Infinite barbarians

This paper discusses an infinite regress that looms behind a certain kind of historical explanation. The movement of one barbarian group is often explained by the movement of others, but those movements in turn call for an explanation. While their explanation can again be the movement of yet another group of barbarians, if this sort of explanation does not stop somewhere we are left with an infinite regress of barbarians. While that regress would be vicious, it cannot be accommodated by several general views about what viciousness in infinite regresses amounts to. This example is additional evidence that we should prefer a pluralist approach to infinite regresses.     

Infinite Reasoning

Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to perform infinite inferences. I argue that we have this ability. My argument looks to our best current theories of inference and considers examples of apparent infinite reasoning. My position is controversial, but if I'm right, our theories of truth, mathematics, and beyond could be transformed. And even if I'm wrong, a more careful consideration of infinite reasoning can only deepen our understanding of thinking and reasoning.     

Boring Infinite Descent

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.     

Avoiding Infinite Masses

The examples of dynamic supertasks analyzed to date in the philosophical literature, in which both determinism and the classical laws of conservation of energy and momentum are violated, all share the important limitation of requiring material systems of infinite mass. This paper demonstrates that this limitation is not necessary. This has important consequences for the scope and meaning of such violations.     

Infinite Prospects*

People with the kind of preferences that give rise to the St. Petersburg paradox are problematic—but not because there is anything wrong with infinite utilities. Rather, such people cannot assign the St. Petersburg gamble any value that any kind of outcome could possibly have. Their preferences also violate an infinitary generalization of Savage's Sure Thing Principle, which we call the Countable Sure Thing Principle, as well as an infinitary generalization of von Neumann and Morgenstern's Independence axiom, which we call Countable Independence. In violating these principles, they display foibles like those of people who deviate from standard expected utility theory in more mundane cases: they choose dominated strategies, pay to avoid information, and reject expert advice. We precisely characterize the preference relations that satisfy Countable Independence in several equivalent ways: a structural constraint on preferences, a representation theorem, and the principle we began with, that every prospect has a value that some outcome could have.     

Leibniz on Infinite Numbers,Infinite Wholes,and Composite Substances

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.     

Infinite Backward Induction Arguments

A large family of paradoxical arguments have been subsumed under the label backward induction arguments. These include the iterated prisonerÕs dilemma, the centipede game, and the surprise test paradox. They are described as backward because they begin by considering a future hypothetical alternative, rule it out, and then rule out each predecessor. Thus they go backward in time ruling out finitely many alternatives. I present examples that go forward in time and eliminate infinitely many alternatives. These pose problems for solutions that focus on common knowledge assumptions.     

Infinite Epistemic Regresses and Internalism

This article seeks to state, first, what traditionally has been assumed must be the case in order for an infinite epistemic regress to arise. It identifies three assumptions. Next it discusses Jeanne Peijnenburg's and David Atkinson's setting up of their argument for the claim that some infinite epistemic regresses can actually be completed and hence that, in addition to foundationalism, coherentism, and infinitism, there is yet another solution (if only a partial one) to the traditional epistemic regress problem. The article argues that Peijnenburg and Atkinson fail to address the traditional regress problem, as they don't adopt all of the three assumptions that underlie the traditional regress problem. It also points to a problem in the notion of making probable that Peijnenburg and Atkinson use in their account of justification.     

The Comically Infinite Man1

A long time ago, I procured a little book edited by Soren Kierkegaard entitled The Sickness Unto Death (1849). What is more, I read it. (I must confess to having been first attracted to it solely by its title). For and as a tribute to Alastair Hannay I was inspired to set down in print this brief (altogether too brief, philosophically speaking) and unsystematic reflection. What struck me most palpably was the suggestion that, although our worldly endeavors and thus our publications are, so to speak, temporally limited, our despair is not. I write on the obligations and privileges of that mood.     

无穷探索与终极答案——从两个有神论兼无神论的小故事看科学与神学的区别

科学承认人类还有很多未知领域,神学才宣称所有问题皆已有答案。科学只给出目前可以验证的答案,对尚验证不了答案的问题继续进行探索;而神学则认为自己给出的是最终的答案,不需要继续探究。用了神学方法,任何动脑筋的事情都可以省了。     

Probability Modals and Infinite Domains

Journal of Philosophical Logic - Recent years have witnessed a proliferation of attempts to apply the mathematical theory of probability to the semantics of natural language probability talk. These...     

An Infinite System with Gravitation

The paper shows a new example of nonuniqueness of the solutionto Newtonian equations of motion for infinite gravitational systems. Unlike otherexamples, the gravitational field presents no singularity, nor are the non-gravitational forcesintroduced in the model singular (in particular, there are no collisions). The result is also ofinterest because it points to an interesting limitation of the elementary (Newtonian) formulationof classical mechanics.     

Arrow Logic and Infinite Counting

We consider arrow logics (i.e., propositional multi-modal logics having three -- a dyadic, a monadic, and a constant -- modal operators) augmented with various kinds of infinite counting modalities, such as 'much more', 'of good quantity', 'many times'. It is shown that the addition of these modal operators to weakly associative arrow logic results in finitely axiomatizable and decidable logics, which fail to have the finite base property.     

Infinite idealization and contextual realism

Synthese - The paper discusses the recent literature on abstraction/idealization in connection with the “paradox of infinite idealization.” We use the case of taking thermodynamics...     

Infinite Populations,Choice and Determinacy

This paper criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow’s theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the Hildenbrand criterion and argue that results based on unrestricted axiom of choice do not meet this criterion. The technically novel part of this paper is a proposal to use a set-theoretic principle known as the axiom of determinacy (\(\mathsf {AD}\)), not as a replacement for Choice, but simply to eliminate applications of set theory violating the Hildenbrand criterion. A particularly appealing aspect of \(\mathsf {AD}\) from the point of view of the research area in question is its game-theoretic character.     

Loke on the Infinite God Objection

In a recent article, Andrew Ter Ern Loke raises several objections to Jacobus Erasmus and Anné Hendrik Verhoef’s exposition and response to the so-called ‘Infinite God Objection’ to the kalām cosmological argument. According to this objection, the argument against the possibility of an actual infinite brings into question the view that God’s knowledge is infinite. Erasmus and Verhoef’s solution to this objection, which Loke criticises, depends on an unusual account of omniscience. In this article, I respond to Loke and show that his objections are unsuccessful.     

Infinite Forcing and the Generic Multiverse

Studia Logica - In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the...     

Infinite Minds: A Philosophical Cosmology

Book Information Infinite Minds: A Philosophical Cosmology. Infinite Minds: A Philosophical Cosmology John Leslie Oxford Clarendon Press 2001 xii + 234 £25 By John Leslie. Clarendon Press. Oxford. Pp. xii + 234. £25,