Analyses of Aristotle (eBook)

TABLE OF CONTENTS 6
ORIGIN OF THE ESSAYS 7
INTRODUCTION 9
CHAPTER 1 ON ARISTOTLE’S NOTION OF EXISTENCE 13
CHAPTER 2 SEMANTICAL GAMES, THE ALLEGED AMBIGUITY OF ‘‘IS’’, AND ARISTOTELIAN CATEGORIES 35
CHAPTER 3 ARISTOTLE’S THEORY OF THINKING AND ITS CONSEQUENCES FOR HIS METHODOLOGY PART I: ARISTOTLE ON THINKING 57
CHAPTER 4 ON THE ROLE OF MODALITY IN ARISTOTLE’S METAPHYSICS1 89
CHAPTER 5 ON THE INGREDIENTS OF AN ARISTOTELIAN SCIENCE 99
CHAPTER 6 ARISTOTELIAN AXIOMATICS AND GEOMETRICAL AXIOMATICS 113
CHAPTER 7 ARISTOTELIAN INDUCTION 123
CHAPTER 8 ARISTOTELIAN EXPLANATIONS ( with Ilpo Halonen) 139
CHAPTER 9 ARISTOTLE’S INCONTINENT LOGICIAN 151
CHAPTER 10 ON THE DEVELOPMENT OF ARISTOTLE’S IDEAS OF SCIENTIFIC METHOD AND THE STRUCTURE OF SCIENCE 165
CHAPTER 11 WHAT WAS ARISTOTLE DOING IN HIS EARLY LOGIC, ANYWAY? A REPLY TO WOODS AND HANSEN 187
CHAPTER 12 CONCEPTS OF SCIENTIFIC METHOD FROM ARISTOTLE TO NEWTON 195
CHAPTER 13 THE FALLACY OF FALLACIES 205
CHAPTER 14 SOCRATIC QUESTIONING, LOGIC AND RHETORIC 231

CHAPTER 6
ARISTOTELIAN AXIOMATICS AND GEOMETRICAL AXIOMATICS (p. 101-102)

Professor Szabo´ deserves credit for calling our attention to the interplay of philosophical and mathematical influences in the development of Greek axiomatics. It is this interplay that lends a special flavor to much of the early as well as some of the later history of the axiomatic method. I believe, however, that in the last analysis the total picture of the early development of axiomatics will turn out to be quite different from the one Szabo´ paints. My reasons for this belief are nevertheless subtler than one might first expect.

Professor Szabo´ finds the true ancestors of the central mathematical methodology of the Greeks, including the axiomatic method, in the Eleatic dialectic. In so doing, Szabo´ prima facie misses a large part of the interdisciplinary interplay with which he is dealing. Most other historians of the axiomatic method would give the pride of place on the philosophical side of the fence to Aristotle, who is sometimes called the first great theoretician of the axiomatic method and whose ideal of a science was by any account explicitly and self-consciously axiomatic. Szabo´ admittedly discusses Aristotle, but gives the Stagirite short shrift, dismissing him as having played no real part in the development of the axiomatic methods actually used in mathematics.

Even if one believes that Aristotle’s actual influence on mathematics was negligible, his views seem to merit close attention by all historians of the axiomatic method. The importance of these views lies of course in the fact that Aristotelian axiomatics is by far the most fully developed object of comparison on the philosophical side with the uses of axiomatic method in Greek mathematics. Szabo´ ’s procedure in downgrading Aristotle’s role might thus seem unfortunate. Isn’t he throwing by the board one of the most important sources of the very development he is dealing with? It appears that some of Szabo´ ’s arguments for assigning Aristotle to the historical limbo he occupies in Szabo´ ’s story are in fact mistaken. Other pieces of evidence he presents are ambivalent, but not nearly as strong grounds for indicting Aristotle as Szabo´ seems to think.

Without trying to present an alternative total picture of the development of the axiomatic method, I will begin by presenting a number of corrections to Professor Szabo´ ’s account, mostly to what he says about Aristotle. All page references not otherwise specified will be to Arpad Szabo´ , The Beginnings of Greek Mathematics, D. Reidel, Dordrecht, 1978 (translation of Anfa¨nge der griechischen Mathematik, R. Oldenbourg Verlag, Mu¨nchen and Wien, 1969), or to Jaakko Hintikka, ‘‘On the ingredients of an Aristotelian science,’’ Nous 6 (1972), 55–69 (Chapter 5 of the present volume). (i) Szabo´ claims (p. 188) that the nontechnical meaning of deiknymi as ‘‘sichtbarmachen’’ is relevant to mathematical and philosophical usage. This claim is made dubious by the fact that Aristotle’s use of deixis, deiknymi, and related terms is already quite sophisticated.

As usual, Aristotle is not consistent, and does not use strict technical terms. However, there is unmistakable contrast in Aristotle between deixis and apodeixis. (See Hintikka, sec. 2) Typically, the former could be used by Aristotle of all and sundry persuasive ‘‘showings,’’ while the latter was used of logical (syllogistic) inferences from appropriate permises. If further evidence is needed, one of the clearest examples of the Aristotelian distinction is found in An.