This article is about the ontological dispute between finitists, who claim that only finitely many numbers exist, and infinitists, who claim that infinitely many numbers exist. Van Bendegem set out to solve the 'general problem' for finitism: how can one recast finite fragments of classical mathematics in finitist terms? To solve this problem Van Bendegem comes up with a new brand of finitism, namely so-called 'apophatic finitism'. In this article it will be argued that apophatic (...) finitism is unable to represent the negative ontological commitments of infinitism or, in other words, that which does not exist according to infinitism. However, there is a brand of infinitism, so-called 'apophatic infinitism', that is able to represent both the positive and the negative ontological commitments of apophatic finitism. Unfortunately, apophatic finitism cannot adopt that way without losing the ability to represent the positive ontological commitments of infinitism. (shrink)

Many philosophers have argued that the past must be finite in duration because otherwise reaching the present moment would have involved something impossible, namely, the sequential occurrence of an actual infinity of events. In reply, some philosophers have objected that there can be nothing amiss in such an occurrence, since actually infinite sequences are ‘traversed’ all the time in nature, for example, whenever an object moves from one location in space to another. This essay focuses on one of the two (...) available replies to this objection, namely, the claim that actual infinities are not traversed in nature because space, time, and other continuous wholes divide into parts only in so far as we divide them in thought, and thus divide into only a finite number of parts. I grant that this reply succeeds in blunting the anti-finitist objection, but I argue that it also subverts the very argument against an eternal past that it was intended to save. (shrink)

Some philosophers contend that the past must be finite in duration, because otherwise reaching the present would have involved the sequential occurrence of an actual infinity of events, which they regard as impossible. I recently developed a new objection to this finitist argument, to which Andrew Ter Ern Loke and Travis Dumsday have replied. Here I respond to the three main points raised in their replies.

All or nothing? Nature in Chinese thought and the apophatic occident -- Nothing and the poetic making of sense -- Immanence: the last word? -- Universalism, or the nothing that is all -- An extra word on originality -- Intercultural dia-logue and its apophatic interstices -- Analytic table of contents.

This pioneering book demonstrates the crucial importance of Wittgenstein's philosophy of mathematics to his philosophy as a whole. Marion traces the development of Wittgenstein's thinking in the context of the mathematical and philosophical work of the times, to make coherent sense of ideas that have too often been misunderstood because they have been presented in a disjointed and incomplete way. In particular, he illuminates the work of the neglected 'transitional period' between the Tractatus and the Investigations.

In this paper a theory of finitistic and frequentistic approximations — in short: f-approximations — of probability measures P over a countably infinite outcome space N is developed. The family of subsets of N for which f-approximations converge to a frequency limit forms a pre-Dynkin system $${{D\subseteq\wp(N)}}$$. The limiting probability measure over D can always be extended to a probability measure over $${{\wp(N)}}$$, but this measure is not always σ-additive. We conclude that probability measures can be regarded as idealizations of (...) limiting frequencies if and only if σ-additivity is not assumed as a necessary axiom for probabilities. We prove that σ-additive probability measures can be characterized in terms of so-called canonical and in terms of so-called full f-approximations. We also show that every non-σ-additive probability measure is f-approximable, though neither canonically nor fully f-approximable. Finally, we transfer our results to probability measures on open or closed formulas of first-order languages. (shrink)

Following the finitist’s rejection of the complete totality of the natural numbers, a finitist language allows only propositional connectives and bounded quantifiers in the formula-construction but not unbounded quantifiers. This is opposed to the currently standard framework, a first-order language. We conduct axiomatic studies on the notion of truth in the framework of finitist arithmetic in which at least smash function $\#$ is available. We propose finitist variants of Tarski ramified truth theories up to rank $\omega $, of Kripke–Feferman truth (...) theory and of Friedman–Sheard truth theory, and show that all of these have the same strength as the finitist arithmetic of one higher level along Grzegorczyk hierarchy. On the other hand, we also show that adding Burgess-style groundedness schema, adjusted to the finitist setting, makes Kripke–Feferman truth theory as strong as primitive recursive arithmetic. Meanwhile, we obtain some basic results on finitist theories of (full and hat) inductive definitions and on the second order axiom of hat inductive definitions for positive operators. (shrink)

It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments considered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle (...) has good reason to resist the traditional arguments offered in favour of the existence of the infinite and that, while there is a lacuna in his own ‘logical’ arguments against actual infinities, his arguments against the existence of infinite magnitude and number are valid and more well grounded than commonly supposed. (shrink)

The ancient doctrine of negative theology or apophasis--the attempt to describe God by speaking only of what cannot be said about the divine perfection and goodness--has taken on new life in the concern with language and its limits that preoccupies much postmodern philosophy, theology, and related disciplines. How does this mystical tradition intersect with the concern with material bodies that is simultaneously a focus in these areas? This volume pursues the unlikely conjunction of apophasis and the body, not for the (...) cachet of the "cutting edge" but rather out of an ethical passion for the integrity of all creaturely bodies as they are caught up in various ideological mechanisms--religious, theological, political, economic--that threaten their dignity and material well-being. The contributors, a diverse collection of scholars in theology, philosophy, history, and biblical studies, rethink the relationship between the concrete tradition of negative theology and apophatic discourses widely construed. They further endeavor to link these to the theological theme of incarnation and more general issues of embodiment, sexuality, and cosmology. Along the way, they engage and deploy the resources of contextual and liberation theology, post-structuralism, postcolonialism, process thought, and feminism. The result not only recasts the nature and possibilities of theological discourse but explores the possibilities of academic discussion across and beyond disciplines in concrete engagement with the well-being of bodies, both organic and inorganic. The volume interrogates the complex capacities of religious discourse both to threaten and positively to draw upon the material well-being of creation. (shrink)

Call an argument a ‘happy sorites’ if it is a sorites argument with true premises and a false conclusion. It is a striking fact that although most philosophers working on the sorites paradox find it at prima facie highly compelling that the premises of the sorites paradox are true and its conclusion false, few (if any) of the standard theories on the issue ultimately allow for happy sorites arguments. There is one philosophical view, however, that appears to allow for at (...) least some happy sorites arguments: strict finitism in the philosophy of mathematics. My aim in this paper is to explore to what extent this appearance is accurate. As we shall see, this question is far from trivial. In particular, I will discuss two arguments that threaten to show that strict finitism cannot consistently accept happy sorites arguments, but I will argue that (given reasonable assumptions on strict finitistic logic) these arguments can ultimately be avoided, and the view can indeed allow for happy sorites arguments. (shrink)

An expression apophatic philosophy can be understood as an appropriate synonym for a more traditional expression apophatic theology. Traditional philosophical views on the mystery of God created besides its mere rational reflection also thought which is over-rational but definitely not antirational. It can be found in texts in the field of mysticism, both religious and philosophical. Classical Greek culture joined with Christian faith. Therefore, we cannot talk about it as an individual entity being separated by these two worlds. (...) Athens can be recognized in Jerusalem – to use expression of Leo Shestov. A symbol joins the mind. Can anybody still follow its directness? Slovenian philosophical field has developed specific understanding for Christian mystical tradition in its high theoretical expression. Most credit can be given to the thought of Gorazd Kocijančič (born 1964), a philosopher, poet, translator and publisher since it confirms axiomatics of the reality of a spiritual world, which does not correspond to any other reality. Radical denial or negation (apo-fasko) as a modus vivendi of philosophy represents at the same time a relationship of a radical openness between mysteries of God and human being. This contribution will mostly focus on Kocijančič’s synthesis from his philosophical trilogy: Razbitje: Sedem radikalnih esejev [Being broken apart: seven radical essays] (2009), Erotika, politika itn. Trije poskusi o duši [Erotics, politics etc.: three essays on the soul] (2011), O nekaterih drugih: Štirje eseji o preobilju [On some others: four essays on superabundance] (2016) and mainly on his relationship towards phenomenological thinking. As he stresses in his preface to his translation of Levinas’ Le temps et l’autre, apophatics goes beyond the phenomenology. (shrink)

The concept of the (full) unfolding of a schematic system is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted ? The program to determine for various systems of foundational significance was previously carried out for a system of nonfinitist arithmetic, ; it was shown that is proof-theoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system (...) of finitist arithmetic, , and for an extension of that by a form of the so-called Bar Rule. It is shown that and are proof-theoretically equivalent, respectively, to Primitive Recursive Arithmetic, , and to Peano Arithmetic,. (shrink)

Dummett's objections to the coherence of the strict finitist philosophy of mathematics are thus, at the present time at least, ill-taken. We have so far no definitive treatment of Sorites paradoxes; so no conclusive ground for dismissing Dummett's response — the response of simply writing off a large class of familiar, confidently handled expressions as semantically incoherent. I believe that cannot be the right response, if only because it threatens to open an unacceptable gulf between the insight into his own (...) understanding available to a philosophically reflective speaker and the conclusions available to one confined to observing the former's linguistic practice; for an observer of our linguistic practice could never justifiably arrive at the conclusion that ‘red’, ‘child’, etc., are governed by inconsistent rules. But the Sorites is not the subject of this paper. The points I hope to have made plausible are: that a generalized intuitionist position cannot be so much as formulated and that even a most local intuitionism, argued for the special case of arithmetic, is hard pressed effectively to stabilize and defend itself; that strict finitism remains the natural outcome of the anti-realism which Dummett has propounded by way of support for the intuitionist philosophy of mathematics; that it is powerfully buttressed by the ideas of the latter Wittgenstein on rule-following; and that there is no extant compelling reason to suppose that its involvement with predicates of surveyability calls its coherence into question. The correct philosophical assessment of strict finitism, and its proper mathematical exegesis, remain absolutely open, almost virgin issues. This is not a situation which philosophers of mathematics should tolerate very much longer. (shrink)

There is a device used particularly in Orthodox Christian theology known as apophatic theology. In this God is spoken of only in 'negating concepts' to emphasize the inability of language to adequately describe the nature of deity. My question is whether there is any way in which this concept, used as it is to underline the 'otherness' of a transcendental god, can be applied to a thealogy of Goddess. This'way of negation' figures prominently in mystical theology, where it is (...) often associated with the 'way of union'. Historically, many mystics believed in the possibility of a true union with God achieved through a vision of divine light. This light leads us towards Sophia, or Wisdom, arguably an ancient goddess predating the Jewish Yhwh. Thus theology and thealogy can be reconciled in this concept of light traced back to its pagan beginnings and apophasty can be applied to an 'intransitive' goddess such as Nisaba-Nidaba. (shrink)

This book intends to show that radical naturalism, nominalism and strict finitism account for the applications of classical mathematics in current scientific theories. The applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry. The fact that so much applied mathematics can be developed within such a weak, strictly finitistic system, is surprising in itself. It also shows that the applications of those classical theories to (...) the finite physical world can be translated into the applications of strict finitism, which demonstrates the applicability of those classical theories without assuming the literal truth of those theories or the reality of infinity. Both professional researchers and students of philosophy of mathematics will benefit greatly from reading this book. (shrink)

In his paper "Finitism", W.W. Tait maintains that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argues that all finitist reasoning is essentially primitive recursive. In this paper, we attempt to show that his thesis "The finitist functions are precisely the primitive recursive functions" is disputable and that another, likewise defended by (...) him, is untenable. The second thesis is that the finitist theorems are precisely the universal closures of the equations that can be proved in PRA. /// En su articulo "Finitism", W.W. Tait sostiene que la dificultad principal para quien quiere comprender la concepción hilbertiana de la matemática finitista es ésta: especificar el sentido de la demostrabilidad de enunciados generales sobre los números naturales sin presuponer totalidades infinitas. Además, Tait argumenta que todo razonamiento finitista es esencialmente primitivo recursivo. En este artículo tratamos de mostrar que su tesis "Las funciones finitistas son precisamente las funciones primitivas recursivas" es discutible y que otra, también defendida por él, resulta insostenible. La segunda tesis es que los teoremas finitistas son precisamente las clausuras universales de las ecuaciones que pueden demostrarse en PRA. (shrink)

Across a variety of religious and philosophical traditions, it is common to think that it is possible that God defies all description. This presents a problem, however, as the claim that God defies all description itself appears to describe God. Drawing on multiple religious and philosophical traditions, this paper proposes an addition to the pragmatic stock of approaches to this problem. The proposal is that apophatic utterances are best interpreted—at least in the first instance—as invitations to engage the world (...) aesthetically and creatively, as an act of faith. Their goal is principally to motivate us to act in ways that will allow us to appreciate the extraordinary or divine, rather than to, say, believe that some proposition regarding the extraordinary or divine is true. (shrink)

In his paper `Finitism', W.W.~Tait maintained that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argued that all finitist reasoning is essentially primitive recursive. In our paper, we attempt to show that his thesis ``The finitist functions are precisely the primitive recursive functions'' is disputable and that another, likewise defended by him, (...) is untenable. The second thesis is that the finitist theorems are precisely those $\Pi^0_1$-sentences that can be proved in. (shrink)

Frege’s theory is inconsistent. However, the predicative version of Frege’s system is consistent. This was proved by Richard Heck in 1996 using a model-theoretic argument. In this paper, we give a finitistic proof of this consistency result. As a consequence, Heck’s predicative theory is rather weak. We also prove the finitistic consistency of the extension of Heck’s theory to $\Delta^{1}_{1}$-comprehension and of Heck’s ramified predicative second-order system.

Puryear develops an objection against a prominent attempt to show that the universe must have a temporal beginning. Here I formulate a reply.

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)

This essay explores the potential for eco-theology as a part of the (Christian) theological tradition that, so far, has only been analyzed to a limited extent with regard to what it might contribute to forms of theology that further more sustainable forms of humankind’s (co-)inhabitation of the world: the tradition of apophatic theology. The question is: ‘can dimensions of the apophatic tradition be identified that can contribute to the development of eco-theology in the Christian tradition by informing the (...) shaping of the relationship between humankind and the rest of creation?’ The question will be answered affirmatively, for which this essay takes the following steps. First, the conditions under which apophatic approaches can inform the relationship between humankind and the rest of creation are considered. Second, a recent discussion of apophatic theology by Rowan Williams is used as a point of departure to discuss apophatic theology, epistemology and representation. Third, the theology of icons, expressive as it is by forms of relational and ‘decentered’ observation, that can be understood as being closely connected with apophatic concerns, is considered as a model for humankind’s relating to the rest of creation. (shrink)

This article introduces finitist set theory (FST) and shows how it can be applied in modeling finite nested structures. Mereology is a straightforward foundation for transitive chains of part-whole relations between individuals but is incapable of modeling antitransitive chains. Traditional set theories are capable of modeling transitive and antitransitive chains of relations, but due to their function as foundations of mathematics they come with features that make them unnecessarily difficult in modeling finite structures. FST has been designed to function as (...) a practical tool in modeling transitive and antitransitive chains of relations without suffering from difficulties of traditional set theories, and a major portion of the functionality of discrete mereology can be incorporated in FST. This makes FST a viable collection theory in ontological modeling. (shrink)

In this paper, building on recent and longstanding work (Warren 2001, Friedman 2013, Glezer 2018), I investigate how the account of the essences or natures of material substances in the Metaphysical Foundations is related to Kant’s demand for the completeness of the system of nature. We must ascribe causal powers to material substances for the properties of those substances to be observable and knowable. But defining those causal powers requires admitting laws of nature, taken as axioms or principles of natural (...) science, which govern anything that can be constructed mathematically or “given as an object of experience” (MAN, 4:474–75). Presenting a complete system of nature requires adumbrating the properties of material substances that can be objects of experience. But it is not possible to show how those properties are involved in explanations of the phenomena without involving the laws that govern interactions between material substances. For Kant, or so I will argue, it is not possible to account for the natures of things, or for the features of laws of nature, independently of each other. The paper will substantiate the Finitist Account (FA) of Kant on laws. The Finitist Account has it that modal judgments about the possible proofs that can be made, or interactions that can be explained, are (1) based on concrete intuitive reasoning and motivated by the desire to avoid appeal to the actual infinite, (2) grounded in finite decision procedures, which are (3) based on reliable systems of axioms or rules of inference. (shrink)

For Martin Heidegger the story of Western philosophy ended basically in egocentrism or the metaphysics of “subjectivity”; however, he acknowledged the possibility of another path in Greece: that of pre-Socratic thinking. Yet, there is a further path he did not acknowledge: the tradition of Orthodox philosophy and theology. The paper focuses on some key works of the prominent contemporary Greek philosopher Christos Yannaras, for a long time professor in Athens. Taking over the notions of “Being” and ontology, Yannaras construes them (...) not as ontic “substances” amenable to epistemic knowledge, but as guideposts to “relational” or participatory experience. His early text On the Absence and Unknowability of God: Heidegger and the Aeropagite explores the affinity between the German thinker and the Orthodox stress on “apophaticism.” For Yannaras, apophaticism profoundly reorients philosophical inquiry; it also has important implications for human “personhood” and “freedom.” Far from denoting individuation, personhood for him means a face standing out toward others, just as freedom transcends self-will in the direction of a relational event. By way of conclusion, I probe the status of relational “community.” For Yannaras, I ask, is community a concretely achieved way of life ; or does it point more apophatically to a future advent? (shrink)

In his paper ‘Wang’s Paradox’, Michael Dummett provides an argument for why strict finitism in mathematics is internally inconsistent and therefore an untenable position. Dummett’s argument proceeds by making two claims: (1) Strict finitism is committed to the claim that there are sets of natural numbers which are closed under the successor operation but nonetheless have an upper bound; (2) Such a commitment is inconsistent, even by finitistic standards. -/- In this paper I claim that Dummett’s argument fails. (...) I question both parts of Dummett’s argument, but most importantly I claim that Dummett’s argument in favour of the second claim crucially relies on an implicit assumption that Dummett does not acknowledge and that the strict finitist need not accept. (shrink)

Plato’s discourse on beauty in the Hippias Major and the Symposium is distinctly apophatic in nature. Plato describes beauty in terms of what it is not (an approach sometimes referred to apophasis, or the via negativa). In this paper, I argue that Platonic apophatic practise in the Hippias Major and the Symposium depicts beauty as an ally to certain aspirations of philosophical discourse. In the first section, I offer some brief prefatory remarks on the nature of apophasis and (...) its presence in Plato’s thinking. In the second section, I provide some background to the dialogue of the Hippias Major and highlight the apophatic nature of the descriptions of beauty offered therein. In the third and final section, I discuss the Symposium, a dialogue within which, in addition to representing beauty apophatically, Plato illustrates how we may become able to achieve some positive, or cataphatic, insights into the nature of beauty. I conclude that the purpose of the Platonic apophatic portrayal of beauty in the Hippias Major and the Symposium is to show that, while beauty cannot be reduced to a series of assertive truth statements, it can nevertheless function as an accessory to other dimensions of philosophical thinking. (shrink)

It can be argued that only the equational theories of some sub-elementary function algebras are finitistic or intuitive according to a certain interpretation of Hilbert's conception of intuition. The purpose of this paper is to investigate the relation of those restricted forms of equational reasoning to classical quantifier logic in arithmetic. The conclusion reached is that Edward Nelson's ‘predicative arithmetic’ program, which makes essential use of classical quantifier logic, cannot be justified finitistically and thus requires a different philosophical foundation, possibly (...) as a restricted form of logicism. (shrink)

I argue that Christian apophaticism provides the most powerful and economical response to the evidential argument from evil for the non-existence of God. I also reply to the objection that Christian apophaticism is incoherent, because it appears to entail the truth of the following contradiction: it is both possible and impossible to know God’s essential properties. To meet this objection, I outline a coherent account of the divine attributes inspired by the theology of the Greek Father’s and St. Gregory Palamas.

I investigate Jean-Luc Marion's phenomenology of love and its relation to ethics. I argue that his phenomenology of love provides a possibility for developing ethics. I rely on the saturated phenomenon of the icon and his phenomenology of love. I establish that the icon provides a rich sense of relation, a need to modify certain appraisals of justice, and provides a descriptive account of the virtue of receptivity. In chapter one, I give an exegesis of Marion's phenomenology of the icon. (...) He argues that the icon moves toward charity, yet does not relate to ethics. Experience of the icon gives transformed vision that voids universal laws and frees the beholder to experience communion. In chapter two, I provide three examples of iconic experience that emphasize the importance of justice for the icon in contrast to Marion's formulation that justice is equivalent to revenge. I argue that Marion attacks justice because of its link to a deontological definition; yet, freed from deontology, Marion makes room for an apophatic way that moves toward a virtue ethic. In chapter three, I look at Marion's reasons for leaving ethics out of the icon. His critique of Kantian ethics, as well as his use and critique of the Levinas suggest that he opposes modern metaphysical ethics. In discussion of the receptivity to the call of the Other, he relies on the virtues. This reliance shows that Marion has room for a descriptive account of ethics, and that ignoring ethics undermines his overall project. I investigate Marion's claim that the icon opens up to charity by reading the apophatic doctrine of the icon in concert with the erotic reduction. I conclude that Marion’s phenomenology can be viewed as a counter-ethics a way to see phenomenology as a virtue practice, to unify reason and love in ethics, to remove the ego from the central concern of ethics, and to see the need for openness and vulnerability to the iconic other. (shrink)

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)

Theology, particularly negative theology (which maintains that we can know only what God is not), has taken the lead historically in developing reflection on the limits of language and the beyond o...

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. (shrink)

This paper is about the specific character of the aesthetic experience of icons. I am arguing for the idea that the aesthetic experience of icons is a necessary condition of their role and function in Christian worship, and, moreover, that this particular aesthetic experience is of an apophatic kind. My arguments will be developed on the background of the Byzantine iconoclastic debate and the apophatic theology of Pseudo-Dionysius the Areopagite. Also, they should present the very debate from the (...) perspective of aesthetics, often overlooked in favour of more theological or ontological issues related to it. (shrink)

The reason for writing the article was the book of the famous analytical theologian Thomas Morris "Our Idea of God", in which he writes Gregory Nazianzin in the camp of pessimistic theologians and criticizes apophatic theology. In our opinion, his criticism is devoid of solid foundations and apophaticism itself is understood one-sidedly, while with a closer examination, the conflict between rational and mystical reason can be at least noticeably reduced. The following text is devoted to the study of this (...) problem field and the illustration of these theses. (shrink)

This essay examines animality through an analysis of Les Chants de Maldoror, an obscure but influential nineteenth-century text by the Comte de Lautréamont. Drawing upon the work of Gaston Bachelard as well as the apophatic tradition in Christian mysticism, Les Chants de Maldoror can be read as a text that complicates the boundary between animality and spirituality, producing an “apophatic animality” that ultimately impacts the poetics of the text itself.

How can the psychotherapist think about not knowing? Is psychoanalysis a contemplative practice? This book explores the possibility that there are resources in philosophy and theology which can help psychoanalysts and psychotherapists think more clearly about the unknown and the unknowable. The book applies the lens of apophasis to psychoanalysis, providing a detailed reading of apophasis in the work of Pseudo-Dionysius and exploring C.G. Jung's engagement with apophatic discourse. Pseudo-Dionysius brought together Greek and biblical currents of negative theology and (...) the via negativa, and the psychology of Jung can be read as a continuation and extension of the apophatic tradition. Henderson discusses the concept of the transcendent function as an apophatic dynamic at the heart of Jung's thought, and suggests that apophasis can provide the key to understanding the family resemblance among the disparate schools of psychoanalysis. Chapters consider: -Jung's discussion of opposites, including his reception of Nicholas of Cusa's concept of the coincidence of opposites -Jung's engagement with Neoplatonism and Pseudo-Dionysius -the work of Jung in relation to Deleuze, Derrida and other writers -how motifs in Pseudo-Dionysius' Ecclesiastical Hierarchy resonate with contemporary psychoanalytic psychotherapy. The in-depth examination of primary sources in this comprehensive volume provides a platform for research into apophasis in the wider field of psychoanalysis. It will prove valuable reading for scholars and analysts of Jungian psychology studying religion and mysticism. (shrink)

This book intends to show that, in philosophy of mathematics, radical naturalism (or physicalism), nominalism and strict finitism (which does not assume the reality of infinity in any format, not even potential infinity) can account for the applications of classical mathematics in current scientific theories about the finite physical world above the Planck scale. For that purpose, the book develops some significant applied mathematics in strict finitism, which is essentially quantifier-free elementary recursive arithmetic (with real numbers encoded as (...) elementary recursive Cauchy sequences of rational numbers). Applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry (sufficient for the basic applications in classical quantum mechanics and general relativity). The fact that so much applied mathematics can be developed within such a weak, strictly finitisitc system is perhaps surprising in itself. It also shows that the applications of those classical theories to the finite physical world can be translated into the applications of strict finitism, which demonstrates the applicability of those classical theories without assuming the literal truth of those theories or the reality of infinity. The first chapter of the book contains an informal introduction to its philosophical motivation and technical strategy. The book is intended for students and researchers in philosophy of mathematics. -/- Contents -/- Preface 1 Introduction 2 Strict Finitism 3 Calculus 4 Metric Space 5 Complex Analysis 6 Integration 7 Hilbert Space 8 Semi-Riemann Geometry References Index. (shrink)

The unfolding of schematic formal systems is a novel concept which was initiated in Feferman , Gödel ’96, Lecture Notes in Logic, Springer, Berlin, 1996, pp. 3–22). This paper is mainly concerned with the proof-theoretic analysis of various unfolding systems for non-finitist arithmetic . In particular, we examine two restricted unfoldings and , as well as a full unfolding, . The principal results then state: is equivalent to ; is equivalent to ; is equivalent to . Thus is proof-theoretically equivalent (...) to predicative analysis. (shrink)

It is reported that in reply to John Wisdom’s request in 1944 to provide a dictionary entry describing his philosophy, Wittgenstein wrote only one sentence: “He has concerned himself principally with questions about the foundations of mathematics”. However, an understanding of his philosophy of mathematics has long been a desideratum. This was the case, in particular, for the period stretching from the Tractatus Logico-Philosophicus to the so-called transitional phase. Marion’s book represents a giant leap forward in this direction. In the (...) preface, Marion provides a diagnosis for why it has taken such a long time to obtain an accurate picture of Wittgenstein’s ideas in the foundations of mathematics. When the Remarks on the Foundations of Mathematics came out in 1956 the reception by specialists, such as Kreisel, was negative. This led many Wittgenstein scholars to set aside the issues in the foundations of mathematics and go on with the business of the day, focusing on Wittgenstein’s philosophy of language and psychology. However, these early negative appraisals were severely limited by two facts. First of all, the nature of the Remarks was a hindrance to an understanding of what Wittgenstein was up to. The book was edited by Wittgenstein’s literary executors by collating together passages from several manuscripts dating from 1937 to 1944, cutting, however, many passages in between remarks. Second, many of the original manuscripts and typescripts of the transitional phase were not available and thus it was virtually impossible to obtain a balanced picture of Wittgenstein’s development. Finally, a stark contraposition between the early and late Wittgenstein did not encourage people to work seriously on the development of Wittgenstein’s position from the Tractatus through the transitional period to the later writings. (shrink)