Aristotle and Greek Geometrical Analysis

Phronesis, 2009

I offer a new interpretation of Aristotle's philosophy of geometry, which he presents in greatest detail in Metaphysics M3. On my interpretation, Aristotle holds that the points, lines, planes, and solids of geometry belong to the sensible realm, but not in a straightforward way. Rather, by considering Aristotle's second attempt to solve Zeno's Runner Paradox in Book VIII of the Physics, I explain how such objects exist in the sensibles in a special way. I conclude by considering the passages that lead Jonathan Lear to his fi ctionalist reading of Met. M3,1 and I argue that Aristotle is here describing useful heuristics for the teaching of geometry; he is not pronouncing on the meaning of mathematical talk.

the development of practical and theoretical geometry by the ancient Greeks was a significant cultural accomplishment, and it proved critical for the evolution of Greek architecture. Interest in geometry in Greece may be divided into three phases. the earliest phase emphasized philosophical and religious applications, such as the speculations of two pre‐Socratic philosophers who used geometric models to articulate their views of the cosmos, thales of Miletos and anaximander of Miletos, both of them active in the first half of the sixth century bce. the second phase was centered in hellenistic alexandria, where the most important compiler, euclid, was active in the third century bce. Our existing corpus of writing about geometry has been preserved from this period. the third phase of exploration of geometry took place in the late hellenistic period, when archimedes of Syracuse and apollonius of perga were prominent among geometers; this period also was remarkable for practical developments in mechanics and engineering (heath 1921: I.345–348, II. 346–352, thomas 1941). the text of Vitruvius illustrates the wide range of interests of such authors; although he regarded himself as an architect and a designer, Vitruvius was, as we would define the role, an engineer (rowland and howe 1999: 21). he wrote his still‐ preserved treatise, De architectura, about 20 bce and dedicated it to the emperor augustus. the development of early Greek geometry took place concurrently with several significant cultural events in the seventh century bce. the reign of psammetichus I (664–610 bce) marked the beginning of increased contact between Greeks and egyptians (hahn 2001: 66–69). During the reign of amasis (570–526 bce), Cyprus fell first under egyptian rule, then that of the assyrians. the capture of the city of Sardis by persia in 547/6 bce opened a direct link between the Greek world, persia, and India (Burkert 2004: 49–55, 70–74). thales observed the solar eclipse of 585 bce, and his book on the cosmos is datable to 547 bce. although individuals such as thales and, later, pythagoras (circa 550–495 bce) could have traveled to egypt and Babylon to gain specialized knowledge, the geometry that existed in egypt and Babylon was not the philosophical geometry of the pre‐Socratics; instead, it featured pragmatic solutions for calculating areas, needed because of the ever‐changing topography due to the annual flooding of the Nile (heath 1921: 122–126). the egyptian rhind Mathematical papyrus documents the interest in area calculations. But the calculations included inaccuracies and approximations, and the level of accuracy was not what we would accept today. Yet the Greeks did not have to go far to meet egyptians or Babylonians as these people were coming to them. It is no coincidence that the early Greek geometers thales and anaximander were from Ionia and that, in the following generation, pythagoras was originally from Samos, just off the Ionian coast.

NTM Zeitschrift für Geschichte der Wissenschaften, Technik und Medizin, 2013

Reading this book should start with the Appendix (Anhang, pp. 395-414), containing the Aristotelian and Platonic Greek sources, and their Latin and German translations, the textual foundations upon which Toth erects his entire interpretive échafaudage. Of these nineteen odd pages, the Greek originals occupy a mere seven (pp. 395-401). The attentive, open-minded reader will not discover any traces whatsoever of non-Euclidean geometry in these sources. Nothing at all. And this should not be surprising: Like Marx, who was no Marxist, Euclid, too, was without party affiliation, he was, of course, no follower of Euclid, he was no Euclidean. Greek geometry was just that, geometry, with no additional qualifiers. Hence, even on the basis of this simple, obvious, semantic consideration, there could not have been any Greek non-Euclidean geometry; but, there is more to it. Tóth's creation is, largely, godlike, de nihilo. He brings to his sources, of course, like all of us, his biases, but, unlike most historians worth their salt, he does not leave his prejudices at the door before entering the textual world; on the contrary, he invests his sources with all of his preconceived ideas, reading

Phronesis 64, pp. 465-513, 2019

There is little agreement about Aristotle's philosophy of geometry. This is due in part to the nature of the textual evidence and in part to differences of opinion over what constitutes a plausible view. I attempt to make Aristotle's position clear by keeping separate the questions " What is Aristotle's philosophy of geometry? " and " Is Aristotle right? " , and by considering the textual evidence in the context of Greek geometrical practice. I find that for Aristotle, plane geometry is about properties of certain sensible objects—specifically, dimensional continuity and certain properties possessed by actual and potential compass-and-straightedge drawings qua quantitative and continuous. For their part, the objects of stereometry are potential sensible three-dimensional figures qua quantitative and continuous.

Oxford Studies in Ancient Philosophy 61, pp. 143-200, 2021

A principal interpretative difficulty for Aristotle’s philosophy of geometry is that he seems to maintain both that sensible things are not the objects of geometry and that geometrical objects are in the sensible world. I offer a philosophical defense of what I take to be Aristotle’s position. Briefly, geometrical objects are certain sensibles (e.g. skillfully produced construction drawings) just insofar as they are quantitative and continuous. A geometrical object, then, is what some scholars call a ‘kooky object’: a sensible object just insofar as it has a certain property or set of properties. I call this the ‘kooky objects interpretation’; it is a refinement of the ‘properties interpretation’ (a view I give elsewhere). I develop the view and argue that it is superior to an alternative reading—the ‘parts interpretation’. I show that the kooky objects interpretation addresses the difficulties motivating the parts interpretation while steering clear of its problems.

Aristotle’s way of conceiving the relationship between mathematics and other branches of scientific knowledge is completely different from the way a contemporary scientist conceives it. This is one of the causes of the fact that we look at the mathematical passage we find in Aristotle’s works with the wrong expectation. We expect to find more or less stringent proofs, while for the most part Aristotle employs mere analogies. Indeed, this is the primary function of mathematics when employed in a philosophical context: not a demonstrative tool, but a purely analogical model. In the case of the geometrical examples discussed in this paper, the diagrams are not conceived as part of a formalized proof, but as a work in progress. Aristotle is not interested in the final diagram but in the construction viewed in its process of development; namely in the figure a geometer draws, and gradually modifies, when he tries to solve a problem. The way in which the geometer makes use of the elements of his diagram, and the relation between these elements and his inner state of knowledge is the real feature which interests Aristotle. His goal is to use analogy in order to give the reader an idea of the states of mind involved in a more general process of knowing.

Historia Mathematica, 2012

This article is a contribution to our knowledge of ancient Greek geometric analysis. We investigate a type of theoretic analysis, not previously recognized by scholars, in which the mathematician uses the techniques of ancient analysis to determine whether an assumed relation is greater than, equal to, or less than. In the course of this investigation, we argue that theoretic analysis has a different logical structure than problematic analysis, and hence should not be divided into Hankel's four-part structure. We then make clear how a comparative analysis is related to, and different from, a standard theoretic analysis. We conclude with some arguments that the theoretic analyses in our texts, both comparative and standard, should be regarded as evidence for a body of heuristic techniques. Ó 2011 Elsevier Inc. All rights reserved.

Teachers Creating Impact, Proceedings of the 55th Annual Conference of the Mathematical Association of Victoria, 6--7 December 2018, Brunswick: MAV, pp. 36-39, 2018

The Greek Anthology is a large collection of very short pieces of writing from Ancient Greece which were complied over several centuries. A translation in English is available in 16 books spread over 5 volumes. The writings in this anthology include inscriptions copied from buildings, poems, and epigrams. However, book XIV is quite different from the other books. Sprinkled throughout book 14 are 45 elementary mathematical problems. Now Pythagoras, Euclid and Archimedes are household names in the history of mathematics in ancient Greece. However, The Greek Anthology is not so well-known in mathematical circles. The purpose of this paper is to describe these ancient problems. Perhaps they can be used to enrich mathematics in the classroom, or to make innovative connections with subjects such as classical studies or ancient history