The article examines the critique of Being and Time formulated by Levinas in Totality and Infinite, a critique centered on Heidegger’s omission of two fundamental forms of being in the world: enjoyment and inhabiting. This omission is symptomatic: as a critique of modernity, Being and Time internalizes and ontologizes the prevalence of the equipmentality that characterizes our era far more than scientific objectivism does. Thus, a certain type of pragmatism would constitute the keystone of Being and Time as (...) a whole: the acknowledgment by Heidegger of this “contamination” would explain, at least partially, his transition to the “history of being”. (shrink)
In the Transcendental Ideal Kant discusses the principle of complete determination: for every object and every predicate A, the object is either determinately A or not-A. He claims this principle is synthetic, but it appears to follow from the principle of excluded middle, which is analytic. He also makes a puzzling claim in support of its syntheticity: that it represents individual objects as deriving their possibility from the whole of possibility. This raises a puzzle about why Kant regarded it as (...) synthetic, and what his explanatory claim means. I argue that the principle of complete determination does not follow from the principle of excluded middle because the externally negated or ?negative? judgement ?Not (S is P)? does not entail the internally negated or ?infinite? judgement ?S is not-P.? Kant's puzzling explanatory claim means that empirical objects are determined by the content of the totality of experience. This entails that empirical objects are completely determinate if and only if the totality of experience has a completely determinate content. I argue that it is not a priori whether experience has such a completely determinate content and thus not analytic that objects obey the principle of complete determination. (shrink)
At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.
This article deals with a question of a most general, comprehensive and profound content as it is the nature of mathematical–logical objects insofar as these are considered objects of knowledge and more specifically objects of formal mathematical theories. As objects of formal theories they are dealt with in the sense they have acquired primarily from the beginnings of the systematic study of mathematical foundations in connection with logic dating from the works of G. Cantor and G. Frege in the last (...) decades of the nineteenth century. Largely motivated by a phenomenologically founded view of mathematical objects/states-of-affairs, I try to consistently argue for their character as objects shaped to a certain extent by intentional forms exhibited by consciousness and the modes of constitution of inner temporality and at the same time as constrained in the form of immanent ‘appearances’ by what stands as their non-eliminable reference, that is, the world as primitive soil of experience and the mathematical intuitions developed in relations of reciprocity within-the-world. In this perspective and relative to my intentions I enter, in the last section, into a brief review of certain positions of G. Sher’s foundational holism and R. Tieszen’s constituted platonism, among others, respectively presented in Sher and Tieszen. (shrink)
Subject and the realisation of ethical responsibility – The Idea of the In finite in Levinas' Totality and Infinity. In Totality and Infinity Emmanuel Levinas writes about the categorical character of the ethical responsibility that the subject owes to the other. The confrontation with the suffering other puts the subject's natural self-interest into question, and brings him/her to realise an ethical responsibility of which s/he cannot unburden himself/herself. The question arises as to what in the constitution of the (...) subject makes him/her susceptible to the realisation of ethical responsibility. This article illustrates that in order to accentuate ethical responsibility as strongly as he does, Levinas needs to take a quasi- metaphysical step. The “trace of the infinite” that “creation” has left on the finite subject, predisposes the subject to the appeal of the other. Levinas' use of words such as “God”, “the Good”, “creation” and “the Idea of Infinity” does not have a theological or a mystical underpinning. These metaphysical concepts are philosophical figures of speech that Levinas borrows from Plato and Descartes. (shrink)
The wrong version of my article ‘Aspects of the Infinite in Kant’ was printed in the last issue of Mind (pp. 205–23). I should like to correct an error that thereby appeared on page 207. In A430–2/B458–60 of the Critique of Pure Reason Kant does not deny that what is (mathematically) infinite should be what I called an actual measurable totality—if, by its measure, we mean ‘the multiplicity of given units which it contains’. His point is simply (...) that what makes it infinite cannot be the fact that its measure is the greatest possible; for there is no such thing. What he has in mind here can be illustrated as follows. Take the infinite multiplicity of hours that have elapsed up to a given moment; then the multiplicity of minutes that have elapsed, and indeed the multiplicity of hours that will have elapsed an hour later, are both greater. No multiplicity is so great that it cannot be increased in this way. (Of course, standard contemporary formal work on the infinite has superseded Kant here. A modern mathematician would want either to quarrel with this or at least to refine it.) He also refuses to allow that an infinite multiplicity is a number (cf. BIII and A526–7/B554–5). In the end, he thinks, there is no saying what it is for something to be (mathematically) infinite without falling back on ‘the true transcendental concept of infinitude,’ namely ‘that the successive synthesis of units required for the enumeration of a quantum can never be completed’. (shrink)
Totality and Infinity , the title of a well-known work by Emmanuel Levinas, takes up a word which readers of Poetic Intention and of many other texts of Édouard Glissant’s will easily recognize: a term sometimes used in a sense that is clearly positive, sometimes in a sense that is not quite as positive, such as when, for instance, he compares “totalizing Reason” to the “Montaigne’s tolerant relativism.” In his final collection of essays, Traité du tout-monde, Poétique IV , (...) Glissant attempts one more time to clarify the sense in which the reader will have to understand his use of the word “totality,” thinking, and rightfully so, that this word might lead to some confusion: “To write is to say the world. The world as totality, which is so dangerously close to the totalitarian.” Of course, here, it will be necessary to try to ascertain whether or not Levinas’s totality and Glissant’s can peacefully coexist, or, rather, whether this word might, in Glissant, have opposite meanings. Where the second word is concerned, “infinity,” any reader of Glissant will know that he locates its source in those societies he calls atavistic, which are grounded in foundational texts that are the bearers of stories of filiation, of legitimacy, societies whose arrogance and whose errors the author never ceases to decry and whose decomposition, in the very times in which we live, he never ceases to announce (even as Glissant recognizes that there was a time when atavistic cultures undoubtedly must have experienced their own period of creolization, and that, conversely, composite cultures undoubtedly often tend to become atavistic). On this level, “totality” and “infinity,” for him, seem to belong to the same world. Thus, and still in Traité du tout-monde , he proposes that "Hebraism, Christianity, Islam are grounded in the same spirituality of the One and to the same belief in a revealed Truth… The thought of the One that has done so much to magnify, as well as to denature. How can one consent to this thought, which transfigures while neither offending nor de-routing the Diverse?" Moreover, it would be interesting, I think, to know how Levinas might react to these words of Glissant’s: “Totality is not that which has often been called the universal. It is the finite and realized quantity of the infinite detail of the real.” This word, “infinite,” is decidedly dangerous: what is an “infinite detail?” Does this word, “infinite,” not always lead to the unknown, to the non-totalizable, to what Levinas would call an “enigma,” to what Glissant would call an “opacity?&rdquo. (shrink)
On standard accounts of responsibility, one is thought to be responsible for one's own actions or affairs. Levinas' philosophy speaks of a responsibility that goes beyond my actions and their consequences to an infinite, irrecusable, asymmetrical responsibility for the other human. In the dissertation, I present a defense of Levinasian responsibility and argue that distinctive of Levinas' thought as an ethics is the manner in which it maintains the absolute and unexceptionable character of responsibility, while simultaneously putting into question (...) every possible ground or justification for this claim. The central chapters of the dissertation consider the connection between the formal structure of transcendence and the deformalization of this notion as responsibility. After discussing the notions of transcendence employed in the phenomenology of Husserl and Heidegger, I examine Levinas' criticism that transcendence in their sense remains an immanence insofar it continues to tie the conditions of intelligibility to modes of understanding and comprehension. Levinas' reconception of transcendence as a relation to exteriority and absolute alterity is then examined as it is developed first in his early writings , and then in Totality and Infinity through the tropes of the idea of infinity, metaphysical desire, and the face. The aim of these chapters is not only to elucidate in detail the conception of transcendence, but to examine why Levinas thinks that transcendence is already an ethical event. While commentators have sometimes claimed that ethics in Levinas' sense is neither normative nor descriptive, I show that the question of normativity--of how ethical claims are binding on us--is central to the problematic of responsibility, though in a way which ultimately forces us to revise our understanding of both normativity and ethics. Levinas contests the possibility of decisive evidence that would prove the truth of our ethical obligations. What I show is how the call for proof or justification always already distorts and passes over the meaning of responsibility. (shrink)
In this article I examine Jean-Luc Marion's two-fold criticism of Emmanuel Levinas’ philosophy of other and self, namely that Levinas remains unable to overcome ontological difference in Totality and Infinity and does so successfully only with the notion of the appeal in Otherwise than Being and that his account of alterity is ambiguous in failing to distinguish clearly between human and divine other. I outline Levinas’ response to this criticism and then critically examine Marion's own account of subjectivity that (...) attempts to go beyond Levinas in its emphasis on a pure or anonymous appeal. I criticize this move as rather problematic and turn instead back to Levinas for a more convincing account of the relations between self, human other, and God. In this context, I also show that Levinas in fact draws quite careful distinctions between human and divine others. (shrink)
Lévinas is one of the most important thinkers of the 20th century and, perhaps, the philosopher who has attempted to think of difference most seriously. In this effort, he encountered the limits of language itself, as well as the difficulties it poses for thinking the other, difference, outside the ..
Se pretende dar cuenta de la crítica a Ser y tiempo que Lévinas formula en Totalidad e infinito, crítica centrada en la omisión de dos formas primordiales de estancia en el mundo: el goce y el habitar. Tal omisión es sintomática: Ser y tiempo, al querer ser una crítica de la modernidad, internaliza ..
In the correspondence with Clarke, Leibniz proposes to construe physical theory in terms of physical (spatio-temporal) relations between physical objects, thus avoiding incorporation of infinite totalities of abstract entities (such as Newtonian space) in physical ontology. It has generally been felt that this proposal cannot be carried out. I demonstrate an equivalence between formulations postulating space-time as an infinitetotality and formulations allowing only possible spatio-temporal relations of physical (point-) objects. The resulting rigorous formulations of physical theory (...) may be seen to follow Leibniz' suggestion quite closely. On the other hand, physical assumptions implicit in the postulation of space-time totalities are made explicit in the reconstruction of the space-time versions from the physical-relation versions. (shrink)
Some issues raised by the notion of surveyability and how it is represented mathematically are explored. Wright considers the sense in which the positive integers are surveyable and suggests that their structure will be a weakly finite, but weakly infinite, totality. One way to expose the incoherence of this account is by applying Wittgenstein's distinction between intensional and extensional to it. Criticism of the idea of a surveyable proof shows the notion's lack of clarity. It is suggested that (...) this concept should be replaced by that of a feasible operation, as strict finitism aims to understand the boundaries of legitimate mathematical knowledge. (shrink)
In spite of differences the thought of Bernays, Dooyeweerd and Gödel evinces a remarkable convergence. This is particularly the case in respect of the acknowledgement of the difference between the discrete and the continuous, the foundational position of number and the fact that the idea of continuity is derived from space (geometry – Bernays). What is furthermore similar is the recognition of what is primitive (and indefinable) as well as the account of the coherence of what is unique, such as (...) when Gödel observes something quasi-spatial character of sets. It is shown that Dooyeweerd’s theory of modal aspects provides a philosophical framework that exceeds his own restrictive understanding of infinity )to the potential infinite) and at the same time makes it possible to account for key insights found in the thought of Bernays and Gödel. When Laugwitz says that discreteness rules within the sphere of the numerical, he says nothing more than what Dooyeweerd had in mind with his idea that discrete quantity, as the meaning-nucleus of the arithmetical aspect, qualifies every element within the structure of the quantitative aspect. And when Bernays says that analysis expresses the idea of the continuum in arithmetical language his mode of speech is equivalent to saying that mathematical analysis could seen as being founded upon the spatial anticipation within the modal structure of the arithmetical aspect. The view of the actual infinite (the at once infinite) in terms of an “as if” approach (Bernays), that is, as appreciated as a regulative hypothesis through which every successively infinite multiplicity of numbers could be envisaged as being giving all at once, as an infinitetotality, provides a sound understanding of the at once infinite and makes it plain why every form of arithmeticism fails. Such attempts have to call upon Cantor’s proof on the non-denumerability of the real numbers – and this proof pre-supposes the use of the at once infinite which, in turn, pre-supposes the (irreducibile) spatial order of simultaneity and the patial whole-parts relation. (shrink)
The general issue addressed in this dissertation is: what do the models of formal model-theoretic semantics represent? In chapter 2, I argue that those of first-order classical logic represent meaning assignments in possible worlds. This motivates an inquiry into what the interpretations of first-order quantified model logic represent, and in Chapter 3 I argue that they represent meaning assignments in possible universes of possible worlds. A possible universe is unpacked as one way model reality might be. The problem arises here (...) as to how we are to understand the distinction between the actual and the possible as it relates to modal reality. ;Along with the development of the main arguments in Chapters 2 and 3, the dissertation assesses the status of semantic accounts or logical properties and relations. Specifically, what does the model-theoretic account of a logically possible situation add to the syntactic account ? ;Proofs of invalidity in terms of the models of formal semantics do not establish that it is possible for the premises to be true and the conclusion false, since a formal model is merely given by a consistent set of sentences. Unless there is some way to generate a non-formal model from a formal one, such proofs do not really go beyond syntactic notions. The dissertation ends by concluding that there is no way to generate a non-formal model from a formal one without relying on logical intuitions that are syntactical. ;Hence efforts to construct a semantic basis for model logic independent of syntactic commitments are misguided. However, in classical logic the independence of the semantic account from the syntactic one is grounded on the intuition that it is metaphysically possible for there to be a denumerably infinitetotality of objects. (shrink)
It is possible that a fair coin tossed infinitely many times will always land heads. So the probability of such a sequence of outcomes should, intuitively, be positive, albeit miniscule: 0 probability ought to be reserved for impossible events. And, furthermore, since the tosses are independent and the probability of heads (and tails) on a single toss is half, all sequences are equiprobable. But Williamson has adduced an argument that purports to show that our intuitions notwithstanding, the probability of an (...)infinite sequence is 0. In this paper, I rebut his argument.No Abstract. (shrink)
The purpose of this note is to contrast a Cantorian outlook with a non-Cantorian one and to present a picture that provides support for the latter. In particular, I suggest that: i) infinite hyperreal numbers are the (actual, determined) infinite numbers, ii) ω is merely potentially infinite, and iii) infinitesimals should not be used in the di Finetti lottery. Though most Cantorians will likely maintain a Cantorian outlook, the picture is meant to motivate the obvious nature of (...) the non-Cantorian outlook. (shrink)
In formal ontology, infinite regresses are generally considered a bad sign. One debate where such regresses come into play is the debate about fundamentality. Arguments in favour of some type of fundamentalism are many, but they generally share the idea that infinite chains of ontological dependence must be ruled out. Some motivations for this view are assessed in this article, with the conclusion that such infinite chains may not always be vicious. Indeed, there may even be room (...) for a type of fundamentalism combined with infinite descent as long as this descent is “boring,” that is, the same structure repeats ad infinitum. A start is made in the article towards a systematic account of this type of infinite descent. The philosophical prospects and scientific tenability of the account are briefly evaluated using an example from physics. (shrink)
Deflationsism about truth is a pot-pourri, variously claiming that truth is redundant, or is constituted by the totality of 'T-sentences', or is a purely logical device (required solely for disquotational purposes or for re-expressing finitarily infinite conjunctions and/or disjunctions). In 1980, Hartry Field proposed what might be called a 'deflationary theory of mathematics', in which it is alleged that all uses of mathematics within science are dispensable. Field's criterion for the dispensability of mathematics turns on a property of (...) theories, called conservativeness. I present some technical results, some of which may be found in Tarski (1936), concerning the logical properties of truth theories; in particular, concerning the conservativeness of adding a truth theory for an object level language to any theory expressed in it. It transpires that various deflationary truth theories behave somewhat differently from the standard Tarskian truth theory. These results suggest that Tarskian theories of truth are not redundant or dispensable. Finally, I hint at an analogy between the behaviour of mathematical theories and of standard (Tarskian) theories of truth with respect to their indispensability to, as Quine would put, our 'scientific world-view'. (shrink)
In this chapter I explain Spinoza's concept of "infinite modes". After some brief background on Spinoza's thoughts on infinity, I provide reasons to think that Immediate Infinite Modes are identical to the attributes, and that Mediate Infinite Modes are merely totalities of finite modes. I conclude with some considerations against the alternative view that infinite modes are laws of nature.
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 collection A is properly included in a collection B then the (...) ‘size’ of A should be less than the ‘size’ of B. 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. Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion. 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 or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano. (shrink)
This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a "pre" form of this thesis that every proof can be presented in everyday statements-only form.
This paper aims to show that a proper understanding of what Leibniz meant by “hypercategorematic infinite” sheds light on some fundamental aspects of his conceptions of God and of the relationship between God and created simple substances or monads. After revisiting Leibniz’s distinction between (i) syncategorematic infinite, (ii) categorematic infinite, and (iii) actual infinite, I examine his claim that the hypercategorematic infinite is “God himself” in conjunction with other key statements about God. I then discuss (...) the issue of whether the hypercategorematic infinite is a “whole”, comparing the four kinds of infinite outlined by Leibniz in 1706 with the three degrees of infinity outlined in 1676. In the last section, I discuss the relationship between the hypercategorematic infinite and created simple substances. I conclude that, for Leibniz, only a being beyond all determinations but eminently embracing all determinations can enjoy the pure positivity of what is truly infinite while constituting the ontological grounding of all things. (shrink)
Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.
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)
Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.
In this article, we evaluate various responses to a noteworthy objection, namely, the infinite God objection to the kalām cosmological argument. As regards this objection, the proponents of the kalām argument face a dilemma—either an actual infinite cannot exist or God cannot be infinite. More precisely, this objection claims that God’s omniscience entails the existence of an actual infinite with God knowing an actually infinite number of future events or abstract objects, such as mathematical truths. (...) We argue, however, that the infinite God objection is based on two questionable assumptions, namely, that it is possible for an omniscient being to know an actually infinite number of things and that there exist an actually infinite number of abstract objects for God to know. (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)
In this paper I argue that the infinite regress of resemblance is vicious in the guise it is given by Russell but that it is virtuous if generated in a (contemporary) trope theoretical framework. To explain why this is so I investigate the infinite regress argument. I find that there is but one interesting and substantial way in which the distinction between vicious and virtuous regresses can be understood: The Dependence Understanding. I argue, furthermore, that to be able (...) to decide whether an infinite regress exhibits a dependence pattern of a vicious or a virtuous kind, facts about the theoretical context in which it is generated become essential. It is precisely because of differences in context that he Russellian resemblance regress is vicious whereas its trope theoretical counterpart is not. (shrink)
Throughout history, almost all mathematicians, physicists and philosophers have been of the opinion that space and time are infinitely divisible. That is, it is usually believed that space and time do not consist of atoms, but that any piece of space and time of non-zero size, however small, can itself be divided into still smaller parts. This assumption is included in geometry, as in Euclid, and also in the Euclidean and non- Euclidean geometries used in modern physics. Of the few (...) who have denied that space and time are infinitely divisible, the most notable are the ancient atomists, and Berkeley and Hume. All of these assert not only that space and time might be atomic, but that they must be. Infinite divisibility is, they say, impossible on purely conceptual grounds. (shrink)
Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.
In ‘Fair Infinite Lotteries’ (FIL), Wenmackers and Horsten use non-standard analysis to construct a family of nicely-behaved hyperrational-valued probability measures on sets of natural numbers. Each probability measure in FIL is determined by a free ultrafilter on the natural numbers: distinct free ultrafilters determine distinct probability measures. The authors reply to a worry about a consequent ‘arbitrariness’ by remarking, “A different choice of free ultrafilter produces a different ... probability function with the same standard part but infinitesimal differences.” They (...) illustrate this remark with the example of the sets of odd and even numbers. Depending on the ultrafilter, either each of these sets has probability 1/2, or the set of odd numbers has a probability infinitesimally higher than 1/2 and the set of even numbers infinitesimally lower. The point of the current paper is simply that the amount of indeterminacy is much greater than acknowledged in FIL: there are sets of natural numbers whose probability is far more indeterminate than that of the set of odd or the set of even numbers. (shrink)
In this paper, I suggest that infinite numbers are large finite numbers, and that infinite numbers, properly understood, are 1) of the structure omega + (omega* + omega)Ө + omega*, and 2) the part is smaller than the whole. I present an explanation of these claims in terms of epistemic limitations. I then consider the importance, part of which is demonstrating the contradiction that lies at the heart of Cantorian set theory: the natural numbers are too large to (...) be counted by any finite number, but too small to be counted by any infinite number – there is no number of natural numbers. (shrink)
Spinoza’s bold claim that there exists only a single infinite substance entails that finite things pose a deep challenge: How can Spinoza account for their finitude and their plurality? Taking finite bodies as a test case for finite modes in general I articulate the necessary conditions for the existence of finite things. The key to my argument is the recognition that Spinoza’s account of finite bodies reflects both Cartesian and Hobbesian influences. This recognition leads to the surprising realization there (...) must be more to finite bodies than their finitude, a claim that goes well beyond the basic substance-monism claim, namely, that anything that is, is in God. This leads to the conclusion, which may seem paradoxical, that finite bodies have both an infinite as well as a finite aspect to them. Finite bodies, I argue, both actively partially determine all the other finite bodies, thereby partially causing their existence insofar as they are finite, as well as are determined by the totality of other bodies. I articulate precisely what this infinite aspect is and how it is distinct from the general substance-monism dictum. (shrink)
This paper compares and contrasts two infinite regress arguments against higher-order theories of consciousness that were put forward by the Buddhist epistemologists Dignāga (ca. 480–540 CE) and Dharmakīrti (ca. 600–660). The two arguments differ considerably from each other, and they also differ from the infinite regress argument that scholars usually attribute to Dignāga or his followers. The analysis shows that the two philosophers, in these arguments, work with different assumptions for why an object-cognition must be cognised: for Dignāga (...) it must be cognised in order to enable subsequent memory of it, for Dharmakīrti it must be cognised if it is to cognise an object. (shrink)
Physicalism is frequently understood as the thesis that everything depends upon a fundamental physical level. This standard formulation of physicalism has a rarely noted and arguably unacceptable consequence—it makes physicalism come out false in worlds which have no fundamental level, for instance worlds containing things which can infinitely decompose into smaller and smaller parts. If physicalism is false, it should not be for this reason. Thus far, there is only one proposed solution to this problem, and it comes from the (...) so-called via negativa account of physicalism. Via negativa physicalism identifies the physical with the non-mental, such that if everything in the world ultimately depends only on non-mental things, then physicalism is true. To deal with the possibility of worlds without a fundamental level, this account says that physicalism is false in worlds with either a fundamental mental level or an infinite descent of mental levels. Here I argue that there could be a world with an infinite descent of all-mental levels, yet in which physicalism might plausibly be true—thus contradicting the sufficient-for-false condition meant to save physicalism from the threat of infinitely decomposable worlds. This leaves the need for a new dependence-based account of physicalism. (shrink)
What if human joy went on endlessly? Suppose, for example, that each human generation were followed by another, or that the Western religions are right when they teach that each human being lives eternally after death. If any such possibility is true in the actual world, then an agent might sometimes be so situated that more than one course of action would produce an infinite amount of utility. Deciding whether to have a child born this year rather than next (...) is a situation wherein an agent may face several alternatives whose effects could well ramify endlessly on such suppositions, for the child born this year would be a different person—one who preferred different things, performed different actions, and had different descendants—from a child born next year. It has recently been suggested that traditional utilitarianism stumbles on such cases of infinite utility. Specifically, utilitarianism seems to require, for its application, that all experience of pleasure and pain cease at some time in the future or asymptotically approach zero.2 If neither of these conditions holds, then the utility produced by each of two alternative actions may turn out to be infinite, and utilitarianism thus loses its ability to discriminate morally between them. (shrink)
Leibniz claims that nature is actually infinite but rejects infinite number. Are his mathematical commitments out of step with his metaphysical ones? It is widely accepted that Leibniz has a viable response to this problem: there can be infinitely many created substances, but no infinite number of them. But there is a second problem that has not been satisfactorily resolved. It has been suggested that Leibniz’s argument against the world soul relies on his rejection of infinite (...) number, and, as such, Leibniz cannot assert that any body has a soul without also accepting infinite number, since any body has infinitely many parts. Previous attempts to address this concern have misunderstood the character of Leibniz’s rejection of infinite number. I argue that Leibniz draws an important distinction between ‘wholes’– collections of parts that can be thought of as a single thing – and ‘fictional wholes’ – collections of parts that cannot be thought of as a single thing, which allows us to make sense of his rejection of infinite number in a way that does not conflict either with his view that the world is actually infinite or that the bodies of substances have infinitely many parts. (shrink)
A multiverse is comprised of many universes, which quickly leads to the question: How many universes? There are either finitely many or infinitely many universes. The purpose of this paper is to discuss two conceptions of infinite number and their relationship to multiverses. The first conception is the standard Cantorian view. But recent work has suggested a second conception of infinite number, on which infinite numbers behave very much like finite numbers. I will argue that that this (...) second conception of infinite number is the correct one, and analyze what this means for multiverses. (shrink)
Kantian transcendental philosophy has shown that we can never decide the question of whether or not the world is infinite in space and time, because, in the field of appearance, the world as a totality of concordant experience "does not exist as [an unconditioned] whole, either of infinite or of finite magnitude."1 However, appearances are encountered in a world, in which one aspect of a thing always invites us to consider others, indicating thereby a road to infinity. (...) According to a discovery of transcendental phenomenology, every single thing contains in itself "a continuum of appearances," which exhibits an "all-sided infinity."2From this an important consequence can be drawn: although the world as a physical .. (shrink)
I have argued for a picture of decision theory centred on the principle of Rationally Negligible Probabilities. Isaacs argues against this picture on the grounds that it has an untenable implication. I first examine whether my view really has this implication; this involves a discussion of the legitimacy or otherwise of infinite decisions. I then examine whether the implication is really undesirable and conclude that it is not.
This paper uses a schema for infinite regress arguments to provide a solution to the problem of the infinite regress of justification. The solution turns on the falsity of two claims: that a belief is justified only if some belief is a reason for it, and that the reason relation is transitive.
In this article the author attempts to establish whether we can find a “theory of appearance” in the philosophy of Jan Patočka. The “appearance” for Patočka is basically composed of two elements. First there is a “primeval movement” which accounts for an infinite possibility of phenomena. The second element is the relation of this movement with an “addressee”, the subjectivity. If we begin to analyse the unity of these two elements we fundamentally come across three problems: what is it (...) that appears, when appearance presupposes a certain totality of appearance; how does this total appearance come forth; and, finally, is this whole “structure of appearance” taken as a free movement, kept once and for all within the boundaries of phenomenology, which is founded on a precise and positive term of “appearance” — or do we have to stipulate a special “experience” as the starting point of a phenomenology, which accepts the abyssal impossibility to control its frame? (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)
In Physics, Aristotle starts his positive account of the infinite by raising a problem: “[I]f one supposes it not to exist, many impossible things result, and equally if one supposes it to exist.” His views on time, extended magnitudes, and number imply that there must be some sense in which the infinite exists, for he holds that time has no beginning or end, magnitudes are infinitely divisible, and there is no highest number. In Aristotle's view, a plurality cannot (...) escape having bounds if all of its members exist at once. Two interesting, and contrasting, interpretations of Aristotle's account can be found in the work of Jaako Hintikka and of Jonathan Lear. Hintikka tries to explain the sense in which the infinite is actually, and the sense in which its being is like the being of a day or a contest. Lear focuses on the sense in which the infinite is only potential, and emphasizes that an infinite, unlike a day or a contest, is always incomplete. (shrink)
I describe the general structure of most infinite regress arguments; introduce some basic vocabulary; present a working hypothesis of the nature and derivation of an infinite regress; apply this working hypothesis to various infinite regress arguments to explain why they fail to entail an infinite regress; describe a common mistake in attempting to derive certain infinite regresses; and examine how infinite regresses function as a premise.
Paradoxes of the Infinite presents one of the most insightful, yet strangely unacknowledged, mathematical treatises of the 19 th century: Dr Bernard Bolzano’s Paradoxien . This volume contains an adept translation of the work itself by Donald A. Steele S.J., and in addition an historical introduction, which includes a brief biography as well as an evaluation of Bolzano the mathematician, logician and physicist.