EXPANDED EDITION (eBook): -/- Infinity Is Not What It Seems...Infinity is commonly assumed to be a logical concept, reliable for conducting mathematics, describing the Universe, and understanding the divine. Most of us are educated to take for granted that there exist infinite sets of numbers, that lines contain an infinite number of points, that space is infinite in expanse, that time has an infinite succession of events, that possibilities are infinite in quantity, and over half of the world’s population believes (...) in a divine Creator infinite in knowledge, power, and benevolence. -/- According to this treatise, such assumptions are mistaken. In reality, to be is to be finite. The implications of this assessment are profound: the Universe and even God must necessarily be finite. -/- The author makes a compelling case against infinity, refuting its most prominent advocates. Any defense of the infinite will find it challenging to answer the arguments laid out in this book. But regardless of the reader’s position, FOREVER FINITE offers plenty of thought-provoking material for anyone interested in the subject of infinity from the perspectives of philosophy, mathematics, science, and theology. -/- This electronic edition of FOREVER FINITE is offered free online for all and is expanded with content that does not appear in the published edition due to print production page limitations. From the Introduction to the Conclusion, Chapters 1–27 are identical to the published print edition, including the page numbering for the chapters. This electronic edition adds an appendix and associated content—specifically, updates to the book’s back matter (68 new endnotes, 17 new bibliographic references, 3 new figure credits, 14 new glossary terms) and front matter (an updated table of contents and this preface note). (shrink)

I defend a new argument for causal finitism, the view that nothing can have an infinite causal history. I begin by defending a number of plausible metaphysical principles, after which I explore a host of novel variants of the Littlewood-Ross and Thomson’s Lamp paradoxes that violate such principles. I argue that causal finitism is the best solution to the paradoxes.

The paper considers a generalization of Peano arithmetic, Hilbert arithmetic as the basis of the world in a Pythagorean manner. Hilbert arithmetic unifies the foundations of mathematics (Peano arithmetic and set theory), foundations of physics (quantum mechanics and information), and philosophical transcendentalism (Husserl’s phenomenology) into a formal theory and mathematical structure literally following Husserl’s tracе of “philosophy as a rigorous science”. In the pathway to that objective, Hilbert arithmetic identifies by itself information related to finite sets and series and quantum (...) information referring to infinite one as both appearing in three “hypostases”: correspondingly, mathematical, physical and ontological, each of which is able to generate a relevant science and area of cognition. Scientific transcendentalism is a falsifiable counterpart of philosophical transcendentalism. The underlying concept of the totality can be interpreted accordingly also mathematically, as consistent completeness, and physically, as the universe defined not empirically or experimentally, but as that ultimate wholeness containing its externality into itself. (shrink)

Consider an infinite series whose items are each explained by their immediate successor. Does such an infinite explanation explain the whole series or does it leave something to be explained? Hume arguably claimed that it does fully explain the whole series. Leibniz, however, designed a very telling objection against this claim, an objection involving an infinite series of book copies. In this paper, I argue that the Humean claim can, in certain cases, be saved from the Leibnizian “infinite book copies” (...) objection, and that this provides an interesting way to defuse some cosmological arguments for the existence of God and to give a non-theistic but complete explanation of the Universe. In the course of my argumentation, I also show that circular explanations can be “self-explanatory” as well: explaining two items by each other can explain the couple of items tout court. (shrink)

Approaching Infinity addresses seventeen paradoxes of the infinite, most of which have no generally accepted solutions. The book addresses these paradoxes using a new theory of infinity, which entails that an infinite series is uncompletable when it requires something to possess an infinite intensive magnitude. Along the way, the author addresses the nature of numbers, sets, geometric points, and related matters. The book addresses the need for a theory of infinity, and reviews both old and new theories of infinity. It (...) discussing the purposes of studying infinity and the troubles with traditional approaches to the problem, and concludes by offering a solution to some existing paradoxes. (shrink)

One of the main issues that dominates Neoplatonism in late antique philosophy of the 3rd–6th centuries A.D. is the nature of the first principle, called the ‘One’. From Plotinus onward, the principle is characterized as the cause of all things, since it produces the plurality of intelligible Forms, which in turn constitute the world’s rational and material structure. Given this, the tension that faces Neoplatonists is that the One, as the first cause, must transcend all things that are characterized by (...) plurality—yet because it causes plurality, the One must anticipate plurality within itself. This becomes the main mo- tivation for this study’s focus on two late Neoplatonists, Proclus (5th cent. A.D.) and Damascius (late 5th–early 6th cent. A.D.): both attempt to address this tension in two rather different ways. Proclus’ attempted solution is to posit intermediate principles (the ‘henads’) that mirror the One’s nature, as ‘one’, but directly cause plurality. This makes the One only a cause of unity, while its production of plurality is mediated by the henads that it produces. Damascius, while appropriating Proclus’ framework, thinks that this is not enough: if the One is posed as a cause of all things, it must be directly related to plurality, even if its causality is mediated through the henads. Damascius then splits Proclus’ One into two entities: (1) the Ineffable as the first ‘principle’, which is absolutely transcendent and has no causal relation; and (2) the One as the first ‘cause’ of all things, which is only relatively transcendent under the Ineffable. -/- Previous studies that compare Proclus and Damascius tend to focus either on the Ineffable or a skeptical shift in epistemology, but little work has been done on the causal framework which underlies both figures’ positions. Thus, this study proposes to focus on the causal frameworks behind each figure: why and how does Proclus propose to assert that the One is a cause, at the same time that it transcends its final effect? And what leads Damascius to propose a notion of the One’s causality that no longer makes it transcendent in the way that a higher principle, like the Ineffable, is? The present work will answer these questions in two parts. In the first, Proclus’ and Damascius’ notions of causality will be examined, insofar as they apply to all levels of being. In the second part, the One’s causality will be examined for both figures: for Proclus, the One’s causality in itself and the causality of its intermediate principles; for Damascius, the One’s causality, and how the Ineffable is needed to explain the One. The outcome of this study will show that Proclus’ framework results in an inner tension that Damascius is responding to with his notion of the One. While Damascius’ own solution implies its own tension, he at least solves a difficulty in Proclus—and in so doing, partially returns to a notion of the One much like Iamblichus’ and Plotinus’ One. (shrink)

It can be shown by means of a paradox that, given the Principle of Sufficient Reason, there is no conjunction of all contingent truths. The question is, or ought to be, how to interpret that result: _Quid sibi velit?_ A celebrated argument against PSR due to Peter van Inwagen and Jonathan Bennett in effect interprets the result to mean that PSR entails that there are no contingent truths. But reflection on parallels in philosophy of mathematics shows it can equally be (...) interpreted either as a proof that there are "too many" contingent truths to combine in a single conjunction or as a proof that the concept _contingent truth_ is indefinitely extensible and there is no such thing as "all contingent truths." Either interpretation would reconcile PSR with contingent truth, but the natural rationales of those interpretations are at odds. This essay argues that the second interpretation is a more satisfactory explanation of why, if PSR is true, there should be no conjunction of all contingent truths. This sheds new light on the nature of the explanatory demand embedded in PSR and uncovers a number of surprising implications for the commitments of rationalism. (shrink)

This article defends the principle of sufficient reason (PSR) from a simple and direct valid argument according to which PSR implies that there is a truth that explains every truth, namely an omni-explainer. Many proponents of PSR may be willing to bite the bullet and maintain that, if PSR is true, then there is an omni-explainer. I object to this strategy by defending the principle that explanation is irreflexive. Then I argue that proponents of PSR can resist the conclusion that (...) there is an omni-explainer by denying that explanation distributes over conjunction. In the final section, I consider a plausible revision of this assumption. I argue there that, given the revised assumption, PSR seems to yield a striking picture of the explanatory structure of the universe. I explain why the resulting model does not appear utterly implausible to me. (shrink)

This book on infinite regress arguments provides (i) an up-to-date overview of the literature on the topic, (ii) ready-to-use insights for all domains of philosophy, and (iii) two case studies to illustrate these insights in some detail. Infinite regress arguments play an important role in all domains of philosophy. There are infinite regresses of reasons, obligations, rules, and disputes, and all are supposed to have their own moral. Yet most of them are involved in controversy. Hence the question is: what (...) exactly is an infinite regress argument, and when is such an argument a good one? (shrink)

This paper argues that considerations about causal origins of the universe do not favour theism over naturalism. Indeed, if the only data that is relevant to the choice between theism and naturalism is data about causal origins, then it turns out that considerations about causal origins favour naturalism over theism.

I spell out a problem with the doctrine of creatio ex nihilo: that, contra the doctrine, it is not possible to efficiently cause something from nothing. This is because an efficient cause requires a material cause in order to have an effect. The material cause supplies the potency that the efficient cause actualises. Because nothingness has no potencies, there is nothing for an efficient cause to actualise. I show that this objection presupposes that the theory of noneism (the proposition that (...) some things do not exist) is false. I postulate that the universe (i.e. the created order) is a non-existent item and so there is no problem with the claim that it was efficiently caused to come from nothing – the universe has no being anyway. After rehearsing the rather strong reasons in favour of the truth of noneism, I deal with two objections that are peculiar to my claim that the universe lacks reality: that creation possesses characteristics that are sufficient to render it existent and that a non-existent object has its properties independent of divine fiat. I show that there are sensible replies to both objections. With regard to the first I show that the possession of such characteristics at most shows that the universe has an ontological status that is equivalent to some reference point. With regard to the second I argue that the Characterisation Principle (i.e. in some world – not necessarily the actual world – an object has the properties that it is characterised as having) entails that non-existent objects possess their properties in virtue of some existent entity and that the only plausible candidate for such an entity is a divine mind of some sort. (shrink)

This dissertation is on infinite regress arguments in philosophy. Its main goals are to explain what such arguments from many distinct philosophical debates have in common, and to provide guidelines for using and evaluating them. Two theories are reviewed: the Paradox Theory and the Failure Theory. According to the Paradox Theory, infinite regress arguments can be used to refute an existentially or universally quantified statement (e.g. to refute the statement that at least one discussion is settled, or the statement that (...) discussions are settled only if there is an agreed-upon criterion to settle them). According to the Failure Theory, infinite regress arguments can be used to demonstrate that a certain solution fails to solve an existentially or universally quantified problem (e.g. to demonstrate that a certain solution fails to settle all discussions, or that it fails to settle even one discussion). In the literature, the Paradox Theory is fairly well-developed, and this dissertation provides the Failure Theory with the same tools. (shrink)

Infinite regress arguments play an important role in many distinct philosophical debates. Yet, exactly how they are to be used to demonstrate anything is a matter of serious controversy. In this paper I take up this metaphilosophical debate, and demonstrate how infinite regress arguments can be used for two different purposes: either they can refute a universally quantified proposition (as the Paradox Theory says), or they can demonstrate that a solution never solves a given problem (as the Failure Theory says). (...) In the meantime, I show that Black’s view on infinite regress arguments (1996, this journal) is incomplete, and how his criticism of Passmore can be countered. (shrink)

A god is a cosmic designer-creator. Atheism says the number of gods is 0. But it is hard to defeat the minimal thesis that some possible universe is actualized by some possible god. Monotheists say the number of gods is 1. Yet no degree of perfection can be coherently assigned to any unique god. Lewis says the number of gods is at least the second beth number. Yet polytheists cannot defend an arbitrary plural number of gods. An alternative is that, (...) for every ordinal, there is a god whose perfection is proportional to it. The n -th god actualizes the best universe(s) in the n -th level of an axiological hierarchy of possible universes. Despite its unorthodoxy, ordinal polytheism has many metaphysically attractive features and merits more serious study. (shrink)

Anselmian theists claim (a) that there is a being than which none greater can be conceived; and (b) that it is knowable on purely—solely, entirely—a priori grounds that there is a being than which none greater can be conceived. In this paper, I argue that Anselmian Theism gains traction by conflating different interpretations of the key description ‘being than which no greater can be conceived’. In particular, I insist that it is very important to distinguish between ideal excellence and maximal (...) possible excellence. At the end of my paper, I illustrate the importance of this distinction by applying my discussion to the recent defence of Anselmian Theism in Nagasawa (Philos Q 58:577–591, 2008). (shrink)

I consider here several versions of finitism or conceptions that try to work around postulating sets of infinite size. Restricting oneself to the so-called potential infinite seems to rest either on temporal readings of infinity (or infinite series) or on anti-realistic background assumptions. Both these motivations may be considered problematic. Quine’s virtual set theory points out where strong assumptions of infinity enter into number theory, but is implicitly committed to infinity anyway. The approaches centring on the indefinitely large and the (...) use of schemata would provide a work-around to circumvent usage of actual infinities if we had a clear understanding of how schemata work and where to draw the conceptual line between the indefinitely large and the infinite. Neither of this seems to be clear enough. Versions of strict finitism in contrast provide a clear picture of a (realistic) finite number theory. One can recapture standard arithmetic without being committed to actual infinities. The major problem of them is their usage of a paraconsistent logic with an accompanying theory of inconsistent objects. If we are, however, already using a paraconsistent approach for other reasons (in semantics, epistemology or set theory), we get finitism for free. This strengthens the case for paraconsistency. (shrink)

On the assumption that theistic religious commitment takes place in the face of evidential ambiguity, the question arises under what conditions it is permissible to make a doxastic venture beyond one’s evidence in favour of a religious proposition. In this paper I explore the implications for orthodox theistic commitment of adopting, in answer to that question, a modest, moral coherentist, fideism. This extended Jamesian fideism crucially requires positive ethical evaluation of both the motivation and content of religious doxastic ventures. I (...) suggest that, even though the existence of horrendous evil does not resolve evidential ambiguity in favour of atheism, there are reasonable value commitments that would preclude those who hold them from satisfying extended Jamesian fideist conditions for committing themselves to classical theism. I then begin a discussion of a possible revisionary theistic alternative (in the Christian tradition) which – one might hope – may meet those conditions. An earlier, shorter, version of this paper was delivered as a keynote address at the APA Pacific 2007 Mini-Conference on Models of God. (shrink)

The Nothing from Infinity paradox arises when the combination of two infinitudes of point particles meet in a supertask and disappear. Corral-Villate claims that my arguments for disappearance fail and concedes that this failure also produces an extreme kind of indeterminism, which I have called plenitudinous. So my supertask at least poses a dilemma of extreme indeterminism within Newtonian point particle mechanics. Plenitudinous indeterminism might be trivial, although easy attempts to prove it so seem to fail in the face of (...) plausible continuity principles. However, the question of its triviality is here moot, since I show that, except in one case, Corral-Villate’s disproofs fail, and with a correction, the original arguments are unrefuted. Consequently, of the two contenders for the outcome of my supertask, the Nothing from Infinity paradox has won out. (shrink)

Erasmus and Verhoef suggest that a promising response to the infinite God objection to the Kalām cosmological argument include showing that abstract objects do not exist; actually infinite knowledge is impossible; and redefining omniscience as : for any proposition p, if God consciously thinks about p, God will either accept p as true if and only if p is true, or accept p as false if and only if p is false. I argue that there is insufficient motivation for showing (...) and and that is problematic as a definition of omniscience. (shrink)

Some theorists believe that there is a plurality of values, and that in many circumstances these values are incommensurable, or at least incomparable. Others believe that all values are reducible to a single super-value, or that even if there is a plurality of irreducible values these values are commensurable. But I will argue that both sides have got it wrong. Values are neither commensurable nor incommensurable, at least not in the way most people think. We are free to believe in (...) incommensurability or not, depending on what particular conception of morality we want to embrace. Incommensurability is accordingly not a theory about value. It is a presupposition that provides a necessary background condition for a certain kind of value to exist. It is therefore not the kind of view that can be morally true or false. As a presupposition, it can only be accepted or rejected on grounds that do not presuppose that morality already exists. Incommensurability is, like the rejection of hard determinism, one of the presuppositions on which morality as we know it happens to be based. (shrink)

Cosmological arguments from contingency attempt to show that there is a necessarily existing god‐like being on the basis of the fact that any concrete things exist at all. Such arguments are built out of the following components: (i) a causal principle that applies to non‐necessary entities of a certain category; (ii) a reason to think that if the causal principle is true, then there would have to be a necessarily existing concrete thing; (iii) a reason to think that the necessarily (...) existing thing would be god‐like. In this essay, I discuss various ways of developing each of these components to produce an argument from contingency, and I point out classic objections and replies along the way. I also make note of some of the most recent developments in arguments from contingency and point out avenues for future research. (shrink)

In this article I concentrate on three issues. First, Graham Oppy’s treatment of the relationship between the concept of infinity and Zeno’s paradoxes lay bare several porblems that must be dealt with if the concept of infinity is to do any intellectual work in philosophy of religion. Here I will expand on some insightful remarks by Oppy in an effort ot adequately respond to these problems. Second, I will do the same regarding Oppy’s treatment of Kant’s first antinomy in the (...) first critique, which deals in part with the question of whether the world had a beginning in time or if time extends infinitely into the past. And third, my examination of these two issues will inform what I have to say regarding a key topic in philosophy of religion: the question regarding the proper relationship between the infinite and the finite in the concept of God. (shrink)

In this article I propose what I call the inverse spaceship paradox. The article's interest lies in the fact that, contrary to what appears to be an implicit agreement in the literature on indeterminism, it shows that coming from infinity can be a perfectly predictable and therefore deterministic process in a classical universe.

I offer an analysis of the Principle of Sufficient Reason and its relevancy for the scientific endeavour. I submit that the world is not, and cannot be, rational—only some brained beings are. The Principle of Sufficient Reason is not a necessary truth nor a physical law. It is just a guiding metanomological hypothesis justified a posteriori by its success in helping us to unveil the mechanisms that operate in Nature.

In the literature, regress arguments often take one of two different forms: either they conclude that a given solution fails to solve any problem of a certain kind (the strong conclusion), or they conclude that a given solution fails to solve all problems of a certain kind (the weaker conclusion). This gives rise to a logical problem: do regresses entail the strong or the weaker conclusion, or none? In this paper I demonstrate that regress arguments can in fact take both (...) forms, and clearly set out the logical difference between them. Throughout the paper, I confine myself to metaphysical examples from the early Russell. Only now that we know they are valid can we start to discuss whether they are sound. (shrink)

Mereological principles are often controversial; perhaps the most stark contrast is between those who claim that Weak Supplementation is analytic—constitutive of our notion of proper parthood—and those who argue that the principle is simply false, and subject to many counterexamples. The aim of this paper is to diagnose the source of this dispute. I’ll suggest that the dispute has arisen by participants failing to be sensitive to two different conceptions of proper parthood: the outstripping conception and the non-identity conception. I’ll (...) argue that the outstripping conception (together with a specific set of definitions for other mereological notions), can deliver the analyticity of Weak Supplementation on at least one sense of ‘analyticity’. I’ll also suggest that the non-identity conception cannot do so independently of considerations to do with mereological extensionality. (shrink)

Mereological principles are often controversial; perhaps the most stark contrast is between those who claim that Weak Supplementation is analytic—constitutive of our notion of proper parthood—and those who argue that the principle is simply false, and subject to many counterexamples. The aim of this paper is to diagnose the source of this dispute. I’ll suggest that the dispute has arisen by participants failing to be sensitive to two different conceptions of proper parthood: the outstripping conception and the non-identity conception. I’ll (...) argue that the outstripping conception, can deliver the analyticity of Weak Supplementation on at least one sense of ‘analyticity’. I’ll also suggest that the non-identity conception cannot do so independently of considerations to do with mereological extensionality. (shrink)

In this short note I argue that, using the type of configurations put forward in a recent paper by Laraudogoitia in this same journal, new paradoxes of infinity of a completely different nature can be formulated.

A popular critique of the kalām cosmological argument is that one argument for its second premise illicitly assumes a finite starting point for the series of past temporal events, thereby begging the question against opponents. Rejecting this assumption, opponents say, eliminates any objections to the possibility that the past is infinitely old and undermines the IFA’s ability to support premise 2. I contend that the plausibility of this objection depends on ambiguities in extant formulations of the IFA and that we (...) may resolve these ambiguities in a way that does not presuppose a finite staring point. I also argue that this disambiguation allows us to construct an argument demonstrating that the concept of an infinite past entails a contradiction. (shrink)

This reply to Gash’s (Found Sci 2014) commentary on Nescolarde-Selva and Usó-Doménech (Found Sci 2014b) answers the questions raised and at the same time opens up new questions.