We provide a new interpretation of Zeno’s Paradox of Measure that begins by giving a substantive account, drawn from Aristotle’s text, of the fact that points lack magnitude. The main elements of this account are (1) the Axiom of Archimedes which states that there are no infinitesimal magnitudes, and (2) the principle that all assignments of magnitude, or lack thereof, must be grounded in the magnitude of line segments, the primary objects to which the notion of linear magnitude applies. Armed (...) with this account, we are ineluctably driven to introduce a highly constructive notion of (outer) measure based exclusively on the total magnitude of potentially infinite collections of line segments. The Paradox of Measure then consists in the proof that every finite or potentially infinite collection of points lacks magnitude with respect to this notion of measure. We observe that the Paradox of Measure, thus understood, troubled analysts into the 1880’s, despite their knowledge that the linear continuum is uncountable. The Paradox was ultimately resolved by Borel in his thesis of 1893, as a corollary to his celebrated result that every countable open cover of a closed line segment has a finite sub-cover, a result he later called the “First Fundamental Theorem of Measure Theory.” This achievement of Borel has not been sufficiently appreciated. We conclude with a metamathematical analysis of the resolution of the paradox made possible by recent results in reverse mathematics. (shrink)

A trope is an abstract particular. Trope theorists maintain that tropes exist and argue that they can solve important philosophical problems, such as explaining the nature of properties. While many contemporary interpreters of Aristotle read him as a trope theorist, few commentators distinguish different versions of trope theory. Which, of any, of these versions did Aristotle hold? Classical trope theorists say that individuals just are bundles of tropes. This essay offers a reading of Categories 2-5 and Metaphysics VII-VIII that aligns (...) Aristotle's view with nonclassical trope theory, according to which objects are more than just bundles of tropes. (shrink)

One of the main issues that dominates Neoplatonism in late antique philosophy of the 3rd–6th centuries A.D. is the nature of the first principle, called the ‘One’. From Plotinus onward, the principle is characterized as the cause of all things, since it produces the plurality of intelligible Forms, which in turn constitute the world’s rational and material structure. Given this, the tension that faces Neoplatonists is that the One, as the first cause, must transcend all things that are characterized by (...) plurality—yet because it causes plurality, the One must anticipate plurality within itself. This becomes the main mo- tivation for this study’s focus on two late Neoplatonists, Proclus (5th cent. A.D.) and Damascius (late 5th–early 6th cent. A.D.): both attempt to address this tension in two rather different ways. Proclus’ attempted solution is to posit intermediate principles (the ‘henads’) that mirror the One’s nature, as ‘one’, but directly cause plurality. This makes the One only a cause of unity, while its production of plurality is mediated by the henads that it produces. Damascius, while appropriating Proclus’ framework, thinks that this is not enough: if the One is posed as a cause of all things, it must be directly related to plurality, even if its causality is mediated through the henads. Damascius then splits Proclus’ One into two entities: (1) the Ineffable as the first ‘principle’, which is absolutely transcendent and has no causal relation; and (2) the One as the first ‘cause’ of all things, which is only relatively transcendent under the Ineffable. -/- Previous studies that compare Proclus and Damascius tend to focus either on the Ineffable or a skeptical shift in epistemology, but little work has been done on the causal framework which underlies both figures’ positions. Thus, this study proposes to focus on the causal frameworks behind each figure: why and how does Proclus propose to assert that the One is a cause, at the same time that it transcends its final effect? And what leads Damascius to propose a notion of the One’s causality that no longer makes it transcendent in the way that a higher principle, like the Ineffable, is? The present work will answer these questions in two parts. In the first, Proclus’ and Damascius’ notions of causality will be examined, insofar as they apply to all levels of being. In the second part, the One’s causality will be examined for both figures: for Proclus, the One’s causality in itself and the causality of its intermediate principles; for Damascius, the One’s causality, and how the Ineffable is needed to explain the One. The outcome of this study will show that Proclus’ framework results in an inner tension that Damascius is responding to with his notion of the One. While Damascius’ own solution implies its own tension, he at least solves a difficulty in Proclus—and in so doing, partially returns to a notion of the One much like Iamblichus’ and Plotinus’ One. (shrink)

Why is there a specific problem with biological individuality? Because the living realm contains a wide range of exotic particular concrete entities that do not easily match our ordinary concept of an individual. Slime moulds, dandelions, siphonophores are among the Odd Entities that excite the ontological zeal of the philosophers of biology. Most of these philosophers, however, seem to believe that these Odd Cases oblige us to refine or revise our common concept of an individual. They think, explicitly or tacitly, (...) that to be a living, evolutionary entity is to be a living individual. In this paper, we explore an alternative proposal: the variety and oddity of the forms of the living realm might be ontologically regimented through an increase in the categorial complexity of the living realm, by admitting, beside living individuals, living non-individuals or by acknowledging, more generally, that the evolutionary development of the living forms is not necessarily a process of building individuals, that life is not necessarily individuals-oriented. We claim that, from an ontological point of view, the spectacle of the living realm obliges us to take aggregativity seriously. (shrink)

Attempting to compare scientific theories requires a philosophical model of meaning. Yet different scientific theories have at times—particularly in early chemistry—pre-supposed disparate theories of meaning. When two theories of meaning are incommensurable, we must say that the scientific theories that rely upon them are meta-incommensurable. Meta- incommensurability is a more profound sceptical threat to science since, unlike first-order incommensurability, it implies complete incomparability.

I show that Aristotelian physics is a correct approximation of Newtonian physics in its appropriate domain, in the same precise sense in which Newton theory is an approximation of Einstein's theory. Aristotelian physics lasted long not because it became dogma, but because it is a very good theory.

In this article, I suggest that reflection on a broadly Aristotelian-cum-Hegelian conception about the determination of the conditions of identity and individuation of objects and properties shows that it entails (what Brandom calls) the Kant–Sellars thesis about modality and identity, one consequence of which is that persons are not identical to human beings. This view is in conflict with the Aristotelian liberal naturalist view to the effect that to be a person is identical to being an individual of a specific (...) animal kind—namely, homo sapiens—characterized by a specific and unique ‘form of life’ (and natural history) which differentiates it from all other animal kinds. I conclude that this novel and unorthodox liberal naturalist view about personhood constitutes an interesting and viable liberal naturalist alternative to more ‘orthodox’ liberal naturalist neo-Aristotelian views like Thompson’s, as it can better accommodate certain counterfactual ‘person-human being decoupling’ scenarios in our modern conception of ourselves-in-the-world. (shrink)

Part is not a univocal term. Uses of parthood and composition that do not obey any supplementation principle have a long philosophical tradition and strong support from contemporary physics. We call such uses potential parts. This paper first shows why potential parts are important and incompatible with supplementation, then provides a formal mereology for such parts inspired by the path-integral approach to quantum electrodynamics.

ABSTRACT ABSTRACT: I show that Aristotelian physics is a correct and nonintuitive approximation of Newtonian physics in the suitable domain in the same technical sense in which Newton’s theory is an approximation of Einstein’s theory. Aristotelian physics lasted long not because it became dogma, but because it is a very good, empirically grounded theory. This observation suggests some general considerations on intertheoretical relationships.

Pt. I. Zeno and the metaphysics of quantity. Zeno's paradox of measure -- Tractarian nominalism -- Logical atoms and combinatorial possibility -- Strict coherence, sigma coherence, and the metaphysics of quantity -- pt. II. Coherent degrees of belief. Higher-order degrees of belief -- A mistake in dynamic coherence arguments? -- Dynamic coherence and probability kinematics -- Updating, supposing, and MAXENT -- The structure of radical probabilism -- Diachronic coherence and radical probabilism -- pt. III. Induction. Carnapian inductive logic for Markov (...) chains -- Carnapian inductive logic and Bayesian statistics -- Bayesian projectability. (shrink)

The Middle Included is a systematic exploration of the meanings of logos throughout Aristotle’s work. It claims that the basic meaning is “gathering,” a relation that holds its terms together without isolating them or collapsing one to the other. This meaning also applies to logos in the sense of human language. Aristotle describes how some animals are capable of understanding non-firsthand experience without being able to relay it, while others relay it without understanding. Aygün argues that what distinguishes human language, (...) for Aristotle, is its ability to both understand and relay firsthand and non-firsthand experiences. This ability is key to understanding the human condition: science, history, news media, propaganda, gossip, utopian fiction, and sophistry, as well as philosophy. Ömer Aygün finds Aristotle’s name for this peculiar but crucial human ability of “gathering” both experiences is logos, and this leads to a claim about the specificity of human rationality and language. (shrink)

Due to the the movements in European thought from the 16th-17th centuries that have generated enough controversy around the ideas of ʻnatural philosophyʼ and ʻexperimental philosophyʼ, I think it would be appropriate to focus our attention on the evolution of the idea of ʻsubstantialismʼ in ʻtranscendental philosophyʼ from the Islamic philosophy. Mulla Sadrā was the most important philosopher and he established a truly hermeneutics on the idea of the ʻBeingʼ. Due tot the new fields of research into the prejudicative hermeneutics, (...) I can say that I will follow in this paper to manage this idea. I will follow the similarities between the Kitab written by Mulla Sadrā and the european idea of ʻnatural philosophyʼ, I will show some similarities between the ″Kitab al-Mashaʻir″ and ″Treatise on Heavens of Kant″, and i will bring those perspectives into a hermeneutical field of interpretation.The ideas of Kitab are very close to Heideggerʼs perspective on Being and aristotelian perspective of non-Being. Because of this reason, we say is necessary to.open a prejudicative perspective to investigate how works the play between Being and non-Being. (shrink)

In an earlier paper I suggested that we can solve the Benacerraf Problem – the problem of explaining how mathematical knowledge is possible on the assumption that the objects of mathematics are abstract and immaterial – by positing efficient causal relations between those abstract objects and our brains. The burden of the paper was to remove the appearance that relations between abstracta and concreta, far from being actual, are inconceivable. This alleged inconceivability has been derived from some putative conditions on (...) efficient causation: physical contact, energy transfer, reciprocal effects on the cause, and the idea that the cause must exist in time if its effects do. I argued that none of these conditions are a priori conditions on efficient causation, and thus do not pose an obstacle to the conceivability or metaphysical possibility of efficient causal relations between platonic objects and brains. In this essay I reply to three published challenges to this proposed solution to the Benacerraf Problem. (shrink)

One of the main challenges of the nonreductionist approach to complex structures and phenomena in philosophy of biology is its defense of the plausibility of the theory of emergence and downward causation. The tension between remaining faithful to the rules of physicalism and physical causal closure, while defending the novelty and distinctiveness of emergents from their basal constituents, makes the argumentation of many proponents of emergentism lacking in coherency and precision. In this article I aim at answering the suggestion of (...) several thinkers to redefine emergence and downward causation in terms of the broader Aristotelian view of causation. In addition, I further develop this interdisciplinary conversation to include theological implications of emergentism, analyzed in reference to Aquinas’ understanding of divine action in terms of the same fourfold division of causes—bringing thus natural science, philosophy, and theology into creative and fruitful dialogue. (shrink)

The accuracy of Aristotle’s definition of place is defended in terms of his form-matter theory. This theory is in turn defended against the objectionable notion that it entails matter is ultimately characterless.

"The world is a finite body, and therefore has finite power." John Philoponus is remembered for using this Aristotelian premise to break ranks with Aristotle and argue that the world is not everlasting. This investigation reconsiders Philoponus’s arguments from finite power, and then explores the aftermath of this line of thinking in the works of three lesser-known Arabic intellectuals active in the generation before Avicenna (d. 1037): Abū l-Ḫayr Ibn Suwār (d. after 1017), Abū al-Ḥasan al-ʿĀmirī (d. 992), and Abū (...) Sahl al-Masīḥī (d. after 1025). Each engaged with this dictum in unique and novel ways, and in so doing anticipated a number of central features of Avicenna’s writings. The history of this argument is of crucial importance for understanding the evolution of natural philosophy and metaphysics in this formative period, away from tedious and simplistic arguments about creation and towards a more robust modal ontology based on intrinsic and extrinsic necessity. (shrink)

The traditional definition per genus et differentiam is argued to be cognitively grounded in perception and in order to avoid needless argument, definitions are stipulated to assert boundaries. An analysis of the notion of perspective shows that a boundary is a composite of two distinctions: similarity that includes and difference that excludes. The concept is applied to the type-token distinction and percepts are shown to be the result of a comparison between a token as representing some phenomenon and a type (...) as representing the kind of object the phenomenon might be. The Principle of Connection is proposed as the mental architecture that gives rise to percepts. The value of a percept is shown to be one of Identical, Similar, Different or Contrary and its perceptual aspects represent the cognitive schema of Essence, Quality, Quantity and Relation. The two notions of similarity and difference are claimed to constitute the necessary and sufficient conditions for definitions. Accordingly, the qualitative elements of a definition signify the genus as representing the similarity of the defined entity in terms of others of its kind and the quantitative elements signify the differentia as representing the difference it has from everything else. Paying proper attention to both qualitative and quantitative aspects will enable formulation of definitions that are ‘good’. (shrink)

Does the act of creation show itself anywhere within the creation? A common contemporary ontology tends to see two possibilities for those who want to defend a notion of creation. The first is to argue that an original set of materials was brought into existence out of nothing by divine action a long time ago. The second, in the tradition of Paley, posits a specific divine action that oversees the development of some of the materials into entities with an end-directedness. (...) Much contemporary energy focuses on the second possibility. The argument of the paper is that the ontology behind both of these possibilities, which limits itself to the notions of a creation of materials and the building of some of the materials into end-directed entities, conceals rather than reveals the idea of creation. The paper tries to show how an Aristotelian sense of nature, with its recognition of internal teleology and original spontaneity, offers a better starting point for coming up against the mystery of divine creative activity. (shrink)

The notion of indivisibles and atoms arose in ancient Greece. The continuum—that is, the collection of points in a straight line segment, appeared to have paradoxical properties, arising from the ‘indivisibles’ that remain after a process of division has been carried out throughout the continuum. In the seventeenth century, Italian mathematicians were using new methods involving the notion of indivisibles, and the paradoxes of the continuum appeared in a new context. This cast doubt on the validity of the methods and (...) the reliability of mathematical knowledge which had been regarded as established by the axiomatic method in geometry expounded by Aristotle’s younger contemporary Euclid. The teaching of indivisibles was banned within the Society of Jesus, the Jesuits. In England, indivisibles were used by the mathematician John Wallis, and there was an acrimonious and extended feud between Wallis and the philosopher Thomas Hobbes over legitimate methods of argument in mathematics. Notions of the infinitesimal were used by Isaac Newton and Gottfried Leibniz, and were attacked by Bishop Berkeley for the vagueness of the concept and the illegitimate reasoning applied to it. This article discusses aspects of these events with reference to the book Infinitesimal by Amir Alexander and to other sources. Also discussed are wider issues arising from Alexander’s book including: the changes in cultural sensibility associated with the growth of new mathematical and scientific knowledge in the seventeenth century, the changes in language concomitant with these changes, what constitutes valid methods of enquiry in various contexts, and the question of authoritarianism in knowledge. More general aims of this article are to widen the immediate mathematical and historical contexts in Alexander’s book, to bridge a gap in conversations between mathematics and the humanities, and to relate mathematical ideas to wider human and contemporary issues. (shrink)

Pan-dispositionalism is one of the major theories in current analytic metaphysics concerning dispositional properties and how they relate to categorical properties. According to pan-dispositionalists, all fundamental properties are dispositional in nature, such that any supposed categorical properties are either unreal or reducible in some way to the dispositional. I argue that if pan-dispositionalism is true then metaphysical naturalism is false. To the extent that one finds pan-dispositionalism a plausible theory, one ought to question the truth of metaphysical naturalism. On the (...) other hand, if one is a committed metaphysical naturalist, one ought to question the truth of pan-dispositionalism. Either way we get a significant result, of interest both to those working in metaphysics and to those working in philosophy of religion. (shrink)

Aristotle’s soul is a first principle (an ‘efficient cause’) of every vital change in an animal, in the way that a craft is a cause of its product’s coming-to-be. We argue that the soul’s causal efficacy cannot therefore be reduced to the formal constitution of vital phenomena, or to discrete interventions into independently constituted processes, but involves the exercise of vital powers. This reading does better justice to Aristotle’s conception of craft as a rational productive disposition; and it captures the (...) soul’s continuous causal role as that which brings about all forms of vital change and underwrites their unity. (shrink)