empirical investigation. In order to support this view, Helmholtz pointed out the empirical origin of geometrical axioms. At the same time, he foreshadowed a conventionalist conception of geometrical axioms as definitions that can be abstracted from our experiences with solid bodies and their free mobility.

It is well known that the calculus of reflections (developed by Hjelmslev, Bachmann et al.) enables the derivation of a large part of Euclidean and non-Euclidean geometry without using assumptions about order and continuity. We show in this article that the calculus of reflections can conversely be used to introduce a relation of order in hyperbolic geometry. Our investigations are based on the famous ‘Endenrechnung’ of Hilbert which was formulated purely in terms of the calculus of reflections by F. Bachmann. We then discuss some implications of these results and show that the calculus of reflections enables (1) the introduction of an order relation in a Pappian projective line and (2) to define an axiom system for hyperbolic planes which seems to be as simple as the famous axiom system of Menger who only used the notion of point-line incidence to axiomatize plane hyperbolic geometry.

A geometric model is introduced which satisfies the Euclidean parallel postulate as well as all of Hilbert's axioms except the Side-Angle-Side axiom. This model provides several teaching opportunities in those Euclidean geometry classrooms that use the axiomatic method. In presenting these models at the same time as the more familiar ℝ2, ℝ3, and Poincare models, students may be less tempted to assume that familiar constructs (lines, trianges, etc.) allow them to rely on familiar assumptions.

A set of axioms for classical absolute geometry is proposed that is accessible to students new to axioms. The metric axioms adopted are the ruler axiom, triangle inequality and the bisector axiom. Angle measure is derived from distance, and all properties needed to establish a consistent system are derived. In particular, the SAS congruence theorem is proved. The proofs are broken into many small steps suitable for the target audience. Some explorations with geometry drawing programmes are shown.

The Swedish mathematician Torsten Brodén (1857–1931) wrote two articles on the foundations of Euclidean geometry. The first was published in 1890, almost a decade before Hilbert's first attempt, and the second was published in 1912. Brodén's philosophical view of the nature of geometry is discussed and his thoughts on axiomatic systems are described. His axiomatic system for Euclidean geometry from 1890 is considered in detail and compared with his later work on the foundations of geometry. The two continuity axioms given are compared to and proved to imply Hilbert's two continuity axioms of 1903.

We formulate and prove an axiomatic characterization of the Riemannian geometry underlying manifolds of conditional models. The characterization holds for both normalized and nonnormalized conditional models. In the normalized case, the characterization extends the derivation of the Fisher information by Cencov while in the nonnormalized case it extends Campbell's theorem. Due to the close connection between the conditional I-divergence and the product Fisher information metric, we provides a new axiomatic interpretation of the geometries underlying logistic regression and AdaBoost

Focusing methodologically on those historical aspects that are relevant to supporting intuition in axiomatic approaches to geometry, the book develops systematic and modern approaches to the three core aspects of axiomatic geometry: Euclidean, non-Euclidean and projective. Historically, axiomatic geometry marks the origin of formalized mathematical activity. It is in this discipline that most historically famous problems can be found, the solutions of which have led to various presently very active domains of research, especially in algebra. The recognition of the coherence of two-by-two contradictory axiomatic systems for geometry (like one single parallel, no parallel at all, several parallels) has led to the emergence of mathematical theories based on an arbitrary system of axioms, an essential feature of contemporary mathematics. This is a fascinating book for all those who teach or study axiomatic geometry, and who are interested in the history of geometry or who want to see a complete proof of one of the famous problems encountered, but not solved, during their studies: circle squaring, duplication of the cube, trisection of the angle, construction of regular polygons, construction of models of non-Euclidean geometries, etc. It also provides hundreds of figures that support intuition. Through 35 centuries of the history of geometry, discover the birth and follow the evolution of those innovative ideas that allowed humankind to develop so many aspects of contemporary mathematics. Understand the various levels of rigor which successively established themselves through the centuries. Be amazed, as mathematicians of the 19th century were, when observing that both an axiom and its contradiction can be chosen as a valid basis for developing a mathematical theory. Pass through the door of this incredible world of axiomatic mathematical theories!

We describe a first order axiom set which yields the classical first order Euclidean geometry of Tarski when used with classical logic, and yields an intuitionistic (or constructive) Euclidean geometry when used with intuitionistic logic. The first order language has a single six place atomic predicate and no function symbols. The intuitionistic system has a computational interpretation in recursive function theory, that is, a realizability interpretation analogous to those given by Kleene for intuitionistic arithmetic and analysis. This interpretation shows the unprovability in the intuitionistic theory of certain “nonconstructive” theorems of the classical geometry.

Three things are presented: How Hilbert changed the original construction postulates of his geometry into existential axioms; In what sense he formalized geometry; How elementary geometry is formalized to present day's standards.

In this paper we give an axiomatization of dierential geometry com- parable to model categories for homotopy theory. Weil functors play a predominant role.

International audience; While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski’s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating “rigorously” with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as “drawn formulas”, and formulas as “written diagrams”, thus suggesting that the former encapsulate propositional information which can be extracted and translated into formulas. In the case of Minkowski diagrams, local geometrical axioms were actually being produced, starting with the diagrams, by a process that was both constrained and fostered by the requirement, brought about by the axiomatic method itself, that geometry ought to be made independent of analysis. This paper aims at making a twofold point. On the one hand, it shows that Minkowski’s diagrammatic methods in number theory prompted Hilbert’s axiomatic investigations into the notion of a straight line as the shortest distance between two points, which start from his earlier work focused on the role of the triangle inequality property in the foundations of geometry, and lead up to his formulation of the 1900 Fourth Problem. On the other hand, it purports to make clear how Hilbert’s assessment of Minkowski’s diagram-based reasoning in number theory both raises and illuminates conceptual compatibility concerns that were crucial to his philosophy of mathematics.

First, Peano’s geometrical calculus theory is a general theory which is of intrinsic mathematical interest and which is also applied to mechanics and to physics. Peano’s contributions, which come from an elaboration of Grassmann’s ideas, consist in an Euclidean interpretation of relative concepts. Moreover, in this context, Peano proves important fundamental theorems of projective geometry. For this reason, Peano’s geometrical calculus has an implicit foundational interest. In our opinion, the protophysical role of Euclidean geometry in Peano’s works is essential and decisive. He distinguishes position geometry from Euclidean geometry, and from a theoretical point of view, it is appropriate. In his ‘Sui fondamenti della geometria’ the congruence theory is well determined and regulated. Classical geometry constitutes the crucial model for the study of the foundations of geometry. Even Hilbert, deep down, takes Euclid into account20. During this period, we have many proposals of systems with different essential or primitive notions and axioms. Hence, we can observe “equivalent theories” for the foundation of elementary geometry, and in this way we have a “theoretical relativism” regarding the choice of primitive elements and fundamental axioms. This is epistemologically and historiographically21 very important22.

Object representation and reasoning in vector based geographic information systems (GIS) is based on Euclidean geometry. Euclidean geometry is built upon Euclid's first postulate, stating that two points uniquely determine a line. This postulate makes geometric constructions unambiguous and thereby provides the foundation for consistent geometric reasoning. It holds for exact coordinate points and lines, but is violated, if points and lines are allowed to have extension. As an example for a point that has extension consider a point feature that represents the city of Vienna in a small scale GIS map representation. Geometric constructions with such a point feature easily produce inconsistencies in the data. The present paper addresses the issue of consistency by formalizing Euclid's first postulate for geometric primitives that have extension.We identify a list of six consequences from introducing extension: These are 'new qualities' that are not present in exact geometric reasoning, but must be taken into account when formalizing Euclid's first postulate for extended primitives. One important consequence is the positional tolerance of the incidence relation ("on"-relation). As another consequence, equality of geometric primitives becomes a matter of degree. To account for this fact, we propose a formalization of Euclid's first postulate in Lukasiewicz t-norm fuzzy logic. A model of the proposed formalization is given in the projective plane with elliptic metric. This is not a restriction, since the elliptic metric is locally Euclidean. We introduce graduated geometric reasoning with Rational Pavelka Logic as a means of approximating and propagating positional tolerance through the steps of a geometric construction process. Since the axioms (postulates) of geometry built upon one another, the proposed formalization of Euclid's first postulate provides one building block of a geometric calculus that accounts for positional tolerance in a consistent way.The novel contribution of the paper is to define geometric reasoning with extended primitives as a calculus that propagates positional tolerance. Also new is the axiomatic approach to positional uncertainty and the associated consistency issue.

This article is devoted to the problem of ontological foundations of three-dimensional Euclidean geometry. Starting from Bertrand Russell’s intuitions concerning the sensual world we try to show that it is possible to build a foundation for pure geometry by means of the so called regions of space. It is not our intention to present mathematically developed theory, but rather demonstrate basic assumptions, tools and techniques that are used in construction of systems of point-free geometry and topology by means of mereology (resp. Boolean algebras) and Whitehead-like connection structures. We list and briefly analyze axioms for mereological structures, as well as those for connection structures. We argue that mereology is a good tool to model so called spatial relations. We also try to justify our choice of axioms for connection relation. Finally, we briefly discuss two theories: Grzegorczyk’s point-free topology and Tarski’s geometry of solids.

In this paper we give probably an exhaustive analysis of the geometry of solids which was sketched by Tarski in his short paper [20, 21]. We show that in order to prove theorems stated in [20, 21] one must enrich Tarski's theory with a new postulate asserting that the universe of discourse of the geometry of solids coincides with arbitrary mereological sums of balls, i.e., with solids. We show that once having adopted such a solution Tarski's Postulate 4 can be omitted, together with its versions 4′ and 4″. We also prove that the equivalence of postulates 4, 4′ and 4″ is not provable in any theory whose domain contains objects other than solids. Moreover, we show that the concentricity relation as defined by Tarski must be transitive in the largest class of structures satisfying Tarski's axioms.We build a model (in three-dimensional Euclidean space) of the theory of so called T*-structures and present the proof of the fact that this is the only (up to isomorphism) model of this theory.Moreover, we propose different categorical axiomatizations of the geometry of solids. In the final part of the paper we answer the question concerning the logical status (within the theory of T*-structures) of the definition of the concentricity relation given by Tarski.

Dynamic geometry tools (e.g., Cinderella, Geometer's Sketchpad, Cabri, Eukleides) visualise geometric objects, allow interactive work, and link formal, axiomatic nature of geometry (most often — Euclidean) with its standard models (e.g., Cartesian model) and corresponding illustrations. These tools are used in teaching and studying geometry, some of them also for producing digital illustrations. The common experience is that dynamic geometry tools significantly help students to acquire knowledge about geometric objects. However, despite the fact that geometry is an axiomatic theory, most (if not all) of these tools concentrate only on concrete models of some geometric constructions and not on their abstract properties — their properties in deductive terms. The user can vary some initial objects and parameters and test if some property holds in all checked cases, but this still does not mean that the given property is valid.

Introduction In the Fall of 1997 Ohio University replaced its traditional “Foundations of Geometry” sequence with one in which the students develop their own sets of axioms and use them to establish some well-known results of plane geometry. By constructing their own axioms, the students gain a sense of both the source and the role of formal axiomatic systems. Since axioms are introduced and developed as needed, the students gain an appreciation of the significance of each axiom as it is added to the set of axioms. As the students start the course by developing their own axioms, the course is not amenable to the traditional lecture approach of developing the material. The students develop their axiom systems while working in structured cooperative groups and making use of a variety of manipulatives and software programs during their discussions. By the end of the sequence, the students have addressed all of the concepts included in the traditional course and more. They also gain, in our belief, a deeper understanding of the material than would be developed in the traditional lecture style course. The General Approach The projects described in this paper were designed for a geometry course taken primarily by prospective middle or high school teachers. The major theme of the projects is to connect experience and abstract mathematics. The early projects are designed to give students experience in working with non-Euclidean geometries while exploring the validity of certain common sense propositions in these geometries.

Publisher Summary In mathematics David Hilbert's Grundlagen der Geometrie was an influential work that led by its axiomatic method to a new thinking in all mathematical fields in the 20th century. In addition, following in the traces of Euclid, it became the classical textbook for geometry in educating mathematicians and mathematics teachers for nearly the whole century. Towards the end of the 19th, century a remarkable change came about in the field of the foundations of geometry. For the first time David Hilbert constructed the axioms in what was subsequently to be their usual sequence. In this manuscript the arrangement of the later Festschrift is already apparent: axioms, proofs of independences, segment arithmetic, Desargues's theorem, Pascal's theorem, and problems concerning constructability. The manuscript Grundlagen der Euklidischen Geometrie (EG) contains an exhaustive discussion of those areas that were mostly treated in brief in the vacation course. The logical meaning of the axioms was studied by construction of arithmetical models. Amongst these were proofs of independence for axioms of the first two groups. The book had some consequences for physics, which gained Hilbert's attention from the 1900s onwards. For during the 1910s, there was a remarkable connection between Einstein and Hilbert.

Modern foundational research got its full start with Cantor’s publications on set theory from 1874 onwards. The tradition which developed in this style was to include as its contributors some of the most famous mathematicians of that time. With the publication of Was sind and was sollen die Zahlen? (‘What are numbers and what should they be?’) in 1887, Dedekind was considered to have given a secure foundation for the theory of natural numbers, based on the laws of logic. Hilbert gave a strictly axiomatic foundation to geometry in his Grundlagen der Geometrie (‘Foundations of geometry’) in 1899, and later extended this to his general proof theory. Zermelo axiomatised set theory. It was a tradition of rigorous proofs with an appeal to logic rather than to intuition. And it was against this tradition that several mathematicians protested, who, by so doing, held views more or less similar to the ones Brouwer was to take later on. This chapter is about these mathematicians. In chronological order, they are: Kronecker, the French semi-intuitionists Borel, Baire and Lebesgue, and Poincare.

Karl Menger was one of the leading and great geometers of our Century. No doubt, much of his work in geometry must be considered as spectacular. Moreover, it initiated and substantially influenced further research in different branches of geometry. The aim of this commentary is to follow the development of his theory Algebra of Geometry, a theory which, in part, also goes under the name of Lattice-theoretic Foundations of Geometry or Geometry and Lattices. Menger was the first to present lattice-theoretic axioms in order to characterize geometrical structures [1]. This fundamental and memorable result is connected with his name in the Foundations of Geometry [2]. It also must be stressed that, although the notion of duality had long been commonplace, Menger was the first geometer to present a self-dual foundation of projective geometry.

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.1

This paper looks into Hilbert’s thought about mathematics and explores its relation which the philosophy of Kant. The focus of the research is in the role of axiomatic thinking and logical analysis in foundational studies. The paper concentrates mainly in Hilbert’s research regarding the foundations of geometry, and follows his main lines of thought up to his programme, which revolves around a consistency proof for the axioms of classical mathematics. A final analysis allows us to conclude that for him mathematics is, in a broad sense, “the science of that which is possible” in this point, Hilbert diverges from Kant, even though he considers that classical mathematics has in its core a content, a view which separates him from the extreme formalism some times ascribed to him.

In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spin^c geometry depending on whether the geometry is ''real'' or not. We attempt to flesh out the details of Connes' ideas. As an illustration we present a proof of his claim, partly extending the validity of the result to pseudo-Riemannian spin manifolds. Throughout we are as explicit and elementary as possible., Comment: Re-tex to get references right. This is a revised version of a previously incorrect version. Changes to the central portion of proof are extensive. 48 pp

Kolmogorov’s axioms of probability theory give an approach, generally accepted at the present time, to the mathematical description of probabilistic-statistical phenomena. The problem of axiomatizing probability theory was formulated in the Sixth problem of D. Hilbert in his famous address of 8 August 1900 at the Second International Congress of Mathematicians in Paris. Hilbert, who included probability theory in physics (as was the general acceptance at the time), formulated the 6th problem as “The mathematical statement of the axioms of physics” (Hilbert 1901): “Closely connected with investigations into the foundations of geometry is the problem of axiomatizing the construction within that same framework of those physical disciplines in which mathematics already plays a distinguished role: in the first place, this is the theory of probability and mechanics.

The parallel postulate V, Euclid’s fifth postulate, seems less natural or convincing than the others. Ever since Euclid’s time, people have felt that it ought to be deducible from Euclid’s other postulates I to IV or from some logically equivalent set of axioms.

The axioms systems of Euclid and Hilbert were intended to provide everything needed for plane geometry without any prior development. The axioms of Hilbert include information about the lines in the plane that implies that each line can be identified with the structure commonly called the “real numbers” and denoted by ℝ. Euclid’s axioms also include information of that kind but the meaning of the “real numbers” may not have been the same in that era as it is now. In both cases, geometry was taken as more fundamental than the real number system. Instead we are going to use ℝ as an ingredient in laying the foundations of hyperbolic plane geometry. (See the Appendix to this chapter for a discussion of the real number system, its properties, its consistency and its uniqueness.) Our axiom system is equivalent to that of Hilbert for the hyperbolic plane, and following the laying of the foundations in this chapter we proceed rigorously with the development of its properties, its consistency and its uniqueness, in later chapters.

Following Tarski, we view n n -dimensional Euclidean geometry as a first-order theory E n {E_n} with an infinite set of axioms about the relations of betweenness (among points on a line) and equidistance (among pairs of points). We show that for k > n k > n , E n {E_n} does not admit a k k -dimensional interpretation in the theory RCF of real closed fields, and we deduce that E n {E_n} cannot be interpreted r r -dimensionally in E s {E_s} , when r ⋅ s > n r \cdot s > n .