I present a discussion of some issues in the ontology of spacetime. After a characterisation of the controversies among relationists, substantivalists, eternalists, and presentists, I offer a new argument for rejecting presentism, the doctrine that only present objects exist. Then, I outline and defend a form of spacetime realism that I call event substantivalism. I propose an ontological theory for the emergence of spacetime from more basic entities. Finally, I argue that a relational theory of pre-geometric entities can give rise (...) to substantival spacetime in such a way that relationism and substantivalism are not necessarily opposed positions, but rather complementary. In an appendix I give axiomatic formulations of my ontological views. (shrink)

Scholars often assert that Plato and Aristotle share the view that discursive thought is internal speech. However, there has been little work to clarify or substantiate this reading. In this paper I show Plato and Aristotle share some core commitments about the relationship of thought and speech, but cash out TIS in different ways. Plato and Aristotle both hold that discursive thinking is a process that moves from a set of doxastic states to a final doxastic state. The resulting judgments (...) can be true or false. Norms govern these final judgments and, in virtue of that, they govern the process that arrives at those judgments. The principal norm is consistency. However, the philosophers differ on the source of this norm. For Plato, persuasiveness and accuracy ground non-contradiction because internal speech is dialogical. For Aristotle, the Principle of Non-Contradiction grounds a Doxastic Thesis that no judgment can contradict itself. For Aristotle, metaphysics grounds non-contradiction because internal speech is monological. (shrink)

I examine the reasons Aristotle presents in Physics VIII 8 for denying a crucial assumption of Zeno’s dichotomy paradox: that every motion is composed of sub-motions. Aristotle claims that a unified motion is divisible into motions only in potentiality (δυνάμει). If it were actually divided at some point, the mobile would need to have arrived at and then have departed from this point, and that would require some interval of rest. Commentators have generally found Aristotle’s reasoning unconvincing. Against David Bostock (...) and Richard Sorabji, inter alia, I argue that Aristotle offers a plausible and internally consistent response to Zeno. I defend Aristotle’s reasoning by using his discussion of what to say about the mobile at boundary instants, transitions between change and rest. There Aristotle articulates what I call the Changes are Open, Rests are Closed Rule: what is true of something at a boundary instant is what is true of it over the time of its rest. By contrast, predications true of something over its period of change are not true of the thing at either of the boundary instants of that change. I argue that this rule issues from Aristotle’s general understanding of change, as laid out in Phys. III. It also fits well with Phys. VI, where Aristotle maintains that there is a first boundary instant included in the time of rest, but not a “first in which the mobile began to change.” I then show how this rule underlies Aristotle’s argument that a continuous motion cannot be composed of actual sub-motions. Aristotle distinguishes potential middles, points passed through en route to a terminus, from actual middles. The Changes are Open, Rests are Closed Rule only applies to actual middles, because only they are boundaries of change that the mobile must arrive at and then depart from. On my reading, Aristotle argues that the instant of arrival, the first instant at which the mobile has come to be at the actual middle, cannot belong to the time of the subsequent motion. If it did, the mobile would already be moving towards the next terminus and thus, per Phys. VI 6, would have already left. But it cannot have moved away from the midpoint at the very same moment it has arrived there. This means that the instant of arrival must be separated from the time of departure by an interval of rest. I show how Aristotle’s reasoning applies generally to rule out any continuous reflexive motion or continuous complex rectilinear motion. On my interpretation, however, the argument does not apply to every change of direction. When, as in the case of projectile motion, multiple movers and their relative powers explain why the mobile changes directions, distinct sub-motions are not involved. Aristotle holds that such motions cannot be continuous, not because they involve intervals of rest, but because they involve multiple causes of motion. My interpretation of the Changes are Open, Rests are Closed Rule allows us to make better sense of Aristotle’s argument than any previous interpretation. (shrink)

ABSTRACT According to an influential line of argument, the past must be finite because no infinite series can be formed by successive addition. The present paper pinpoints the non sequitur at the heart of this argument, disentangles the ambiguities that disguise it, and dismantles the misleading picture of ‘traversing the infinite’ that gives the argument so much of its allure. Finally, the paper critically explores the related argument that a beginningless series of past events is impossible because there could be (...) no explanation of its having been ‘completed’ at one time rather than another. (shrink)

In this paper I investigate central temporal and spatial notions in the second part of Plato’s Parmenides and argue that also these notions, and not only the metaphysical ones usually discussed in the literature, can be understood as a response to positions and problems put on the table by Parmenides and Zeno. Of the spatial notions examined in the dialogue, I look at the problems raised for possessing location and shape, while with respect to temporal notions, I focus on the (...) discussion of ‘being in time’ and exaiphnês (the latter notion will be shown to be also an important influence for Aristotle’s Physics). In displaying a paradoxical character and taking up crucial notions from Parmenides, the second part of the Parmenides seems to display some resemblance to Zeno’s paradoxes. I will show, however, that in contrast to Zeno, the second part of Plato’s Parmenides also demonstrates Parmenides’ One to be problematic on its own terms. Furthermore, the dialogue presents not only important problems that these spatial and temporal notions seem to lead us into, but also establishes some positive features necessary to think of time and space. In this way, Plato’s Parmenides can be seen as contributing to the development of temporal and spatial conceptualisations in ancient Greek thought. (shrink)

Perhaps the most pressing challenge for singularism—the predominant view that definite plurals like ‘the students’ singularly refer to a collective entity, such as a mereological sum or set—is that it threatens paradox. Indeed, this serves as a primary motivation for pluralism—the opposing view that definite plurals refer to multiple individuals simultaneously through the primitive relation of plural reference. Groups represent one domain in which this threat is immediate. After all, groups resemble sets in having a kind of membership-relation and iterating: (...) we can have groups of groups, groups of groups of groups, etc. Yet there cannot be a group of all non-self-membered groups. In response, we develop a potentialist theory of groups according to which we always can, but do not have to, form a group from any sum. Modalizing group-formation makes it a species of potential, as opposed to actual or completed, infinity. This allows for a consistent, plausible, and empirically adequate treatment of natural language plurals, one which is motivated by the iterative nature of syntactic and semantic processes more generally. (shrink)

We think of a boundary whenever we think of an entity demarcated from its surroundings. There is a boundary (a line) separating Maryland and Pennsylvania. There is a boundary (a circle) isolating the interior of a disc from its exterior. There is a boundary (a surface) enclosing the bulk of this apple. Sometimes the exact location of a boundary is unclear or otherwise controversial (as when you try to trace out the margins of Mount Everest, or even the boundary of (...) your own body). Sometimes the boundary lies skew to any physical discontinuity or qualitative differentiation (as with the border of Wyoming, or the boundary between the upper and lower halves of a homogeneous sphere). But whether sharp or blurry, natural or artificial, for every object there appears to be a boundary that marks it off from the rest of the world. Events, too, have boundaries — at least temporal boundaries. Our lives are bounded by our births and by our deaths; the soccer game began at 3pm sharp and ended with the referee's final whistle at 4:45pm. It is sometimes suggested that even abstract entities, such as concepts or sets, have boundaries of their own, and Wittgenstein could emphatically proclaim that the boundaries of our language are the boundaries of our world. Whether all this boundary talk is coherent, however, and whether it reflects the structure of the world or simply the organizing activity of our mind, are matters of deep philosophical controversy. (shrink)

I argue that Descartes' Second Causal Proof of God in the Third Meditation evidences, and commits him to, the belief that time is "strongly discontinuous" -- that is, that there is actually a gap between each consecutive moment of time. Much of my article attempts to reconcile this interpretation, the "received view," with Descartes' statements about time, space, and matter in his other writings, including his correspondence with various philosophers.

This paper looks at Gregory of Nyssa's so-called "bundle theory" sensible individuals and matter (as recently argued by Gerd Van Riel and Thomas Wauters in a 2020 article) amidst the broader context of Gregory's view of created beings and his reception of Neoplatonist, Stoic, and Aristotelian conceptions of particulars and matter. I argue that Gregory's position is closer to an Aristotelian position, despite the parallels to Plotinus and other contemporaneous bundle theory positions: in arguing against prime matter, and insofar as (...) he emphasizes the identity of each kind of created being with its material substrate—understood as a "bundle" of properties—Gregory inadvertently upholds a reading of Aristotle that refutes an ontological notion of prime matter. (shrink)

I offer a novel solution to Zeno’s paradox of The Arrow by introducing nilpotent infinitesimal lengths of time. Nilpotents are nonzero numbers that yield zero when multiplied by themselves a certain number of times. Zeno’s Arrow goes like this: during the present, a flying arrow is moving in virtue of its being in flight. However, if the present is a single point in time, then the arrow is frozen in place during that time. Therefore, the arrow is both moving and (...) at rest. In “Zeno’s Arrow, Divisible Infinitesimals, and Chrysippus,” White suggests using an infinitesimal value as the length of the present. Contra Zeno, this allows the arrow to be moving in the present, rather than frozen in place. In this paper, I follow the basic outline of White’s solution but argue that his solution suffers from arbitrariness and a related theoretical artificiality in relation to the system of infinitesimals he invokes, viz. in relation to the hyperreal infinitesimals of nonstandard analysis. After arguing that any solution to the paradox must satisfy certain theoretical requirements, I examine White’s solution alongside two nilpotent solutions. One of these solutions is inspired by F.W. Lawvere’s Smooth Infinitesimal Analysis and the other is inspired by Paolo Giordano’s ring of Fermat Reals. I argue that Giordano’s nilpotents supply the best answer to Zeno’s paradox. (shrink)

One of the principal objections to a tensed or dynamic theory of time is the ancient puzzle about the extent of the present. Three alternative conceptions of the extent of the present are considered: an instantaneous present, an atomic present, and a non-metrical present. The first conception is difficult to reconcile with the objectivity of temporal becoming posited by a dynamic theory of time. The second conception solves that problem, but only at the expense of making change discontinuous. The third (...) conception is the most plausible: that "the present" is a non-metrical notion which must be completed by the mention of some event or interval in order to have a measure, in which case what is present varies with one's context. (shrink)

_ Source: _Volume 61, Issue 1, pp 43 - 59 Contemporary debate on Plato’s cosmogony often assumes that the ‘literal’ reading of the _Timaeus_ yields an account of creation, while the view that the cosmos always existed is non-literal. In antiquity, Taurus has been seen as a forerunner of the ‘non-literal’ interpretation. This paper shows, on the contrary, that Taurus’ argument for the sempiternity of the cosmos is a literalist one, relying on a strict linguistic analysis of _Timaeus_ 28b6-8.

On Zeno Beach there are infinitely many grains of sand, each half the size of the last. Supposing Aristotle denied the possibility of Zeno Beach, did he have a good argument for the denial? Three arguments, each of ancient origin, are examined: the beach would be infinitely large; the beach would be impossible to walk across; the beach would contain a part equal to the whole, whereas parts must be lesser. It is attempted to show that none of these arguments (...) was Aristotle’s. Indeed, perhaps Aristotle’s finitism applied to magnitude only, not plurality. (shrink)

We explore the better known paradoxes of Zeno including modern variants based on infinite processes, from the point of view of standard, classical analysis, from which there is still much to learn (especially concerning the paradox of division), and then from the viewpoints of non-standard and non-classical analysis (the logic of the latter being intuitionist).The standard, classical or Cantorian notion of the continuum, modeled on the real number line, is well known, as is the definition of motion as the time (...) derivative of distance (we are not concerned with position and motion in more than one dimension, since Zeno wasn't). The real number line consists of its points, the distance between distinct points being positive and finite. The standard, classical derivative relies on the classical notion of limit, which does not use infinitesimals. (shrink)

Perhaps the real paradox of Zeno's Arrow is that, although entirely stationary, it has, against all odds, successfully traversed over two millennia of human thought to trouble successive generations of philosophers. The prospects were not good: few original Zenonian fragments survive, and our access to the paradoxes has been for the most part through unsympathetic commentaries. Moreover, like its sister paradoxes of motion, the Arrow has repeatedly been dismissed as specious and easily dissolved. Even those commentators who have taken it (...) seriously have propounded solutions with which they profess themselves to be perfectly satisfied. So my question is: will Zeno's Arrow survive into the millennium just begun? (shrink)

Proclus defends the Platonic view that the heavens consist in (the highest gradations) of all four elements. He attacks Aristotle's view that the heavens consist in a distinct, fifth element.

The paradigmatic examples of what we call nowadays ‘mere Cambridge changes’ are relational properties. If someone is on the left of a table at t − 1 and on the right of this table at t, the table does not undergo a physical change, but it has nonetheless new relational properties. What kind of relation lies behind this kind of change? Should we abandon the definition of identity as a set of permanent properties through time? This concern with identity and (...) change was already present in Aristotle's Physics 5 and 7 and medieval commentators tackled the problem with some important refinements due to their metaphysical discussions about the nature of relations. John of Jandun's discussion of this topic, at the beginning of the fourteenth century, is particularly interesting. First, he defines self-identity as a relation of reason, which means for him that it is not a real relation. Second, he distinguishes two kinds of relational changes: those involving real relations and the acquisition a qualitative propert... (shrink)

Hartshorne and Reese thought that in the Timaeus Plato wasn’t quite a panentheist—though he would have been if he’d been consistent. More recently, Cooper has argued that while Plato’s World Soul may have inspired panentheists, Plato’s text does not itself describe a form of panenetheism. In this paper, I will reconsider this question not only by examining closely the Timaeus but by thinking about which features of current characterizations of panentheism are historically accidental and how the core of the doctrine (...) might most fruitfully be understood. I’ll argue that there is a polytheistic view that deserves to be called panentheistic and that Plato’s Timaeus describes such a view. (shrink)

What is time? Neither the numbering of the motion of things nor their schema, but their way of being. In language, time shows itself as tense. But every verb has both tense and aspect. So what is aspect? Irreducible to tense, it is the way in which anything is at any time whatsoever. Thus the way things are, their being, is not merely temporal – for it is just as aspectual.

Of the many problems which evolutionary theodicy tries to address, the ones of animal suffering and extinction seem especially intractable. In this essay, I show how C. D. Broad's growing block conception of time does much to ameliorate the problems. Additionally, I suggest it leads to another way of understanding the soul. Instead of it being understood as a substance, it is seen as a history—a history which is resurrected in the end times. Correspondingly, redemption, I argue, should not be (...) seen as an event which redeems some future portion of time. God's triumph is over all of history, not just some future temporal portion. (shrink)

In this paper, we consider Andrew Loke’s recent contributions to the successive addition argument. Although he claims to develop the discussion, we conclude that he fails to provide anything that goes beyond the position critiqued by Fellipe Leon. When analysing Loke’s position, we find that his proposals either directly collapse back into those critiqued by Leon, or beg the relevant question at hand. We conclude with some speculations about why this sort of mistake may have arisen.

Despite the language we saw in the previous chapter, which allowed for time apprehension by perception and marking, in Physics iv 14, Aristotle famously argues that time is dependent on nous.

Writing about the history of science and the history of philosophy involves assumptions about the role of context and about the relationships between past and present ideas. Some historians emphasize the context, concentrating on the intellectual, personal, and social factors that affect the way earlier thinkers have approached their subject. Analytic philosophers take a critical approach, considering the logic and merit of the arguments of past thinkers almost as though they are engaging in contemporary debates. Some philosophers use the ideas (...) of historical figures to support their own philosophical agendas. Scholarly studies of the French natural philosopher Pierre Gassendi (1592–1655) exemplify many of these .. (shrink)

Terms for consciousness, used with a cognitive meaning, emerged as count nouns in the 17th century. This transformation repeats an evolution that had taken place in late antiquity, when related vocabulary, used in the sense of conscience, went from being mass nouns designating states to count nouns designating faculties possessed by every individual. The reified concept of consciousness resulted from the rejection of the Scholastic-Aristotelian theory of mind according to which the mind is not a countable thing, but a pure (...) potentiality. This rejection was motivated by an acute sense of the mind’s fallible subjectivity. While conditioned by recent historical events, the 17th century’s pervasive sense of subjectivity also reveals a heavy debt to Hellenistic philosophy, which had been recently rediscovered. But whereas Hellenistic thought, mistrustful of theoria, only reifies conscience, early modern thinking, more mistrustful of praxis and seeking its grounding in theoria, goes a step further and reifies consciousness. Partly modeled on theological ideas, the resulting concept of consciousness is plagued by paradoxes that have becomes notorious for their intractability. But essentially the same model of consciousness underwrites contemporary theory, embroiling contemporary debates in the same controversies that dominated the 17th century. Sidestepping these difficulties by returning to the Scholastic-Aristotelian theory of mind would be a tall order, but it is not impossible. Alfred North Whitehead's theory of consciousness offers an example. His novel theory of time enables Whitehead to rehabilitate the Aristotelian concept of passive mind in a wholly naturalistic way. (shrink)

According to classical theism, impassibility is said to be systematically connected to divine attributes like timelessness, immutability, simplicity, aseity, and self-sufficiency. In some interesting way, these attributes are meant to explain why the impassible God cannot suffer. I shall argue that these attributes do not explain why the impassible God cannot suffer. In order to understand why the impassible God cannot suffer, one must examine the emotional life of the impassible God. I shall argue that the necessarily happy emotional life (...) of the classical God explains why the impassible God cannot suffer. (shrink)

In this paper, we address an infamous argument against divisibility that dates back to Zeno. There has been an incredible amount of discussion on how to understand the critical notions of divisibility, extension, and infinite divisibility that are crucial for the very formulation of the argument. The paper provides new and rigorous definitions of those notions using the formal theories of parthood and location. Also, it provides a new solution to the paradox of divisibility which does not face some threats (...) that can possibly undermine the standard Lebesgue measure solution to such a paradox. (shrink)

In the eleventh book of the Confessiones Augustine puts forward several considerations about the nature of time. The received view is that he held that only the present exists, while the past and the future do not exist. This received view has recently been attacked by Paul Helm and Katherin Rogers, who have offered alternative interpretations according to which Augustine held that the present has no privileged ontological status, and that past, present and future all equally exist. The aim of (...) this paper is to defend the received view vis-à-vis these attacks. I argue that the received view fits best Augustine’s view of God’s eternity and relation to time and should therefore be preferred. (shrink)

According to three-dimensionalism, objects persist in time by being wholly present at each time they exist; on the contrary, four-dimensionalism asserts that objects persist by having different temporal parts at different times or that they are instantaneous temporal parts of four-dimensional aggregates. Le Poidevin has argued that four-dimensionalism better accommodates two common assumptions concerning persistence and continuity; namely, that time itself is continuous and that objects persist in time in a continuous way. To this purpose, he has offered two independent (...) arguments, each of which moves from a different understanding of continuity. In this paper, both of Le Poidevin’s arguments will be discussed and rejected. (shrink)

In our recent book (Abraham and Roy 2010) we have repurposed a mathematical model for the quantum vacuum as a model of consciousness. In this model, discrete space and time are derived from a discrete cellular dynamical network. As our model is essentially atomistic, we included in our book a short support chapter on atomism. In this aticle we expand on the few pages of that chapter devoted to the history of atomism, to place the current revival of atomism in (...) a larger context. (shrink)

Proclus' interpretation of the Timaeus confronts the question of whether the living being that is the Platonic cosmos perceives itself. Since sense perception is a mixed blessing in the Platonic tradition, Proclus solves this problem by differentiating different gradations of perception. The cosmos has only the highest kind. This paper contrasts Proclus' account of the world's perception of itself with James Lovelock's notion that the planet Earth, or Gaia, is aware of things going on within itself. This contrast illuminates several (...) key differences between contemporary theories of perception and the neoplatonic world view. In particular, it argues that the neoplatonists had a radically different view of these matters because they assigned the property of truth not only to representations, but to objects as well. (shrink)

This page is dedicated to an analysis of the first section of Parmenides' Poem, the Way of Truth, with a selection of critical judgments by the most important commentators and critics. In the Annotated Bibliography I list the main critical editions (from the first printed edition of 1573 to present days) and the translations in English, French, German, Italian and Spanish, with a selection of studies on Parmenides; in future, a section will be dedicated to an examination of some critical (...) variants of the Greek text, with particular attention to corrections to the Diels-Kranz (abbreviated DK) edition of the Fragmente der Vorsokratiker. The fragments of Parmenides' Poem are cited according to Diels-Kranz numbering system as adopted in the 6th edition, Berlin 1952; the Poem is divided into three parts: the Proem fr. I, 1-32; the Way of Truth (Alethéia) from fr. II to VIII, 49, and the Way of Mortal Opinion (Doxa) fr. VIII, 50 to XIX, 3. (shrink)

In the process of making bedsteads, Plato claimed, makers look towards the ‘idea’ of the bed. But what is that idea? Two candidates come to mind: shape and purpose. The fact that we identify objects of very different shape, not even involving a bedstead, as beds seems to render purpose conceptually superior. But, then, what is a bed’s purpose? An obvious response appears tobe: lying down and sleeping. Yet, first, beds are not needed for that. Secondly, precisely when a bed (...) is slept on, it is not perceived as a design object. Thirdly, beds constitute a kind of existential furniture, associated with giving birth, making love, suffering illness and dying. All such issues humans really care about, however, are strikingly indifferent towards the design of beds. As elsewhere, the maxim that form follows function fails. Design thinking might rather make headway when beds are conceived of as peculiar spaces within spaces. (shrink)

Timelines is an inquiry into the nature of time, both as an apparent feature of the external physical world and as a fundamental feature of our experience of ourselves in the world. The principal argument of Timelines is that our coventional ideas about time are largely mistaken and that what we think of as independent physical time is actually our calibration of a certain relation between events. Namely, the relation between time-keeping events and the causal sequential differences of physical processes (...) which are real and independent of us. (shrink)