Ontological separation plays a key role in Aristotle’s metaphysical project: substances alone are ontologically χωριστόν. The standard view identifies Aristotelian ontological separation with ontological independence, so that ontological separation is a non-symmetric relation. I argue that there is strong textual evidence that Aristotle employs an asymmetric notion of separation in the Metaphysics—one that involves the dependence of other entities on the independent entity. I argue that this notion allows Aristotle to prevent the proliferation of substance-kinds and thus to secure the (...) unity of his metaphysical system. (shrink)

This chapter examines Plato's recurrent response to the “Growing Argument” in the dialogue that exhibits Plato's most extensive evaluation of the concept of “number,” Phaedo. There, Plato illustrates mathematical Pythagorean argumentative techniques in the figures of Socrates's interlocutors Simmias and Cebes, and critiques them according to whether or not they exhibit a proper methodological rigor. The proposition of Forms and teleological causation generates new ways of thinking about number. When Socrates finally develops the most complete analysis of number that can (...) be found in Plato's oeuvre, in the final argument, he does so chiefly in order to lay the groundwork for a proof of the soul's immortality. The Phaedo thus illustrates Plato's most circumspect appropriation of mathematical Pythagorean concepts. (shrink)

Review of Substance and Essence in Aristotle: an Interpretation of Metaphysics VII-IX, by Charlotte Witt (Cornell University Press: 1989).

The opening argument in the Metaphysics M.2 series targeting separate mathematical objects has been dismissed as flawed and half-hearted. Yet it makes a strong case for a point that is central to Aristotle’s broader critique of Platonist views: if we posit distinct substances to explain the properties of sensible objects, we become committed to an embarrassingly prodigious ontology. There is also something to be learned from the argument about Aristotle’s own criteria for a theory of mathematical objects. I hope to (...) persuade readers of Metaphysics M.2 that Aristotle is a more thoughtful critic than he is often taken to be. (shrink)

The second highest level of the divided line in Plato’s Republic (510b-511a) appears to be about the entities of mathematics—entities such as particular (though non-physical) triangles. It differs from the highest level in two respects. It involves reasoning from hypotheses, and it uses visible images. This article defends the traditional view that the passage is indeed about these mathematical ‘intermediates’; and tries to show how the apparently different features of the second level are related, by focussing on Plato’s need to (...) give an account of how we can speak of many particulars of the same kind, without assuming that they are imperfect copies, in the way sensible things can be imperfect copies, of forms. (shrink)

Books M and N of Aristotle's Metaphysics receive relatively little careful attention. Even scholars who give detailed analyses of the arguments in M-N dismiss many of them as hopelessly flawed and biased, and find Aristotle's critique to be riddled with mistakes and question-begging. This assessment of the quality of Aristotle's critique of his predecessors (and of the Platonists in particular), is widespread. The series of arguments in M 2 (1077a14-b11) that targets separate mathematical objects is the subject of particularly strong (...) criticism by Annas and Ross. Two related arguments in this series (1077a14-20 and 1077a24-31) will serve as cases in point. The principal charges made against these arguments (that Aristotle misunderstands or misrepresents his opponents' views, and that he engages in question-begging because he presupposes his own metaphysical views) are frequently made against Aristotle's critique of Platonist positions more generally. If, as I argue, these charges are false for our two test case arguments, then there is good reason to think that they might also be false when they are leveled against the other arguments in this M 2 series. And, although presenting an argument for this is beyond the scope of this paper, this suggests that these two charges are more often than not false when applied to Aristotle's critique of Platonist mathematical views in M-N and beyond. (shrink)

In a new English translation by Christopher Rowe, this great classic of moral philosophy is accompanied here by an extended introduction and detailed lin-by-line commentary by Sarah Broadie. Assuming no knowledge of Greek, her scholarly and instructive approach will prove invaluable for students reading the text for the first time. This thorough treatment of Aristotle's text will be an indispensable resource for students, teachers, and scholars alike.

Both literalism, the view that mathematical objects simply exist in the empirical world, and fictionalism, the view that mathematical objects do not exist but are rather harmless fictions, have been both ascribed to Aristotle. The ascription of literalism to Aristotle, however, commits Aristotle to the unattractive view that mathematics studies but a small fragment of the physical world; and there is evidence that Aristotle would deny the literalist position that mathematical objects are perceivable. The ascription of fictionalism also faces a (...) difficult challenge: there is evidence that Aristotle would deny the fictionalist position that mathematics is false. I argue that, in Aristotle's view, the fiction of mathematics is not to treat what does not exist as if existing but to treat mathematical objects with an ontological status they lack. This form of fictionalism is consistent with holding that mathematics is true. (shrink)

I offer a new interpretation of Aristotle's philosophy of geometry, which he presents in greatest detail in Metaphysics M 3. 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 fictionalist 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. (shrink)

Charlotte Witt extracts from this text a coherent and provocative view about sensible substance by focusing on Aristotle's account of form or essence.

Aristotle holds that individual substances are ontologically independent from nonsubstances and universal substances but that non-substances and universal substances are ontologically dependent on substances. There is then an asymmetry between individual substances and other kinds of beings with respect to ontological dependence. Under what could plausibly be called the standard interpretation, the ontological independence ascribed to individual substances and denied of non-substances and universal substances is a capacity for independent existence. There is, however, a tension between this interpretation and the (...) asymmetry between individual substances and the other kinds of entities with respect to ontological independence. I will propose an alternative interpretation: to weaken the relevant notion of ontological independence from a capacity for independent existence to the independent possession of a certain ontological status. (shrink)

Whether aristotle wrote a work on mathematics as he did on physics is not known, and sources differ. this book attempts to present the main features of aristotle's philosophy of mathematics. methodologically, the presentation is based on aristotle's "posterior analytics", which discusses the nature of scientific knowledge and procedure. concerning aristotle's views on mathematics in particular, they are presented with the support of numerous references to his extant works. his criticism of his predecessors is added at the end.

Dancy (philosophy, Florida State U.) presents two new interpretations of the evidence regarding the metaphysical ideas of two important figures in Plato's Academy, Eudoxus and Speusippus, and of Aristotle's reaction to those ideas.

This book examines Aristotle's critical reaction to the mathematical cosmology of Plato's Academy, and traces the aporetic method by which he developed his own ...

Was Plato a Pythagorean? Plato's students and earliest critics thought so, but scholars since the nineteenth century have been more skeptical. With this probing study, Phillip Sidney Horky argues that a specific type of Pythagorean philosophy, called "mathematical" Pythagoreanism, exercised a decisive influence on fundamental aspects of Plato's philosophy. The progenitor of mathematical Pythagoreanism was the infamous Pythagorean heretic and political revolutionary Hippasus of Metapontum, a student of Pythagoras who is credited with experiments in harmonics that led to innovations in (...) mathematics. The innovations of Hippasus and other mathematical Pythagoreans, including Empedocles of Agrigentum, Epicharmus of Syracuse, Philolaus of Croton, and Archytas of Tarentum, presented philosophers like Plato with novel ways to reconcile empirical knowledge with abstract mathematical theories. Plato and Pythagoreanism demonstrates how mathematical Pythagoreanism established many of the fundamental philosophical questions Plato dealt with in his central dialogues, including Cratylus, Phaedo, Republic, Timaeus, and Philebus. In the process, it also illuminates the historical significance of the mathematical Pythagoreans, a group whose influence on the development of philosophical and scientific methods has been obscured since late antiquity. The picture that results is one in which Plato inherits mathematical Pythagorean method only to transform it into a powerful philosophical argument about the essential relationships between the cosmos and the human being. (shrink)

Kit Fine; XIV*—Ontological Dependence, Proceedings of the Aristotelian Society, Volume 95, Issue 1, 1 June 1995, Pages 269–290, https://doi.org/10.1093/aristote.

Analyzes the poem "On Nature" by Parmenides, arguing that is actually a philosophical argument disguised as Homer-like mythological journey. Original.

Cleary discusses the origin, development, and use of the many senses of priority as a central thesis in Aristotle’s metaphysics. Cleary contends that one of the most revealing problems for the ambiguity of Aristotle’s relationship to Platonism is that of the ontological status of mathematical objects. In support of his claim, Cleary analyzes a curious passage from Aristotle’s _Topics, _where he appears to accept a schema of priorities that makes mathematical entities more substantial than sensible things. How does Aristotle try (...) to reconcile the ordering of things dictated by sciences like mathematics and dialectic with the ordering of sense experience upon which his own physics and metaphysics are based? To find the answer, Cleary reviews three different outlines of the many senses of priority given by Aristotle himself and found in _Categories _12-13, _Metaphysics _Delta 11, and _Metaphysics _Theta 8. Cleary suggests there is an implicit hierarchy for Aristotle that leads him to posit the Prime Mover at its apex as complete actuality and, therefore, as the focus for the concept of priority. Having reviewed Aristotle’s treatment of the many uses of priority, Cleary demonstrates how the concept is used in some typical arguments by Aristotle for his mature metaphysical positions. (shrink)

CHAPTER ONE LIFE The extant evidence about Speusippus' life is scanty, and little of it is reliable. The reasons are not difficult to discover : the greater ...

In various texts, Aristotle assigns priority to form, in its role as a principle and cause, over matter and the matter-form compound. Given the central role played by this claim in Aristotle's search for primary substance in the Metaphysics, it is important to understand what motivates him in locating the primary causal responsibility for a thing's being what it is with the form, rather than the matter. According to Met. Theta.8, actuality [ energeia / entelecheia ] in general is prior (...) to potentiality [ dunamis ] in three ways, viz., in definition, time and substance. I propose an explicitly causal reading of this general priority claim, as it pertains to the matter-form relationship. The priority of form over matter in definition, time and substance, in my view, is best explained by appeal to the role of form as the formal, efficient and final cause of the matter-form compound, respectively, while the posteriority of matter to form according to all three notions of priority is most plausibly accounted for by the fact that the causal contribution of matter is limited to its role as material cause. When approached from this angle, the work of Met. Theta.8 can be seen to lend direct support to the more specific and explicitly causal priority claim we encounter in Met. Z.17, viz., that form is prior to matter in its role as the principle and primary cause of a matter-form compound's being what it is. (shrink)

Many have argued against the claim that Plato posited the mathematical objects that are the subjects of Metaphysics M and N. This paper shifts the burden of proof onto these objectors to show that Plato did not posit these entities. It does so by making two claims: first, that Plato should posit the mathematical Intermediates because Forms and physical objects are ill suited in comparison to Intermediates to serve as the objects of mathematics; second, that their utility, combined with Aristotle’s (...) commentary on Plato’s posit of Intermediates, provides good reason to conclude that Plato did posit them. (shrink)

Commentaries on Aristotle's writings have been produced since the 2nd century AD. This edition contains Greek commentaries on his work from the 3rd to the 8th centuries AD by, among others, Alexander of Aphrodiensias, Themistios, Joh. Philoponus, Simplicius in Greek.