Eleatic Ontology in Aristotle: Introduction

The introduction summarizes the six new papers collected in Volume 1, Tome 5: Eleatic Ontology and Aristotle. The papers take a fresh look at virtually every aspect of Aristotle’s engagement with Eleaticism. They are particularly concerned with Aristotle’s responses to Parmenidean monism, the Eleatic rejection of change, and Zeno’s paradoxes. The contributions also focus on the ways in which Aristotle developed several of his own theories in metaphysics and natural science partly in reaction to Eleatic puzzles and arguments.

What about Plurality? Aristotle’s Discussion of Zeno’s Paradoxes

While Aristotle provides the crucial testimonies for the paradoxes of motion, topos, and the falling millet seed, surprisingly he shows almost no interest in the paradoxes of plurality. For Plato, by contrast, the plurality paradoxes seem to be the central paradoxes of Zeno and Simplicius is our primary source for those. This paper investigates why the plurality paradoxes are not examined by Aristotle and argues that a close look at the context in which Aristotle discusses Zeno holds the answer to this question.

Aristotle, Eleaticism, and Zeno’s Grains of Millet

This paper explores how Aristotle rejects some Eleatic tenets in general and some of Zeno’s views in particular that apparently threaten the Aristotelian “science of nature.” According to Zeno, it is impossible for a thing to traverse what is infinite or to come in contact with infinite things in a finite time. Aristotle takes the Zenonian view to be wrong by resorting to his distinction between potentiality and actuality and to his theory of mathematical proportions as applied to the motive power and the moved object (Ph. VII.5). He states that some minimal parts of certain magnitudes (i.e., continuous quantities) are perceived, but only in potentiality, not in actuality. This being so, Zeno’s view that a single grain of millet makes no sound on falling, but a thousand grains make a sound must be rejected. If Zeno’s paradoxes were true, there would be no motion, but if there is no motion, there is no nature, and hence, there cannot be a science of nature. What Aristotle noted in the millet seed paradox, I hold, is that it apparently casts doubt on his theory of mathematical proportions, i.e., the theory of proportions that holds between the moving power and the object moved, and the extent of the change and the time taken. This approach explains why Aristotle establishes an analogy between the millet seed paradox, on the one hand, and the argument of the stone being worn away by the drop of water (Ph. 253b15–16) and the hauled ship, on the other.

The myth of the three crises in foundations of mathematics Part 1

The story of “the three crises in foundations of mathematics” is widely popular in Russian publications on the philosophy of mathematics. This paper aims at evaluating this story against the background of the contemporary scholarship in the history of mathematics. The conclusion is that it should be considered as a specimen of modern myth-making activity brought to the fore by an unconscious tendency to model the whole history of mathematics on the pattern of the foundational crisis of the first decades of the 20th century. What is more, the consideration of the specific role and character of the foundations in both early Greek mathematics and 18th-century mathematics gives an occasion to raise a more general question regarding the true meaning of the historicity of mathematics. The first part of this paper deals with the point whether there was a foundational crisis in pre-Euclidean Greek mathematics caused by the discovery of incommensurable magnitudes and Zeno’s paradoxes. The result is negative: we have no direct historical evidence of such a crisis; as for secondary considerations, they also mainly count against it. The idea of the first crisis in foundations of mathematics has emerged as a result of the unjustified transference of the modern grasp of foundational issues and the modern “mentalité de crise” to the ancient past.

L’infini entre deux bouts. Dualités, univers algébriques, esquisses, diagrammes

The article affixes a resolutely structuralist view to Alain Badiou’s proposals on the infinite, around the theory of sets. Structuralism is not what is often criticized, to administer mathematical theories, imitating rather more or less philosophical problems. It is rather an attitude in mathematical thinking proper, consisting in solving mathematical problems by structuring data, despite the questions as to foundation. It is the mathematical theory of categories that supports this attitude, thus focusing on the functioning of mathematical work. From this perspective, the thought of infinity will be grasped as that of mathematical work itself, which consists in the deployment of dualities, where it begins the question of the discrete and the continuous, Zeno’s paradoxes. It is, in our opinion, in the interval of each duality ― “between two ends”, as our title states ― that infinity is at work. This is confronted with the idea that mathematics produces theories of infinity, infinitesimal calculus or set theory, which is also true. But these theories only give us a grasp of the question of infinity if we put ourselves into them, if we practice them; then it is indeed mathematical activity itself that represents infinity, which presents it to thought. We show that tools such as algebraic universes, sketches, and diagrams, allow, on the one hand, to dispense with the “calculations” together with cardinals and ordinals, and on the other hand, to describe at leisure the structures and their manipulations thereof, the indefinite work of pasting or glueing data, work that constitutes an object the actual infinity of which the theory of structures is a calculation. Through these technical details it is therefore proposed that Badiou envisages ontology by returning to the phenomenology of his “logic of worlds”, by shifting the question of Being towards the worlds where truths are produced, and hence where the subsequent question of infinity arises.

Paradox as a Guide to Ground

I will use paradox as a guide to metaphysical grounding, a kind of non-causal explanation that has recently shown itself to play a pivotal role in philosophical inquiry. Specifically, I will analyze the grounding structure of the Predestination paradox, the regresses of Carroll and Bradley, Russell's paradox and the Liar, Yablo's paradox, Zeno's paradoxes, and a novel omega plus one variant of Yablo's paradox, and thus find reason for the following: We should continue to characterize grounding as asymmetrical and irreflexive. We should change our understanding of the transitivity of grounding in a certain sense. We should require foundationality in a new, generalized sense, that has well-foundedness as its limit case. Meta-grounding is important. The polarity of grounding can be crucial. Thus we will learn a lot about structural properties of grounding from considering the various paradoxes. On the way, grounding will also turn out to be relevant to the diagnosis (if not the solution) of paradox. All the paradoxes under consideration will turn out to be breaches of some standard requirement on grounding, which makes uniform solutions of large groups of these paradoxes more desirable. In sum, bringing together paradox and grounding will be shown to be of considerable value to philosophy.1

Applying the immobility theory to thoroughly solve the three Zeno’s paradoxes

- Applying the law of conservation of time to solve the Achilles and the tortoise paradox.- Applying the smallest unit of time T_min in the universe to solve the Dichotomy paradox.- Applying the disappearing property of matter when moving to solve the Arrow paradox.

Infinity

The paradoxes of Zeno in antiquity might seem like sophisms, but, as it has turned out, they shed light on mathematical infinity and have led to many derivative ideas in mathematics and science. Zeno’s paradoxes portray movement in terms of discrete points on a number line, where a move from A to B is done in separate (discrete) steps. But the gap in between is continuous. So, to resolve the paradoxes a basic distinction between discrete and continuous is required—a gap that was filled with the calculus much later. The infinity concept became in the nineteenth century the basis for the discovery of new numbers—by the German mathematician Georg Cantor—which seemed at the time to be counterintuitive. This chapter looks at the paradox of infinity and at Cantor’s ingenious discoveries.

When Science Confronts Philosophy: Three Case Studies

This paper examines three cases of the clash between science and philosophy: Zeno’s paradoxes, the Frame Problem, and a recent attempt to experimentally refute skepticism. In all three cases, the relevant science claims to have resolved the purported problem. The sciences, construing the term broadly, are mathematics, artificial intelligence, and psychology. The goal of this paper is to show that none of the three scientific solutions work. The three philosophical problems remain as vibrant as ever in the face of robust scientific attempts to dispel them. The paper concludes by examining some consequences of this persistence.