Results 1  10
of
34
A formal system for Euclid's Elements
, 2009
"... We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning. ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
(Show Context)
We present a formal system, E, which provides a faithful model of the proofs in Euclid’s Elements, including the use of diagrammatic reasoning.
Transitive closure and the mechanization of mathematics
 Thirtyfive Years of Automating Mathematics
, 2003
"... Abstract. We argue that the concept of transitive closure is the key for understanding nitary inductive denitions and reasoning, and we provide evidence for the thesis that logics which are based on it (in which induction is a logical rule) are the right logical framework for the formalization and ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
(Show Context)
Abstract. We argue that the concept of transitive closure is the key for understanding nitary inductive denitions and reasoning, and we provide evidence for the thesis that logics which are based on it (in which induction is a logical rule) are the right logical framework for the formalization and mechanization of Mathematics. We investigate the expressive power of languages with the most basic transitive closure operation TC. We show that with TC one can dene all recursive predicates and functions from 0, the successor function and addition, yet with TC alone addition is not denable from 0 and the successor function. However, in the presence of a pairing function, TC does suÆce for having all types of nitary inductive denitions of relations and functions. This result is used for presenting a simple version of Feferman's framework FS 0, demonstrating that TClogics provide in general an excellent framework for mechanizing formal systems. An interesting side eect of these results is a simple characterization of recursive enumerability and a new, concise version of Church thesis. We end with a use of TC for a formalization of Set Theory which is based on purely syntactical considerations, and re
ects real mathematical practice. 1.
Empirical foundation of space and time
 EPSA; Epistemology and Methodology: Launch of the European Philosophy of Science Association Proceedings of the First Conference of the European Philosophy of Science Association
"... I will sketch a possible way of empirical/operational definition of space and time tags of physical events, without logical or operational circularities and with a minimal number of conventional elements. As it turns out, the task is not trivial; and the analysis of the problem leads to a few surpri ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
(Show Context)
I will sketch a possible way of empirical/operational definition of space and time tags of physical events, without logical or operational circularities and with a minimal number of conventional elements. As it turns out, the task is not trivial; and the analysis of the problem leads to a few surprising conclusions. 1
AXIOMS, ALGEBRAS, AND TOPOLOGY
"... This work explores the interconnections between a number of different perspectives on the formalisation of space. We begin with an informal discussion of the intuitions that motivate these formal representations. ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
This work explores the interconnections between a number of different perspectives on the formalisation of space. We begin with an informal discussion of the intuitions that motivate these formal representations.
Axiomatizing geometric constructions
, 2008
"... In this survey paper, we present several results linking quantifierfree axiomatizations of various Euclidean and hyperbolic geometries in languages without relation symbols to geometric constructibility theorems. Several fragments of Euclidean and hyperbolic geometries turn out to be naturally occu ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
In this survey paper, we present several results linking quantifierfree axiomatizations of various Euclidean and hyperbolic geometries in languages without relation symbols to geometric constructibility theorems. Several fragments of Euclidean and hyperbolic geometries turn out to be naturally occurring only when we ask for the universal theory of the standard plane (Euclidean or hyperbolic), that can be expressed in a certain language containing only operation symbols standing for certain geometric constructions.
Vienna Circle and Logical Analysis of Relativity Theory
, 2009
"... 1 introduction In this paper we present some of our school’s results in the area of building up relativity theory (RT) as a hierarchy of theories in the sense of logic. We use plain firstorder logic (FOL) as in the foundation of mathematics (FOM) and we build on experience gained in FOM. The main a ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
1 introduction In this paper we present some of our school’s results in the area of building up relativity theory (RT) as a hierarchy of theories in the sense of logic. We use plain firstorder logic (FOL) as in the foundation of mathematics (FOM) and we build on experience gained in FOM. The main aims of our school are the following: We want to base the theory on simple, unambiguous axioms with clear meanings. It should be absolutely understandable for any reader what the axioms say and the reader can decide about each axiom whether he likes it. The theory should be built up from these axioms in a straightforward, logical manner. We want to provide an analysis of the logical structure of the theory. We investigate which axioms are needed for which predictions of RT. We want to make RT more transparent logically, easier to understand, easier to change, modular, and easier to teach. We want to obtain deeper understanding of RT. Our work can be considered as a casestudy showing that the Vienna
Multidimensional Mereotopology with Betweenness ∗
"... Qualitative reasoning about commonsense space often involves entities of different dimensions. We present a weak axiomatization of multidimensional qualitative space based on ‘relative dimension ’ and dimensionindependent ‘containment ’ which suffice to define basic dimensiondependent mereotopolog ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Qualitative reasoning about commonsense space often involves entities of different dimensions. We present a weak axiomatization of multidimensional qualitative space based on ‘relative dimension ’ and dimensionindependent ‘containment ’ which suffice to define basic dimensiondependent mereotopological relations. We show the relationships to other meoreotopologies and to incidence geometry. The extension with betweenness, a primitive of relative position, results in a firstorder theory that qualitatively abstracts ordered incidence geometry. 1
OTTER proofs in Tarskian geometry
"... Abstract. We report on a project to use OTTER to find proofs of the theorems in Tarskian geometry proved in Szmielew’s part (Part I) of [9]. These theorems start with fundamental properties of betweenness, and end with the development of geometric definitions of addition and multiplication that perm ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We report on a project to use OTTER to find proofs of the theorems in Tarskian geometry proved in Szmielew’s part (Part I) of [9]. These theorems start with fundamental properties of betweenness, and end with the development of geometric definitions of addition and multiplication that permit the representation of models of geometry as planes over Euclidean fields, or over realclosed fields in the case of full continuity. They include the four challenge problems left unsolved by Quaife, who two decades ago found some OTTER proofs in Tarskian geometry (solving challenges issued in [15]). Quaife’s four challenge problems were: every line segment has a midpoint; every segment is the base of some isosceles triangle; the outer Pasch axiom (assuming inner Pasch as an axiom); and the first outer connectivity property of betweenness. These are to be proved without any parallel axiom and without even linecircle continuity. These are difficult theorems, the first proofs of which were the heart of Gupta’s Ph. D. thesis under Tarski. OTTER proved them all in 2012. Our success, we argue, is due to improvements in techniques of automated deduction, rather than to increases in computer speed and memory. The theory of Hilbert (1899) can be translated into Tarski’s language, interpreting lines as pairs of distinct points, and angles as ordered triples of noncollinear points. Under this interpretation, the axioms of Hilbert either occur among, or are easily deduced from, theorems in the first 11 (of 16) chapters of Szmielew. We have found Otter proofs of all of Hilbert’s axioms from Tarski’s axioms (i.e. through Satz 11.49 of Szmielew, plus Satz 12.11). Narboux and Braun have recently checked these same proofs in Coq. 1