Notes on axiomatic geometry

Handbook of Spatial Logics, 2007

2015

Similar remarks appear in Russell’s Principles of Mathematics, in Principia Mathematica, and in the 1906 “The Theory of Implication.” The sentiment expressed in these passages is a strange one, especially given its setting in the first decade of the twentieth century. The modern method of demonstrating independence had become, by this point, standard fare, having been applied already in geometry, arithmetic, analysis, and class theory. And despite Russell’s concerns, the method was soon to be applied (by Bernays) to demonstrate the independence of Russell’s own fundamental principles of logic as expressed in Principia Mathematica.

Beiträge Zur Algebra Und Geometrie / Contributions To Algebra And Geometry, 2020

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.

Handbook of Spatial Logics, 2007

We survey models and theories of geometric structures of parallelism, orthogonality, incidence, betweenness and order, thus gradually building towards full elementary geometry of Euclidean spaces, in Tarski's sense. Besides the geometric aspects of such structures we look at their logical (first-order and modal) theories and discuss logical issues such as: expressiveness and definability, axiomatizations and representation results, completeness and decidability, and interpretations between structures and theories.

2022

Despite the efforts undertaken to separate scientific reasoning and metaphysical considerations, despite the rigor of construction of mathematics, these are not, in their very foundations, independent of the modalities, of the laws of representation of the world. The OdC shows that the logical Facts Exist neither more nor less than the Facts of the world which are Facts of Knowledge. Mathematical facts are representation facts. The primary objective of this article is to integrate the subject into mathematics as a mode of emergence of meaning and then to evaluate its consequences.

The present work is devoted to the exploration of some formal possibilities suggesting, since some years, the possibility to elaborate a new, whole geometry, relative to the concept of “opposition”. The latter concept is very important and vast (as for its possible applications), both for philosophy and science and it admits since more than two thousand years a standard logical theory, Aristotle’s “opposition theory”, whose culminating formal point is the so called “square of opposition”. In some sense, the whole present enterprise consists in discovering and ordering geometrically an infinite amount of “avatars” of this traditional square structure (also called “logical square” or “Aristotle’s square”). The results obtained here go even beyond the most optimistic previous expectations, for it turns out that such a geometry exists indeed and offers to science many new conceptual insights and formal tools. Its main algorithms are the notion of “logical bi-simplex of dimension m” (which allows “opposition” to become “n-opposition”) and, beyond it, the notions of “Aristotelian pq-semantics” and “Aristotelian pq-lattice” (which allow opposition to become p-valued and, more generally, much more fine-grained): the former is a game-theoretical device for generating “opposition kinds”, the latter gives the structure of the “opposition frameworks” containing and ordering the opposition kinds. With these formal means, the notion of opposition reaches a conceptual clarity never possible before. The naturalness of the theory seems to be maximal with respect to the object it deals with, making this geometry the new standard for dealing scientifically with opposition phenomena. One question, however, philosophical and epistemological, may seem embarrassing with it: this new, successful theory exhibits fundamental logical structures which are shown to be intrinsically geometrical: the theory, in fact, relies on notions like those of “simplex”, of “n-dimensional central symmetry” and the like. Now, despite some appearances (that is, the existence, from time to time, of logics using some minor spatial or geometrical features), this fact is rather revolutionary. It joins an ancient and still unresolved debate over the essence of mathematics and rationality, opposing, for instance, Plato’s foundation of philosophy and science through Euclidean geometry and Aristotle’s alternative foundation of philosophy and science through logic. The geometry of opposition shows, shockingly, that the logical square, the heart of Aristotle’s transcendental, anti-Platonic strategy is in fact a Platonic formal jungle, containing geometrical-logical hyper-polyhedra going into infinite. Moreover, this fact of discovering a lot of geometry inside the very heart of logic, is also linked to a contemporary, raging, important debate between the partisans of “logic-inspired philosophy” (for short, the analytic philosophers and the cognitive scientists) and those, mathematics-inspired, who begin to claim more and more that logic is intrinsically unable to formalise, alone, the concept of “concept” (the key ingredient of philosophy), which in fact requires rather geometry, for displaying its natural “conceptual spaces” (Gärdenfors). So, we put forward some philosophical reflections over the aforementioned debate and its deep relations with questions about the nature of concepts. As a general epistemological result, we claim that the geometrical theory of oppositions reveals, by contrast, the danger implicit in equating “formal structures” to “symbolic calculi” (i.e. non-geometrical logic), as does the paradigm of analytic philosophy. We propose instead to take newly in consideration, inspired by the geometry of logic, the alternative paradigm of “structuralism”, for in it the notion of “structure” is much more general (being not reduced to logic alone) and leaves room to formalisations systematically missed by the “pure partisans” of “pure logic”.

This paper is a continuation of earlier work by the author on the connection between the logic KR and projective geometry. It contains a new, simplified construction of KR model structures; as a consequence, it extends the previous results to a much more extensive class of projective spaces and the corresponding modular lattices.

2008

Incidence spatial geometry is based on three-sorted structures consisting of points, lines and planes together with three intersort binary relations between points and lines, lines and planes and points and planes. We introduce an equivalent one-sorted geometrical structure, called incidence spatial frame, which is suitable for modal considerations. We are going to prove completeness by SD-Theorem. Extensions to projective, affine and hyperbolic geometries are also considered. Un sistema axiomático para la geometría espacial de incidencia Resumen. La geometría espacial de incidencia está construida por medio de estructuras trisurtidas formadas por puntos, rectas y planos con relaciones binarias de interconexión, para cada dos de estos elementos. En este trabajo introducimos una estructura monosurtida, que denominamos marco espacial de incidencia y que resulta adecuada para un tratamiento modal. Probaremos la completitud del sistema por medio del SD-teorema. Las extensiones a los casos proyectivo, afín e hiperbólico son también considerados.

The American Mathematical Monthly, 2010

Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid's, which handled some basic concepts awkwardly and imprecisely. They would introduce precision required for generalization and application to new, delicate problems in higher mathematics. Their work was controversial: they departed from tradition, criticized standards of rigor, and addressed fundamental questions in philosophy. This paper follows the problem, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and an argument that one of those is optimal. It's about axioms, definitions, and definability, and emphasizes contributions of Mario Pieri (1860-1913) and Alfred Tarski (1901. By following this thread of ideas and personalities to the present, the author hopes to kindle interest in a fascinating research area and an exciting era in the history of mathematics.

Logic and Logical Philosophy, 2014

We suggest possible approaches to point-free geometry based on multi-valued logic. The idea is to assume as primitives the notion of a region together with suitable vague predicates whose meaning is geometrical in nature, e.g. 'close', 'small', 'contained'. Accordingly, some first-order multivalued theories are proposed. We show that, given a multi-valued model of one of these theories, by a suitable definition of point and distance we can construct a metrical space in a natural way. Taking into account that interesting metrical approaches to geometry exist, this looks to be promising for a point-free foundation of the notion of space. We hope also that this way to face point-free geometry provides a tool to illustrate the passage from a naïve and 'qualitative' approach to geometry to the 'quantitative' approach of advanced science.

Problems of education in the 21st century, 2024

Proceedings of the 10th Asian Logic Conference, 2009

Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean "constructions" to "constructive mathematics" leads to the development of a first-order theory ECG of the "Euclidean Constructive Geometry", which can serve as an axiomatization of Euclid rather close in spirit to the Elements of Euclid. ECG is axiomatized in a quantifier-free, disjunction-free way. Unlike previous intuitionistic geometries, it does not have apartness. Unlike previous algebraic theories of geometric constructions, it does not have a test-for-equality construction. We show that ECG is a good geometric theory, in the sense that with classical logic it is equivalent to textbook theories, and its models are (intuitionistically) planes over Euclidean fields. We then apply the methods of modern metamathematics to this theory, showing that if ECG proves an existential theorem, then the object proved to exist can be constructed from parameters, using the basic constructions of ECG (which correspond to the Euclidean straightedge-and-compass constructions). In particular, objects proved to exist in ECG depend continuously on parameters. We also study the formal relationships between several versions of Euclid's parallel postulate, and show that each corresponds to a natural axiom system for Euclidean fields. 1

Journal of Philosophical Logic, 1988

arXiv (Cornell University), 2021

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the homogeneity of space), all directions are the same to us (the isotropy of space) and all units of length we use to create geometric figures are the same to us (the scale invariance of space). On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl's system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: (i) it supports the thesis that Euclidean geometry is a priori, (ii) it supports the thesis that in modern mathematics the Weyl's system of axioms is dominant to the Euclid's system because it reflects the a priori underlying symmetries, (iii) it gives a new and promising approach to learn geometry which, through the Weyl's system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics.