Aggregative consequentialism and several other popular moral theories are threatened with paralysis: when coupled with some plausible assumptions, they seem to imply that it is always ethically indifferent what you do. Modern cosmology teaches that the world might well contain an infinite number of happy and sad people and other candidate value-bearing locations. Aggregative ethics implies that such a world contains an infinite amount of positive value and an infinite amount of negative value. You can affect only a finite amount (...) of good or bad. In standard cardinal arithmetic, an infinite quantity is unchanged by the addition or subtraction of any finite quantity. So it appears you cannot change the value of the world. Modifications of aggregationism aimed at resolving the paralysis are only partially effective and cause severe side effects, including problems of “fanaticism”, “distortion”, and erosion of the intuitions that originally motivated the theory. Is the infinitarian challenge fatal? (shrink)

Isn't probability 1 certainty? If the probability is objective, so is the certainty: whatever has chance 1 of occurring is certain to occur. Equivalently, whatever has chance 0 of occurring is certain not to occur. If the probability is subjective, so is the certainty: if you give credence 1 to an event, you are certain that it will occur. Equivalently, if you give credence 0 to an event, you are certain that it will not occur. And so on for other (...) kinds of probability, such as evidential probability. The formal analogue of this picture is the regularity constraint: a probability distribution over sets of possibilities is regular just in case it assigns probability 0 only to the null set, and therefore probability 1 only to the set of all possibilities. (shrink)

Williams and J. Fraser have recently argued that effective field theory methods enable scientific realists to make more reliable ontological commitments in quantum field theory than those commonly made. In this paper, I show that the interpretative relevance of these methods extends beyond the specific context of QFT by identifying common structural features shared by effective theories across physics. In particular, I argue that effective theories are best characterized by the fact that they contain intrinsic empirical limitations, and I extract (...) from their structure one central interpretative constraint for making more reliable ontological commitments in different subfields of physics. While this is in principle good news, this constraint still raises a challenge for scientific realists in some contexts, and I bring the point home by focusing on Williams’s and J. Fraser’s defense of selective realism in QFT. (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)

By “Brentanian inner consciousness” I mean the conception of inner consciousness developed by Franz Brentano. The aim of this paper is threefold: first, to present Brentano’s account of inner consciousness; second, to discuss this account in light of the mereology outlined by Brentano himself; and third, to decide whether this account incurs an infinite regress. In this regard, I distinguish two kinds of infinite regress: external infinite regress and internal infinite regress. I contend that the most plausible reading of Brentano’s (...) account is the so-called fusion thesis, and I argue that internal infinite regress turns out to be inherent to Brentanian inner consciousness. (shrink)

Popper functions allow one to take conditional probabilities as primitive instead of deriving them from unconditional probabilities via the ratio formula P=P/P. A major advantage of this approach is it allows one to condition on events of zero probability. I will show that under plausible symmetry conditions, Popper functions often fail to do what they were supposed to do. For instance, suppose we want to define the Popper function for an isometrically invariant case in two dimensions and hence require the (...) Popper function to be rotationally invariant and defined on pairs of sets from some algebra that contains at least all countable subsets. Then it turns out that the Popper function trivializes for all finite sets: P=1 for all A ) if B is finite. Likewise, Popper functions invariant under all sequence reflections can’t be defined in a way that models a bidirectionally infinite sequence of independent coin tosses. (shrink)

Abstractabstract:Kant's treatment of infinity seems to be plagued by two contradictions. First, the Transcendental Aesthetic claims that space is an infinite given magnitude, whereas the First Antinomy argues that the spatial world cannot be infinite. Second, the Transcendental Aesthetic claims that the representation of infinite space belongs to sensibility, but the third Critique seems to argue, instead, that infinity is an Idea of reason. This paper resolves these apparent contradictions by noting that Kant groups his various conceptions of space into (...) three kinds: (1) merely subjectively given space, (2) objectively given space, and (3) objective space as a mere Idea. Attending to these three conceptions of space illustrates that the Transcendental Aesthetic, First Antinomy, and third Critique refer to different conceptions of infinite space and thus do not contradict one another, illuminating the importance of Kant's various conceptions of space for his critical project. (shrink)

In this paper, we study the relationship between the cofinalityc(Sym(ω)) of the infinite symmetric group and the minimal cardinality $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{b} $$ of an unbounded familyF of ω ω.

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)

Although Peano’s negative attitude toward infinitesimals—particularly, geometric infinitesimals—is widely documented, his conception of a single infinite cardinality and, more generally, his views on the infinite are less known. In this article, we reconstruct the evolution of Peano’s ideas on these questions and formulate several hypotheses about their underlying motivations.

The singularity arising from the violation of the Lipschitz condition in the simple Newtonian system proposed recently by Norton (2003) is so fragile as to be completely and irreparably destroyed by slightly relaxing certain (infinite) idealizations pertaining to elastic phenomena in this model. I demonstrate that this is also true for several other Lipschitz-indeterministic systems, which, unlike Norton's example, have no surface curvature singularities. As a result, indeterminism in these systems should rather be viewed as an artefact of certain infinite (...) idealizations essential for these models, depriving them of much of their intended metaphysical import. (shrink)

According to Friedrich Schleiermacher, religiosity is rooted in feeling (Gefühl). As a result of our engagement with the world, on which we depend and which we can influence, we have both a sense of dependence and of freedom. Schleiermacher speculated that a sense of absolute dependence in reflective beings with self-consciousness (human beings) gave rise to religion. Using insights from contemporary philosophy of biology and cognitive science, I seek to naturalize Schleiermacher's ideas. I moreover show that this naturalization is in (...) line with Schleiermacher's outlook on biology, as he already had evolutionary considerations in mind when he wrote the Christian Faith (1830). While Schleiermacher rejects natural theology in a narrow sense (proofs for the existence of God), his project is natural theological in a broader sense, as it roots religion in experiences that we can examine using naturalistic theories. (shrink)

We examine the Dual Ramsey Theorem and two related combinatorial principles VW(k,l) and OVW(k,l) from the perspectives of reverse mathematics and effective mathematics. We give a statement of the Dual Ramsey Theorem for open colorings in second order arithmetic and formalize work of Carlson and Simpson [1] to show that this statement implies ACA 0 over RCA 0 . We show that neither VW(2,2) nor OVW(2,2) is provable in WKL 0 . These results give partial answers to questions posed by (...) Friedman and Simpson [3]. (shrink)

This paper offers a defense of Charles Parsons’ appeal to mathematical intuition as a fundamental factor in solving Benacerraf’s problem for a non-eliminative structuralist version of Platonism. The literature is replete with challenges to his well-known argument that mathematical intuition justifies our knowledge of the infinitude of the natural numbers, in particular his demonstration that any member of a Hilbertian stroke string ω-sequence has a successor. On Parsons’ Kantian approach, this amounts to demonstrating that for an “arbitrary” or “vaguely represented” (...) string of strokes, we can always “add” one more stroke. Critics have contested the cogency of a notion of an arbitrary object, our capacity to vaguely represent a definite object, and the role of spatial and temporal representation in the demonstration that we can “add” one more. The bulk of this paper is devoted to demonstrating how to meet all extant criticisms of his key argument. Critics have also suggested that Parsons’ whole approach is misbegotten because the appeal to mathematical intuition inevitably falls short of providing a complete solution to Benacerraf’s problem. Since the natural numbers are essentially and exclusively characterized by their structural properties, they cannot be identified with any particular model of arithmetic, and thus a notion of intuition will fail to capture the universality of arithmetic, its applicability to all entities. This paper also explains why we should not reject appeal to mathematical intuition even though it is not itself sufficient to fully “close the gap” on Benacerraf’s challenge. (shrink)

This paper offers an analysis of Kant’s account of the mathematical sublime with reference to his claim that ‘Nature is thus sublime in those of its appearances the intuition of which brings with them the idea of its infinity’. In undertaking this analysis I challenge Paul Crowther’s interpretation of this species of aesthetic experience, and I reject his interpretation as not being reflective of Kant’s actual position. I go on to show that the experience of the mathematical sublime is necessarily (...) connected with the progression of the imagination in its move towards the infinite. View HTML Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.Kant’s Mathematical Sublime and the Role of the Infinite: Reply to CrowtherVolume 20, Issue 1Simon D. Smith DOI: https://doi.org/10.1017/S1369415414000302Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. Kant’s Mathematical Sublime and the Role of the Infinite: Reply to CrowtherVolume 20, Issue 1Simon D. Smith DOI: https://doi.org/10.1017/S1369415414000302Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. Kant’s Mathematical Sublime and the Role of the Infinite: Reply to CrowtherVolume 20, Issue 1Simon D. Smith DOI: https://doi.org/10.1017/S1369415414000302Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation. (shrink)

Much recent academic literature on the afterlife has been focused on the justice of eternity and whether a good God could allow a person to experience eternal suffering in Hell. Two primary escapes are typically suggested to justify never-ending punishment for sinners: the traditional view focuses the blame for an individual’s condemnation away from God onto the sinner’s freely chosen actions; the universalist position denies the eternality of the punishment on the grounds that God’s inescapable love and eventual victory over (...) evil will bring all souls into His presence. I propose a third option that hinges on the possibility of Heaven itself being experienced as eternal punishment to demonstrate that if God’s presence is both the blessedness of Heaven for some and the agony of Hell for others, then the biblical affirmation of the universal restoration of all with the eternal punishment of some need not remain paradoxical. (shrink)

Loveland and Meyer have studied necessary and sufficient conditions for an infinite binary string x to be recursive in terms of the programsize complexity relative to n of its n-bit prefixes xn. Meyer has shown that x is recursive iff ∃c, ∀n, K(xn/n) ≤ c, and Loveland has shown that this is false if one merely stipulates that K(xn/n) ≤ c for infinitely..

One of the principal lines of argument deployed by the friends of the kalām cosmological argument against the possibility of a beginningless series of events is a quite general argument against the possibility of an actual infinite. The principal thesis of the present paper is that if this argument worked as advertised, parallel considerations would force us to conclude, not merely that a series of discrete, successive events must have a first member, but also that such a series must have (...) a final member. Anyone who thinks that an endless series of events is possible must therefore reject this popular line of argument against the possibility of an actual infinite. (shrink)

Suppose you found that the universe around you was infinite—that it extended infinitely far in space or in time and, as a result, contained infinitely many persons. How should this change your moral decision-making? Radically, it seems, according to some philosophers. According to various recent arguments, any moral theory that is ’minimally aggregative’ will deliver absurd judgements in practice if the universe is (even remotely likely to be) infinite. This seems like sound justification for abandoning any such theory. -/- My (...) goal in this thesis is simple: to demonstrate that we need not abandon minimally aggregative theories, even if we happen to live in an infinite universe. I develop and motivate an extension of such theories, which delivers plausible judgements in a range of realistic cases. I show that this extended theory can overcome key objections—both old and new—and that it succeeds where other proposals do not. With this proposal in hand, we can indeed retain minimally aggregative theories and continue to make moral decisions based on what will promote the good. (shrink)

A solution of the sourceless Einstein's equation with an infinite value for the cosmological constant Λ is discussed by using Inönü–Wigner contractions of the de Sitter groups and spaces. When Λ→∞, spacetime becomes a four-dimensional cone, dual to Minkowski space by a spacetime inversion. This inversion relates the four-cone vertex to the infinity of Minkowski space, and the four-cone infinity to the Minkowski light-cone. The non-relativistic limit c→∞ is further considered, the kinematical group in this case being a modified Galilei (...) group in which the space and time translations are replaced by the non-relativistic limits of the corresponding proper conformal transformations. This group presents the same abstract Lie algebra as the Galilei group and can be named the conformal Galilei group. The results may be of interest to the early Universe Cosmology. (shrink)