Wittgenstein's conception of infinity can be seen as continuing the tradition of the potential infinite that begins with Aristotle. Transfinite cardinals in set theory might seem to render the potential infinite defunct with the actual infinite now given mathematical legitimacy. But Wittgenstein's remarks on set theory argue that the philosophical notion of the actual infinite remains philosophical and is not given a mathematical status as a result of set theory. The philosophical notion of the actual infinite is not to be (...) found in the mathematics of set theory, only in a certain associated philosophy – what Wittgenstein calls a certain kind of “prose”. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

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.

Personal time, as opposed to external time, has a certain role to play in the correct account of death and immortality. But saying exactly what that role is, and what role remains for external time, is not straightforward. I formulate and defend accounts of death and immortality that specify these roles precisely.

The medieval distinction between categorematic and syncategorematic words is usually given as the distinction between words which have signification or meaning in isolation from other words and those which have signification only when combined with other words . Some words, however, are classified as both categorematic and syncategorematic. One such word is Latin infinita ‘infinite’. Because infinita can be either categorematic or syncategorematic, it is possible to form sophisms using infinita whose solutions turn on the distinction between categorematic and syncategorematic (...) uses of infinita. As a result, medieval logicians were interested in identifying correct logical rules governing the categorematic and syncategorematic uses of the term. In this paper, we look at 13th–15th-century logical discussions of infinita used syncategorematically and categorematically. We also relate the distinction to other medieval distinctions with which it has often been conflated in modern times, and show how and where these conflations go wrong. (shrink)

Hilary Putnam’s BIV argument first occurred to him when ‘thinking about a theorem in modern logic, the “Skolem–Löwenheim Theorem”’ (Putnam 1981: 7). One of my aims in this paper is to explore the connection between the argument and the Theorem. But I also want to draw some further connections. In particular, I think that Putnam’s BIV argument provides us with an impressively versatile template for dealing with sceptical challenges. Indeed, this template allows us to unify some of Putnam’s most enduring (...) contributions to the realism/antirealism debate: his discussions of brains-in-vats, of Skolem’s Paradox, and of permutations. In all three cases, we have an argument which does not merely defeat the sceptic; it also shows us that we must reject some prima facie plausible philosophical picture. (shrink)

This thesis offers a naturalist revision of Alain Badiou’s philosophy. This goal is pursued through an encounter of Badiou’s mathematical ontology and theory of truth with contemporary trends in philosophy of mathematics and philosophy of science. I take issue with Badiou’s inability to elucidate the link between the empirical and the ontological, and his residual reliance on a Heideggerian project of fundamental ontology, which undermines his own immanentist principles. I will argue for both a bottom-up naturalisation of Badiou’s philosophical approach (...) to mathematics and a top-down naturalisation. Articulating my particular understanding of what realism and naturalism should commit us to, I propose a creative fusion of Badiou’s attention to metamathematical results with a structural-informational metaphysics, proposing a ‘matherialism’ uniting the more daring speculative insights of the former with the naturalist and empiricist commitments motivating the latter. (shrink)

Argument-forms exist which are valid over finite but not infinite domains. Despite understanding of this by formal logicians, philosophers can be observed treating as valid arguments which are in fact invalid over infinite domains. In support of this claim I will first present an argument against the classical pragmatist theory of truth by Mark Johnston. Then, more ambitiously, I will suggest the fallacy lurks in certain arguments for physicalism taken for granted by many philosophers today.

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

Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That series (...) rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology. (shrink)

(No abstract is available for this citation).

I consider a serious objection to the knowledge account of assertion and develop a response. In the process I introduce important new data on prompting assertion, which all theorists working in the area should take note of.

Einstein considered general covariance to characterize the novelty of his General Theory of Relativity (GTR), but Kretschmann thought it merely a formal feature that any theory could have. The claim that GTR is ``already parametrized'' suggests analyzing substantive general covariance as formal general covariance achieved without hiding preferred coordinates as scalar ``clock fields,'' much as Einstein construed general covariance as the lack of preferred coordinates. Physicists often install gauge symmetries artificially with additional fields, as in the transition from Proca's to (...) Stueckelberg's electromagnetism. Some post-positivist philosophers, due to realist sympathies, are committed to judging Stueckelberg's electromagnetism distinct from and inferior to Proca's. By contrast, physicists identify them, the differences being gauge-dependent and hence unreal. It is often useful to install gauge freedom in theories with broken gauge symmetries (second-class constraints) using a modified Batalin-Fradkin-Tyutin (BFT) procedure. Massive GTR, for which parametrization and a Lagrangian BFT-like procedure appear to coincide, mimics GTR's general covariance apart from telltale clock fields. A generalized procedure for installing artificial gauge freedom subsumes parametrization and BFT, while being more Lagrangian-friendly than BFT, leaving any primary constraints unchanged and using a non-BFT boundary condition. Artificial gauge freedom licenses a generalized Kretschmann objection. However, features of paradigm cases of artificial gauge freedom might help to demonstrate a principled distinction between substantive and merely formal gauge symmetry. (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)

This paper reconsiders certain of Kierkegaard's criticisms of Hegel's theoretical philosophy in the light of recent interpretations of the latter. The paper seeks to show how these criticisms, far from being merely parochial or rhetorical, turn on central issues concerning the nature of thought and what it is to think. I begin by introducing Hegel's conception of "pure thought" as this is distinguished by his commitment to certain general requirements on a properly philosophical form of inquiry. I then outline Hegel's (...) strategy for resolving a crucial problem he takes himself to face. For his account of the nature of thought depends upon the idea of a form of inquiry in which nothing whatsoever is presupposed; but this idea appears basically paradoxical inasmuch as the mere act of beginning to inquire in a certain way embodies an assumption about how it is appropriate to begin. Turning to Kierkegaard, I consider a key objection to the effect that Hegel's strategy for resolving this paradox depends on the incoherent idea of a purely reflexive act of thinking. Finally, I draw out some central features of the alternative account of "situated" thought and inquiry which Kierkegaard presents as distinctively Socratic. (shrink)

Putnam argues that, by ‘reinterpretation’, the Axiom of Constructibility can be saved from empirical refutation. This paper contends that this argument fails, a failure which leaves Putnam's sweeping appeal to the Lowenheim –Skolem Theorem inadequately motivated.

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

Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation or theorem (...) and its proof. An attempt is made to meet directly some of Wittgenstein’s critical comments on the mathematical treatment of infinity and irrational numbers. (shrink)

It is usually supposed that, with his dichotomy paradox, Zeno gave birth to the modern so-called supertask debate – the debate of whether carrying out an infinite sequence of actions or operations...

In our wakeful conscious lives, the experience of time and dynamic temporal phenomena—such as continuous motion and change—appears to be ubiquitous. How is it that temporality is woven into our conscious experience? Is it through perceptual experience presenting a series of instantaneous states of the world, which combine together—in a sense which would need to be specified—to give us experience of dynamic temporal phenomena? In this paper, I argue that this is not the case. -/- Several authors have recently proposed (...) dynamic snapshot models of temporal experience— such as Prosser and Arstila, building upon Le Poidevin— according to which perceptual experience has no temporal content of a non‐zero extent. I argue that there is an absence of motivation for such a view; I develop and defend the claim that perceptual experience minimally presents something of some non‐zero temporal extent as such. (shrink)

Many historical and philosophical studies treat infinity as an exclusively quantitative notion, whose proper domain of application is mathematics and physics. The main aim of this paper is to disentangle, by critically examining, three notions of infinity in the early modern period, and to argue that one—but only one—of them is quantitative. One of these non-quantitative notions concerns being or reality, while the other concerns a particular iterative property of an aggregate. These three notions will emerge through examination of three (...) central figures in the period: Locke, Descartes, and Leibniz. (shrink)

ABSTRACTTwo distinct domains of philosophic enquiry are selected in order to disclose the core dynamics and concerns of a particular mode of “aesthetic skepsis”. Aspects of philosophy of cosmology and philosophy of infinity are considered in ways that serve to discipline the diminution of “belief” and the cultivation of creativity. The journey begins with a skeptic ego that is phenomenologically “empty” but wedded to a rhetoric of “darkness and light.” The result is a skepsis that needs to recapture and reconfigure (...) aesthetics and art in order to represent and communicate itself. “Skepoiesis” is a “subtle knot”: it actively manifests as a “need for art” as well as enquiries into the problematics of philosophical aesthetics from the standpoint of creative skepsis. (shrink)

Change seems missing in Hamiltonian General Relativity's observables. The typical definition takes observables to have $0$ Poisson bracket with \emph{each} first-class constraint. Another definition aims to recover Lagrangian-equivalence: observables have $0$ Poisson bracket with the gauge generator $G$, a \emph{tuned sum} of first-class constraints. Empirically equivalent theories have equivalent observables. That platitude provides a test of definitions using de Broglie's massive electromagnetism. The non-gauge ``Proca'' formulation has no first-class constraints, so everything is observable. The gauge ``Stueckelberg'' formulation has first-class constraints, (...) so observables vary with the definition. Which satisfies the platitude? The team definition does; the individual definition does not. Subsequent work using the gravitational analog has shown that observables have not a 0 Poisson bracket, but a Lie derivative for the Poisson bracket with the gauge generator $G$. The same should hold for General Relativity, so observables change locally and correspond to 4-dimensional tensor calculus. Thus requiring equivalent observables for empirically equivalent formulations helps to address the problem of time. (shrink)

This paper examines Alain Badiou’s critical engagement with religion. It is argued that there are two central points at which religion enters the scene of Badiou’s philosophy. First, in his critique, the ‘motif of finitude’ Badiou repeatedly refers to religion, claiming that ‘the obsession with finitude is a remnant of the tyranny of the sacred’. Second, Badiou stages his attempt to regenerate philosophy against the proclamation of its end as a confrontation with the religion, through philosophy’s detachment from the poetization (...) of philosophy and its reattachment to mathematics. By examining these two points, the paper aims to encircle Badiou’s notion of religion and thus clarify the role it plays in his philosophical system. This paper suggests that his notion of religion is a by-product of his polemics and therefore needs to be ‘extracted’ from his writings. Hence, the ‘critique of religion’ present in Badiou’s work is not straightforward and explicit critique. Another central aspect of Badiou’s critique of religion is related to his effort to separate the concept of truth from the category of meaning, which he understands as the emblem of religion. The paper sheds light on this matter and its setting in Jacques Lacan’s considerations on religion. (shrink)

I suggest a way in which metaphysics might cure us of our desire for immortality. Supposing that time is composed of instants, or even that time could be composed of instants, leads to the conclusion that there is nothing good that immortality offers, nothing we might reasonably want, that is in principle unavailable to a mere mortal. My argument proceeds in three stages. First, I suggest a necessary condition for a feature to ground the desirability of a life or a (...) portion thereof. Second, after distinguishing between three different features that could plausibly be meant by ’immortality’, I argue that if time could be composed of instants, only one of those features satisfies the necessary condition, and it evidently fails to ground the desirability of a life. Third, I argue that no feature that entails any of the three features grounds the desirability of a life either. I conclude with reflections on what this means for our longing for immortality. (shrink)

In his 2006 paper `Pantheists in Spite of Themselves: God and Infinity in Contemporary Theology,’ William Lane Craig examines the work of Wolfhart Pannenberg, Philip Clayton, and F. LeRon Shults, whose conceptions of God are influenced by Hegel. Craig shows that these thinkers’ Hegelian formulations lead to monism, despite their attempts to avoid it. He then attempts to refute Hegelian thinking by appealing to Cantor. I argue that that this refutation fails because Cantor and Hegel are far more amicable than (...) Craig realizes, as Small’s and Drozdek’s work shows. (shrink)

If mysticism, as Coventry Patmore defines it, is 'the science of ultimates,' in what way would mysticism explain the possibility of a profound relationship between ultimate reality as infinite and proximate reality as finite ? This paper attempts to address that question through the lens of Evelyn Underhill’s philosophy of mysticism. The paper fundamentally works at framing two of Hegel’s triadic patterns of dialectic against the being-becoming binary as engaged by Underhill. This application helps unveil the relation of transcendence with (...) immanence, a relation that is crucial for a structuring of the infinite-finite mystical intimacy. (shrink)

Famously, Galilei made the ontological claim that the book of nature is written in the language of mathematics. Probably, if only implicitly, most contemporary natural scientists share his view. This paper, in contradistinction, argues that nature is only partly written in the language of mathematics; partly, it is written in the language of functions and partly in a very simple purely qualitative language, too. During the argumentation, three more specific but in themselves interesting theses are put forward: first (in Section (...) 3), there are more shapes than real numbers; second (in Section 4), the metrological notion ‘amount of substance’ can profitably be exchanged for ‘number of entities’; third (in Section 5), prototypical concepts will always be scientifically important. (shrink)

If proofs are nothing more than truth makers, then there is no force in the standard argument against classical logic (there is no guarantee that there is either a proof forA or a proof fornot A). The standard intuitionistic conception of a mathematical proof is stronger: there are epistemic constraints on proofs. But the idea that proofs must be recognizable as such by us, with our actual capacities, is incompatible with the standard intuitionistic explanations of the meanings of the logical (...) constants. Proofs are to be recognizable in principle, not necessarily in practice, as shown in section 1. Section 2 considers unknowable propositions of the kind involved in Fitch''s paradox:p and it will never be known thatp. It is argued that the intuitionist faces a dilemma: give up strongly entrenched common sense intuitions about such unknowable propositions, or give up verificationism. The third section considers one attempt to save intuitionism while partly giving up verificationism: keep the idea that a proposition is true iff there is a proof (verification) of it, and reject the idea that proofs must be recognizable in principle. It is argued that this move will have the effect that some standard reasons against classical semantics will be effective also against intuitionism. This is the case with Dummett''s meaning theoretical argument. At the same time the basic reason for regarding proofs as more than mere truth makers is lost. (shrink)

In 'General Propositions and Causality' Ramsey rejects his earlier view that universal generalizations are infinite conjunctions, arguing that they are not genuine propositions at all. We argue that his new position is unstable. The issues about infinity that lead Ramsey to the new view are essentially those underlying Wittgenstein's rule-following considerations. If they show that generalizations are not genuine propositions, they show that there are no genuine propositions. The connection raises interesting historical questions about the direction of influence between Ramsey (...) and Wittgenstein, the origin of the rule-following argument, and the influence of writers such as Brouwer. (shrink)

In this paper a model in particle dynamics of a well-known supertask is constructed. As a consequence, a new and simple result about the failure of determinism of classical particle dynamics can be proved which is related to the non-existence of boundary conditions at spatial infinity. This result is much more accessible to the non-technical reader than similar ones in the scientific literature.

Ways and words about infinity have frequently hidden a continuing paradox inspired by Zeno. The basic puzzle is the tortoise's – Mr T's – Extension Challenge, the challenge being how any extension, be it in time or space or both, moving or still, can yet be of an endless number of extensions. We identify a similarity with Mr T's Deduction Challenge, reported by Lewis Carroll, to the claim that a conclusion can be validly reached in finite steps. Rejecting common solutions (...) to both challenges, we use a Wittgensteinian approach for dissolution, noting also that other paradoxes, such as the Surprise Examination and Yablo's, trap us inconsistently into similar endlessnesses. In so doing, we encounter a hopping flea, generate a proportionality paradox and consider the knight's move in chess. (shrink)