Orna Harari

This paper examines the widely accepted contention that geometrical constructions serve in Greek mathematics as proofs of the existence of the constructed figures. In particular, I consider the following two questions: first, whether the evidence taken from Aristotle's philosophy does support the modern existential interpretation of geometrical constructions; and second, whether Euclid's Elements presupposes Aristotle's concept of being. With regard to the first question, I argue that Aristotle's ontology cannot serve as evidence to support the existential interpretation, since Aristotle's ontological discussions address the question of the relation between the whole and its parts, while the modern discussions of mathematical existence consider the question of the validity of a concept. In considering the second question, I analyze two syllogistic reformulations of Euclidean proofs. This analysis leads to two conclusions: first, it discloses the discrepancy between Aristotle's view of mathematical objects and Euclid's practice, whereby it will cast doubt on the historical and theoretical adequacy of the existential interpretation. Second, it sets the conceptual background for an alternative interpretation of geometrical constructions. I argue, on the basis of this analysis that geometrical constructions do not serve in the Elements as a means of ascertaining the existence of geometrical objects, but rather as a means of exhibiting spatial relations between geometrical figures.

The absence of explicit discussions of the conformity of mathematical proofs to Aristotelian demonstrations in the earlier tradition seems puzzling from the perspective of contemporary studies of Aristotle’s theory of demonstration. Th e formal discrepancies between Aristotelian demonstrations and mathematical proofs seem so obvious to us that it is diffi cult to understand how the conformity between mathematical proofs and Aristotelian demonstrations was ever taken for granted. In this chapter I attempt to bring to light the presuppositions that led ancient thinkers to regard the conformity of mathematical proofs to Aristotelian demonstrations as self-evident.

A fragment preserved in Simplicius’ commentary on the Physics, written in the sixth century CE, suggests that the peripatetic philosopher and commentator, Alexander of Aphrodisias (third century CE), questioned this claim, arguing that the lines that geometers can use in their constructions should be shorter than the diameter of the universe. In this article I examine the motivation behind this claim, showing that it is not grounded in Alexander’s understanding of mathematics and its objects, as a comparison with other ancient commentators suggests, but that exceeding the limits of the universe is in his view logically impossible.

In this study I examine the sole detailed evidence we have for Simplicius’ view of sign-based, i.e. tekmeriodic proofs, thereby questing the widespread assumption that he espouses Philoponus’ account of these proofs. Specifically, I argue that (1) it is more plausible to understand the signs on which Simplicius bases his tekmeriodic proofs as refutable, (2) he grounds the epistemic worth of these proofs in the evidential strength of their premises rather than in their validity, (3) unlike Philoponus, he conceives of the argument that leads to the principles of natural philosophy, which tekmeriodic proofs are aimed to prove, as inductive, and (4) he evaluates these proofs against Plato’s un-hypothetical science, hence denying natural philosophy the autonomy from metaphysics that Philoponus’ account of tekmeriodic proofs grants.

I argue that Aristotle’s late-antique commentators read into his theory of demonstration the notion of tekmeriodic proofs in attempt to integrate into the theory of demonstration the assumption that the principles of demonstration should be evident. In so doing, I trace the late antique commentators’ view to Alexander of Aphrodisias’ discussion of the principles of demonstration, showing how his assumption that the principles of demonstration should be evident underlies their notions of tekmeriodic proofs.

I examine the question why in Proclus’ view genetic processes provide demonstrative explanations, in light of the interpretation of Aristotle’s theory of demonstration in late antiquity. I show that in this interpretation mathematics is not an explanatory science in the strict sense because its objects, being immaterial, do not admit causal explanation. Placing Proclus’ account of demonstrative explanation in this context, I argue that this account is aimed at answering the question whether mathematical proofs provide causal explanation as opposed to grounds. I show further that Proclus can answer this question in the affirmative due to his realist view of mathematical objects and the priority he ascribes to causal relations over logical relations.*

This study brings to light Alexander of Aphrodisias’ modification of the notion of continuity found in Aristotle’s _Physics_ V.3. It shows that Alexander’s interpretation of the definitions of contact and contiguity is based on his assumption that continuity is equivalent to unity and therefore he interprets _Physics_ V.3 in terms of a strong notion of continuity which holds for continuous wholes whose motion is one. It shows further that this sense is incompatible with Aristotle’s account of continuous motion but useful for explaining the beginning of change. Its possible atomistic implications are avoided because the efficacy of the cause of change, which varies from case to case, determines the actual divisions of a continuum.

The debate between Alexander and Galen on Aristotle’s argument in _Physics_ VII.1 that anything that moves is moved by something is currently understood as restricted to the question of the validity of this argument. This article questions this understanding and argues that (1) this debate directly bears on the tenability of the thesis that Aristotle’s argument establishes, and (2) it is rooted in an even more fundamental disagreement over hylomorphism and efficient causality. In substantiating these conclusions, the article examines Galen’s argument against Aristotle in light of his notion of _dunamis_ and his account of antecedent causes. It also analyses fragments from Alexander’s lost commentary on the _Physics_, where the validity of Aristotle’s argument in _Physics_ VII.1 is directly discussed and shows that he addresses Galen’s criticism through the view found in his _De anima_ and elsewhere that substantial forms are causal factors and that external causes are the primary efficient causes.

Through an examination of Eduard Zeller’s and Michael Frede’s historiographical approaches, I question the assumption that by placing philosophy in its historical context we can avoid anachronism. I argue that an exegetical approach that studies texts and places the discussions found therein in their philosophical contexts can minimize anachronism and offer a historically sensitive understanding. I clarify this claim through three case studies: the relation between John Philoponus’ philosophy and theology, Aristotle’s account of relatives, and Alexander of Aphrodisias’ notion of mixture. In so doing, I distinguish contextualization that facilitates judgment from contextualization that facilitates understanding and show how the latter furthers our understanding of philosophical views in different ways.

I examine Alexander of Aphrodisias’ theory of action, addressing the question how his view that human actions are determined by reason accounts for the capacity of doing otherwise. Calling into question the standard view that Alexander frees agents from internal determination, I argue that (1) the capacity of doing otherwise is a consequence of determination by reason, since it enables agents to do something different from what they would have done had they followed external circumstances; and (2) this capacity is compatible with causal determination by reason because as a case of potentiality for opposites, it grants agents the qualified possibility of doing otherwise insofar as their nature as human beings is concerned – a possibility which remains also when their actions are causally determined by reason and by their internal disposition. I show further that these elements of Alexander’s theory of action are ultimately based on his conception of the soul, specifically on his commitment to Aristotle’s view that the human soul is not purely rational, as the Stoics hold, but has nonrational and rational parts.

In this article I explain three puzzling features of Simplicius’ use of syllogistic reconstructions in his commentary on Aristotle’s Physics: (1) Why does he reconstruct Aristotle’s non-argumentative remarks? (2) Why does he identify the syllogistic figure of an argument but does not explicitly present its reconstruction? (3) Why in certain lemmata does he present several reconstructions of the same argument? Addressing these questions, I argue that these puzzling features are an expression of Simplicius’ assumption that formal reasoning underlies Aristotle’s prose, hence they reflect his attempt to capture as faithfully as possible Aristotle’s actual mode of reasoning. I show further that, as a consequence of this seemingly descriptive use of syllogistic reconstructions, logic serves Simplicius not only as an expository and clarificatory tool of certain interpretations or philosophical views, but also motivates and shapes his exegetical stances and approach.