The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the last hundred years. This article (...) explains the distinction and why it has proved to be one of the great organizing themes of mathematics. (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)

It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments considered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle has (...) good reason to resist the traditional arguments offered in favour of the existence of the infinite and that, while there is a lacuna in his own ‘logical’ arguments against actual infinities, his arguments against the existence of infinite magnitude and number are valid and more well grounded than commonly supposed. (shrink)

In Physics VI.9 Aristotle addresses Zeno's four paradoxes of motion and amongst them the arrow paradox. In his brief remarks on the paradox, Aristotle suggests what he takes to be a solution to the paradox.In two famous papers, both called 'A note on Zeno's arrow', Gregory Vlastos and Jonathan Lear each suggest an interpretation of Aristotle's proposed solution to the arrow paradox. In this paper, I argue that these two interpretations are unsatisfactory, and suggest an alternative interpretation. In particular, I (...) claim that what seems on the face of it to be Aristotle's solution to the paradox raises two puzzles to which my interpretation, as opposed to Lear's and Vlastos's, provides an adequate response. (shrink)

Naturalism in the philosophy of mathematics is the view that philosophy cannot legitimately gainsay mathematics. I distinguish between reinterpretation and reconstruction naturalism: the former states that philosophy cannot legitimately sanction a reinterpretation of mathematics (i.e. an interpretation different from the standard one); the latter that philosophy cannot legitimately change standard mathematics (as opposed to its interpretation). I begin by showing that neither form of naturalism is self-refuting. I then focus on reinterpretation naturalism, which comes in two forms, and examine the (...) only available argument for it. I argue that this argument, the so-called Failure Argument, itself fails. My overall conclusion is that although there is no self-refutation argument against reinterpretation naturalism, there are as yet no good reasons to accept it. Naturalism in mathematics The consistency of mathematical naturalism The failure argument Objections to the failure argument Philosophy as the default. (shrink)

James Thomson envisaged a lamp which would be turned on for 1 minute, off for 1/2 minute, on for 1/4 minute, etc. ad infinitum. He asked whether the lamp would be on or off at the end of 2 minutes. Use of “internal set theory” (a version of nonstandard analysis), developed by Edward Nelson, shows Thomson's lamp is chimerical; its copy within set theory yields a contradiction. The demonstration extends to placing restrictions on other “infinite tasks” such as Zeno's paradoxes (...) of motion and Kant's First Antinomy. Resolution of such logical-philosophical problems leads to some very general constraints which must be placed upon the syntax of physical theories. In particular, at some scale space and time would appear granular. The suitability of internal set theory for analyzing phenomena is examined, using a paper by Alper and Bridger (1997) to frame the discussion. (shrink)

Let a smooth experience be an experience with perfectly gradual changes in phenomenal character. Consider, as examples, your visual experience of a blue sky or your auditory experience of a rising pitch. Do the phenomenal characters of smooth experiences have continuous or discrete structures? If we appeal merely to introspection, then it may seem that we should think that smooth experiences are continuous. This paper (1) uses formal tools to clarify what it means to say that an experience is continuous (...) or discrete, and (2) develops a discrete model of the phenomenal characters of smooth experiences. As a result, I'll argue that introspection leaves open whether smooth experiences are continuous or discrete. Yet I’ll also argue—perhaps surprisingly—that the discrete theory may better fit our introspective evidence. (shrink)

This volume announces a new era in the philosophy of God. Many of its contributions work to create stronger links between the philosophy of God, on the one hand, and mathematics or metamathematics, on the other hand. It is about not only the possibilities of applying mathematics or metamathematics to questions about God, but also the reverse question: Does the philosophy of God have anything to offer mathematics or metamathematics? The remaining contributions tackle stereotypes in the philosophy of religion. The (...) volume includes 35 contributions. It is divided into nine parts: 1. Who Created the Concept of God; 2. Omniscience, Omnipotence, Timelessness and Spacelessness of God; 3. God and Perfect Goodness, Perfect Beauty, Perfect Freedom; 4. God, Fundamentality and Creation of All Else; 5. Simplicity and Ineffability of God; 6. God, Necessity and Abstract Objects; 7. God, Infinity, and Pascal's Wager; 8. God and (Meta-)Mathematics; and 9. God and Mind. (shrink)

Infinite regress arguments are a powerful tool in Aristotle, but this style of argument has received relatively little attention. Improving our understanding of infinite regress arguments has become pressing since recent scholars have pointed out that it is not clear whether Aristotle’s infinite regress arguments are, in general, effective or indeed what the logical structure of these arguments is. One obvious approach would be to hold that Aristotle takes infinite regress arguments to be per impossibile arguments, which derive an infinite (...) sequence. Due to his finitism, Aristotle then rejects such a sequence as impossible. This paper argues that this obvious approach does not work, even for its most amenable cases. The paper argues instead that infinite regress arguments involve domain-specific infinities, and so there is not a general finitism which underpins infinite regress arguments in Aristotle, but rather domain-specific reasons that there cannot be an infinite number of entities in each domain in which Aristotle invokes an infinite regress argument. (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)

In this excellent book Sebastien Gandon focuses mainly on Russell's two major texts, Principa Mathematica and Principle of Mathematics, meticulously unpicking the details of these texts and bringing a new interpretation of both the mathematical and the philosophical content. Winner of The Bertrand Russell Society Book Award 2013.

The model of a closed linear measure space, which can be used to model Aristotle’s treatment of motion (kinesis), can be analogically extended to the qualitative ‘spaces’ implied by his theory of contraries in Physics I and in Metaphysics Iota, and to the dimensionless ‘space’ of the unity of matter and form discussed in book Eta of the Metaphysics. By examining Aristotle’s remarks on contraries, the subject of change, continuity, and the unity of matter and form, Aristotle’s thoughts on motion, (...) on contraries, and on the unity of substance can be brought together under a single interpretive framework, which can provide an approach to interpreting his ontology. (shrink)

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)

In Physics 4.11, Aristotle says that changes are continuous because magnitude is continuous. I suggest that this is not Aristotle’s considered view, and that in Generation and Corruption 2.10 Aristotle argues that this leads to the unacceptable consequence that alterations can occur discontinuously. Physics 6.4 was written to amend this theory, and to argue that changes are continuous because changing bodies are so. I also discuss the question of Aristotle’s consistency on the possibility of discontinuous alterations, such as freezing.

In this paper, I advance an original view of the structure of space called Infinitesimal Gunk. This view says that every region of space can be further divided and some regions have infinitesimal size, where infinitesimals are understood in the framework of Robinson’s nonstandard analysis. This view, I argue, provides a novel reply to the inconsistency arguments proposed by Arntzenius and Russell, which have troubled a more familiar gunky approach. Moreover, it has important advantages over the alternative views these authors (...) suggested. Unlike Arntzenius’s proposal, it does not introduce regions with no interior. It also has a much richer measure theory than Russell’s proposal and does not retreat to mere finite additivity. (shrink)

This essay proposes a comprehensive blueprint for the hylomorphic foundations of cosmology. The key philosophical explananda in cosmology are those dealing with global processes and structures, the regularity of global regularities, and the existence of the global as such. The possibility of elucidating these using alternatives to hylomorphism is outlined and difficulties with these alternatives are raised. Hylomorphism, by contrast, provides a sound philosophical ground for cosmology insofar as it leads to notions of cosmic essence, the unity of complex essences, (...) and globally emergent properties. These are used as the basis to account for the aforementioned cosmological explananda and to resolve two problems in the philosophy of cosmology: the meta-law dilemma and the uniqueness of the universe. In summary, cosmology needs hylomorphism because it is able to ground cosmology’s efforts as a scientific inquiry. It can do so because hylomorphism philosophically accounts for changing substances and aggregates of substances, the various scales of law-governed behavior measured by the natures of those substances, and how those substances as parts relate to the universe as a whole. (shrink)

Essays on Aristotle's Sea-Battle, Lazy Argument, Argument Reaper, Diodorus' Master Argument -/- The book is devoted to the ancient logical theories, reconstruction of their semantic proprieties and possibilities of their interpretation by modern logical tools. The Ancient arguments are frequently misunderstood in modern interpretations since authors usually have tendency to ignore their historical proprieties and theoretical background what usually leads to a quite inappropriate picture of the argument’s original form and mission. Author’s primary intention was to draw attention to the (...) complexity of some historical arguments and to the theoretical context in which arguments were created, circulated, developed, and finally tuned. Four well-known ancient arguments – with a common central subject related to the future contingencies problem – are reconstructed from available historical sources: “The Sea Battle”, which is drawn from Aristotle’s treatise De Interpretatione; two arguments, usually ascribed to the Stoics, “The Lazy Argument” and “The Reaper”; “The Master Argument” of the Megarian philosopher Diodorus. Arguments are linguistically and semantically detaily analyzed, formally presented by reflecting some relevant corresponding hypotheses based on physical or logical theories of their ancient authors, and finally covered by appropriate logical tools familiar to a modern reader. Two appendices are added at the closing part of the book. One covering some assumptions relevant for understanding of rival streams in ancient theories of meaning related to the nature of names and naming; the other is devoted to the ancient understanding of logical proposition and attempts to find an adequate Latin translation of the Greek delicate philosophical term “ἀξίωμα”. (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)

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)

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)

The arrow paradox is an argument purported to show that objects do not really move. The two main metaphysics of motion, the At–At theory of motion and velocity primitivism, solve the paradox differently. It is argued that neither solution is completely satisfactory. In particular it is contended that there are no decisive arguments in favor of the claim that velocity as it is constructed in the At–At theory is a truly instantaneous property, which is a crucial assumption to solve the (...) paradox. If so the At–At theory faces the threat that most of our physical theories turn out to be non-Markovian. Finally it is considered whether all those threats and paradoxes are dispelled if only a new metaphysics of persistence is taken into account, namely four-dimensionalism. (shrink)

The author tends to emphasize that there are at least three reasons to analyze Callimachus\' epigram about Diodorus : First of all, the date of this epigram shows us that it represents the earliest information about Diodorus doctrine. Second, another support of its authenticity could be found in fact that this epigram expressing part of the atmosphere following, and also remaining after, discussing the Diodorian topics. Third, its philosophical relevance, usually minimized in classical literature, could be found in those facts (...) that it could show the way out in many today dilemmas about his philosophical claims and support some of our contemporary assumptions about its logical conception, as well as that of space, time, and meaning of statements. The author defends a position that it is necessary to develop well-grounded and methodologically relevant base covering the historical reconstruction and the interpretation of ancient logical theories. (shrink)

Calosi-Fano_Pre-socratic-discrete-kinematics2.