Aristotle: Mathematical Objects

Nos Segundos Analíticos I. 14, 79a16-21 Aristóteles afirma que as demonstrações matemáticas são expressas em silogismos de primeira figura. Apresento uma leitura da teoria da demonstração científica exposta nos Segundos Analíticos I (com maior ênfase nos capítulo 2-6) que seja consistente com o texto aristotélico e explique exemplos de demonstrações geométricas presentes no Corpus. Em termos gerais, defendo que a demonstração aristotélica é um procedimento de análise que explica um dado explanandum por meio da conversão de uma proposição previamente estabelecida. (...) Em uma estrutura silogística, a proposição previamente estabelecida é a premissa maior e o termo mediador deve ser comensurado ao explanandum. O conjunto da premissa maior e da premissa menor (o explanans) é coextensivo ao explanandum, mas há uma assimetria intensional entre o explanans e o explanandum, de modo que apenas o primeiro explique o último. Por fim, defendo que o elemento identificado no termo mediador deve ser o mais apropriado para explicar precisamente o que certo explanandum é. (shrink)

This paper examines Aristotle's discussion of the priority of actuality to potentiality in geometry at Metaphysics Θ9, 1051a21–33. Many scholars have assumed what I call the "geometrical construction" interpretation, according to which his point here concerns the relation between an inquirer's thinking and a geometrical figure. In contrast, I defend what I call the "geometrical analysis" interpretation, according to which it concerns the asymmetrical relation between geometrical propositions in which one is proved by means of the other. His argument as (...) so construed is ultimately based on the asymmetrical relation between the corresponding geometrical facts. Then I explore this ontological priority in geometry by drawing attention to a parallel passage, Posterior Analytics II.11, 94a24–35, where Aristotle explains the relation between the same geometrical propositions in connection to material causation. (shrink)

Aristotelês (Ἀριστοτέλης) wurde 384 v.u.Z. in Stageira (im Grenzgebiet zwischen Thrakien und Makedonien) geboren. Er trat 367 in Platons „Akademie“ ein und blieb ihr Mitglied bis zu Platons Tod im Jahre 348/347. Im Jahre 343 wurde er am makedonischen Königshof Lehrer des damals 13-jährigen Alexander (später ;der Große‘ genannt). 336 kehrte er nach Athen zurück und wurde schon bald darauf Leiter des Lykeions. Zum Gebäude gehörte eine Wandelhalle (Peripatos, περίπατος). Die Mitglieder dieser Schule wurden daher „Peripatetiker“ genannt (περιπατέω = herumgehen).

In the first argument of Metaphysics Μ.2 against the Platonist introduction of separate mathematical objects, Aristotle purports to show that positing separate geometrical objects to explain geometrical facts generates an ‘absurd accumulation’ of geometrical objects. Interpretations of the argument have varied widely. I distinguish between two types of interpretation, corrective and non-corrective interpretations. Here I defend a new, and more systematic, non-corrective interpretation that takes the argument as a serious and very interesting challenge to the Platonist.

Aristotle analyses a large range of objects as composites of matter and form. But how exactly should we understand the relation between the matter and form of a composite? Some commentators have argued that forms themselves are somehow material, that is, forms are impure. Others have denied that claim and argued for the purity of forms. In this paper, I develop a new purist interpretation of Metaphysics Z.10-11, a text central to the debate, which I call 'hierarchical purism'. I argue (...) that hierarchical purism can overcome the difficulties faced by previous versions of purism as well as by impurism. Roughly, on hierarchical purism, each composite can be considered and defined in two different ways: From the perspective of metaphysics, composites are considered only insofar as they have forms and defined purely formally. From the perspective of physics, composites are considered insofar as they have forms and matter and defined with reference to both. Moreover, while the metaphysical definition is a definition in the strict sense of 'definition', the physical definition is a definition in a loose sense. Analogous points hold for intelligible composites and geometry. Finally, neither sort of definitional practice implies that, for Aristotle, forms are impure. (shrink)

In Metaphysics M.2, 1077a9-14, Aristotle appears to argue against the existence of Platonic Forms on the basis of there being certain universal mathematical proofs which are about things that are ‘beyond’ the ordinary objects of mathematics and that cannot be identified with any of these. It is a very effective argument against Platonism, because it provides a counter-example to the core Platonic idea that there are Forms in order to serve as the object of scientific knowledge: the universal of which (...) theorems of universal mathematics are proven in Greek mathematics is neither Quantity in general nor any of the specific quantities, but Quantity-of-type-x. This universal cannot be a Platonic Form, for it is dependent on the types of quantity over which the variable ranges. Since for both Plato and Aristotle the object of scientific knowledge is that F which explains why G holds, as shown in a ‘direct’ proof about an arbitrary F (they merely disagree about the ontological status of this arbitrary F, whether a Form or a particular used in so far as it is F), Plato cannot maintain that Forms must be there as objects of scientific knowledge - unless the mathematics is changed. (shrink)

Aristotle ends Metaphysics books M–N with an account of how one can get the impression that Platonic Form-numbers can be causes. Though these passages are all admittedly polemic against the Platonic understanding, there is an undercurrent wherein Aristotle seems to want to explain in his own terms the evidence the Platonist might perceive as supporting his view, and give any possible credit where credit is due. Indeed, underlying this explanation of how the Platonist may have formed his impression, we discover (...) Aristotle's own understanding of what we today might call the mathematical structure of nature. Mathematicals are, to Aristotle, abstractions of order, symmetry, and definiteness, which are in turn aspects of the formal cause of goodness and beauty. This is a lens through which an Aristotelian could view modern mathematical physics as an important aspect of natural philosophy, and which is foundational for an Aristotelian philosophy of mathematics. (shrink)

: Metaphysics B considers two sets of views that hypostatize mathematicals. Aristotle discusses the first in his B.2 treatment of aporia 5, and the second in his B.5 treatment of aporia 12. The former has attracted considerable attention; the latter has not. I show that aporia 12 is more significant than the literature suggests, and specifically that it is directly addressed in M.2 – an indication of its importance. There is an immediate problem: Aristotle spends most of M.2 refuting the (...) view that mathematicals are separate; but the B.5 mathematicals are inherent. I argue that the B.5 inherence is compatible with the M.2 separateness, and further, that aporia 12 is more than just a puzzle about the hypostatization of mathematicals. It is also a puzzle about the ontological status of mathematicals relative to sensibles. (shrink)

This paper reconstructs the relationship between the now, motion, and number in Aristotle to clarify the nature of the now, and, thereby, the relationship between motion and time. Although it is clear that for Aristotle motion, and, more generally, change, are prior to time, the nature of this priority is not clear. But if time is the number of motion, then the priority of motion can be grasped by examining his theory of number. This paper aims to show that, just (...) as numbers are generated by the soul, time is not presupposed by motion, but emerges through the soul’s articulation of motion. Time is co-constituted by the soul and motion. The now is the key to understanding both the contribution of motion and soul to the being of time. The now is part of the soul’s articulation of motion, and sets the stage for an act that distinguishes a unit from its underlying motion. The now, then, sets up an abstraction by which the soul generates the temporal number from motion. Reconstructing this account of abstraction allows us to formulate more strongly Aristotle’s claim to the ontological dependence of time on motion. The paper then gives a systematic overview of the relationships between the now and number in order to address the question of whether the now might be extended. It closes with an examination of the possibility that motion depends on time, and how universal time is possible. (shrink)

For Aristotle, the shape of a physical body is perceptible per se (DA II.6, 418a8-9). As I read his position, shape is thus a causal power, as a physical body can affect our sense organs simply in virtue of possessing it. But this invites a challenge. If shape is an intrinsically powerful property, and indeed an intrinsically perceptible one, then why are the objects of geometrical reasoning, as such, inert and imperceptible? I here address Aristotle’s answer to that problem, focusing (...) on the version of it that he presents in De caelo III.8. I argue that if we grant that Aristotle conceived of the shape of a sensible body as some kind of causal power, then the satisfactory resolution of that challenge pushes us to interpret him as having conceived of it as being, more specifically, an impure power—that is, as a property that is not only intrinsically powerful but also, in some way, intrinsically non-powerful as well. This is a notable result not only insofar as it illuminates Aristotle’s conception of shape but also insofar as it contributes to our knowledge of Aristotle’s theory of dunameis and his ontology more broadly. (shrink)

Aristotle’s philosophy of geometry is widely interpreted as a reaction against a Platonic realist conception of mathematics. Here I argue to the contrary that Aristotle is concerned primarily with the methodological question of how universal inferences are warranted by particular geometrical constructions. His answer hinges on the concept of abstraction, an operation of “taking away” certain features of material particulars that makes perspicuous universal relations among magnitudes. On my reading, abstraction is a diagrammatic procedure for Aristotle, and it is through (...) imagined variation of diagrams that one can universalize spatial inferences. In order to shift the discussion of Aristotle’s work on geometry from ontological to methodological questions, I argue in the first section that geometrical properties for Aristotle are necessary accidents of the particulars, canonically diagrams, in which they inhere. I then consider a line of research that shows how diagrams in ancient geometry are not illustrations of the textual proof but are in part constitutive of geometrical inference, an insight I claim can shed light on Aristotle’s philosophical project. Next I substantiate this assertion by giving an alternative interpretation of abstraction according to which it is a diagrammatic procedure that allows a part of a particular diagram to stand in for a universal. In the last section, I consider Aristotle’s account of mathematical cognition, arguing that there is evidence for the view that imagination is necessary both for the construction of figures, and for their presentation as abstractions subject to universal inference. (shrink)

In some influential histories of ancient philosophy, teleological explanation and mechanistic explanation are assumed to be directly opposed and mutually exclusive alternatives. I contend that this assumption is deeply flawed, and distorts our understanding both of teleological and mechanistic explanation, and of the history of mechanistic philosophy. To prove this point, I shall provide an overview of the first systematic treatise on mechanics, the short and neglected work Mechanical Problems, written either by Aristotle or by a very early member of (...) his school. I will argue that the work is thoroughly Aristotelian in methodology, and that taking it seriously can deepen our understanding of Aristotle’s discussion of animal and human self-motion in the Physics and On the Movement of Animals. (shrink)

My thesis is an exposition and defence of Aristotle’s philosophy of mathematics. The first part of my thesis is an exposition of Aristotle’s cryptic and challenging view on mathematics and is based on remarks scattered all over the corpus aristotelicum. The thesis’ central focus is on Aristotle’s view on numbers rather than on geometrical figures. In particular, number is understood as a countable plurality and is always a number of something. I show that as a consequence the related concept of (...) counting is based on units. In the second part of my thesis, I verify Aristotle’s view on number by applying it to his account of time. Time presents itself as a perfect test case for this project because Aristotle defines time as a kind of number but also considers it as a continuum. Since numbers and continuous things are mutually exclusive this observation seems to lead to an apparent contradiction. I show why a contradiction does not arise when we understand Aristotle properly. In the third part, I argue that the ontological status of mathematical objects, dubbed as materially [hulekos, ÍlekÀc] by Aristotle, can only be defended as an alternative to Platonism if mathematical objects exist potentially enmattered in physical objects. In the fourth part, I compare Aristotle’s and Plato’s views on how we obtain knowledge of mathematical objects. The fifth part is an extension of my comparison between Aristotle’s and Plato’s epistemological views to their respective ontological views regarding mathematics. In the last part of my thesis I bring Frege’s view on numbers into play and engage with Plato, Aristotle and Frege equally while exploring their ontological commitments to mathematical objects. Specifically, I argue that Frege should not be mistaken for a historical Platonist and that we find surprisingly many similarities between Frege and Aristotle. After having acknowledged commonalities between Aristotle and Frege, I turn to the most significant differences in their views. Finally, I defend Aristotle’s abstractionism in mathematics against Frege’s counting block argument. This whole project sheds more light on Aristotle’s view on mathematical objects and explains why it remains an attractive view in the philosophy of mathematics. (shrink)

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)

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)

The article examines Greek philosopher Aristotle's understanding of mathematical numbers as pluralities of discreet units and the relations of unity and multiplicity. Topics discussed include Aristotle's view that a mathematical number has determinate properties, a contrast between Aristotle and French philosopher René Descartes in terms of their understanding of number and Aristotle's description of ways to understand eidetic numbers.

By examining Klein’s discussion of the difference between Plato and Aristotle regarding the ontology of number, this article aims to spells out the significanceof that debate both in itself and for the development of the later mathematical sciences. This is accomplished by explicating and expanding Klein’s account of the differences that exist in the understanding of number presented by these two thinkers. It is ultimately argued that Klein’s analysis can be used to show that the transition from the ancient to (...) the modern number concept has some roots in this disagreement between Plato and Aristotle regarding number. This, in turn, sets up that dispute as an essential part of the background to the more general break between ancient and modern conceptuality, the uncovering of whichis Klein’s main concern. (shrink)

Bowin begins with an apparent paradox about Aristotelian infinity: Aristotle clearly says that infinity exists only potentially and not actually. However, Aristotle appears to say two different things about the nature of that potential existence. On the one hand, he seems to say that the potentiality is like that of a process that might occur but isn't right now. Aristotle uses the Olympics as an example: they might be occurring, but they aren't just now. On the other hand, Aristotle says (...) that infinity "exists in actuality as a process that is now occurring" (234). Bowin makes clear that Aristotle doesn't explicitly solve this problem, so we are left to work out the best reading we can. His proposed solution is that "infinity must be...a per se accident...of number and magnitude" (250). (Bryn Mawr Classical Review 2008.07.47). (shrink)

I argue here that a properly Platonic theory of the nature of number is still viable today. By properly Platonic, I mean one consistent with Plato's own theory, with appropriate extensions to take into account subsequent developments in mathematics. At Parmenides 143a-4a the existence of numbers is proven from our capacity to count, whereby I establish as Plato's the theory that numbers are originally ordinal, a sequence of forms differentiated by position. I defend and interpret Aristotle's report of a Platonic (...) distinction between form and mathematical numbers, arguing that mathematical numbers alone are cardinals, by reference to certain non-technical features of a set-theoretical approach and other considerations in philosophy of mathematics. Finally I respond to the objections that such a conception of number was unavailable in antiquity and that this theory is contradicted by Aristotle's report in Metaph . XIII that Platonic numbers are collections of units. I argue that Aristotle reveals his own misinterpretation of the terms in which Plato's theory was expressed. (shrink)

This dissertation argues that Aristotle intended his so-called theory of abstraction to serve primarily as the resolution of a special problem in the philosophy of mathematics; i.e., the ontological status of mathematical objects. My general approach is dictated by the view that Aristotle's 'theories' must be understood in terms of the particular problems that he is trying to resolve. Thus, most of my dissertation is devoted to examining his treatment of the problem which I show to be relevant to the (...) theory of abstraction. In fact, it contains a methodological study of Aristotle's characteristic manner of treating a philosophical problem. ;My first chapter shows: how a problem must arise for Aristotle about the mode of being of mathematical objects; why this is a peculiarly difficult problem for him to resolve; and that he recognizes it as an appropiate problem for First Philosophy. The second chapter selects two relevant puzzles) from among those listed in Metaphysics Beta and examines Aristotle's review of the difficulties associated with each puzzle. This even-handed review is distinguished from his elenctic treatment of the same general problem in Mu 1-2, where his purpose is to refute the Platonists and to prepare the way for his own solution in Mu 3. ;My final chapter elucidates this positive solution and establishes its connection with the terminology of abstraction. I show that the crucial issue for Aristotle is the precise mode of being of mathematical objects, given that they are the products of a dialectical method of subtraction ). Thus, I argue that he makes these entities doubly dependent, in contrast with the independent substances posited by the Platonists. In one respect mathematical quantities are potentially in sensible substances but, precisely as they are defined, they are actualized only through the abstractive activity of the mathematician. To support my interpretation, I show that such a complex solution is paralleled in Aristotle's account of the mode of being of the infinite. Furthermore, I show that this solution "saves the phenomena", in the sense that it resolves the outstanding difficulties associated with the problem. (shrink)

Miracles are not ordinarily looked for in the area of the Aristotelian Metaphysics. Least of all, perhaps, would one expect to be presented with a readable and lucid translation of Books M and N. Yet that is what has been given in the present scholarly volume. The difficult and often puzzling Greek text of the two books has been neatly rendered in attractive, idiomatic, precise, smooth flowing English.

Das historische Buch konnen zahlreiche Rechtschreibfehler, fehlende Texte, Bilder, oder einen Index. Kaufer konnen eine kostenlose gescannte Kopie des Originals durch den Verlag. 1907. Nicht dargestellt. Auszug:... I. TEIL DIE PROBLEME DER GRUNDWISSENSCHAFT ndem wir nunmehr an die von uns zu behandelnde Wissenschaft herantreten, gilt es zunachst uns klar zu werden uber die Fragen, uber die wir eine Entscheidung zu treffen haben. Es sind zum Teil solche, uber welche die Denker vor uns abweichende Ansichten geaussert haben; wir mussen aber auch (...) an etwaige weitere denken, die sie ubersehen haben mochten. Wer einer Sache recht auf den Grund kommen will, fur den ist das erste Erfordernis dies, dass er den Problemen scharf ins Gesicht sehe. Denn die nachher zu erlangende Einsicht hangt an der Losung der vorher ins Auge gefassten Probleme; wer den Knoten nicht kennt, der kann ihn auch nicht losen. Solch ein Knoten in dem Objekte aber ist es, den das Problem darstellt, wie es sich fur das Nachdenken auftut. Dem Denken namlich, sofern es die Schwierigkeit gewahr wird, ergeht es ganz ahnlich wie den Leuten, die sich durch einen festen Knoten eingeschnurt fuhlen: beide konnen keinen Schritt vorwarts tun. Darum ist es notig, zuvorderst alle Schwierigkeiten ins Auge gefasst zu haben; ist es schon aus diesem Grunde geboten, so ist ein weiterer Grund der, dass man, wenn man ohne diese Erwagung der Probleme an die Untersuchung herantritt, dem Wanderer gleicht, der nicht weiss, wohin er seinen Weg zu richten hat; und dass man ausserdem im andern Fall nicht einmal, wenn man etwas gefunden hat, zu erkennen imstande ware, ob das Gefundene auch das ist, was man sucht, oder nicht. Denn in jenem Fall ist das Ziel nicht klar; wohl aber ist es dem klar, der zuvor die Probleme sich zum Bewusstsein gebracht hat. Daz. (shrink)