This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary (...) for this to work. Case-branching occurs when a construction renders a diagram un-representative. The roles of diagrams in reductio arguments, and of objection in probing a demonstration, are discussed. (shrink)

Proposition I.1 is, by far, the most popular example used to justify the thesis that many of Euclid’s geometric arguments are diagram-based. Many scholars have recently articulated this thesis in different ways and argued for it. My purpose is to reformulate it in a quite general way, by describing what I take to be the twofold role that diagrams play in Euclid’s plane geometry (EPG). Euclid’s arguments are object-dependent. They are about geometric objects. Hence, they cannot be diagram-based unless diagrams (...) are supposed to have an appropriate relation with these objects. I take this relation to be a quite peculiar sort of representation. Its peculiarity depends on the two following claims that I shall argue for: ( i ) The identity conditions of EPG objects are provided by the identity conditions of the diagrams that represent them; ( ii ) EPG objects inherit some properties and relations from these diagrams. (shrink)

An examination of the emergence of the phenomenon of deductive argument in classical Greek mathematics.

In the last two decades there has been renewed interest in visualization in logic and mathematics. Visualization is usually understood in different ways but for the purposes of this article I will take a rather broad conception of visualization to include both visualization by means of mental images as well as visualizations by means of computer generated images or images drawn on paper, e.g. diagrams etc. These different types of visualization can differ substantially but I am interested in offering a (...) characterization of visualization that is as broad as possible. The article describes and explains (1) the way in which visual thinking fell into disrepute, (2) the renaissance of visual thinking in mathematics over recent decades, (3) the ways in which visual thinking has been rehabilitated in epistemology of mathematics and logic. (shrink)

This book offers a general introduction to the geometrical studies of Gottfried Wilhelm Leibniz and his mathematical epistemology. In particular, it focuses on his theory of parallel lines and his attempts to prove the famous Parallel Postulate. Furthermore it explains the role that Leibniz’s work played in the development of non-Euclidean geometry. The first part is an overview of his epistemology of geometry and a few of his geometrical findings, which puts them in the context of the 17th-century studies on (...) the foundations of geometry. It also provides a detailed mathematical and philosophical commentary on his writings on the theory of parallels, and discusses how they were received in the 18th century as well as their relevance for the non-Euclidean revolution in mathematics. The second part offers a collection of Leibniz’s essays on the theory of parallels and an English translation of them. While a few of these papers have already been published (in Latin) in the standard Leibniz editions, most of them are transcribed from Leibniz’s manuscripts written in Hannover, and published here for the first time. The book provides new material on the history of non-Euclidean geometry, stressing the previously neglected role of Leibniz in these developments. (shrink)

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late 19th century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis received much attention in the 19th century. They helped to instigate what Hans Hahn called a ‘crisis of intuition’, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this ‘crisis’ as follows : " Mathematicians had for (...) a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof … "Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called ‘logicization’, ‘when the discipline requires nothing but purely logical fundamental concepts and propositions for its development’. On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals …. (shrink)

The law of gravitation, an example of physical law The relation of mathematics to physics The great conservation principles Symmetry in physical law The distinction of past and future Probability and uncertainty: the quantum mechanical view of nature Seeking new laws.

This chapter provides a survey of issues about diagrams in traditional geometrical reasoning. After briefly refuting several common philosophical objections, and giving a sketch of diagram-based reasoning practice in Euclidean plane geometry, discussion focuses first on problems of diagram sensitivity, and then on the relationship between uniform treatment and geometrical generality. Here, one finds a balance between representationally enforced unresponsiveness (to differences among diagrams) and the intellectual agent's contribution to such unresponsiveness that is somewhat different from what one has come (...) to expect in modern logic. Finally, challenges and opportunities for further work are indicated. (shrink)

We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.

Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously. With respect to the proofs in the Elements in particular, the received view is that Euclid's reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory. The central difficulty concerns the generality of the theory. How can inferences made from a particular diagrams license general mathematical results? After surveying the history behind the received (...) view, this essay provides a contrary analysis by introducing a formal account of Euclid's proofs, termed Eu. Eu solves the puzzle of generality surrounding Euclid's arguments. It specifies what diagrams Euclid's diagrams are, in a precise formal sense, and defines generality-preserving proof rules in terms of them. After the central principles behind the formalization are laid out, its implications with respect to the question of what does and does not constitute a genuine picture proof are explored. (shrink)

A number of people have recently argued for a structural approach to accounting for the applications of mathematics. Such an approach has been called "the mapping account". According to this view, the applicability of mathematics is fully accounted for by appreciating the relevant structural similarities between the empirical system under study and the mathematics used in the investigation ofthat system. This account of applications requires the truth of applied mathematical assertions, but it does not require the existence of mathematical objects. (...) In this paper, we discuss the shortcomings of this account, and show how these shortcomings can be overcome by a broader view of the application of mathematics: the inferential conception. (shrink)

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. (shrink)

This long-awaited work by prominent Harvard psychologist Stephen Kosslyn integrates a twenty-year research program on the nature of high-level vision and mental ...

Visual thinking in mathematics is widespread; it also has diverse kinds and uses. Which of these uses is legitimate? What epistemic roles, if any, can visualization play in mathematics? These are the central philosophical questions in this area. In this introduction I aim to show that visual thinking does have epistemically significant uses. The discussion focuses mainly on visual thinking in proof and discovery and touches lightly on its role in understanding.

Twentieth-century developments in logic and mathematics have led many people to view Euclid’s proofs as inherently informal, especially due to the use of diagrams in proofs. In _Euclid and His Twentieth-Century Rivals_, Nathaniel Miller discusses the history of diagrams in Euclidean Geometry, develops a formal system for working with them, and concludes that they can indeed be used rigorously. Miller also introduces a diagrammatic computer proof system, based on this formal system. This volume will be of interest to mathematicians, computer (...) scientists, and anyone interested in the use of diagrams in geometry. (shrink)

Diagrams have distinctive characteristics that make them an effective medium for communicating research findings, but they are even more impressive as tools for scientific reasoning. Focusing on circadian rhythm research in biology to explore these roles, we examine diagrammatic formats that have been devised to identify and illuminate circadian phenomena and to develop and modify mechanistic explanations of these phenomena.