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)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

ABSTRACTStephen Puryear argues that William Lane Craig's view, that time as duration is logically prior to the potentially infinite divisions that we make of it, involves the idea that time is prior to any parts we conceive within it. He objects that PWT entails the Priority of the Whole with respect to Events, and that it subverts the argument, used by proponents of the Kalam Cosmological Argument such as Craig, against an eternal past based on the impossibility of traversing an (...) actual infinite sequence of events. I argue that proponents of KCA can affirm that time is not discrete, nor is it continuous with actual infinite number of parts or points, but rather that it is a continuum with various parts yet without an actual infinite number of parts or points. I defend this view, and I reply to Puryear's other objections. (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)

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)

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)

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.

Concepts of infinity have been subjects of dispute since antiquity. The main problems of this paper are: is the mind able to acquire a concept of infinity? and: how are concepts of infinity acquired? The aim of this paper is neither to say what the meanings of the word “infinity” are nor what infinity is and whether it exists. However, those questions will be mentioned, but only in necessary extent.

At the end of the 19th century, Josiah Royce participated in what has come to be called the great debate (Royce, 1897; Armour, 2005).1 The great debate concerned issues in metaphysical theology, and, since metaphysics was primarily idealistic, it dealt considerably with the relations between the divine Self and lesser selves. After the great debate, Royce developed his idealism in his Gifford Lectures (1898-1900). These were published as The World and the Individual. At the end of the first volume, Royce (...) added a Supplementary Essay. The Essay, continuing themes from the great debate, developed an intriguing mathematical model of the relations between the divine Self and lesser selves. The second volume went on to.. (shrink)

(No abstract is available for this citation).

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)

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.

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 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.

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.

This paper considers a recent criticism of the physical possibility of supertasks which involves Achilles’s staccato run. It is held that the criticism fails and that the underlying fallacy can be linked with interesting developments in the modern literature on physical supertasks.

Searching for common ground in philosophy, science and theology, it seems to us that it would be reasonable to maintain the position of realistic pragmatism that Charles Sanders Peirce had called pragmaticism. In the pragmaticist manner, we typify the knowledge and select the types of knowledge that might be useful for understanding the problems that are of interest to us. We pose a question of how it would be possible to obtain practically useful information about reality, first from the perspective (...) of natural sciences, and then from that of theology; that is, to diversify the ways of knowledge and just maybe, to move toward a productive dialogue between science and religion. (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)

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, 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)

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.

There is a famous quip of F.P. Ramsey's, which is my second epigraph. According to a widespread legend, the quip is a criticism of Wittgenstein's treatment in the Tractatus of what cannot be said. The remark is indeed Ramsey's, but he didn't mean what he is taken to mean in the legend. His quip, looked at in context, means something quite different. The legend is sometimes taken to provide support for a reading of the Tractatus according to which the nonsensical (...) propositions of the book were intended to convey what cannot be said. But, since the legend has no basis in reality, it provides no evidence in favor of any such reading of the Tractatus. The quip has great interest if it is read in the context of Ramsey's discussion of generality; it is closely related to issues of importance in the development of Wittgenstein's thought. (shrink)

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)

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)

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)

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.

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)

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)

This article is about the history of logic in Australia. Douglas Gasking (1911?1994) undertook to translate the logical terminology of John Anderson (1893?1962) into that of Ludwig Wittgenstein's (1921) Tractatus. At the time Gilbert Ryle (1900?1976), and more recently David Armstrong, recommended the result to students; but it is reasonable to have misgivings about Gasking as a guide to either Anderson or Wittgenstein. The historical interest of the debate Gasking initiated is that it yielded surprisingly little information about Anderson's traditional (...) (syllogistic or Aristotelian) logic and its relation to classical (first-order predicate or Russellian) logic, the ostensible topic; but the materials now exist to interpret Anderson's logic in classical logic, possibly as an algebra of classes. This would be of little interest to contemporary logicians, but it might shed some light on Anderson's philosophy. (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)

How much value can our decisions create? We argue that unless our current understanding of physics is wrong in fairly fundamental ways, there exists an upper limit of value relevant to our decisions. First, due to the speed of light and the definition and conception of economic growth, the limit to economic growth is a restrictive one. Additionally, a related far larger but still finite limit exists for value in a much broader sense due to the physics of information and (...) the ability of physical beings to place value on outcomes. We discuss how this argument can handle lexicographic preferences, probabilities, and the implications for infinite ethics and ethical uncertainty. (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)

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)