One of the common claims of the eternalists is that the "actual" infinite is possible and the universe is eternal. They are trying to refute the Kalam argument. What I wanted to show in this paper is that the "actual" infinite is impossible for logical reasons, and I have shown further that infinity has an effect and application over time, and that there is no way to deny the beginning of the universe for existence. The paper points (...) out the problems of infinity and points to the beginning of the universe. (shrink)

Navimitto proposes that reality unfolds within a fundamental continuum of increasingly complex yet never infinite resolutions. Dimensions, space and time, particles and concepts all emerge step-by-step as these resolutions progress, recursively generating more nuanced angles, deeper differences and stronger distinctions. Yet resolution never actually reaches infinity according to Navimitto. It remains finite, cycling endlessly through this progression and simplifying back down again. There are infinite resolutions, but all remain finite steps. This gives rise to a kind of looping or (...) circular dynamics, with higher resolutions converging on lower ones. Reality is thus infinitely rich in possibility and infinitely dynamic, yet never transcends the finite. Infinities are approached but never reached. All concepts, from space and time to density and dimensionality, cycle through this endless progression and simplification. They take on new forms and properties at each point, as resolution changes, yet remain fundamentally defined by this looping cycle. Some of the most extreme and paradoxical phenomena, like singularities, black holes and the Big Bang, emerge from complicating this cycle. They represent infinitelimits and vastly strange warping, yet remain cycles embedded within the overall progression. Navimitto suggests we perceive such extremes from limited perspectives at any given point in the cycle. But a higher view sees only endless dynamics between simplicity and complexity. Navimitto thus proposes a deeply strange yet simple perspective - that reality infinitely generates itself in an endless cycle of recursion between limits. Space, time, matter and all we know emerge from and collapse back into fundamental simplicity and infinitely nuanced complexity, forever progressing yet never transcending the finite. We perceive edges and beginnings, but there are only cycles within cycles, forever reinventing perspective yet never changing the whole. Infinite possibility, finite reality. Endless dynamics, simple cycling. Strange simplicity, infinitely nuanced complexity. Navimitto aims to resolve paradox by finding unity across seeming extremes. It suggests we have but glimpsed the depths of what reality contains and what existence means. There remain far stranger things than we yet have comprehended within the finite yet infinitely rich possibilities of being. -/- Navomitto: A Multi-Dimensional Framework for Understanding Reality and Perception authur: Dr Mostafa Rahmani -/- navomitto’s principles: -/- top to down view: -/- 1-there is illusion. illusion is multi dimentional and made of dimentions. iluusion is all aspects of zero to infinity. -/- 2-there is dimention. each dimention describe a unique concept. these concepts can be anything. it can be mathematic concept like lenght or abstract concept like love or anything else. each dimension is unique in its own way but it can be seen as an interaction of infinite dimentions (entanglement with all of illusion). Some interactions are stronger and some weaker. These interactions creates physics. dimention exists in different resolutions. -/- 3-there is resolution. resolution determines how many angle there is in that specific resolution of that dimention. resolution can be anything from zero to infinity. so each concept like lenght has infinite resolutions. in each resolution there is angles. resolusion 0 of all dimentions are literally the same and it is potentiality itself. Each resolution can be seen as a slice of moment in illusion that each angle of that resolution represent a branch of illusion (in parallel universe view). -/- 4-there is angle. angle defines the content of that concept in that specific resolution. in resolution infinity there are infinite angles from zero to infinity. in resolution 0 there is no angle. -/- 5-angle 0 is illusion resolution 0. angle 1 of each resolusion is dimention resolution 1 of that concept. angle A of resolution R (R>=A) is dimention resolution A of that concept in scale 1/R. -/- 6-there is scale. scale mean the bigness of illusion in an angle. each angle in resolusion R has scale 1/R -/- 7-there is power. power is an aspect of reality that means how much quantity there is (when percieved) in an angle of each concept. power of each angle A in resolusion R is (2A-1)/2*R -/- down to top view: -/- 1-there is resoulution 0. illusion resolution 0 is resoulution 0 of all dimentions. illusion resolution 0 is illusion itself at resolution 0. resolution 0 contain no angle. -/- 2-there is dimention resoulution 1. next resolution is created when one illusion is added to previous resolution. so in dimention resolution 1, one illusion is added to illusion resolution 0 in previous resolution and create one angle. each concept has its own resolution 1. combination of resolution 1 of each dimention create infinite resolution 1. each resolution 1 contain 1 angle (angle 1). -/- 4-there is dimention resolution infinity. dimention resolution infinity consist of the dimention resolution before infinity plus one illusion. illusion resolution infinity is the illusion itself at resolution infinity. -/- 5- awareness is one of illusions's dimentions. the laws of nature, each physical object or any event or each mental construct or anything unknown are embeded in their particular dimentions. -/- 6-illusion is the cause for each angle and and each angle is the cause for illusion. -/- overall views: -/- 1-in topdown view reality is continues but in down to top view reality is quantized. -/- 2-brain by just seeing some dimentions of illusion, can identify order within the chaotic, complex and paradoxical aspects of the illusion, and create logical frameworks for understanding and interpreting and reasoning. Product of brain is a low resolution picture of an image with higher resolution (matter). So many different angles in high resolution image (matter) can be associated with one angle of a low resolution image (mental constructs). This is the cause for collapsing of a wavefunction that is illusory. -/- 3-illusion resolution infinity is a rescaled version of illusion resolution 0. if you compress resolution 1 in a way that it be to the size of an angle in resolution infinity, in this transformation resolution infinity will change to resolution 0. so resolusion 0 has the potential to be rescaled to any resolution and any dimention. -/- 4-all angles and no_angle exist in illusion. -/- 5-anti illusion is embeded in illusion. -/- 6-black holes are some region in space dimention that reaches infinite resolution in some dimentions (like density). In other words, it is some dimentions (like density) in resolution infinty that are in relation with some other lower resolution dimentions (like space). -/- 7-each resolution has the same distance from resolution 0 and resolution infinity. -/- 8-down to up view representing Presentism and up to down view representing Eternalism. -/- 9-free will means having abillity for changing some dimentions (like position dimention) based on resolution 0. -/- 10-different dimensions evolve at different rates, with some dimensions changing rapidly while others change slowly or based on interactions sometimes dimention changing rapidly while othertime this dimention change slowly. (shrink)

Bowin begins with an apparent paradox about Aristotelian infinity: Aristotle clearly says that infinity exists only potentially and not actually. However, Aristotle appears to say two different things about the nature of that potential existence. On the one hand, he seems to say that the potentiality is like that of a process that might occur but isn't right now. Aristotle uses the Olympics as an example: they might be occurring, but they aren't just now. On the other (...) hand, Aristotle says that infinity "exists in actuality as a process that is now occurring" (234). Bowin makes clear that Aristotle doesn't explicitly solve this problem, so we are left to work out the best reading we can. His proposed solution is that "infinity must be...a per se accident...of number and magnitude" (250). (Bryn Mawr Classical Review 2008.07.47). (shrink)

In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more well-known dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s well-known rejection of the actual infinite as stated in Lorenzen (1957) was not a major (...) motivation for operative logic and mathematics. In this article, we claim that this is in fact not the case. Rather, we argue for a shift that happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operativism is concerned with a critique of the Cantorian notion of set and related questions about the notion of countability and uncountability; only later, his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (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)

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)

The seventeenth century was an important period in the conceptual development of the notion of the infinite. In 1643, Evangelista Torricelli (1608-1647)—Galileo’s successor in the chair of mathematics in Florence—communicated his proof of a solid of infinite length but finite volume. Many of the leading metaphysicians of the time, notably Spinoza and Leibniz, came out in defense of actual infinity, rejecting the Aristotelian ban on it, which had been almost universally accepted for two millennia. Though it would be (...) another two centuries before the notion of the actually infinite was rehabilitated in mathematics by Dedekind and Cantor (Cauchy and Weierstrass still considered it mere paradox), their impenitent advocacy of the concept had significant reverberations in both philosophy and mathematics. In this essay, I will attempt to clarify one thread in the development of the notion of the infinite. In the first part, I study Spinoza’s discussion and endorsement, in the Letter on the Infinite (Ep. 12), of Hasdai Crescas’ (c. 1340-1410/11) crucial amendment to a traditional proof of the existence of God (“the cosmological proof” ), in which he insightfully points out that the proof does not require the Aristotelian ban on actual infinity. In the second and last part, I examine the claim, advanced by Crescas and Spinoza, that God has infinitely many attributes, and explore the reasoning that motivated both philosophers to make such a claim. Similarities between Spinoza and Crescas, which suggest the latter’s influence on the former, can be discerned in several other important issues, such as necessitarianism, the view that we are compelled to assert or reject a belief by its representational content, the enigmatic notion of amor Dei intellectualis, and the view of punishment as a natural consequent of sin. Here, I will restrict myself to the issue of the infinite, clearly a substantial topic in itself. (shrink)

Aristotle is said to have held that any kind of actual infinity is impossible. I argue that he was a finitist (or "potentialist") about _magnitude_, but not about _plurality_. He did not deny that there are, or can be, infinitely many things in actuality. If this is right, then it has implications for Aristotle's views about the metaphysics of parts and points.

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 order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of (...) the concept of ‘denumerability’ as it is presented in set theory as well as his philosophic refutation of Cantor’s Diagonal Argument and the implications of such a refutation onto the problems of the Continuum Hypothesis and Cantor’s Theorem. Throughout, the discussion will be placed within the historical and philosophical framework of the Grundlagenkrise der Mathematik and Hilbert’s problems. (shrink)

Descartes' philosophy contains an intriguing notion of the infinite, a concept labeled by the philosopher as indefinite. Even though Descartes clearly defined this term on several occasions in the correspondence with his contemporaries, as well as in his Principles of Philosophy, numerous problems about its meaning have arisen over the years. Most commentators reject the view that the indefinite could mean a real thing and, instead, identify it with an Aristotelian potential infinite. In the first part of this article, (...) I show why there is no numerical infinity in Cartesian mathematics, as such a concept would be inconsistent with the main fundamental attribute of numbers: to be comparable with each other. In the second part, I analyze the indefinite in the context of Descartes' mathematical physics. It is my contention that, even with no trace of infinite in his mathematics, Descartes does refer to an actual indefinite because of its application to the material world within the system of his physics. This fact underlines a discrepancy between his mathematics and physics of the infinite, but does not lead a difficulty in his mathematical physics. Thus, in Descartes' physics, the indefinite refers to an actual dimension of the world rather than to an Aristotelian mathematical potential infinity. In fact, Descartes establishes the reality and limitlessness of the extension of the cosmos and, by extension, the actual nature of his indefinite world. This indefinite has a physical dimension, even if it is not measurable. La filosofía de Descartes contiene una noción intrigante de lo infinito, un concepto nombrado por el filósofo como indefinido. Aunque en varias ocasiones Descartes definió claramente este término en su correspondencia con sus contemporáneos y en sus Principios de filosofía, han surgido muchos problemas acerca de su significado a lo largo de los años. La mayoría de comentaristas rechaza la idea de que indefinido podría significar una cosa real y, en cambio, la identifica con un infinito potencial aristotélico. En la primera parte de este artículo muestro por qué no hay infinito numérico en las matemáticas cartesianas, en la medida en que tal concepto sería inconsistente con el principal atributo fundamental de los números: ser comparables entre sí. En la segunda parte analizo lo indefinido en el contexto de la física matemática de Descartes. Mi argumento es que, aunque no hay rastro de infinito en sus matemáticas, Descartes se refiere a un indefinido real a causa de sus aplicaciones al mundo material dentro del sistema de su física. Este hecho subraya una discrepancia entre sus matemáticas y su física de lo infinito, pero no implica ninguna dificultad en su física matemática. Así pues, en la física de Descartes, lo indefinido se refiere a una dimensión real del mundo más que a una infinitud potencial matemática aristotélica. De hecho, Descartes establece la realidad e infinitud de la extensión del cosmos y, por extensión, la naturaleza real de su mundo indefinido. Esta indefinición tiene una dimensión física aunque no sea medible. (shrink)

Physical boundaries and the earliest topologists. Topology has a relatively short history; but its 19th century roots are embedded in philosophical problems about the nature of extended substances and their boundaries which go back to Zeno and Aristotle. Although it seems that there have always been philosophers interested in these matters, questions about the boundaries of three-dimensional objects were closest to center stage during the later medieval and modern periods. Are the boundaries of an object actually existing, less-than-three-dimensional parts of (...) the object—that is, are solids bounded by two-dimensional surfaces, surfaces by one-dimensional “edges” or “physical lines”, edges by dimensionless “simples”? If not, how does a perfectly spherical object manage to touch a perfectly flat object—what part of the sphere is in immediate contact with the plane, if the sphere has no unextended parts? But if such parts be admitted, are we not then saddled with “actual infinities” of simples, lines, and surfaces spread throughout each continuous object—the boundaries of all the object’s internal parts? Does it help to say that these internal boundaries exist only “potentially”? (shrink)

In Aristotle’s writings regarding the biology of embryology, especially in the Generation of Animals, he contends that the mother’s menstrual fluids provide the material for the generation of the offspring, and the father’s form determines its formation as a member of that species (e.g. human). The katamenia (menstrual fluids) of the mother are said to be potentially all the body parts of the offspring, though actually none of them. So, the fluids are potentially the offspring. But are they a first (...) potentiality or second potentiality (first actuality) of a human, in the terminology of De Anima II? In this paper I will argue that katamenia are a first potentiality of a human. My first argument is that katamenia do not have the potential for human activities such as thinking, but rather the potential of becoming something having the potential for those activities. I answer the objection that katamenia are not even a first potentiality, by appealing to an important text contending that for any x whose source of becoming is external, x is potentially y if nothing in x with respect to matter needs to be changed in order for an external principle to make x into y. (shrink)

The current definition of Constructive mathematics as “mathematics within intuitionist logic” ignores two fundamental issues. First, the kind of organization of the theory at issue. I show that intuitionist logic governs a problem-based organization, whose model is alternative to that of the deductive-axiomatic organization, governed by classical logic. Moreover, this dichotomy is independent of that of the kind of infinity, either potential or actual, to which respectively correspond constructive mathematical and classical mathematical tools. According to this view (...) a mathematical theory is based on the choices regarding these two dichotomies. As an example of this kind of foundation, arithmetic is rationally re-founded on constructive mathematical tools and the model of the problem-based organization. In conclusion, constructive mathematics is not only mathematics making use of constructive tools in intuitionist logic but also organized according to around a basic problem, solved by a method discovered using intuitionist logic. (shrink)

The book aims to examine how a Trinitarian Theism can be formulated through the elaboration of a Relational Ontology and a Trinitarian Metaphysics, in the context of a hyperphatic epistemology. This metaphysics has been proposed by some supporters of the so-called Open Theism as a solution to the numerous dilemmas of Classical Theism. The hypothesis they support is that the Trinitarian nature of God, reflected in a world of multiplicity, relationality, substance and relations, demands that we think of God as (...) dynamic, internally multiple and relational. However, if the expression «God is love» – understood as the formula of the Trinity – is the key to a new theism, it leaves a problem open: how can it be translated philosophically? The research develops on two different levels: first, it assesses whether the expression should be translated into the Trinitarian paradigm, and the aporias it generates; secondly, it tries to assess whether this paradigm (eminently relational) has a correspondence in ontology: is there a satisfactory Relational Metaphysics already available? The suspicion is that such Trinitarian-Relational Ontology, invoked by many as a solution to the incongruities of classical theism, has yet to be formulated in a satisfactorily manner, despite the existence of various attempts to formulate Relational Ontologies. In order to provide an evaluation of all these attempts and to outline some possible new perspectives, the thesis consists of five chapters. It is the aim of the present dissertation to evaluate such attempts, and to outline some possible new solutions. In the chapters some points have been established: 1. there must be at least one irreducible real relation (non-reductionist realism); 2. the relation must be equally fundamental to the substance: this means that it is both external and internal; 3. this relation must be able to account for contingent causality; 4. holism is a plausible position; 5. the Trinity is a good theistic model of the divine, apophatic but rational. The last chapter then returns to ontological questions: several Relational Ontologies are examined – including ontic structural realism and process (or eventist) ontology – together with their applications in philosophy of religion (e.g.: the Relational God, Process Trinitarianism, the Entangled God). This assessment shows how all these ontologies postulate real transcendental relations (subsistent relations), inside the substances or inside the powers, inside the tropes or the structures, describing them as monads or processes (or actual occasions). These relations are the same we need to describe the Trinity. Therefore, they are either possible for both domains – ontology and theology – or they are both impossible. It has been opted for the second conclusion. But they are both impossible and inevitable: the fundamental entities of the world and their interactions (causation) must be described as real transcendental relations because each ontology transforms entities or relations into real transcendental relationships at some point. Thus, neither relationalism nor substantialism are convincing: from the fall of these two dogmas (or, rather, from the fall of their naive interpretations) we can realise that the fundamental reality is something that lies between processism (relationalism) and substantialism. It is impossible to completely substitute the notion of substance with the notion of event, process, structure or relation, both in speaking of God and in speaking about the entities of fundamental ontology. Neither monism nor pluralism can be affirmed in their pure forms. The hypothesis proposed, then, is that the notion of gunk-junk is the only one that can translate the relationality hitherto sought in an ontological model. The central part of the chapter describes the merits and defects of an eventist-infinitive ontology (EIO) based on the concept of gunk, and its potentiality to generate new theistic accounts. Through the notion of infinityings it seems to be possible to find some solutions for the questions left open by the causal relation, and therefore to defend at least the existence of one relation (external and internal). Each fundamental entity is described, in this ontology, through four transcendentals: ‘entity’, ‘relation’, ‘unity’ and ‘multiplicity’. The co-primarity between substance and relations (borrowed from the notion of relatio subsistens) differentiates EIO from the process philosophies precisely because it does not pretend to eliminate the substantial principle, or the category of substance, but wants to think of it with the transcendental of relationality. Of course, EIO poses a challenge to the role, the method and the explicative capacity of metaphysics, because what we can state of the fundamental reality is antinomic. EIO tries to assume the antinomy as a result, to make it the basis of a kind of apophatic ontology. If our best ontology is gunky, then it is possible to confirm what has been said in chap. 4: the ‘how’ of individual substances is unknown to us at least as much as the ‘how’ of God. Even in the world we have the mystery of a distinction that is not division. However, the convergence of these antinomies can find an ultimate explanation in a theistic metaphysics (EIM): God creates inside of Himself and his Trinitarian substance is “contracted” into the worldly entities. They are made contingent and ontologically different by this self-limitation, but it is still the infinity of God that makes space in Himself for something new, even though He is totally present in every entity. It is not absurd to think that the substances of the world keep track of the divine nature, even in His “contraction”. Among the characteristics of the divine nature that each created entity maintains we have the subsistent relationality, the pericoretic indwelling (the infinite gunk-junk), the fact of being always one-and-many. If EIO and EIM are accepted as a good compromise between relationalism and substantialism, even in their apophaticity, then, on this basis, a Trinitarian Philosophy is possible. The picture of reality that emerges represents the world as multiple, substantial, contingent and intrinsically relational, forcing us to postulate the transcendental of relationality and multiplicity. Such transcendental leads us to think of the world and God (and their relations) in a Trinitarian way – With the necessary acknowledgement that this is a reasonable but apophatic discourse. (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)

Vetter (2015) develops a localised theory of modality, based on potentialities of actual objects. Two factors play a key role in its appeal: its commitment to Hardcore Actualism, and to Naturalism. Vetter’s commitment to Naturalism is in part manifested in her adoption of Aristotelian universals. In this paper, we argue that a puzzle concerning the identity of unmanifested potentialities cannot be solved with an Aristotelian conception of properties. After introducing the puzzle, we examine Vetter’s attempt at amending the Aristotelian (...) conception in a way that avoids the puzzle, and conclude that this amended version is no longer to be considered naturalistic. Potentiality theory cannot be both actualist and naturalist. We then argue that, if naturalism is to be abandoned by the actualist, there are good reasons to adopt a Platonist conception of universals, for they offer a number of theoretical advantages and allow us to avoid some of the problems facing Vetter’s theory. (shrink)

In the early twentieth century, many philosophers in America thought that time should be taken seriously in one way or another. George P. Adams (1882-1961) argued that the past, present and future are all real but only the present is actual. I call this theory ‘actualist eternalism’. In this paper, I articulate his novel brand of eternalism as one piece of his metaphysical system and I explain how he argued for the view in light of the best explanations of (...) temporal experience and the present. I argue that his exploitation of analogies between time and modality offer some lessons for current debates about time such as the importance of providing a temporal epistemology. I also extract what I call the temporal boundary problem and argue that it gives rise to an unaddressed challenge for presentists and growing block theorists. (shrink)

Hillel Steiner has recently attacked the notion of inalienable rights, basing some of his arguments on the Hohfeldian analysis to show that infinite arrays of legal positions would not be associated with any inalienable rights. This essay addresses the nature of the Hohfeldian infinity: the main argument is that what Steiner claims to be an infinite regress is actually a wholly unproblematic form of infinite recursion. First, the nature of the Hohfeldian recursion is demonstrated. It is shown that infinite (...) recursions of legal positions ensue regardless of whether inalienable rights exist or not. Second, the alleged problems that this might pose for the analysis are discussed. The conclusion is that one should not worry about the recursion as long as one understands correctly the role of the Hohfeldian analysis in normative reasoning. (shrink)

In two rarely discussed passages – from unpublished notes on the Principles of Philosophy and a 1647 letter to Chanut – Descartes argues that the question of the infinite extension of space is importantly different from the infinity of time. In both passages, he is anxious to block the application of his well-known argument for the indefinite extension of space to time, in order to avoid the theologically problematic implication that the world has no beginning. Descartes concedes that we (...) always imagine an earlier time in which God might have created the world if he had wanted, but insists that this imaginary earlier existence of the world is not connected to its actual duration in the way that the indefinite extension of space is connected to the actual extension of the world. This paper considers whether Descartes’s metaphysics can sustain this asymmetrical attitude towards infinite space vs. time. I first consider Descartes’s relation to the ‘imaginary’ space/time tradition that extended from the late scholastics through Gassendi and More. I next examine carefully Descartes’s main argument for the indefinite extension of space and explain why it does not apply to time. Most crucially, since duration is merely conceptually distinct from enduring substance, the end or beginning of the world entails the end or beginning of real time. In contrast, extension does not depend on any enduring substance besides itself. (shrink)

Infinity exists as a concept but has no existence in actuality. For infinity to have existence in actuality either time or space have to already be infinite. Unless something is already infinite, the only way to become infinite is by an 'infinity leap' in an infinitely small moment, and this is not possible. Neither does infinitely small have an existence since anything larger than zero is not infinitely small. Therefore infinity has no existence in actuality.

The familiar complaint that some ambitious proposal is infeasible naturally invites the following response: Once upon a time, the abolition of slavery and the enfranchisement of women seemed infeasible, yet these things were actually achieved. Presumably, then, many of those things that seem infeasible in our own time may well be achieved too and, thus, turn out to have been perfectly feasible after all. The Appeal to History, as we call it, is a bad argument. It is not true that (...) if some desirable state of affairs was actually achieved, then it was feasible that it was achieved. “Actual” does not imply “feasible,” as we put it. Here is our objection. “Feasible” implies “not counterfactually fluky.” But “actual” does not imply “not counterfactually fluky.” So, “actual” does not imply “feasible.” While something like the Flukiness Objection is sometimes hinted at in the context of the related literature on abilities, it has not been developed in any detail, and both premises are inadequately motivated. We offer a novel articulation of the Flukiness Objection that is both more precise and better motivated. Our conclusions have important implications, not only for the admissible use of history in normative argument, but also by potentially circumscribing the normative claims that are applicable to us. (shrink)

According to the orthodox view, the goodness of a life depends exclusively on the things that actually happened within it, such as its pleasures and pains, the satisfaction of its subject’s preferences, or the presence of various objective goods and bads. In this paper, I argue that the goodness of a life also depends on what could have happened, but didn’t. I then propose that this view helps us resolve ethical puzzles concerning the standards for a life worth living for (...) animals and the significance of a life’s shape. (shrink)

The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristotelian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian voluntarist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in Cantor’s thought with (...) reference to his main philosophical-mathematical treatise, the Grundlagen (1883) as well as with reference to his article, “Über die verschiedenen Standpunkte in bezug auf das aktuelle Unendliche” (“Concerning Various Perspectives on the Actual Infinite”) (1885). (shrink)

There exists a huge number of numerical methods that iteratively construct approximations to the solution y(x) of an ordinary differential equation (ODE) y′(x) = f(x,y) starting from an initial value y_0=y(x_0) and using a finite approximation step h that influences the accuracy of the obtained approximation. In this paper, a new framework for solving ODEs is presented for a new kind of a computer – the Infinity Computer (it has been patented and its working prototype exists). The new computer (...) is able to work numerically with finite, infinite, and infinitesimal numbers giving so the possibility to use different infinitesimals numerically and, in particular, to take advantage of infinitesimal values of h. To show the potential of the new framework a number of results is established. It is proved that the Infinity Computer is able to calculate derivatives of the solution y(x) and to reconstruct its Taylor expansion of a desired order numerically without finding the respective derivatives analytically (or symbolically) by the successive derivation of the ODE as it is usually done when the Taylor method is applied. Methods using approximations of derivatives obtained thanks to infinitesimals are discussed and a technique for an automatic control of rounding errors is introduced. Numerical examples are given. (shrink)

In this paper I introduce the idea of a higher-order modal logic—not a modal logic for higher-order predicate logic, but rather a logic of higher-order modalities. “What is a higher-order modality?”, you might be wondering. Well, if a first-order modality is a way that some entity could have been—whether it is a mereological atom, or a mereological complex, or the universe as a whole—a higher-order modality is a way that a first-order modality could have been. First-order modality is modeled in (...) terms of a space of possible worlds—a set of worlds structured by an accessibility relation, i.e., a relation of relative possibility—each world representing a way that the entire universe could have been. A second-order modality would be modeled in terms of a space of spaces of (first-order) possible worlds, each space representing a way that (first-order) possible worlds could have been. And just as there is a unique actual world which represents the way that things actually are, there is a unique actual space which represents the way that first-order modality actually is. -/- One might wonder what the accessibility relation itself is like. Presumably, if it is logical or metaphysical modality that is being dealt with, it is reflexive; but is it also symmetric, or transitive? Especially in the case of metaphysical modality, the answer is not clear. And whichever of these properties it may or may not have, could that itself have been different? Could at least some rival modal logics represent different ways that first-order modality could have been? -/- To be clear, the idea behind my proposal is not just that some things which are possible or necessary might not have been so at the first order, as determined by the actual accessibility relation, but also that the actual accessibility relation, and hence the nature or structure of actual modality, could have been different at some higher order of modality. Even if the accessibility relation is actually both symmetric and transitive, perhaps it could (second-order) have been otherwise: There is a (second-order) possible space of worlds in which it is different, where it fails to be symmetric, or transitive. We must, therefore, introduce the notion of a higher-order accessibility relation, one that in this case relates spaces of first-order worlds. The question then arises as to whether that relation is symmetric, or transitive. We can then consider third-order modalities, spaces of spaces of spaces of possible worlds, where the second-order accessibility relation differs from how it actually is. I can see no reason why there should be a limit to this hierarchy of higher-order modalities, any more than I can see a reason why there should be a limit to the hierarchy of higher-order properties. There will thus be an infinity of orders, one for each positive integer, and each order will have an accessibility relation of its own. To keep things as clear as possible, a space of first-order points (i.e., of possible worlds) shall be called a galaxy, a space of second-order points, a universe, and a space of any higher order, a cosmos. However, to keep things as simple as possible, in what follows I will deal with but a single cosmos, and hence will not deal with modalities higher than the third order. -/- The accessibility relation is not the only thing that might be thought to vary between spaces of worlds: Perhaps the contents of the spaces can vary as well. While I presume that the contents of the worlds themselves remain constant—it makes doubtful sense to suppose that in one space some entity e exists in a world w and in another space e doesn’t exist in that same world w—we may suppose that different spaces differ as to which worlds they contain, just as different worlds may differ as to which objects they contain. Thus we might have a higher-order analogue of a variable-domain modal logic. There seem, then, to be three ways in which spaces can differ: First, as to the properties of the accessibility relation; second, as to which worlds the relation relates; and third, as to which worlds or spaces are parts of their domains. -/- The paper will be structured as follows. In Section 2 I provide some reasons why one might want to pursue this kind of project in the first place. In Section 3 I outline the syntax and semantics of my proposed logic. Section 4 covers semantic tableaux for this system; and after giving the rules for their construction, I construct a few of them myself to establish some logical consequences of the system and give the reader a feel for how it works. In Sections 5, 6 and 7 I explore some of its potential philosophical implications for areas besides logic, namely the philosophy of language; metaphysics, including the metaphysics of modality and the philosophy of time, and finally the philosophy of religion, before concluding the paper in Section 8. (shrink)

This study seeks to critically explore the link between sovereignty, violence and war in Giorgio Agamben’s Homo Sacer series and Thomas Hobbes’s Leviathan. From a brief rereading of Leviathan’s main arguments that explicitly revolves around the Aristotelian distinction between actuality/ potentiality, it will conclude that Hobbesian pre-contractual violence is primarily based on what Hobbes terms “anticipatory reason” and the problem of future contingency. Relying on Foucauldian insights, it will be emphasized that the assumption of certain potentialities suffices in leading to (...) Hobbes’s well-known conclusion that the state of nature is a “condition of Warre.” In a second step, this study considers some of Agamben’s arguments to account for how pre-contractual violence as envisioned by Hobbes cannot be rendered impotent through the integration of a sovereign. In specific, Agamben’s claims shed light on an irreducible, inextricable entwining between the state of nature and the state of law as “Siamese twins” (Virno). On a meta-level, Agamben thus implicitly shows how the “Hobbesian problem” cannot be merely reduced to a “problem of order” (Parsons). With regard to the current functioning of the stratagems of financial markets and the mechanisms of future-colonization underpinning global politics, this study finally argues that Hobbes ought to be reevaluated in particular regarding the problem of the future in his account. Partly responding to Agamben’s critical investigations, I suggest that a careful exploration of what will be coined “the prospects of an actualization of the potentiality not-to-be” might serve as a first theoretical step towards a productive form of criticism. (shrink)

This paper examines Aristotle's discussion of the priority of actuality to potentiality in geometry at Metaphysics Θ9, 1051a21–33. Many scholars have assumed what I call the "geometrical construction" interpretation, according to which his point here concerns the relation between an inquirer's thinking and a geometrical figure. In contrast, I defend what I call the "geometrical analysis" interpretation, according to which it concerns the asymmetrical relation between geometrical propositions in which one is proved by means of the other. His argument as (...) so construed is ultimately based on the asymmetrical relation between the corresponding geometrical facts. Then I explore this ontological priority in geometry by drawing attention to a parallel passage, Posterior Analytics II.11, 94a24–35, where Aristotle explains the relation between the same geometrical propositions in connection to material causation. (shrink)

Several ways used to rank countries with respect to medals won during Olympic Games are discussed. In particular, it is shown that the unofficial rank used by the Olympic Committee is the only rank that does not allow one to use a numerical counter for ranking – this rank uses the lexicographic ordering to rank countries: one gold medal is more precious than any number of silver medals and one silver medal is more precious than any number of bronze medals. (...) How can we quantify what do these words, more precious, mean? Can we introduce a counter that for any possible number of medals would allow us to compute a numerical rank of a country using the number of gold, silver, and bronze medals in such a way that the higher resulting number would put the country in the higher position in the rank? Here we show that it is impossible to solve this problem using the positional numeral system with any finite base. Then we demonstrate that this problem can be easily solved by applying numerical computations with recently developed actual infinite numbers. These computations can be done on a new kind of a computer – the recently patented Infinity Computer. Its working software prototype is described briefly and examples of computations are given. It is shown that the new way of counting can be used in all situations where the lexicographic ordering is required. (shrink)

The entire world community, since 2019, affected by the global pandemic COVID-19. The pandemic caused by this virus, led not only to significant human losses worldwide, but also imposed significant restrictions on the socio-cultural life of the population and radically changed the trends of the global economy and the further functioning of socio-economic systems. Now, huge economic losses have been recorded, which affected almost all sectors of the national economy and the state in the short, medium and long term. However, (...) it is important to consider individual economic development forecasts and measures developed by the governments of the world’s leading countries to overcome the negative effects of the COVID-19 pandemic. This will allow to form a real vision of the possible course of economic processes that will directly affect the living standards of the population and the restoration of socio-economic systems. To further restore the socio-economic system it is necessary to developing modern strategies and forecasts to ensure the effective functioning of economic entities through innovation, digitalization, marketing and use of competitive advantages in the consumer markets in conditions of limited resources, development promising sectors of the national economy, etc. The purpose of writing this scientific monograph is to substantiate the theoretical and methodological foundations, the formation new strategies for restoring socio-economic systems and overcoming the negative consequences of the caused by the COVID-19 pandemic, taking into account changes and challenges in the modern world. The object of the authors’ research is the process of forming new approaches, strategies and mechanisms for managing socio-economic systems in the context of the COVID-19 pandemic, eliminating the negative consequences in the activities of economic entities. The subject of research is socio-economic, organizational and institutional processes of formation and effective implementation of approaches, strategies and mechanisms for managing socio-economic systems; stabilization of the functioning of economic entities; introduction of innovative processes and digital technologies; implementation of best practices in the managing of socio-economic systems using world experience in various sectors of the economy caused by the COVID-19 pandemic. (shrink)

Apparency and actuality are discussed with respect to science, genetics, evolution, cognition and perception. Rules for a (sub)set of actual/allowable sets of entities and objects from a total set that is based on differences and contrasts extracted from perceptual experience of the world elucidate tenable combinations/recombinations from a total of the transparent(history/past dependant) and the apparent as they can be construed to comprise the actual/real faces of perception. The elements of dreams, language, symbolisms, abstractions are proposed to encompass (...) to encompass a total set of plausible apparencies. Nature from which both internal and external life is mirrored at the simplest structural level is propose to be constructed similarly-to physically possess also concepts and abstractions as states that reflect differences/combinations/recombinations parametrically that are emerged from the parameters associated with particular circumstances. This view is dependant upon, for its' conceptual soundness, a total construction of the world that descends, rather than ascends, in structural complexity and diversity and which is innately divided and compartmentalized in sequence from a more complex uniquely existing state describable singularly as a place of force and proximities form which all processes ensue. Integration of sound, light, into sensory experience and language is proposed to be a natural process of the accomplishment of new proximities/states from total apparent plausibility’s (past and actual combined) verses the actually(past and present) existing. Two potential important aspects of this scheme exist with regards to evolution theory: 1) a very relative dynamic state of change and emergence is plausible 2) a consistent identity of objects and entities is resolvable that is not based on apparent observations or partial history of events but on a set of total states and processes occurred during the progression to an actual condition. This is suggested to be unique for each and every actual situation as an identifier and is proposed to account for allowances with respect to new potentials. "Self" in this sense can assume more involved relations with self more than with the more distant, hence alien, otherwise arrived at proximal components, components of the environment, inert or otherwise. An example is presented with respect to solar energies and their toxicity to cause disturbances/disease upon internal exposure-as an unnaturally arrived imbalance/proximity between the transparent (history dependant) and the apparent-together comprising, in this situation, a disturbed actual state that can encompass also distortions in the perception of time. Time is proposed to be delineated as a contrast of proximal and distal factors mutually involved with both transparent (genealogically) oriented parameters, and empirically apparent factors that together comprise the actual, as well as those more immediate proximal and distal components(e.g. solar radiation, light-the less structured ambient verse those elements that comprise visual perceptions) as they influence the senses. Each entity species in this scheme possesses its’ own characteristic sense of time along with contemporary factors that influence the perception of time state, state of health and disturbance. (shrink)

What I will argue here is that the ethical potentiality of the human being that Aristotle cites in the Nicomachean Ethics refers to the general, rational capacity for someone to appropriate and develop their own specific, natural capacities which make them human; the name of this ability is called virtue, which, when expressed in actions, we call good. To separate out the concepts at work here demands an exegesis of the two kinds of dunamis in Metaphysics Theta, that is, dunamis (...) as causal power of change and dunamis as a potential way of being for an existent capacity. Once we grasp this, we can clarify the nature of virtue as the active, complete state of a capacity in human being which defines their function even though it must be acquired, and it need not ever be actualized. The concepts of habit (ethos) and practical judgment (phronesis) give Aristotle the means to solve the problem of how such a potentiality is acquired and how it is actualized, and hence we will end with a discussion of them. -/- . (shrink)

In this paper, we deal with the computation of Lie derivatives, which are required, for example, in some numerical methods for the solution of differential equations. One common way for computing them is to use symbolic computation. Computer algebra software, however, might fail if the function is complicated, and cannot be even performed if an explicit formulation of the function is not available, but we have only an algorithm for its computation. An alternative way to address the problem is to (...) use automatic differentiation. In this case, we only need the implementation of the algorithm that evaluates the function in terms of its analytic expression in a programming language, but we cannot use this if we have only a compiled version of the function. In this paper, we present a novel approach for calculating the Lie derivative of a function, even in the case where its analytical expression is not available, that is based on the In finity Computer arithmetic. A comparison with symbolic and automatic differentiation shows the potentiality of the proposed technique. (shrink)

An actual cause of some token effect is itself a token event that helped to bring about that effect. The notion of an actual cause is different from that of a potential cause – for example a pre-empted backup – which had the capacity to bring about the effect, but which wasn't in fact operative on the occasion in question. Sometimes actual causes are also distinguished from mere background conditions: as when we judge that the struck (...) match was a cause of the fire, while the presence of oxygen was merely part of the relevant background against which the struck match operated. Actual causation is also to be distinguished from type causation: actual causation holds between token events in a particular, concrete scenario; type causation, by contrast, holds between event kinds in scenario kinds. (shrink)

It is modernly debated whether application of the free will has potential to cause harm to nature. Power possessed to the discourse, sensory/perceptual, physical influences on life experience by the slow moving machinery of change is a viral element in the problems of civilization; failed resolution of historical paradox involving mind and matter is a recurring source of problems. Reference is taken from the writing of Euclid in which a oneness of nature as an indivisible point of thought is (...) made prerequisite in criteria of interpretation to demonstrate that contemporary scientific methodologies alternately ensue from the point of empirically centered induction. A qualification for conceptualizations is proposed that involves a physically describable form bound to energy in addition to contemporary notions of energy bound to form and a visually based mathematical-physical form is elaborated and discussed with respect to biological and natural processes. (shrink)

Do essence-based accounts of necessity and Vetter’s potentiality-based account of possibility in fact lead to the same result, viz., a single derived notion of necessity that is interdefinable with possibility or vice versa? And does each approach independently have the ability to reach its desired goal without having to rely on the primitive notion utilized by the other? In this essay, I investigate these questions and Vetter’s responses to them. Contrary to the “separatist” position defended by Vetter, I argue that (...) there are reasons to favor “Combination”, according to which an essence-based account of necessity should be combined with a potentiality-based account of possibility, or vice versa. According to this alternative approach, essentialism and potentialism should be regarded as allies, rather than as competitors, in a theory of derived modality, since both notions are needed in order to give a full account of necessity and possibility. Keywords: possibility, necessity, potentiality, essence, interdefinability, actuality, dependence, artifact-functions. (shrink)

The “Human Brain Project” (HBP) is a large-scale European neuroscience and information communication technology (ICT) project that has been a matter of heated controversy since its inception. With its aim to simulate the entire human brain with the help of supercomputing technologies, the HBP plans to fundamentally change neuroscientific research practice, medical diagnosis, and eventually the use of computers itself. Its controversial nature and its potential impacts render the HBP a subject of crucial importance for critical studies of science (...) and society. In this paper, we provide a critical exploratory analysis of the potential mid- to long-term impacts the HBP and its ICT infrastructure could be expected to have, provided its agenda will indeed be implemented and executed to a substantive degree. We analyse how the HBP aspires to change current neuroscientific practice, what impact its novel infrastructures could have on research culture, medical practice and the use of ICT, and how, given a certain degree of successful execution of the project’s aims, potential clinical and methodological applications could even transform society beyond scientific practice. Furthermore, we sketch the possibility that research such as that projected by the HBP may eventually transform our everyday world, even beyond the scope of the HBP’s explicit agenda, and beyond the isolated ‘application’ of some novel technological device. Finally, we point towards trajectories for further philosophical, historical and sociological research on the HBP that our exploratory analysis might help to inspire. Our analysis will yield important insights regardless of the actual success of the HBP. What we drive at, for the most part, is the broader dynamics of scientific and technological development of which the HBP agenda is merely one particularly striking exemplification. (shrink)

In Nicomachean Ethics VII Aristotle describes akrasia as a disposition. Taking into account that it is a disposition, I argue that akrasia cannot be understood on an epistemological basis alone, i.e., it is not merely a problem of knowledge that the akratic person acts the ways he does, but rather one is akratic due to a certain kind of habituation, where the person is not able to activate the potential knowledge s/he possesses. To stress this point, I focus on (...) the gap between potential knowledge and its activation, whereby I argue that the distinction between potential and actual knowledge is at the center of the problem of akrasia. I suggest that to elaborate on this gap, we must go beyond the limits of Nicomachean Ethics to Metaphysics IX, where we find Aristotle’s discussion of the distinction between potentiality and actuality. I further analyze the gap between potential and actual knowledge by means of Aristotle’s discussion of practical syllogism, where I argue that akrasia is a result of a conflict in practical reasoning. I conclude my paper by stressing that for the akratic person the action is determined with respect to the conclusion of the practical syllogism, where the conclusion is produced by means of a ‘conflict’ between the universal opinion which is potential and the particular opinion which is appetitive. (shrink)

Medicine is a space to meet the weak and the defenseless. Its environment,professional object and the agents that participate directly or indirectly in the exercise of medicine have a potential generator of culture of life opposing the prevailing culture of death in today's world. In this paper we have described how the exercise of a profession, in this case of medicine, has the potential to evangelize culture. To do this, we delved into the concept of culture, its current (...) context, the generalities of the evangelization of culture, andthen to consider different angles in which medicine demonstrates its potential to evangelize culture. This potential is highlighted to sensitize young people who want to be doctors, who by being fully integrated, scientifically sound and based on ethical, moral and evangelical principles, have a significant impact on society. On the other hand, by generating a culture of solidarity, always seeking to heal escape from pain and suffering, attending mainly to the poorest and most needy. Finally, by generating spaces and initiatives to transmit ideas,reflections, knowledge and disseminating values in the exercise of the profession. -/- La medicina es un espacio para salir al encuentro del débil y el indefenso. Su entorno, objeto profesional y los agentes que participan directa o indirectamente en el ejercicio de la medicina posee un potencial generador de cultura de vida oponiéndose a la cultura de muerte imperante en el mundo de hoy. En este trabajo hemos descrito de qué manera el ejercicio de una profesión, en este caso de la medicina, tiene potencial para evangelizar la cultura. Para ello, se ahondó en el concepto de cultura, su contexto actual, las generalidades de la evangelización de la cultura, para luego al considerar diferentes ángulos en los que la medicina evidencia su potencial para evangelizar la cultura. Se destaca dicho potencial para sensibilizar a los jóvenes que quieren ser médicos, quienes, al formarse integralmente, científicamente sólidos y fundamentados en principios éticos, morales yevangélicos, ejercen un impacto significativo en la sociedad. Por otro lado, al generar una cultura de solidaridad, buscando salir siempre al encuentro del dolor y el sufrimiento,atendiendo principalmente a los más pobres y necesitados. Finalmente, al generar espacios e iniciativas para trasmitir ideas, reflexiones, conocimientos y difundiendo valores en el ejercicio de la profesión. (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)

At first sight the Theory of Computation i) relies on a kind of mathematics based on the notion of potential infinity; ii) its theoretical organization is irreducible to an axiomatic one; rather it is organized in order to solve a problem: “What is a computation?”; iii) it makes essential use of doubly negated propositions of non-classical logic, in particular in the word expressions of the Church-Turing’s thesis; iv) its arguments include ad absurdum proofs. Under such aspects, it is (...) like many other scientific theories, in particular the first theories of both mechanical machines and heat machines. A more accurate examination of Theory of Computation shows a difference from the above mentioned theories in its essentially including an odd notion, “thesis”, to which no theorem corresponds. On the other hand, arguments of each of the other theories conclude a doubly negative predicate which then, by applying the inverse translation of the ‘negative one’, is translated into the corresponding affirmative predicate. By also taking into account three criticisms to the current Theory of Computation a rational re-formulation of it is sketched out; to Turing-Church thesis of the usual theory corresponds a similar proposition, yet connecting physical total computation functions with constructive mathematical total computation functions. -/- . (shrink)

William Lane Craig has argued that there cannot be actual infinities because inverse operations are not well-defined for infinities. I point out that, in fact, there are mathematical systems in which inverse operations for infinities are well-defined. In particular, the theory introduced in John Conway's *On Numbers and Games* yields a well-defined field that includes all of Cantor's transfinite numbers.

Many philosophers have argued that the past must be finite in duration because otherwise reaching the present moment would have involved something impossible, namely, the sequential occurrence of an actual infinity of events. In reply, some philosophers have objected that there can be nothing amiss in such an occurrence, since actually infinite sequences are ‘traversed’ all the time in nature, for example, whenever an object moves from one location in space to another. This essay focuses on one of (...) the two available replies to this objection, namely, the claim that actual infinities are not traversed in nature because space, time, and other continuous wholes divide into parts only in so far as we divide them in thought, and thus divide into only a finite number of parts. I grant that this reply succeeds in blunting the anti-finitist objection, but I argue that it also subverts the very argument against an eternal past that it was intended to save. (shrink)

Of Aristotle’s core terms, potency (dunamis) and actuality (energeia) are among the most important. But when we attempt to understand what they mean, we face the following problem: their primary meaning is movement, as a source (dunamis) or as movement itself (energeia). We therefore have to understand movement in order to understand them. But the structure of movement is itself articulated using these terms: it is the activity of a potential being, as potent. This paper examines this hermeneutic circle, (...) and works out a strategy for reading Aristotle based on his conception of our epistemological predicament. This hermeneutic approach helps us gain access to the phenomena of movement and its sources (potency, and energeia). The paper closes with a review of the conceptual resources we deploy to think about movement: homogeneity, space and time, impulse, relativity, the blend of sameness and difference, and being and non-being. Showing that Aristotle uses none of these clears the landscape for a fresh inquiry into his account of movement. (shrink)

Some philosophers contend that the past must be finite in duration, because otherwise reaching the present would have involved the sequential occurrence of an actual infinity of events, which they regard as impossible. I recently developed a new objection to this finitist argument, to which Andrew Ter Ern Loke and Travis Dumsday have replied. Here I respond to the three main points raised in their replies.

According to an influential idea in the philosophy of set theory, certain mathematical concepts, such as the notion of a well-order and set, are indefinitely extensible. Following Parsons (1983), this has often been cashed out in modal terms. This paper explores instead an extensional articulation of the idea, formulated in higher-order logic, that flat-footedly formalizes some remarks of Zermelo. The resulting picture is incompatible with the idea that the entire universe can be well-ordered, but entirely consistent with the idea that (...) the sets of any set-theoretic universe can be. (shrink)

Famously, the Church–Fitch paradox of knowability is a deductive argument from the thesis that all truths are knowable to the conclusion that all truths are known. In this argument, knowability is analyzed in terms of having the possibility to know. Several philosophers have objected to this analysis, because it turns knowability into a nonfactive notion. In addition, they claim that, if the knowability thesis is reformulated with the help of factive concepts of knowability, then omniscience can be avoided. In this (...) article we will look closer at two proposals along these lines :557–568, 1985; Fuhrmann in Synthese 191:1627–1648, 2014a), because there are formal models available for each. It will be argued that, even though the problem of omniscience can be averted, the problem of possible or potential omniscience cannot: there is an accessible state at which all truths are known. Furthermore, it will be argued that possible or potential omniscience is a price that is too high to pay. Others who have proposed to solve the paradox with the help of a factive concept of knowability should take note :53–73, 2010; Spencer in Mind 126:466–497, 2017). (shrink)

According to the Free Will Explanation of a traditional view of hell, human freedom explains why some people are in hell. It also explains hell’s punishment and finality: persons in hell have freely developed moral vices that are their own punishment and that make repentance psychologically impossible. So, even though God continues to desire reconciliation with persons in hell, damned persons do not want reconciliation with God. But this moral vice explanation of hell’s finality is implausible. I argue that God (...) can and would make direct or indirect alterations in their character to give them new motivational reasons that re-enable their freedom to repent. Subsequently, I argue that it is probable that each damned person will be saved eventually, because there is a potential infinity of opportunities for free repentance. Thus, if the Free Will Explanation’s descriptions of hell and divine love are correct, it is highly probable that each person in hell escapes to heaven. (shrink)

In this paper, I present a categorial theory of meaning which asserts that the meaning of a sentence is the function from the actualization of some potentiality or the potentiality of some actuality to the truth of the sentence. I argue that it builds on the virtues of David Lewis’s Possible World Semantics but advances beyond problems that Lewis’s theory faces with its distinctly Aristotelian turn toward actuality and potentiality.

Potentialism is the view that the universe of sets is inherently potential. It comes in two main flavours: height-potentialism and width-potentialism. It is natural to think that height and width potentialism are just aspects of a broader phenomenon of potentialism, that they might both be true. The main result of this paper is that this is mistaken: height and width potentialism are jointly inconsistent. Indeed, I will argue that height potentialism is independently committed to an ultimate background universe of (...) sets, an ultimate V, up to its height. (shrink)