Consistency, Models, and Soundness

This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. E. J. Brouwer and D. Hilbert (1899). Part 2 is mainly devoted to Hilbert’s proof theory of the 1920s (1922–1931). I begin with an account of his early attempt to prove directly, and thus not by reduction or by constructing a model, the consistency of (a fragment of) arithmetic. In subsequent sections, I give a kind of overview of Hilbert’s metamathematics of the 1920s and try to shed light on a number of difficulties to which it gives rise. One serious difficulty that I discuss is the fact, widely ignored in the pertinent literature on Hilbert’s programme, that his language of finitist metamathematics fails to supply the conceptual resources for formulating a consistency statement qua unbounded quantification. Along the way, I shall comment on W. W. Tait’s objection to an interpretation of Hilbert’s finitism by Niebergall and Schirn, on G. Gentzen’s allegedly finitist consistency proof for Peano Arithmetic as well as his ideas on the provability and unprovability of initial cases of transfinite induction in pure number theory. Another topic I deal with is what has come to be known as partial realizations of Hilbert’s programme, chiefly advocated by S. G. Simpson. Towards the end of this essay, I take a critical look at Wittgenstein’s views about (in)consistency and consistency proofs in the period 1929–1933. I argue that both his insouciant attitude towards the emergence of a contradiction in a calculus and his outright repudiation of metamathematical consistency proofs are unwarranted. In particular, I argue that Wittgenstein falls short of making a convincing case against Hilbert’s programme. I conclude with some philosophical remarks on consistency proofs and soundness and raise a question concerning the consistency of analysis.     

Shaping the Enemy: Foundational Labelling by L.E.J. Brouwer and A. Heyting

The use of the three labels (logicism, formalism, intuitionism) to denote the three foundational schools of the early twentieth century are now part of literature. Yet, neither their number nor their adoption has been stable over the twentieth century. They were not introduced by the founding fathers of each school: namely, neither Frege nor Russell spoke of ‘logicism’; and even Hilbert did not use the word ‘formalism’ to introduce his foundational programs. At a certain point, only Brouwer used the label ‘intuitionism’ in his scientific production to personify his philosophy of mathematics and he used the label ‘formalism’ for Hilbert’s foundational viewpoint. Starting with Brouwer, the origin of the use of the three labels to represent a foundational meaning, will be analysed in this paper. Thereafter, the role that Brouwer’s pupil Arend Heyting had in the production and use of foundational labels will be considered. On the basis of the comparison of the attitudes of these two scholars I will finally advance the thesis that not only the creation but also the use of labels, far from being a mere gesture of academic reference to literature, can be a sign of the cultural operation each scholar wanted to do.     

A Theory of Infinitary Relations Extending Zermelo’s Theory of Infinitary Propositions

An idea attributable to Russell serves to extend Zermelo’s theory of systems of infinitely long propositions to infinitary relations. Specifically, relations over a given domain \({\mathfrak{D}}\) of individuals will now be identified with propositions over an auxiliary domain \({\mathfrak{D}^{\mathord{\ast}}}\) subsuming \({\mathfrak{D}}\). Three applications of the resulting theory of infinitary relations are presented. First, it is used to reconstruct Zermelo’s original theory of urelements and sets in a manner that achieves most, if not all, of his early aims. Second, the new account of infinitary relations makes possible a concise characterization of parametric definability with respect to a purely relational structure. Finally, based on his foundational philosophy of the primacy of the infinite, Zermelo rejected Gödel’s First Incompleteness Theorem; it is shown that the new theory of infinitary relations can be brought to bear, positively, in that connection as well.     

The Trinity in Qur’anic Idiom: Q 4.171 and the Christian Arabic Presentation of the Trinity as God,his Word,and his Spirit

ABSTRACT

Since the early centuries of Islam, the Qur’an’s deep imprint on Arabophone Christians has been evident, not only in their evocation of qur’anic language, but also in their creative employment of the text in constructing their own orthodox Christian Arabic theology. This article investigates the presentation of the Trinity as ‘God, his Word, and his Spirit’ in Christian Arabic theological tracts in the early centuries of Islam. It argues that Q 4.171 played a foundational role in constructing a distinct Christian Arabic Trinitarian theology and that Arabophone Christian writers discerned in it the nucleus of what could be developed as an orthodox Trinitarian theology. It traces the development of the Christian Arabic Trinitarian formulation in four works by Arabophone authors: John Damascene’s On Heresies 100; On the Triune Nature of God; the interreligious disputation in the court of the ?Abbāsid Caliph al-Ma?mūn attributed to the theologian Theodore Abū Qurra; and the apologetic letter by ?Abd al-Masī? al-Kindī. This article also makes observations on the implications of the Christian Arabic theological project for interreligious encounter in the early Islamic centuries.     

Nietzsche’s pragmatic genealogy of justice

This paper analyses the connection between Nietzsche’s early employment of the genealogical method and contemporary neo-pragmatism. The paper has two goals. On the one hand, by viewing Nietzsche’s writings in the light of neo-pragmatist ideas and reconstructing his approach to justice as a pragmatic genealogy, it seeks to bring out an under-appreciated aspect of his genealogical method which illustrates how genealogy can be used to vindicate rather than to subvert, and accounts for Nietzsche’s lack of historical references. On the other hand, by highlighting what Nietzsche has to offer neo-pragmatism, it seeks to contribute to neo-pragmatism’s conception of genealogy. The paper argues that Nietzsche and the neo-pragmatists share a naturalistic concern and a pragmatist strategy in responding to it. The paper then shows that Nietzsche avoids a reductive form of functionalism by introducing a temporal axis, but that this axis should be understood as a developmental model rather than as historical time. This explains Nietzsche’s failure to engage with history. The paper concludes that pragmatic genealogy can claim a genuinely Nietzschean pedigree.     

Hume's Foundational Project in the Treatise

In the Introduction to the Treatise Hume very enthusiastically announces his project to provide a secure and solid foundation for the sciences by grounding them on his science of man. And Hume indicates in the Abstract that he carries out this project in the Treatise. But most interpreters do not believe that Hume's project comes to fruition. In this paper, I offer a general reading of what I call Hume's ‘foundational project’ in the Treatise, but I focus especially on Book 1. I argue that in Book 1 much of Hume's logic is put in the service of the other sciences, in particular, mathematics and natural philosophy. I concentrate on Hume's negative thesis that many of the ideas central to the sciences are ideas that we cannot form. For Hume, this negative thesis has implications for the sciences, as many of the texts I discuss make evident. I consider and criticize different proposals for understanding these implications: the Criterion of Meaning and the ‘Inconceivability Principle’. I introduce what I call Hume's ‘No Reason to Believe’ Principle, which I argue captures more adequately the link Hume envisions between his logic, in particular his examination of ideas, and the other sciences.     

The return of the repressed. On Robert N. Bellah,Norman O. Brown,and religion in human evolution

As much as Robert Bellah’s final work, Religion in Human Evolution, has been studied and dissected, no critic underlined the importance of psychoanalysis for its main argument and its theoretical framework. The paper shows the influence exerted by a controversial interpreter of Freud, Norman O. Brown, on Bellah’s ideas, intellectual profile, and writing style in the late-1960s and early 1970s. While in search for a new intellectual voice, Bellah was struck by Brown’s work and began to make intensive use of his book, Love’s Body, both in his teaching and in his research of the early 1970s, during his so-called “symbolic realism” period. While Bellah abandoned Brown’s ideas and style in the mid-1970s, some of the basic intuitions he had during that period still survived as one of the major theoretical intuitions of Religion and Human Evolution.     

Beneficence,Interests, and Wellbeing in Medicine: What It Means to Provide Benefit to Patients

Abstract

Beneficence is a foundational ethical principle in medicine. To provide benefit to a patient is to promote and protect the patient’s wellbeing, to promote the patient’s interests. But there are different conceptions of wellbeing, emphasizing different values. These conceptions of wellbeing are contrary to one another and give rise to dissimilar ideas of what it means to benefit a patient. This makes the concept of beneficence ambiguous: is a benefit related to the patient’s goals and wishes, or is it a matter of objective criteria that constitute wellbeing? This paper suggests a unified conception of wellbeing for use in medicine to determine what counts as a benefit. Two components of wellbeing are identified: (1) objective functioning/health and (2) the patient’s view of her own good. The paper explores how to apply, balance, and weigh these components in clinical situations to determine what counts as a benefit to a patient.     

Habermas,Religion, and Public Life

Abstract

Although many scholars view Jürgen Habermas as the most important philosopher and social theorist since Weber, his account of religion has been relatively neglected. This may be a reflection of the fact that he appears to offer quite distinct views of religion, but it may also reflect the strong secularist assumptions of his early work. In this paper, Habermas’s early and later views of religion are outlined and the change in his understanding of religion is mapped—from being one of the sources of modernity’s inner problems to being a principal source of the passions and motivations underpinning cultural life today. I argue that the relative neglect of Habermas’s work on religion is unfortunate, as it offers a creative and important attempt to move beyond the secular assumptions that have characterised much modern study of religion, providing it with a central role in the mediation of the costs and benefits of modernity.     

Local axioms in disguise: Hilbert on Minkowski diagrams

While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski??s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating ??rigorously?? with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as ??drawn formulas??, and formulas as ??written diagrams??, thus suggesting that the former encapsulate propositional information which can be extracted and translated into formulas. In the case of Minkowski diagrams, local geometrical axioms were actually being produced, starting with the diagrams, by a process that was both constrained and fostered by the requirement, brought about by the axiomatic method itself, that geometry ought to be made independent of analysis. This paper aims at making a twofold point. On the one hand, it shows that Minkowski??s diagrammatic methods in number theory prompted Hilbert??s axiomatic investigations into the notion of a straight line as the shortest distance between two points, which start from his earlier work focused on the role of the triangle inequality property in the foundations of geometry, and lead up to his formulation of the 1900 Fourth Problem. On the other hand, it purports to make clear how Hilbert??s assessment of Minkowski??s diagram-based reasoning in number theory both raises and illuminates conceptual compatibility concerns that were crucial to his philosophy of mathematics.     

Revisioning Fordham's ‘Defences of the self’ in light of modern relational theory and contemporary neuroscience

This paper explores the evolution of Michael Fordham's ideas concerning ‘defences of the self’, including his application of this concept to a group of ‘difficult’ adult patients in his famous 1974 paper by the same name. After tracing the relevance of Fordham's ideas to my own discovery of a ‘self‐care system’ in the psychological material of early trauma patients (Kalsched 1996 ), I describe how Fordham's seminal notions might be revisioned in light of contemporary relational theory as well as early attachment theory and affective neuroscience. These revisionings involve an awareness that the severe woundings of early unremembered trauma are not transformable through interpretation but will inevitably be repeated in the transference, leading to mutual ‘enactments’ between the analytic partners and, hopefully, to a new outcome. A clinical example of one such mutual enactment between the author and his patient is provided. The paper concludes with reflections on the clinical implications of this difficult case and what it means to become a ‘real person’ to our patients. Finally, Jung's alchemical views on transference are shown to be useful analogies in our understanding of the necessary mutuality in the healing process with these patients.     

Killing the father,Parmenides: On Lacan’s anti-philosophy

This paper examines the historical claims about philosophy, dating back to Parmenides, that we argue underlie Jacques Lacan’s polemical provocations in the mid-1970s that his position was an “anti-philosophie”. Following an introduction surveying the existing literature on the subject, in part ii, we systematically present the account of classical philosophy Lacan has in mind when he declares psychoanalysis to be an antiphilosophy after 1975, assembling his claims about the history of ideas in Seminars XVII and XX in ways earlier contributions of this subject have not systematically done. In part iii, focusing upon Lacan’s remarkable reading of Descartes’ break with premodern philosophy—but touching on Lacan’s readings of Hegel and (in a remarkable confirmation of Lacan’s “Parmenidean” conception of philosophy) the early Wittgenstein—we examine Lacan’s positioning of psychoanalysis as a legatee of the Cartesian moment in the history of western ideas, nearly-contemporary with Galileo’s mathematization of physics and carried forwards by Kant’s critical philosophy and account of the substanceless subject of apperception. In different terms than Slavoj ?i?ek, we propose that it is Lacan’s famous avowal that the subject of the psychoanalysis is the subject first essayed by Descartes in The Meditations on First Philosophy as confronting an other capable of deceit (as against mere illusion or falsity) that decisively measures the distance between Lacan’s unique “antiphilosophy” and the forms of later modern linguistic and cultural relativism whose hegemony Alain Badiou has decried, at the same time as it sets Lacan’s antiphilosophy apart from the Parmenidean legacy for which thinking and being could be the same.

     

A critical discussion of Prior’s philosophical and tense-logical analysis of the ideas of indeterminism and human freedom

This paper is a critical discussion of A.N. Prior’s contribution to the modern understanding of indeterminism and human freedom of choice. Prior suggested that these ideas should be conceived in terms of his tense logic. It can be demonstrated that his approach provides an attractive formalization that makes it possible to discuss indeterminism and human freedom of choice in a very precise manner and in a broader metaphysical context. It is also argued that Prior’s development of this approach was closely linked to his very personal struggles with fundamental religious and metaphysical questions. In his opinion, holding the doctrine of divine foreknowledge together with the doctrine of human freewill gives rise to difficult logical and philosophical problems. It appears that Prior, rather early on, decided to support what was later known as the Peircean solution, although he also analysed what he considered to be the most important alternative solution—the Ockhamist solution. This paper offers a discussion of some of Prior’s published books and papers as well as some of the papers in his Nachlass.

     

From the clarity of ideas to the validity of judgments: Kant's farewell to epistemic perfectionism

Against the standard interpretation of Kant's ‘Copernican revolution’ as the prioritization of epistemology over ontology, I argue in this paper that his critique of traditional metaphysics must be seen as a farewell to the perfectionism on which early modern rationalist ontology and epistemology are built. However, Kant does not simply replace ‘perfection’ with another fundamental concept of normativity. More radically, Kant realizes that it is not simply ideas but only the relation of ideas that can be subject to norms, and thus he shifts the focus from the reality of ideas to the validity of judgments. Section 1 of this paper clarifies the pre-Kantian role of the concept of perfection and examines Kant's critical response to that concept. Section 2 identifies Kant's point of departure from the Cartesian ‘way of ideas.’ Section 3 explains the key problem of his novel account of epistemic normativity. I conclude that Kant's anti-perfectionism must be seen as the driving force behind his ‘Copernican revolution’ in order to fully appreciate his mature account of epistemic normativity.     

Difference – is it hated or desired? Reflections on the totalitarian state of mind

In an age of polarized political views and growing nationalism it is vital that the psychoanalytic profession offers its contribution. The author makes a link between early infant development and social and political behaviour. Psychoanalytic, Jungian and Relational ideas are explored. Starting from Freud and his theory of ‘minor differences’, a dichotomy between closeness and separateness is investigated. The writer argues that difference is at the centre of human identity and human development and explores why we struggle to accept it. The totalitarian political system is described as one that eliminates difference. A case study is offered as an illustration of a patient’s struggle to move from a symbiotic, undifferentiated state towards object relating and individuation .     

论张积中的三教观

自魏晋以来,儒释道三家思想并存发展成为了中国文化的显著特征之一。清末太谷学派作为流传于民间的一个学术派别,其三教观一直存在诸多争议。学派北宗传人张积中在山东肥城黄崖山聚徒讲学近十年,门人弟子遍及山左,现存遗著十九种,为太谷二传弟子中著述最多者。从张积中的三教观来看,他并非倡导三教合一,其三教观的核心是以弘扬儒家文化为基本理念,同时汲取佛道思想来完善其理论体系的太谷圣功之学,当属宋明"新儒家"思想在民间进行儒学传播的新尝试。     

Rousseau on Sex-Roles,Education and Happiness

Over the last decade, philosophers of education have begun taking a renewed interest in Rousseau’s educational thought. This is a welcome development as his ideas are rich with educational insights. His philosophy is not without its flaws, however. One significant flaw is his educational project for females, which is sexist in the highest degree. Rousseau argues that females should be taught to “please men…and make [men’s] lives agreeable and sweet.” The question becomes how could Rousseau make such strident claims, especially in light of his far more insightful ideas concerning the education of males. This paper attempts to make sense of Rousseau’s ideas on the education of females. While I maintain that Rousseau’s project for Sophie ought to be rejected, I argue that we should try to understand how this otherwise insightful thinker could make such surprising claims. Is it a bizarre inconsistency in his philosophical reasoning or an expression of his unabashed misogyny, as so many have claimed? I argue that it is neither. Rather, it is a product of his conception of human happiness and his belief in the irreducible role human sexual relations has in achieving and prolonging that happiness. For Rousseau, sex, love and happiness are inextricably connected, and he believes that men and women will be happiest when they inhabit certain sex roles—not because sex roles are valuable in themselves, but because only through them can either men or women hope to be happy.