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)

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)

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)

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)

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)

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)

A formal theory of oppositions and opposites is proposed on the basis of a non- Fregean semantics, where opposites are negation-forming operators that shed some new light on the connection between opposition and negation. The paper proceeds as follows. After recalling the historical background, oppositions and opposites are compared from a mathematical perspective: the first occurs as a relation, the second as a function. Then the main point of the paper appears with a calculus of oppositions, by means of a (...) non-Fregean semantics that redefines the logical values of various sorts of sentences. A num- ber of topics are then addressed in the light of this algebraic semantics, namely: how to construct value-functional operators for any logical opposition, beyond the classical case of contradiction; Blanché's "closure problem", i.e., how to find a complete structure connecting the sixteen binary sentences with one another. All of this is meant to devise an abstract theory of opposition: it encompasses the relation of consequence as subalternation, while relying upon the use of a primary "proto- negation" that turns any relatum into an opposite. This results in sentential negations that proceed as intensional operators, while negation is broadly viewed as a difference-forming operator without special constraints on it. (shrink)

There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake (...) when adjudicating issues concerning the identity of neo-logicist abstracts — so-called ‘Caesar questions’. (shrink)

One over Many: The Unitary Pluralism of Plato’s World is a tightly integrated collection of essays by the author, some newly drafted for the present work, some previously published as journal articles, all conceived and developed in pursuit of corrective intervention in Plato’s metaphysics. The book replaces the standard view of Plato as a metaphysical dualist with an alternative interpretation providing greater explanatory power through the paradigm of unitary pluralism in a single reality built on ontological diversity.

One of the manuscripts of Buridan’s Summulae contains three figures, each in the form of an octagon. At each node of each octagon there are nine propositions. Buridan uses the figures to illustrate his doctrine of the syllogism, revising Aristotle's theory of the modal syllogism and adding theories of syllogisms with propositions containing oblique terms (such as ‘man’s donkey’) and with ‘propositions of non-normal construction’ (where the predicate precedes the copula). O-propositions of non-normal construction (i.e., ‘Some S (some) P is (...) not’) allow Buridan to extend and systematize the theory of the assertoric (i.e., non-modal) syllogism. Buridan points to a revealing analogy between the three octagons. To understand their importance we need to rehearse the medieval theories of signification, supposition, truth and consequence. (shrink)

In this paper, I explore the origins of the ‘problem of universals’. I argue that the problem has come to be badly formulated and that consideration of it has been impeded by falsely supposing that Platonic Forms were ever intended as an alternative to Aristotelian universals. In fact, the role that Forms are supposed by Plato to fulfill is independent of the function of a universal. I briefly consider the gradual mutation of the problem in the Academy, in Alexander of (...) Aphrodisias, and among some of the major Neoplatonic commentators on Aristotle, including Porphyry and Boethius. (shrink)

In this paper I propose a set-theoretical interpretation of the logical square of opposition, in the perspective opened by generalized quantifier theory. Generalized quantifiers allow us to account for the semantics of quantificational Noun Phrases, and of other natural language expressions, in a coherent and uniform way. I suggest that in the analysis of the meaning of Noun Phrases and Determiners the square of opposition may help representing some semantic features responsible to different logical properties of these expressions. I will (...) conclude with some consideration on scope interactions between quantifiers. (shrink)

Frege argues that number is so unlike the things we accept as properties of external objects that it cannot be such a property. In particular, (1) number is arbitrary in a way that qualities are not, and (2) number is not predicated of its subjects in the way that qualities are. Most Aristotle scholars suppose either that Frege has refuted Aristotle's number theory or that Aristotle avoids Frege's objections by not making numbers properties of external objects. This has led some (...) to conclude that Aristotle's accounts of arithmetical and geometrical objects differ substantially. I close this supposed gap by showing that Aristotle's arithmetical objects, like geometrical objects, are just certain sensible things qua certain properties they in fact possess. Specifically, numbers are pluralities qua quantitative or relational properties like ten units or ten. I show that this view is resistant to the Fregean concerns about arbitrariness and numerical predication. (shrink)

In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, (...) and we show that the two approaches are equivalent. (shrink)

A novel rereading of the relationship between ethics and ontology in Aristotle.

We critically reconsider an old hypothesis of the role of the dekad in Pythagorean philosophy. Unlike Zhmud, we claim that: 1) the dekad did play a role in Philolaus’ astronomical system, and 2) Aristotle did not project Plato’s theory of the ten eidetic numbers onto the Pythagoreans. We claim that the dekad, as the τέλειος ἀριθμός, should be understood in Philolaus’ philosophy as completeness and the basis of counting in Greek – as in most other languages – in a decimal (...) system. Additionally, we argue that the number ten is not even a candidate for the τέλειος ἀριθμός in Plato’s philosophy. As a final result of our discussion, we compare and contrast Philolaus’, Plato’s, and Speusippus’ accounts of completeness in relation to numbers. (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)

At Metaphysics A 5 986a22-b2, Aristotle refers to a Pythagorean table, with two columns of paired opposites. I argue that 1) although Burkert and Zhmud have argued otherwise, there is sufficient textual evidence to indicate that the table, or one much like it, is indeed of Pythagorean origin; 2) research in structural anthropology indicates that the tables are a formalization of arrays of “symbolic classification” which express a pre-scientific world view with social and ethical implications, according to which the presence (...) of a principle on one column of the table will carry with it another principle within the same column; 3) a close analysis of Aristotle's arguments shows that he thought that the table expresses real causal relationships; and 4) Aristotle faults the table of opposites with positing its principles as having universal application and with not distinguishing between those principles that are causally prior and those that are posterior. Aristotle's account of scientific explanation and his own explanations that he developed in accordance with this account are in part the result of his critical encounter with this prescientific Pythagorean table. (shrink)

I discuss Aristotle’s opening argument against Platonic Forms in _Metaphysics_ A.9, ‘the Razor’, which criticizes the introduction of Forms on the basis of an analogy with a hypothetical case of counting things. I argue for a new interpretation of this argument, and show that it involves two interesting objections against the introduction of Forms as formal causes: one concerns the completeness and the other the adequacy of such an explanatory project.

: 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)

Plato’s nephew Speusippus has been widely accepted as the historical person behind the mask of the anti-hedonists in Phlb. 42b–44c. This hypothesis is supported by, inter alia, the link between Socrates’ char- acterization of them as δυσχερεῖς and the frequent references of δυσχέρεια as ἀπορία to Speusippus in Aristotle’s Metaphysics MN. This study argues against assigning any privileged status to Speusippus in the assimilation of δυσχέρεια with ἀπορία. Instead, based on a comprehensive survey of how δυσχερ- words were used in (...) classical antiquity, the semantic shift of δυσχέρεια can be explained in an alternative way. (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)

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)

Philosophical accounts of emergence have been explicated in terms of logical relationships between statements (derivation) or static properties (function and realization). Jaegwon Kim is a modern proponent. A property is emergent if it is not explainable by (or reducible to) the properties of lower level components. This approach, I will argue, is unable to make sense of the kinds of emergence that are widespread in scientific explanations of complex systems. The standard philosophical notion of emergence posits the wrong dichotomies, confuses (...) compositional physicalism with explanatory physicalism, and is unable to represent the type of dynamic processes (self-organizing feedback) that both generate emergent properties and express downward causation. (shrink)

Many have argued that Plato’s intermediates are not independent entities. Rather, they exemplify the incapacity of discursive thought to cognizing Forms. But just what does this incapacity consist in? Any successful answer will require going beyond the intermediates themselves to another aspect of Plato’s mathematical thought - his attribution of a quasi-numerical structure to Forms. For our purposes, the most penetrating account of eidetic numbers is Jacob Klein’s, who saw clearly that eidetic numbers are part of Plato’s inquiry into the (...) ontological basis for all counting: the existence of a plurality of formal elements, distinct yet combinable into internally articulate unities. However, Klein’s study of the Sophist reveals such articulate unities as imperfectly countable and therefore opaque to διάνοια. And only this opacity, I argue, successfully explains the relationship of intermediates to Forms. (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)

In the 'Hippias Major' Socrates uses a counter-example to oppose Hippias‘s view that parts and wholes always have a "continuous" nature. Socrates argues, for example, that even-numbered groups might be made of parts with the opposite character, i.e. odd. As Gadamer has shown, Socrates often uses such examples as a model for understanding language and definitions: numbers and definitions both draw disparate elements into a sum-whole differing from the parts. In this paper I follow Gadamer‘s suggestion that we should focus (...) on the parallel between numbers and definitions in Platonic thought. However, I offer a different interpretation of the lesson implicit in Socrates‘s opposition to Hippias. I argue that, according to Socrates, parts and sum-wholes may share in essential attributes; yet this unity or continuity is neither necessary, as Hippias suggests, nor is it impossible, as Gadamer implies. In closing, I suggest that this seemingly minor difference in logical interpretation has important implications for how we should understand the structure of human communities in a Platonic context. (shrink)

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)

It is a notion commonly acknowledged that in his work Timaeus the Athenian philosopher Plato (_c_. 429–347 BC) laid down an early chemical theory of the creation, structure and phenomena of the universe. There is much truth in this acknowledgement because Plato’s “chemistry” gives a description of the material world in mathematical terms, an approach that marks an outstanding advancement over cosmologic doctrines put forward by his predecessors, and which was very influential on western culture for many centuries. In the (...) present article, I discuss inter-transformations among Plato’s four types (fire, air, water, and earth) as well as the interpretation they received in the literature. I find that scientists and scholars generally emphasized (and often misunderstood) mathematical aspects of these “reactions” over the philosophical ones. I argue that Plato’s “chemistry” in fact bears on crucial topics of his philosophical system, such as Forms, Becoming, causation and teleology. I propose that consideration of these doctrines help to understand not only the sense of his “chemical” reactions, but also the reason why their stoichiometry is by surface balance and is restricted only to types that come to be and pass away but not to those that provoke the inter-transformations. (shrink)

Aristotle’s classification of ideal number in Metaphysics M 6 has often been considered an unfair presentation of Plato’s actual views. I take another look at the passage and argue that Aristotle is a more careful critic than has been usually recognised. In particular, I argue that much of the scholarly discussion on the passage has failed to take account of Aristotle’s deeper concern, namely, the conditions necessary for numbers to be ordinal. I then set Aristotle’s critique within the broader context (...) of his concern to separate Form from Number, and thus to separate metaphysical questions from mathematical ones. The latter half of the article shows some of the consequences of mixing mathematics with metaphysics from a systematic and historical point of view. (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.

A formalização dos raciocínios a que Aristóteles se consagra nos Primeiros Analíticos é restrita, como sabemos, a proposições da forma categórica. Embora reco- nheça certas especificidades formais nas proposições encerrando predicados relacionais, Aristóteles parece reservar-lhes um estatuto secundário, considerando-as, em alguma medida, redutíveis a proposições categóricas. Neste artigo, pretendo examinar algumas passagens de Categorias 7 que possam lançar alguma luz sobre o estatuto que Aristóte- les confere às atribuições relativas, visando melhor precisar as razões que o teriam conduzido a negligenciar (...) um tratamento formal mais rigoroso das proposições relacionais nos Primeiros Analíticos. (shrink)

Seweryn Blandzi Plato and the Protestant Principle of Autarchy of the Scripture (sola scriptura)The author gives reasons why the new holistic Tübingen-interpretation of Plato (H. Krämer, K. Gaiser, Th. A. Szlezák), which combines the Dialogues with his unwritten teaching is still difficult to accept (especially in Germany). The discovery (on the basis of indirect testimonies of Aristotle and his commentators) that there was a separate oral (“exoteric”) metaphysics of principles, which was parallel to dialogues but more valuable (timiotera) in content, (...) and which Plato did not popularize in his writings, has been an antithesis to the traditional romantic paradigm (particularly strong in Germany) of F. Schlegel-Schleiermacher, founded on the conviction of the primacy of autarchy of the written (“exoteric”) text in general. This conviction leads us to the Protestant postulate of Biblical exegesis treated as self-sufficient (sola scriptura), interpreted exclusively on its own basis (sui ipsius interpres), and Schleiermacher transposes this requirement onto Plato’s writings. On the other hand, F. Schlegel, who emphasizes the fragmentary and asystematic character of Plato’s thought in dialogues, sees Plato as a thinker who constantly moves toward infinity, unable ever to reach it. With respect to the dialogues, the reader finds himself in a similar situation as the limited individuality of finite being to infinity. For the romantics (inspired by the idealistic philosophy of subjectivity and identity), man’s greatness as a finite being manifests itself in creating works which reflect infinity. The infinity of the Absolute manifests itself in the absolute freedom and the power of creation, which man – so to speak – meets halfway, with his infinite reflection, creativity, invention, interpretation as an expression of spirit’s freedom. Hence the lasting attraction of this traditional approach, which is unlike the new paradigm that imposes great requirements and constraints on the reader (studying doxographic literature, finding in it testimonies of Plato’s disciples regarding the intra-academic teaching). When Plato’s literary dialogue is treated as a kind of an “open text,” it thus justifies all the possible “hermeneutic” attempts at free and creative interpretations. Keywords: Plato, Hans Krämer, Friedrich Daniel Ernst Schleiermacher, Friedrich Schlegel, romantic paradigm, sola scriptura, infinity of reflection, new paradigm, unwritten teaching, metaphysics of principles. (shrink)