What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the current debate on (...) Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem. (shrink)

In Book K of the Metaphysics Aristotle raises a problem about a very persistent concern of Greek philosophy, that of the relation between the one and the many , but in a rather peculiar context. He asks: ‘What on earth is it in virtùe of which mathematical magnitudes are one? It is reasonable that things around us [i.e. sensible things] be one in virtue of [their] ψνχ or part of their ψνχ, or something else; otherwise there is not one but (...) many, the thing is divided up. But [mathematical] objects are divisible and quantitative. What is it that makes them one and holds them together?’. (shrink)

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)

In this paper, we examine Sextus Empiricus' treatise Against the geometers . We first set this treatise in the overall context of the sceptic's polemics against the liberal arts. After a discussion of Sextus' attitude to the quadrivium , we discuss the structure, the sources and the target of the Against the geometers . It appears that Euclid is not Sextus' source, and neither he, nor the professional geometers, seem to be Sextus' main targets. Of course, Sextus never really makes (...) clear his precise target, but his attacks are rather directed against geometry as a means of modelling the physical world, thus ruining the support geometry was intended to bring to the physical part of dogmatic philosophy. (shrink)

Aristotle explicitly speaks of intelligible matter in three passages only, all from the Metaphysics, in the context of the analysis of definition as the formula that expresses the essence: Metaph. Z10, 1036 a8-11; Metaph.Z11, 1037 a5; Metaph.H6, 1045 a34-36 and 45 b1. In the case of the occurrences of Z10 and Z11, there is almost unanimous consensus that Aristotle uses the expression in a technical way, to indicate the matter of that particular type of objects that are intelligible compounds, of (...) which mathematical objects are paradigmatic instances. By contrast, there is no agreement on how to understand the expression in the case of H6. Here, ‘intelligible matter’ would denote, as stated by some interpreters, the generic element of the definition. The aim of this paper is to show that ὕλη νοητή has the same technical meaning in all the three explicit occurrences in the Metaphisics. Contrary to the interpretation that there is a shift in meaning from Z10-11 to H6, my goal is to illustrate that in both contexts, Aristotle uses the expression ‘intelligible matter’ to designate the matter of mathematical objects insofar as they are paradigmatic intelligible compounds. (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)

Avicenna believed in mathematical finitism. He argued that magnitudes and sets of ordered numbers and numbered things cannot be actually infinite. In this paper, I discuss his arguments against the actuality of mathematical infinity. A careful analysis of the subtleties of his main argument, i. e., The Mapping Argument, shows that, by employing the notion of correspondence as a tool for comparing the sizes of mathematical infinities, he arrived at a very deep and insightful understanding of the notion of mathematical (...) infinity, one that is much more modern than we might expect. I argue, moreover, that Avicenna’s mathematical finitism is interwoven with his literalist ontology of mathematics, according to which mathematical objects are properties of existing physical objects. (shrink)

Some authors have proposed that Avicenna considers mathematical objects, i.e., geometric shapes and numbers, to be mental existents completely separated from matter. In this paper, I will show that this description, though not completely wrong, is misleading. Avicenna endorses, I will argue, some sort of literalism, potentialism, and finitism.

La filosofía de las matemáticas de Aristóteles es una investigación acerca de tres asuntos diferentes pero complementarios: el lugar epistemológico de las matemáticas en el organigrama de las ciencias teoréticas o especulativas; el estudio del método usado por el matemático para elaborar sus doctrinas, sobre todo la geometría y la aritmética; y la averiguación del estatuto ontológico de las entidades matemáticas. Para comprender lo peculiar de la doctrina aristotélica es necesario tener en cuenta que su principal interés está en poner (...) en relación la matemática con la filosofía primera u ontología y la física, y salvaguardar el campo de conocimiento propio de cada una de ellas. (shrink)

This paper aims to reassess the philosophical puzzle of the “applicability of mathematics to physical sciences” as a misunderstood disciplinary interplay. If the border isolating mathematics from the empirical world is based on appropriate criteria, how does one explain the fruitfulness of its systematic crossings in recent centuries? An analysis of the evolution of the criteria used to separate mathematics from experimental sciences will shed some light on this question. In this respect, we will highlight the historical influence of three (...) major disciplinary paradigms. According to the Aristotelian classification of the sciences, the separation of mathematics from physics is based on their respective objects of study. The Baconian system distinguishes these sciences by the type of knowledge involved in each field. Finally, the Whewellian disciplinary layout categorises these disciplines by their respective methods. In this paper, we argue that the cascading effect of such disciplinary categorisations—based successively on ontological, epistemological, and heuristic criteria—resulted in a profound redefinition of the border between mathematics and physics, with the consequence of obscuring the foundations of the interplay between these fields. Our approach intends to put forward such a mechanism as a constitutive piece of the puzzle of the applicability of mathematics. (shrink)

In a series of recent papers, Emily Katz has argued that on Aristotle's view mathematical sciences are in an important respect no different from most natural sciences: They study sensible substances, but not qua sensible. In this paper, I argue that this is only half the story. Mathematical sciences are distinctive for Aristotle in that they study things ‘from’, ‘through’ or ‘in’ abstraction, whereas natural sciences study things ‘like the snub’. What this means, I argue, is that natural sciences must (...) study properties as they occur in the subjects from which they are originally abstracted, even where they reify these properties and treat them as subjects. The objects of mathematical sciences, on the other hand, can be studied as if they did not really occur in an underlying subject. This is because none of the properties of mathematical objects depend on their being in reality features of the subjects from which they are abstracted, such as bodies and inscriptions. Mathematical sciences are in this way able to study what are in reality non-substances as if they were substances. (shrink)

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)

Although there are many questions to be asked about philosophy of mathematics, the fundamental questions to be asked will be questions about what the mathematical object is in view of being and what the mathematical reasoning is in view of knowledge. It is clear that other problems will develop in parallel within the framework of the answers to these questions. For this reason, when we approach Aristotle's philosophy of mathematics over these two basic problems, we come up with the concept (...) of abstraction. In our work, I will try to explain the mathematical abstraction that Aristotle has developed to understand mathematical philosophy. (shrink)

I argue that intelligible matter, for Aristotle, is what makes mathematical objects quantities and divisible in their characteristic way. On this view, the intelligible matter of a magnitude is a sensible object just insofar as it has dimensional continuity, while that of a number is a plurality just insofar as it consists of indivisibles that measure it. This interpretation takes seriously Aristotle's claim that intelligible matter is the matter of mathematicals generally – not just of geometricals. I also show that (...) intelligible matter has the same meaning in all three places where it is explicitly invoked: Z.10, Z.11, and H.6. Since the H.6 passage involves a mathematical definition, this requires determining what the mathematician defines and how she defines it. I show that, as with natural scientific definitions, there must be a matterlike element in mathematical definitions. This element is not identical with, but rather refers to, intelligible matter. (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)

There is little agreement about Aristotle’s philosophy of geometry, partly due to the textual evidence and partly part to disagreement over what constitutes a plausible view. I keep separate the questions ‘What is Aristotle’s philosophy of geometry?’ and ‘Is Aristotle right?’, and consider the textual evidence in the context of Greek geometrical practice, and show 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. (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)

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)

This article considers whether and how there can be for Aristotle a genuine science of ‘pure’ psychology, of the soul as such, which amounts to considering whether Aristotle’s model of science in the Posterior Analytics is applicable to the de Anima.

At Aristotle,MetaphysicsE.1, 1026a14, Schwegler’s conjectural emendation of the manuscript reading ἀχώριστα to χωριστά has been widely adopted. The objects of physical science are therefore here ‘separate’, or ‘independently existent’. By contrast, the manuscripts make them ‘not separate’, construed by earlier commentators as dependent on matter. In this paper, I offer a new defense of the manuscript reading. I review past defenses based on the internal consistency of the chapter, explore where they have left supporters of the emendation unpersuaded, and attempt (...) to strengthen their appeal. I challenge Schwegler’s central case, developed by Ross and others, that the construction μὲν ἀλλ’ οὐκ demands an implausible ‘logical antithesis’ between inseparability and mobility. This is arguably the fundamental obstacle to the manuscript reading, and counterexamples have not to date convinced emenders. I offer a new, systematic review of Aristotle’s use of the phrase, including relevant cases from theRhetoric, to show how its usual meaning in Aristotle supports the transmitted text; I also reply to possible objections. Finally, I explore the implications of this defense for the classification of the sciences in E.1. (shrink)

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 contrasts episteme and doxa through the key notions of universal and necessary. These notions have played a central role in Aristotle’s characterization of scientific knowledge in the previous chapters of APo. They are not spelled out in APo I.33, but work as a sort of reminder that packs an adequate characterization of scientific knowledge and thereby gives a highly specified context for Aristotle’s contrast between episteme and doxa. I will try to show that this context introduces a contrast in (...) terms of explanatory claims: on the one hand, episteme covers those claims which capture explanatory connections that are universal and necessary and thereby deliver scientific understanding; on the other hand, doxa covers the explanatory attempts that fail at doing so. (shrink)

This article focusses on the generality of the entities involved in a geometric proof of the kind found in ancient Greek treatises: it shows that the standard modern translation of Greek mathematical propositions falsifies crucial syntactical elements, and employs an incorrect conception of the denotative letters in a Greek geometric proof; epigraphic evidence is adduced to show that these denotative letters are ‘letter-labels’. On this basis, the article explores the consequences of seeing that a Greek mathematical proposition is fully general, (...) and the ontological commitments underlying the stylistic practice. (shrink)

The present PhD dissertation aims to examine the relation between modality and change in Aristotle’s metaphysics. -/- On the one hand, Aristotle supports his modal realism (i.e., worldly objects have modal properties - potentialities and essences - that ground the ascriptions of possibility and necessity) by arguing that the rejection of modal realism makes change inexplicable, or, worse, banishes it from the realm of reality. On the other hand, the Stagirite analyses processes by means of modal notions (‘change is the (...) actuality of what is virtual insofar as it is virtual’). In other words: to grasp what change is, one has to resort to the modal idiom of potentialities, while the fact that there is change is indicative of the fact that nature is full of modal properties. -/- Aristotle’s modal and kinetic realism finds a negative in the figure of the Megaric Diodorus Kronus. The polemical situation of Greek philosophy has indeed the dialectical advantage of not opposing the Aristotelian position to a sui generis straw man. Both in its reduction of modal judgements to temporal quantifications and in its dissolution of the reality of the state-of-being-in-motion in favour of a cinematographic view of processes, the philosophical figure of reductionist antirealism embodied by Diodorus constitutes an alternative that takes the opposite view of Aristotle’s realism. -/- The present study is structured as a discussion between these two positions. In the face of Diodorus’ antirealist challenges, Aristotle articulates modal and kinetic considerations: the answers he gives to Diodorus’ puzzles thus provide valuable insight into his metaphysics. Moreover, the examination of Aristotle’s and Diodorus’ metaphysics, because of their insight and originality, will not fail to interest the philosopher concerned with the metaphysical foundations of modern physics, insofar as the entanglement between modalities and processes is nowadays at the core of mechanics (phase spaces, path integrals, etc.). (shrink)

This book investigates Aristotle’s views on abstraction and explores how he uses it. In this work, the author follows Aristotle in focusing on the scientific detail first and then approaches the metaphysical claims, and so creates a reconstructed theory that explains many puzzles of Aristotle’s thought. Understanding the details of his theory of relations and abstraction further illuminates his theory of universals. Some of the features of Aristotle’s theory of abstraction developed in this book include: abstraction is a relation; perception (...) and knowledge are types of abstraction; the objects generated by abstractions are relata which can serve as subjects in their own right, whereupon they can appear as items in other categories. The author goes on to look at how Aristotle distinguishes the concrete from the abstract paronym, how induction is a type of abstraction which typically moves from the perceived individuals to universals and how Aristotle’s metaphysical vocabulary is "relational.’ Beyond those features, this work also looks at how of universals, accidents, forms, causes and potentialities have being only as abstract aspects of individual substances. An individual substance is identical to its essence; the essence has universal features but is the singularity making the individual substance what it is. These theories are expounded within this book. One main attraction in working out the details of Aristotle’s views on abstraction lies in understanding his metaphysics of universals as abstract objects. This work reclaims past ground as the main philosophical tradition of abstraction has been ignored in recent times. It gives a modern version of the medieval doctrine of the threefold distinction of essence, made famous by the Islamic philosopher, Avicenna. (shrink)

Although there are many questions to be asked about philosophy of mathematics, the fundamental questions to be asked will be questions about what the mathematical object is in view of being and what the mathematical reasoning is in view of knowledge. It is clear that other problems will develop in parallel within the framework of the answers to these questions. For this rea­ son, when we approach Aristotle's philosophy of mathematics over these two basic problems, we come up with the (...) concept of abstraction. In our work, I will try to explain the mathematical abstraction that Aristotle has developed to understand mathematical philosophy. (shrink)