Space and Time

A Priori and A Posteriori Studies

Vincenzo Fano, Francesco Orilia, Giovanni Macchia, Vincenzo Fano, Francesco Orilia, Giovanni Macchia

• Publisher: De Gruyter
• Year: 2014
• Print ISBN:
  9783110345490
• eBook ISBN:
  9783110369144
• Topic: Philosophy
• Subtopic: Philosophical Metaphysics

This collection focuses on the ontology of space and time. It is centred on the idea that the issues typically encountered in this area must be tackled from a multifarious perspective, paying attention to both a priori and a posteriori considerations. Several experts in this area contribute to this volume:
G. Landini discusses how Russell's conception of time features in his general philosophical perspective; D. Dieks proposes a middle course between substantivalist and relationist accounts of space-time; P. Graziani argues that it is necessary to provide an account of the "synthetic procedures" implicit in the recourse to diagrams in Euclid's Elements, while E. Mares comes to the conclusion that in Euclid's Elements we should treat the parallel postulate as empirical and the postulate that space is continuous as a priori.
M. Arsenijevi?/M. Adži? present an important formal result concerning two theories of the infinite two-dimensional continua, which sheds new light on the current dispute between gunkologists and pointilists; F. Orilia discusses two problems for presentism, one regarding the duration of the present and the other related to Zeno's paradoxes. A. Iacona delves deep into logical matters by focusing on the so-called T×W modal frames in order to deal with the deteterminism-indeterminism controversy. D. Mancuso outlines a non-standard temporal model compatible with time travel, and V. Fano/G. Macchia discuss time travels in the light of an important foundational principle of modern cosmology, Weyl's Principle.

A STRUCTURAL AND FOUNDATIONAL ANALYSIS OF EUCLID’ S PLANE GEOMETRY:

THE CASE STUDY OF CONTINUITY

PIERLUIGI GRAZIANI

University of Urbino
Department of Basic Sciences and Foundations
pierluigi.graziani@uniurb. it

ABSTRACT. In this paper I offer a structural²² and foundational analysis of Euclid’s Plane Geometry. These objectives require the analysis of some characteristics of Euclid’s plane geometry and of the logic of his argumentations. In particular, I will focus my analysis on the continuity problem as a case study. So, the aims of this paper are: (a) to show that it is necessary to analyse the status of the continuity postulate in the Elements; (b) to contribute to an understanding of how the Elements contain particular cases of the continuity postulate that are certainly used by Euclid, even though they are not explicitly stated; (c) to contribute to an understanding of how these particular cases really play a fundamental role, due to the centrality of diagrammatic configurations in the Elements; (d) to contribute to an understanding of how the study of these issues can also be important for developing modern synthetic-constructive axiomatizations of Euclid’s geometry; (e) to present a Sequent Calculus for Euclid’s Plane Geometry.

1. A famous example

We can start by considering a (famous) example in which the continuity problem is of great relevance.

Problem I.1 of Euclid’s Elements requires us to ‘construct’ an equilateral triangle on a ‘given finite straight line’, or segment, in modern parlance. Let’s see the Euclidean solution:

Protasis : To construct an equilateral triangle on a given finite straight line.

Ekthesis : Let AB be the given finite sstraight line.

Diorismos : It is required to construct an equilateral triangle on the straight line AB.

Kataskeuee: Describe the circle BCD with center A and radius AB. Again describe the circle ACE with center B and radius BA. Join the straight lines CA and CB from point C at which the circles cut one another to points A and B. (Postt. 3; Post.1).²³

Appodeixis:: Since point A is the center of circle CDB, therefore AC equals AB. Again, since point B is the center of ccircle CAE, therefore BC equals BA. (I. Def. 15). But AC was proved equal to AB, therefore each of the straight lines AC and BC equals AB.. And things which equal a same thing also equaal one another, therefore AC also equals BC. (C.N. 1).

Sumperasma: Therefore the three straight lines AC, AB, and BC equal one another.²⁴

So, to achieve his result, Euclid takes this segment to be AB, then d scribes two circles with centres in the two extremities A and B of this segment, respectively, and takes for granted that these circles intersect each other in a point C. This is not licensed by his postulates. Hence, either his argument is flawed, or it is warranted on other grounds.

According to a classical view ‘the Principle of Continuity’ provides an appropriate ground for this argument,, insofar as it ensures ‘the actual existence of points of intersection’ of lines.

Michael Friedman²⁵ has remarked, however, that in the Elements “the notion of ‘continuity’ […] is not logicallly analyzed” and thus there is no room for a “valid syllogistic inference of the form: C1 is continuous, C2 is continuous, then C exists”, where C1 and C2 are the two circles involved in this proposition.

But we can see circles cutting one another in the diagrammatic configuration !

This example is very famous because it has played a central role in the construction of the classical interpretation of continuity in Euclid. Here are some quotes.

Thomas Heath:²⁶

It is a commonplace that Euclid has no right to assume, without premising some postulate, that the two circles w...

Table of contents

• Philosophische Analyse / Philosophical Analysis
• Title Page
• Copyright Page
• Table of Contents
• EDITORS’ INTRODUCTION
• ON RUSSELL’ S METAPHYSICS OF TIME
• WEAK DISCERNIBILITY AND THE IDENTITY OF SPACETIME POINTS
• A STRUCTURAL AND FOUNDATIONAL ANALYSIS OF EUCLID’ S PLANE GEOMETRY: THE CASE STUDY OF CONTINUITY
• CAN THE MATHEMATICAL STRUCTURE OF SPACE BE KNOWN A PRIORI? - A TALE OF TWO POSTULATES
• GUNKOLOGY AND POINTILISM: TWO MUTUALLY SUPERVENING MODELS OF THE REGION-BASED AND THE POINT-BASED THEORY OF THE INFINITE TWO-DIMENSIONALCONTINUUM
• THIS MOMENT AND THE NEXT MOMENT
• T×W EPISTEMIC MODALITY
• TOWARDS A THEORY OF MULTIDIMENSIONAL TIME TRAVEL
• GÖDELIAN TIME TRAVEL AND WEYL’S PRINCIPLE
• About the authors