Hilbert's ‘Verunglückter Beweis’, the first epsilon theorem,and consistency proofs

In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's programme, were working on consistency proofs for arithmetical systems. One proposed method of giving such proofs is Hilbert's epsilon-substitution method. There was, however, a second approach which was not reflected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's first epsilon theorem and a certain ‘general consistency result’ due to Bernays. An analysis of the form of this so-called ‘failed proof’ sheds further light on an interpretation of Hilbert's programme as an instrumentalist enterprise with the aim of showing that whenever a ‘real’ proposition can be proved by ‘ideal’ means, it can also be proved by ‘real’, finitary means.     

The fairness of Hell

The Christian conception of Hell as everlasting punishment for past sins is confronted with two charges of unfairness. The first is the inequity of an eternal punishment. The never‐ending punishment seems disproportionate to the finite sin (Kershnar, Lewis, Adams). A second and related problem is that the boundary between sins that send one for all eternity to Hell and those sins that are slightly less bad that are compatible with an eternity in Heaven is arbitrary and thus it is unfair that sinners so alike are treated differently (Sider). Hell, as traditionally conceived, is then claimed to be incompatible with God's traditional attributes such as his commitment to justice, omniscience and omnipotence. The unfairness can be avoided by appealing to God's foreknowledge and a debt/atonement theory of punishment. My view is analogous to refusing to parole the unrepentant. If a wrongdoer is eternally defiant, then he can never be released from Hell for his debt won't ever be paid if he isn't reformed and reconciled with the wronged. So it doesn't matter that his initial sin was a finite wrong not deserving of infinite punishment nor a sin no worse than that of the penitent in Heaven.     

Extensions of the Finitist Point of View

Hilbert developed his famous finitist point of view in several essays in the 1920s. In this paper, we discuss various extensions of it, with particular emphasis on those suggested by Hilbert and Bernays in Grundlagen der Mathematik (vol. I 1934, vol. II 1939). The paper is in three sections. The first deals with Hilbert's introduction of a restricted ω -rule in his 1931 paper ‘Die Grundlegung der elementaren Zahlenlehre’. The main question we discuss here is whether the finitist (meta-)mathematician would be entitled to accept this rule as a finitary rule of inference. In the second section, we assess the strength of finitist metamathematics in Hilbert and Bernays 1934. The third and final section is devoted to the second volume of Grundlagen der Mathematik. For preparatory reasons, we first discuss Gentzen's proposal of expanding the range of what can be admitted as finitary in his esssay ‘Die Widerspruchsfreiheit der reinen Zahlentheorie’ (1936). As to Hilbert and Bernays 1939, we end on a ‘critical’ note: however considerable the impact of this work may have been on subsequent developments in metamathematics, there can be no doubt that in it the ideals of Hilbert's original finitism have fallen victim to sheer proof-theoretic pragmatism.     

On the difficult case of loving life: Plato's Symposium and Nietzsche's eternal recurrence

A simple but significant historical fact has been overlooked in interpretations of Nietzsche's eternal recurrence. In making eternal recurrence the standard for the affirmation and love of life, Nietzsche accepts an understanding of love developed in Plato's Symposium: love means ‘wanting to possess the good forever’. I argue that Plato develops two distinct types of love, which remain in tension with one another. I then show that a corresponding tension arises in Nietzsche's work when we consider eternal recurrence as the love of life. By making love central in the phrase ‘love of life’, and by allowing Plato's thoughts on love to inform the love of life that Nietzsche expresses in the thought of eternal recurrence, I show that Nietzsche's dramatic presentations of the eternal recurrence do not present us with a test, but in revealing an incompatibility between loving something in life and loving life in its entirety, they present the tragic conflict in the task of life affirmation.     

New Perspectives on an Old Debate: Friedrich Schleiermacher's Essay on Election

Abstract: This article analyses Schleiermacher's essay on election, which has not received the attention it deserves. I argue that it presents a significant advance in the debate about unconditional grace and the relation between redemption and predestination. While Schleiermacher defends the Augustinian‐Calvinist view of election, he criticizes the ideas of double predestination and eternal damnation (for the unbelievers). Thus, he wants to provide a new basis for Lutherans and Calvinists to settle or, at least, reconsider their longstanding controversy over the matter. After the publication of the essay in 1819, Schleiermacher's doctrine of election remained unchanged, except for minor modifications.     

Purity and danger in the solar temple

Based on research conducted in Québec, this study explores the shape of the social life, apocalyptic ideology and authority structure of the Ordre du Temple Solaire (OTS) or Solar Temple within the framework of Mary Douglas's typology of ‘group and grid’. The pollution fears and purity rituals of this controversial new religious movement are analysed as an important factor in their decision to orchestrate a religiously‐motivated mass suicide/homicide, explained in their suicide documents as a ‘transit’ (a magical feat of soul travel) to the Star Sirius. Douglas's insights into how the human body becomes a ‘natural symbol’ for small, persecuted groups, mirroring the social body and the vulnerability of its exits and entrances vis‐à‐vis the surrounding culture, are applied to the alternative patterns of sexuality and parenting in the OTS. It is suggested that the magical aspect of the mass suicide expressed a concern for purity and for protecting the boundaries of their community. It is also suggested that the ritual homicides in Morin Heights resemble the ‘witch‐hunts’ characteristic of Douglas's ‘small society’ that conceives of itself as the perfect, impermeable vessel.     

Escaping Heaven

In response to the problem of Hell, Buckareff and Plug (Relig Stud 41:39–54, 2005; Relig Stud 45:63–72, 2009) have recently proposed and defended an ‘escapist’ conception of Hell. In short, they propose that the problem of Hell does not arise because God places an open-door policy on Hell. In this paper, I expose a fundamental problem with this conception of Hell—namely, that if there’s an open door policy on Hell, then there should be one on Heaven too. I argue that a coherent conception of Heaven cannot have such a policy. Hence, escapism is not an adequate response to the problem of Hell.     

Existential Import in Cartesian Semantics

The paper explores the existential import of universal affirmative in Descartes, Arnauld and Malebranche. Descartes holds, inconsistently, that eternal truths are true even if the subject term is empty but that a proposition with a false idea as subject is false. Malebranche extends Descartes’ truth-conditions for eternal truths, which lack existential import, to all knowledge, allowing only for non-propositional knowledge of contingent existence. Malebranche's rather implausible Neoplatonic semantics is detailed as consisting of three key semantic relations: illumination by which God's ideas cause mental terms, creation by which God's ideas cause material substances by a kind of ‘ontic privation’, and sensation in which brain events occasion states of mental awareness. In contrast, Arnauld distinguishes two types of propositions – necessary and contingent – with distinct truth-conditions, one with and one without existential import. Arnauld's more modern semantics is laid out as a theory of reference that substitutes earlier causal accounts with one that adapts the medieval notion of objective being. His version anticipates modern notions of intentional content and appeals in its ontology only to substances and their modes.     

Why God Lied to Me: Salvationist Theism and Justice

It is widely assumed that God is either incapable of lying to humans or utterly unwilling to do so. However, there appear to be compelling reasons for God to intentionally deceive that are rooted in the traditional conception of God as an agent of salvation for humanity. A terroristic threat like eternal damnation ("hell") illustrates these reasons. God's love for human beings as wayward members of a divine family in concert with the obvious moral and cognitive limitations many humans suffer provide sufficient reason for God to deploy (or allow uncorrected) the threat of eternal damnation. A proper understanding of justice supports the contention that eternal damnation is contrary to justice, and therefore divinely inspired threats of eternal damnation are deceitful.     

From Buddha's speech to Buddha's essence: philosophical discussions of Buddha‐vacana in India and China1

This is a comparative study of the discourses on the nature of sacred language found in Indian Abhidharma texts and those written by 7th century Chinese Buddhist scholars who, unlike the Indian Buddhists, questioned ‘the essence of the Buddha’s teaching'. This issue labeled fo‐chiao t'i lun, the theory of ‘the essence of the Buddha’s teaching', was one of the topics on which Chinese Yogācāra scholars have shown a keen interest and served as the inspiration for extensive intellectual dialogues in their texts. It is in Hsüan‐tsang's massive and organized translation works, begun in 648, that various previous translations of the term buddhavacana from Indian Abhidharma texts were given the unified translation of fo‐chiao. (Fo‐chiao literally means “the Buddha's teachings,” and is the term used in the modern period for “Buddhism.”) By combining fo‐chiao with the term t'i, meaning ‘essence’ or ‘substance’ throughout his translations, Hsüan‐tsang attempted to define ‘the essence of the Buddha’s teaching'. In Indian Abhidharma texts, the nature of the Buddha's word was either ‘sound’ (?abdha), the oral component of speech, or ‘name’ (nāma), the component of language that conveys meaning, or some combination of the two. From the time of Hsüan‐tsang's translation, however, discourse on the nature of sacred language was no longer relegated to the category of language or of epistemological investigation, but became grounded in the Chinese discussion investigating the ‘essence’ or ‘substance’ of the Buddha's teaching, and even of ‘Buddhism’ itself. As such, it sought to transcend the distinction between language and meaning. This gradual but explicit process of inquiry into the nature of ‘the Buddha’s word' was a necessary antecedent to the transition to a ‘Chinese’ Buddhism.     

How to Change the Past in One‐Dimensional Time

The possibility of changing the past by means of time‐travel appears to depend on the possibility of distinguishing the past as it is ‘before’ and ‘after’ the time‐travel. So far, all the metaphysical models that have been proposed to account for the possibility of past‐changing time‐travels operate this distinction by conceiving of time as multi‐dimensional, and thus by significantly inflating our metaphysics of time. The aim of this article is to argue that there is an intuitive sense in which past‐changing time‐travels are metaphysically possible also in one‐dimensional time.     

I. Marx's two‐fold character of labour

Ulrich Steinvorth ('Marx's Analysis of Commodity Exchange’, Inquiry, Vol. 19 [1976]) and C. J. Arthur ('Labour: Marx's Concrete Universal’, Inquiry, Vol. 21 [1978]) rely on the two‐fold character of labour in arguing that the mysteries of money and profit have been correctly interpreted by Marx. However, Marx's own arguments for his distinction between abstract and concrete labour are faulty, as is his identification of labour and material products. They also claim that the exchange of commodities and distribution of resources in capitalist society validate Marx's theory that the determination of value by labour‐time is the ‘secret’ behind capitalist crises. These claims are insufficiently justified, and provide no additional reason for accepting the two‐fold character of labour.     

Bulgakov's Account of Creation: Neglected Aspects,Critics and Contemporary Relevance

The article focuses on a central, yet neglected dimension of the ‘Sophia Debate’ in twentieth‐century Russian Orthodox theology: Bulgakov's panentheistic account of creation and its critique by Nikolai Lossky. Bulgakov understood the doctrine of creation to be negatively defined as creatio ex nihilo and positively defined as creatio ex Deo. Bulgakov's sophiology seeks to relate God and the world through the intermediate concept of Sophia, balancing an account of God's being in the world with an account of the world's eternal foundation in God. Lossky objected that Bulgakov's account underemphasizes novelty, contingency and the free character of creation. Lossky's objections notwithstanding, Bulgakov's version of panentheism – especially its trinitarian, antinomian and kenotic dimensions – finds significant points of contact with contemporary accounts of creation.     

Eschatology: Eternal Now or Cosmic Future?

Paul Tillich's eternal now is the ground from which all things emerge and perish in each and every moment. A Tillichean eschatology involves the gathering of all things finite into the eternity of the present moment, into God. Salvation is present moment. But is the “eternal now” enough? This essay offers biblical and theological critiques of Tillich's present eschatology and posits an eschatology that combines Tillich's “eternal now” with Wolfhart Pannenberg's “end‐oriented eschatology.” The result is an eschatology that recognizes the eternal now in which all things (including all time) belong to God yet with an eye toward the God‐given possibilities of the next moment, the future. The end of being is not cessation; rather, it is the fulfillment of time, the consummation of all things.     

Hegel's notion of Aufheben

The paper is an attempt to make sense of Hegel's notion of aufheben. The double meaning of aufheben and its alleged ‘rise above the mere “either‐or”; of understanding’ have been taken, by some, to constitute a criticism of the logic of either‐or. It is argued, on the contrary, that Hegel's notion of aufheben, explicated in its primary and philosophical context, turns out to be a substantiation of that logic. The intelligibility of the formula of either‐or depends, for example, on the categories of Being and Not‐Being. But if these categories are regarded as particular finite determinations themselves subject to the formula of either‐or, then the formula, far from being intelligible, ‘falls apart’. Hegel is arguing, in other words, that if we are to substantiate the logic of either‐or, we must, at the same time, ‘rise above’ that logic. The role of aufheben is then considered in the special sciences. Here it is argued that we must distinguish between empirical transitions, governed by the finite determinations of things, and logical or dialectical transitions, governed by considerations of the intelligibility of the notions involved. Applying the notion of aufheben to the former transitions suggests wrongly that empirical transitions have an objective or philosophic necessity. Finally, the place of ‘immanent transformation’ in the context of aufheben is examined. It is concluded that if there is to be a transformation, then a distinction must be drawn between thought and its content, but then the transformation cannot be regarded as immanent.     

Heidegger's Generative Thesis

Abstract: For William Blattner, Heidegger's phenomenology fails to demonstrate how a nonsuccessive temporal manifold can ‘generate’ the appropriate sequence of world‐time Nows. Without this he cannot explain the ‘derivative’ status of ordinary time. In this article I show that it is only Blattner's reconstruction that makes failure inevitable. Specifically, Blattner is wrong in the way he sets out the explanatory burden, arguing that the structure of world‐time must meet the traditional requirements of ordinary time logic if the derivation is to succeed. He takes this to mean: mundane ‘tasks’, the contents of world‐time nows, must form a transitive series, importing back into world‐time the very structure that Heidegger says is derived by its levelling‐off. I argue, instead, that world‐time nows, seen at the level of lived content, can be quite ‘irrational’ but this is perfectly consistent with the generative thesis. Adapting Blattner's useful suggestion that temporality is sequence building or ‘iterative’ I show that iteration does not manifest itself at the level of tasks but at the ‘existential’ level of my involvement in a task. Depriving that involvement of its expressive content is what accounts for the levelling‐off of the world‐time now and thus the derivation of the ordinary concept of time.     

Test‐Specificity of the Advantage of Retaking Cognitive Ability Tests

In selection settings, when people retake the same cognitive ability tests, scores are generally positively biased. Our approach aimed to test whether these previous exposure effects are test‐specific or transferable to other tests. We compared the differences between scores for first‐time test takers and retakers for 2 kinds of material: ‘old’ tests, known only to the retakers, and ‘new’ tests, unknown to both groups. The current study used data collected during 2 sessions –S and S+1 – of a selection process for entry into the French national air transport pilot training system, with at least 500 first‐time test takers and 130 retakers in each session. For Session S, on average, retakers scored higher on the ‘old’ tests, but not on the ‘new’ tests. Moreover, the material that was new to retakers at Session S was old at Session S+1, and the finding for old tests could be replicated at Session S+1. The finding that the acquired skills that led to higher scores on old tests were only test‐specific is discussed.