Do actual infinities exist or are they impossible? Does mathematical practice require the existence of actual infinities, or are potential infinities enough? Contrasting points of view are examined in depth, concentrating on Aristotle’s ancient arguments against actual infinities. In the long 19th century, we consider Cantor’s successful rehabilitation of the actual infinite within his set theory, his views on the continuum, Zeno's paradoxes, and the domain principle, criticisms by Frege, and the axiomatisation of set (...) theory by Zermelo, as well as Zermelo’s assertion of the primacy of potential infinity in mathematics. (shrink)

InMetaphysicsΘ.6, 1048b14-17, Aristotle treats the problem of what it is for the infinite to exist potentially, i.e. to be potentially actual. According to my interpretation, Aristotle argues that to exist potentially is for the infinite to have a potentiality which cannot be actualized in reality but only in thought, because it is a potentiality the process of whose actualization cannot be brought to an end.

It is commonly assumed that Aristotle denies any real existence to infinity. Nothing is actually infinite. If, in order to resolve Zeno’s paradoxes, Aristotle must talk of infinity, it is only in the sense of a potentiality that can never be actualized. Aristotle’s solution has been both praised for its subtlety and blamed for entailing a limitation of mathematic. His understanding of the infinite as simply indefinite (the “bad infinite” that fails to reach its accomplishment), his conception (...) of the cosmos and even its prime mover as finite (in the sense of autarchic/self-contained) have been contrasted with the subsequent claim of God as ens infinitum (understood as a “positive” infinity). The goal of this essay is to reexamine the major texts (notably De caelo) and to demonstrate that (1) Aristotle’s claim according to which there is no actual infinite concerns only substances, not processes. (2) That Aristotle does not deny an actual infinite as such. (3) That when considering time and God (qua eternal) Aristotle acknowledges an actual infinite. (shrink)

It is well known that Leibniz advocated the actual infinite, but that he did not admit infinite collections or infinite numbers. But his assimilation of this account to the scholastic notion of the syncategorematic infinite has given rise to controversy. A common interpretation is that in mathematics Leibniz’s syncategorematic infinite is identical with the Aristotelian potential infinite, so that it applies only to ideal entities, and is therefore distinct from the actual (...) infinite that applies to the actual world. Against this, I argue in this paper that Leibniz’s actual infinite, understood syncategorematically, applies to any entities that are actually infinite in multitude, whether numbers, actual parts of matter, or monads. It signifies that there are more of them than can be assigned a number, but that there is no infinite number or collection of them, which notion involves a contradiction. Similarly, to say that a magnitude is actually infinitely small in the syncategorematic sense is to say that no matter how small a magnitude one takes, there is a smaller, but there are no actual infinitesimals. In geometry one may calculate with expressions apparently denoting such entities, on the understanding that they are fictions, standing for variable magnitudes that can be made arbitrarily small, so as to produce demonstrations that there is no error in the resulting expressions. (shrink)

Part of the kalam cosmological argument draws upon the claim that an actual infinite cannot exist. Classical theists also maintain both that some individuals will earn eternal life and that God infallibly foreknows the future. The claim that these latter two theses do not require that an actual infinite exists because God possesses an intuitive, rather than propositional intellect, is examined and rejected. Although the future is potential, rather than actual, classical theism requires that (...) the future be, in a sense, actually infinite. (shrink)

The paper begins with a more carefully stated version of ontologically neutral (ON) logic, originally introduced in (Hailperin, 1997). A non-infinitistic semantics which includes a definition of potential infinite validity follows. It is shown, without appeal to the actual infinite, that this notion provides a necessary and sufficient condition for provability in ON logic.

The general concept of mental disorder specified in the fifth edition of the Diagnostic and Statistical Manual of Mental Disorders is definitional in character: a mental disorder might be identified with a harmful dysfunction. The manual also contains the explicit claim that each individual mental disorder should meet the requirements posed by the definition. The aim of this article is two-fold. First, we shall analyze the definition of the superordinate concept of mental disorder to better understand what necessary criteria actually (...) characterize such a concept. Second, we shall consider the concepts of some individual mental disorders and show that they are in tension with the definition of the superordinate concept, taking pyromania and narcissistic personality disorder as case studies. Our main point is that an unexplained and not-operationalized dysfunction requirement that is included in the general definition, while being systematically violated by the diagnostic criteria of specific mental disorders, is a logical error. Then, either we unpack and operationalize the dysfunction requirement, and include explicit diagnostic criteria that can actually meet it, or we simply drop it. (shrink)

The paper begins with a more carefully stated version of ontologically neutral (ON) logic, originally introduced in (Hailperin, 1997). A non-infinitistic semantics which includes a definition of potential infinite validity follows. It is shown, without appeal to the actual infinite, that this notion provides a necessary and sufficient condition for provability in ON logic.

Dr. Vukov analyzing patients with disorders of consciousness, proposed that medical well-regarded policy recommendations cannot be justified by looking solely to patients’ actual levels of consciousness (minimally conscious state – MCS versus vegetative state – VS), but that they can be justified by looking to patients’ potential for consciousness. One objective way to estimate this potential (actual physical possibility) is to consider a neurophysiologically informed strategy. Ideally such strategy would utilize objective brain activity markers of (...) consciousness/unconsciousness. The Operational Architectonics (OA) theory of brain-mind functioning is an example of such a strategy. Besides being mathematically simple, neurophysiologically accurate, and compatible with the cognitive/phenomenal perspectives on consciousness, application of OA strategy to quantitative EEG analysis of patients in VS and MCS revealed that the absence of consciousness in VS is paralleled by impairment of overall EEG operational architecture. Specifically, neuronal assemblies become smaller, their life span shortened, and they became highly unstable and functionally disconnected (desynchronized). At the same time, fluctuating (minimal) awareness in patients in MCS was paralleled by partial restoration of EEG operational architecture (increased size, life span, and stability of neuronal assemblies, together with an increased number and strength of functional connections among them), approaching the level found in healthy fully conscious participants. Moreover OA strategy to EEG analysis allowed predicting the emergence of consciousness in VS patients after six years following the brain trauma. Therefore, the OA methodology could reliably predict which patients may regain consciousness, and thus determine which patients may have the potential for consciousness. (shrink)

A form of quantification logic referred to by the author in earlier papers as being 'ontologically neutral' still made use of the actual infinite in its semantics. Here it is shown that one can have, if one desires, a formal logic that refers in its semantics only to the potential infinite. Included are two new quantifiers generalizing the sentential connectives, equivalence and non-equivalence. There are thus new avenues opening up for exploration in both quantification logic and (...) semantics of the infinite. (shrink)

This paper investigates the ontological presuppositions of quantifier logic. It is seen that the actual infinite, although present in the usual completeness proofs, is not needed for a proper semantic foundation. Additionally, quantifier logic can be given an adequate formulation in which neither the notion of individual nor that of a predicate appears.

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the (...) right modal logic is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the (...) right modal logic is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

The concept of minimal risk has been used to regulate and limit participation by adolescents in clinical trials. It can be understood as setting an absolute standard of what risks are considered minimal or it can be interpreted as relative to the actual risks faced by members of the host community for the trial. While commentators have almost universally opposed a relative interpretation of the environmental risks faced by potential adolescent trial participants, we argue that the ethical concerns (...) against the relative standard may not be as convincing as these commentators believe. Our aim is to present the case for a relative standard of environmental risk in order to open a debate on this subject. We conclude by discussing how a relative standard of environmental risk could be defended in the specific case of an HIV vaccine trial among adolescents in South Africa. (shrink)

1. Approximations of arbitrarily large but finite systems are often mistaken for infinite idealizations in statistical and thermal physics. The problem is illustrated by thermodynamically reversible processes. They are approximations of processes requiring arbitrarily long, but finite times to complete, not processes requiring an actual infinity of time.2. The present debate over whether phase transitions comprise a failure of reduction is confounded by a confusion of two senses of “level”: the molecular versus the thermodynamic level and the (...) few component versus the many component level. (shrink)

In Physics, Aristotle starts his positive account of the infinite by raising a problem: “[I]f one supposes it not to exist, many impossible things result, and equally if one supposes it to exist.” His views on time, extended magnitudes, and number imply that there must be some sense in which the infinite exists, for he holds that time has no beginning or end, magnitudes are infinitely divisible, and there is no highest number. In Aristotle's view, a plurality cannot (...) escape having bounds if all of its members exist at once. Two interesting, and contrasting, interpretations of Aristotle's account can be found in the work of Jaako Hintikka and of Jonathan Lear. Hintikka tries to explain the sense in which the infinite is actually, and the sense in which its being is like the being of a day or a contest. Lear focuses on the sense in which the infinite is only potential, and emphasizes that an infinite, unlike a day or a contest, is always incomplete. (shrink)

There are two distinct but interrelated questions concerning Aristotle’s account of infinity that have been the subject of recurring debate. The first of these, what I call here the interpretative question, asks for a charitable and internally coherent interpretation of the limited pieces of text where Aristotle outlines his view of the ‘potential’ (and not ‘actual’) infinite. The second, what I call here the philosophical question, asks whether there is a way to make Aristotle’s notion of the (...) potential infinite coherent and rigorous with modern tools that can stand as a rival to the widely-accepted view of the infinite as characterized in a mathematical theory of sets. In this paper, I argue that the theoretical roles that Aristotle intends his account of the potential infinite to fulfill lead to a deep and irresoluble tension that can help explain the persistence of debates on both of these questions. I do so by turning to the places where Aristotle attempts to argue for or against the existence of particular infinite processes to show that he slides between different underlying notions of when changes are possible. Making these underlying notions clear can help us better understand the role of Aristotle’s account in the history of philosophy, the possible pitfalls for a contemporary theory of the potential infinite, and what each of these debates might learn from each other. (shrink)

I conceive of Spinoza’s substance monism as a response to Aristotle’s prohibition against actual infinity for one key reason: nature, being all things, is necessarily infi nite. Spinoza encapsulates his substance monism with the phrase, “Deus sive Natura,” implying that there is only one infinite substance, which also possesses an infi nity of attributes, of which we are but modes. These logical delineations of substance never actually break up God’s reality. Aristotle’s well-known argument against the reality of an (...) actual infinity in his Physics prohibits the existence of an actually infinite bodily substance because it would necessarily “destroy” (Physics 204b26-27) all other elements or bodies. On Aristotle’s view, there is a fundamental and concrete distinction between things: each substance is primarily a this (Categories 3b10). I maintain that Spinoza’s rationalism and radicalization of the principle of sufficient reason lends him greater explanatory potential than Aristotle to justify the (non) existence of actual infinity. (shrink)

The paper discusses some different conceptions of the infinity, from Aristotle to Georg Cantor (1845-1918) and beyond. The ancient distinction between actual and potential infinity is explained, along with some arguments against the possibility of actually infinite collections. These arguments were eventually rejected by most philosophers and mathematicians as a result of Cantor’s elegant and successful theory of actually infinite collections.

The purpose of this note is to contrast a Cantorian outlook with a non-Cantorian one and to present a picture that provides support for the latter. In particular, I suggest that: i) infinite hyperreal numbers are the (actual, determined) infinite numbers, ii) ω is merely potentially infinite, and iii) infinitesimals should not be used in the di Finetti lottery. Though most Cantorians will likely maintain a Cantorian outlook, the picture is meant to motivate the obvious nature (...) of the non-Cantorian outlook. (shrink)

In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Here Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of (...) infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism. (shrink)

Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting (...) of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover. (shrink)

In this paper, I present a puzzle involving special relativity and the random selection of real numbers. In a manner to be specified, darts thrown later hit reals further into a fixed well-ordering than darts thrown earlier. Special relativity is then invoked to create a puzzle. I consider four ways of responding to this puzzle which, I suggest, fail. I then propose a resolution to the puzzle, which relies on the distinction between the potential infinite and the (...) class='Hi'>actual infinite. I suggest that certain structures, such as a well-ordering of the reals, or the natural numbers, are examples of the potential infinite, whereas infinite integers in a nonstandard model of arithmetic are examples of the actual infinite. (shrink)

In this paper I argue for the superiority of a critical realist understanding of interdisciplinarity over a mainstream understanding of it. I begin by exploring the reasons for the failure of mainstream researchers to achieve interdisciplinarity. My main argument is that mainstream interdisciplinary researchers tend to hypostatize facts, fetishize constant conjunctions of events and apply to open systems an epistemology designed for closed systems. I also explain how mainstream interdisciplinarity supports oppression and gross inequality. I argue that mainstream interdisciplinarity is (...) not true interdisciplinarity and refer to it accordingly as ‘condisciplinarity’. By way of example, I examine the condisciplinarity of the World Health Organization’s ecological model applied to the issue of men’s violence against women. Specifically, I argue that critical realist interdisciplinarity is preferable because it acknowledges inter alia the empirical, actual and real layers of reality, which allows it to develop depth-explanations of phenomena. In practice, this means that critical realist interdisciplinarity can potentially provide explanations that, compared to condisciplinarity, are broader (include more of the human and non-human context) and deeper (include for example individuals’ conscious and unconscious psychological motivations). In the World Health Organization’s example of the causes of men’s violence against women, condisciplinarity resulted in the absence of historical, global and unconscious aspects of the problem. It is also restricted the analysis to reductive, constant-conjunction based theories of the causes of the problem, specifically ‘risk factors’, thereby providing a relatively shallow explanation for the problem. (shrink)

Human language has the characteristic of being open and in some cases polysemic. The word “infinite” is used often in common speech and more frequently in literary language, but rarely with its precise meaning. In this way the concepts can be used in a vague way but an argument can still be structured so that the central idea is understood and is shared with to the partners. At the same time no precise definition is given to the concepts used (...) and each partner makes his own reading of the text based on previous experience and cultural background. In a language dictionary the first meaning of “infinite” agrees with the etymology: what has no end. We apply the word infinite most often and incorrectly as a synonym for “very large” or something that we do not perceive its completion. In this context, the infinite mentioned in dictionaries refers to the idea or notion of the “immeasurably large” although this is open to what the individual’s means by “immeasurably great.” Based on this linguistic imprecision, the authors present a non Cantorian theory of the potential and actual infinite. For this we have introduced a new concept: the homogon that is the whole set that does not fall within the definition of sets established by Cantor. (shrink)

Recent work on Kant’s conception of space has largely put to rest the view that Kant is hostile to actual infinity. Far from limiting our cognition to quantities that are finite or merely potentially infinite, Kant characterizes the ground of all spatial representation as an actually infinite magnitude. I advance this reevaluation a step further by arguing that Kant judges some actual infinities to be greater than others: he claims, for instance, that an infinity of miles (...) is strictly smaller than an infinity of earth-diameters. This inequality follows from Kant’s mereological conception of magnitudes (quanta): the part is (analytically) less than the whole, and an infinity of miles is equal to only a part of an infinity of earth-diameters. This inequality does not, however, imply that Kant’s infinities have transfinite and unequal sizes (quantitates). Because Kant’s conception of size (quantitas) is based on the Eudoxian theory of proportions, infinite magnitudes (quanta) cannot be assigned exact sizes. Infinite magnitudes are immeasurable, but some are greater than others. (shrink)

Resumen El modelo de ciudadanía pretendidamente universal inaugurado con la filosofía moderna junto al predominio de la razón instrumental derivada de la economía neoclásica, confluyen para definir una posición social inferior para las mujeres. La confluencia de ambas construcciones filosóficas, políticas y económicas configura un espacio económico de desigualdad de género que debe ser repensado críticamente. Frente a esta situación, nos proponemos abordar el potencial crítico de la economía feminista a la hora de subvertir la desigualdad generada por la preeminencia (...) del modelo del homo oeconomicus en el mundo actual y la falta de un sujeto político que atienda a las particularidades de otros grupos.The model of universal citizenship inaugurated with modern philosophy link to the predomíname of instrumental rationality derived from neoclassical economics come together to define a disadvantaged social position of women. The confluence of these philosophical, political and economic constructions builds an economic space of gender inequality that must be approached critically. Faced with this situation, we intend to address the critical potential of feminist economics to subverting the inequality generated by the preeminence of the homo oeconomicus model in economic space, and the lack of a political subject that addresses the particularles of other groups. (shrink)

Resumen El modelo de ciudadanía pretendidamente universal inaugurado con la filosofía moderna junto al predominio de la razón instrumental derivada de la economía neoclásica, confluyen para definir una posición social inferior para las mujeres. La confluencia de ambas construcciones filosóficas, políticas y económicas configura un espacio económico de desigualdad de género que debe ser repensado críticamente. Frente a esta situación, nos proponemos abordar el potencial crítico de la economía feminista a la hora de subvertir la desigualdad generada por la preeminencia (...) del modelo del homo oeconomicus en el mundo actual y la falta de un sujeto político que atienda a las particularidades de otros grupos.The model of universal citizenship inaugurated with modern philosophy link to the predomíname of instrumental rationality derived from neoclassical economics come together to define a disadvantaged social position of women. The confluence of these philosophical, political and economic constructions builds an economic space of gender inequality that must be approached critically. Faced with this situation, we intend to address the critical potential of feminist economics to subverting the inequality generated by the preeminence of the homo oeconomicus model in economic space, and the lack of a political subject that addresses the particularles of other groups. (shrink)

Philosophers have long discussed whether time really passes. Simultaneously, they have also discussed whether time could have begun from an infinite past. This paper clarifies the relationship between the reality of time’s passage and an infinite past. I assert that time cannot have an infinite past if time really passes. This argument is based on a proposition that an infinite series of events cannot be completed if time really passes. A seemingly strong objection to this proposition (...) is that no movement is possible if an infinite series cannot be completed. However, movement is clearly possible; thus, an infinite series can be completed. I argue that although it is sure that an infinite series can be completed in the static view, an infinite series cannot be completed (and thus, time could not have begun from an infinite past) in the dynamic view. (shrink)

In England, higher education institutions, together with the schools whose staff they train, are being required to incorporate synthetic phonics as one of the key approaches to the teaching of reading. Yet even if synthetic phonics can be identified as one of the component ‘skills’ of reading, an assumption vigorously contested in this paper, it does not follow that it can or should be taught explicitly and independently of reading for meaning. Imposing such a ‘method’ is, at a deep level, (...) incompatible with teachers acting as teachers and potentially constitutes a kind of abuse of early readers. Moreover, because of the conceptual difficulties confronting any attempt to specify synthetic phonics as an approach, there cannot exist, in principle, empirical research that actually supports its value. Current policy in England since the Rose Report, at least, assumes the opposite. All this cannot be dismissed as a little local difficulty for England concerning the teaching of reading English with all its spelling irregularities: there are broader implications for the relationship between research and teaching, together with continuing tendencies to impose ‘teacher proof’ approaches on schools and their staff in many countries. (shrink)

In this paper we attempt to clear out the ground concerning the Aristotelian notion of density. Aristotle himself appears to confuse mathematical density with that of mathematical continuity. In order to enlighten the situation we discuss the Aristotelian notions of infinity and continuity. At the beginning, we deal with Aristotle’s views on the infinite with respect to addition as well as to division. In the sequel, we focus our attention to points and discuss their status with respect to the (...) actuality–potentiality distinction. Then we focus our attention to the nature of continuity, which Aristotle tends to consider as leading to infinite divisibility and potential density of the extended. (shrink)

This study assesses the disconnect between television news directors' and journalism students' perceptions of issues in media ethics. Responses from 60 news directors and 166 students enrolled in ethics courses at three universities offer insight into what issues practitioners actually face, what issues students think they will face, and how serious each group perceives potential ethical dilemmas to be. Both groups agree that ethics is best learned on the job.

Most interpreters hold that Kant rejects actually infinite tota synthetica as conceptually impossible. This view is attributed to Kant to relieve him of the charge that the first antinomy’s thesis argument presupposes transcendental idealism. I argue that important textual evidence speaks against this view, and Kant in fact affirms the conceptual possibility of actually infinite tota synthetica. While this means the first antinomy may not be decisive as an indirect argument for idealism, it gives us a better account (...) of how our ideas of the unconditioned generate the antinomies, and it allows us to see important and often overlooked elements in Kant’s account of the infinite. (shrink)

SummaryAlthough desired family size is often different from actual family size, the dynamics of this difference are not well understood. This paper examines the patterns and determinants of the difference between desired and actual number of children among women aged 15–49 years using pooled data from the 1990, 1999 and 2003 Nigeria Demographic and Health Surveys. The results show that more than two-thirds of the sample have unmet fertility desires. It was found that early and late childbearing increased (...) the odds of unmet fertility desires. Also, women with low levels of education, from poor households, rural residents as well as those who had experienced child death were at a higher risk of unmet fertility desires in the multivariate context. The study highlights the policy and programme implications of the findings. (shrink)

In an article in this issue of the Hastings Center Report, Aaron Wightman and his coauthors attempt to address health care providers’ moral distress about acceding to parents’ requests to provide life‐sustaining medical treatment to children who have profound cognitive disabilities. They propose combining John Arras's relational potential standard and care ethics, and they argue that the capacity for caring relationships can provide an independent moral justification for honoring such requests. This combination is, however, unstable. Wightman et al.'s language (...) of potential and capacity opens the possibility of substantial misinterpretation. They rely on epistemological and prognostic uncertainty to argue that reciprocity and participation may be present in the parent‐child relationship even when the child's engagement cannot be observed. The terminology suggests that these are characteristics that can be gained or lost rather than characteristics of being born within certain social practices. In contrast, Eva Feder Kittay illuminates family membership as an important social relation. Her articulation of the independent moral value of parenting stands on its own without being conjoined to Arras's position. (shrink)

In a series of much discussed articles and books, William Lane Craig defends the view that the past could not consist in a beginningless series of events. In the present paper, I cast a critical eye on just one part of Craig's case for the finitude of the past – viz. his philosophical argument against the possibility of actually infinite sets of objects in the ‘real world’. I shall try to show that this argument is unsuccessful. I shall also (...) take a close look at several considerations that are often thought to favour the possibility of an actual infinite, arguing in each case that Craig's response is inadequate. (shrink)

Aristotle's defense of Dunamis -- Power and potentiality -- Rational and nonrational powers -- The priority of actuality -- Ontological hierarchy, normativity, and gender.

The era of AI-based decision-making fast approaches, and anxiety is mounting about when, and why, we should keep “humans in the loop” (“HITL”). Thus far, commentary has focused primarily on two questions: whether, and when, keeping humans involved will improve the results of decision-making (making them safer or more accurate), and whether, and when, non-accuracy-related values—legitimacy, dignity, and so forth—are vindicated by the inclusion of humans in decision-making. Here, we take up a related but distinct question, which has eluded the (...) scholarship thus far: does it matter if humans appear to be in the loop of decision-making, independent from whether they actually are? In other words, what is stake in the disjunction between whether humans in fact have ultimate authority over decision-making versus whether humans merely seem, from the outside, to have such authority? Our argument proceeds in four parts. First, we build our formal model, enriching the HITL question to include not only whether humans are actually in the loop of decision-making, but also whether they appear to be so. Second, we describe situations in which the actuality and appearance of HITL align: those that seem to involve human judgment and actually do, and those that seem automated and actually are. Third, we explore instances of misalignment: situations in which systems that seem to involve human judgment actually do not, and situations in which systems that hold themselves out as automated actually rely on humans operating “behind the curtain.” Fourth, we examine the normative issues that result from HITL misalignment, arguing that it challenges individual decision-making about automated systems and complicates collective governance of automation. (shrink)

One of the principal lines of argument deployed by the friends of the kalām cosmological argument against the possibility of a beginningless series of events is a quite general argument against the possibility of an actual infinite. The principal thesis of the present paper is that if this argument worked as advertised, parallel considerations would force us to conclude, not merely that a series of discrete, successive events must have a first member, but also that such a series (...) must have a final member. Anyone who thinks that an endless series of events is possible must therefore reject this popular line of argument against the possibility of an actual infinite. (shrink)

One of the principal lines of argument deployed by the friends of the kalām cosmological argument against the possibility of a beginningless series of events is a quite general argument against the possibility of an actual infinite. The principal thesis of the present paper is that if this argument worked as advertised, parallel considerations would force us to conclude, not merely that a series of discrete, successive events must have a first member, but also that such a series (...) must have a final member. Anyone who thinks that an endless series of events is possible must therefore reject this popular line of argument against the possibility of an actual infinite. (shrink)

In The Metaphysics Book Theta, Chapter four, Aristotle claims that to state that “some X is possible but X will never be” is a mistake. In effect, he collapses the possible into the actual. This view conflicts with the existence of dispositions which I argue exist, as they are indispensable to science. In Theta Chapter three, Aristotle sets out a test of possibility whereby we assume that some entity exists and then see if an impossibility ensues. I apply this (...) test to Aristotle’s theory and show that it entails the impossibility of dispositions. Given the clear existence of dispositions, Aristotle’s conflation of the possible with the actual fails his own test of possibility and must be wrong. (shrink)

This book offers a discussion of the kalām cosmological argument, and presents a defence of a version of that argument after critically evaluating three of the most important versions of the argument. It argues that, since the versions of the kalām cosmological argument defended by Philoponus (c. 490–c. 570), al-Ghazālī (1058– 1111), and the contemporary philosopher, William Lane Craig, all deny the possibility of the existence of an actual infinite, these arguments are incompatible with Platonism and the view (...) that God foreknows an endless future. This conclusion, however, is not a problem for the proponents of the kalām cosmological argument, for the book shows how the argument can be defended without denying the possibility of the actual infinite. -/- In order to offer a comprehensive analysis of Philoponus and al-Ghazālī’s cosmological arguments, the book draws on recent English translations of some of their works. Next, the book advances a detailed argument against the popular argument based on the impossibility of an actual infinite. Finally, the book offers a unique defence of the kalām cosmological argument by defending philosophical arguments for a beginning of time that do not deny the actual infinite, evaluating which hypothesis best explains the discoveries of modern cosmology, and offering an argument in support of the premise that, if the universe came into existence, then God brought it into existence. (shrink)