Husserl's Two Notions Of Completeness

In this paper I discuss Husserl's solution of the problem of imaginary elements in mathematics as presented in the drafts for two lectures hegave in Göttingen in 1901 and other related texts of the same period,a problem that had occupied Husserl since the beginning of 1890, whenhe was planning a never published sequel to Philosophie der Arithmetik(1891). In order to solve the problem of imaginary entities Husserl introduced,independently of Hilbert, two notions of completeness (definiteness in Husserl'sterminology) for a formal axiomatic system. I present and discuss these notionshere, establishing also parallels between Husserl's and Hilbert's notions ofcompleteness.     

The Pragmatism of Hilbert's Programme

It is shown that David Hilbert's formalistic approach to axiomaticis accompanied by a certain pragmatism that is compatible with aphilosophical, or, so to say, external foundation of mathematics.Hilbert's foundational programme can thus be seen as areconciliation of Pragmatism and Apriorism. This interpretation iselaborated by discussing two recent positions in the philosophy ofmathematics which are or can be related to Hilbert's axiomaticalprogramme and his formalism. In a first step it is argued that thepragmatism of Hilbert's axiomatic contradicts the opinion thatHilbert style axiomatical systems are closed systems, a reproachposed by Carlo Cellucci. In the second section the question isdiscussed whether Hilbert's pragmatism in foundational issuescomes close to an a-philosophical ``naturalism in mathematics' assuggested by Penelope Maddy. The answer is ``no', because forHilbert philosophy had its specific tasks in the general projectto found mathematics. This is illuminated in the concludingsection giving further evidence for Hilbert's foundationalapriorism by discussing his ``axiom of the existence of mind' andrelating it to the ``one and only axiom' of the German algebraistof logic, Ernst Schröder, postulating the inherence of signs onthe paper.     

Die Mathematisierung der Erfahrung: Vorg?nger zu Carnaps “Aufbau”

At present, there is a renewed interest in thoseaspects of Rudolf Carnap's LogischerAufbau der Welt that cannot just be reduced tothe tradition of logical positivism. There is, however,one of these aspects which seems to have beenneglected by the historical analyses of theAufbau's background: what may be called aprogram for the mathematization of sense experience,as developed by Carnap in the most difficult partof his work, Chapter IV/A. It is the program of applying settheory and topology to the reconstruction of phenomenal structures,quite independently of any other general philosophical aim.I claim that there are three forerunners of this program:Henri Poincaré's essayPourquoi l'espace a trois dimensions (1912), Chapter 4of Bertrand Russell's Our Knowledge of the ExternalWorld (1914), and Jean Nicod's La géométriedans le monde sensible (1923). This article provides acomparison of these different instances of theabove-mentioned program.     

Russell's Influence On Carnap's Aufbau

This paper concerns the debate on the nature of Rudolf Carnap'sproject in his 1928 book The Logical Structure of the Worldor Aufbau. Michael Friedman and Alan Richardson haveinitiated much of this debate. They claim that the Aufbauis best understood as a work that is firmly grounded inneo-Kantian philosophy. They have made these claims in oppositionto Quine and Goodman's ``received view' of the Aufbau. Thereceived view sees the Aufbau as an attempt to carry out indetail Russell's external world program. I argue that both sidesof this debate have made errors in their interpretation ofRussell. These errors have led these interpreters to misunderstandthe connection between Russell's project and Carnap's project.Russell in fact exerted a crucial influence on Carnap in the1920s. This influence is complicated, however, due to the factthat Russell and Carnap disagreed on many philosophical issues. Iconclude that interpretations of the Aufbau that ignoreRussell's influence are incomplete.     

Hilbert and the internal logic of mathematics

Hilbert's programme is shown to have been inspired in part by what we can call Kronecker's programme in the foundations of an arithmetic theory of algebraic quantities.While finitism stays within the bounds of intuitive finite arithmetic, metamathematics goes beyond in the hope of recovering classical logic. The leap into the transfinite proved to be hazardous, not only from the perspective of Gödel's results, but also from a Kroneckerian point of view.Hilbert's rare admission of a Kroneckerian influence does not constitute the basis of such a reconstruction; it is rather Kronecker's mathematical practice which is seen as a forerunner of Hilbert's endeavour in the foundations of mathematics.I am indebted to an anonymous referee for many helpful critical remarks.     

Attitudes and illusions: Herbert Leyendecker’s phenomenology of perception

In this paper, I discuss aspects of Herbert Leyendecker’s 1913 doctoral dissertation, Towards the Phenomenology of Deceptions (Zur Phänomenologie der Täuschungen), which he defended in 1913 at the University of Munich. Leyendecker was a member of the Munich and Göttingen Phenomenological Circles. In my discussion of his largely neglected views, I explore the connection between his ideas concerning “attitudes” (Einstellungen), e.g., of searching for, observing, counting, or working with objects, and the central topic of his text, perceptual illusions, thematized by Leyendecker as a kind of perceptual “deception” (Täuschung). Indeed, Leyendecker argues that a change of attitude is a necessary aspect of an illusion. I argue that Leyendecker’s use of the notion of attitude in accounting for illusions is problematic; yet I also suggest that his ideas are not devoid of philosophical interest, in relation to current debates.

     

The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program

After a brief flirtation with logicism around 1917, David Hilbertproposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays andWilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for everstronger and more comprehensive areas of mathematics, and finitisticproofs of consistency of these systems. Early advances in these areaswere made by Hilbert (and Bernays) in a series of lecture courses atthe University of Göttingen between 1917 and 1923, and notably in Ackermann's dissertation of 1924. The main innovation was theinvention of the -calculus, on which Hilbert's axiom systemswere based, and the development of the -substitution methodas a basis for consistency proofs. The paper traces the developmentof the ``simultaneous development of logic and mathematics' throughthe -notation and provides an analysis of Ackermann'sconsistency proofs for primitive recursive arithmetic and for thefirst comprehensive mathematical system, the latter using thesubstitution method. It is striking that these proofs use transfiniteinduction not dissimilar to that used in Gentzen's later consistencyproof as well as non-primitive recursive definitions, and that thesemethods were accepted as finitistic at the time.     

On interpreting Gödel's Second Theorem

Summary In this paper I have considered various attempts to attribute significance to G2.²⁵ Two of these attempts (Beth-Cohen and the position maintaining that G2 shows the failure of Hilbert's Program), I have argued, are literally false. Two others (BCR and Resnik's Interpretation), I have argued, are groundless.I would like to thank Dale Gottlieb, Stephen Barker, Tim McCarthy, Philip Kitcher, Michael Resnik and Richmond Thomason for extensive and helpful discussion of this work.     

Hilbert's Program Revisited

After sketching the main lines of Hilbert's program, certain well-known andinfluential interpretations of the program are critically evaluated, and analternative interpretation is presented. Finally, some recent developments inlogic related to Hilbert's program are reviewed.     

What It Is to Pretend

Pretense is a topic of keen interest to philosophers and psychologists. But what is it, really, to pretend? What features qualify an act as pretense? Surprisingly little has been said on this foundational question. Here I defend an account of what it is to pretend, distinguishing pretense from a variety of related but distinct phenomena, such as (mere) copying and practicing. I show how we can distinguish pretense from sincerity by sole appeal to a person's beliefs, desires, and intentions – and without circular recourse to an ‘intention to pretend’ or to a sui generis mental state of ‘imagining.’     

Thomas Aquinas on hylomorphism and the in-act principle

In Summa Theologiae I.76.1 Aquinas presents an argument for the hylomorphic union of body and soul that he attributes to Aristotle. Aquinas builds on Aristotle’s original argument, however, offering his own short but powerful line of reasoning in support of one of the main premises. This additional argument involves an appeal to the principle that nothing acts except insofar as it is in act. This principle has roots in the thought of Aristotle, but is not explicitly used by him. It is, however, fundamental for Aquinas and pervasive throughout his work. In this paper I examine the principle and its implications for Aquinas’ version of the argument. Furthermore, I argue that the principle is foundational to Aquinas’ criticisms of Averroes’ account of the intellective soul and that its inclusion renders Aquinas’ version of the argument incompatible with Averroes’ view.     

Levinasian Reflections on Somaticity and the Ethical Self

In this article, I attempt to bring some conceptual clarity to several key terms and foundational claims that make up Levinas's body‐based conception of ethics. Additionally, I explore ways that Levinas's arguments about the somatic basis of subjectivity and ethical relatedness receive support from recent empirical research. The paper proceeds in this way: First, I clarify Levinas's use of the terms “sensibility”, “subjectivity”, and “proximity” in Otherwise than Being: or Beyond Essence. Next, I argue for an interpretation of Levinas's thought that I suggest is buttressed by recent experimental work in both developmental psychology and neuroscience. I provide examples of research that I suggest opens up Levinas's phenomenological analysis in new and interesting ways. I also urge the importance of Levinas's phenomenological analysis in contextualizing the ethical significance of these empirical findings.     

Jesus ACTED UP: Learning with Robert Goss

In this essay, I discuss my responses to Bob Goss’ Jesus ACTED UP as I re-read it 20 years after its original discussion. Bob's work is foundational for subsequent queer theology even as it is bound to the particular context of the early 1990s. Next, I place the book in the larger context of what I call the “queerification” of Christianity. The Lesbian, Gay, Bisexual, Transgender, Queer equality movement has made significant gains in terms of opening the church to openly queer clergy and access to marriage because of the contributions of scholar/activists such as Bob Goss. I conclude by looking ahead at the need for future explorations that can agitate while also reflecting the nuance required in relation to the more complex understandings of gender and sexuality that have emerged, the growing attention to intersectionality and the instability and complexity of personal identities, and the more subtle but no less deadly forms of homohatred we now face — perhaps in the form of collaborative, even co-authored work.     

Russellian and Wittgensteinian Atomism

One difference between Russell's logical atomism in The Philosophy of Logical Atomism and Wittgenstein's in the Tractatus is that Russell's doctrine is explicitly epistemological, whereas Wittgenstein's is not; another difference is that Wittgenstein gives an a priori argument for the doctrine of logical atomism whereas Russell gives no such argument. I argue that these two differences are instructively connected: Russell's focus on epistemology prevents him from being able to give a motivated argument for the truth of logical atomism. Furthermore, I argue that this is not just a contingent failure of Russell's system: no primarily epistemological atomism can avail itself of Wittgenstein's style of a priori argument for the truth of atomism. An important suggestion of the argument, illuminating with respect to the subsequent history of analytic philosophy, is that Russell's logical atomism already contains the seeds of verificationism in a nascent form, whereas Wittgenstein's atomism has no tendency toward verification.     

Hilbert's epsilon as an operator of indefinite committed choice

Hilbert and Bernays avoided overspecification of Hilbert's ε-operator. They axiomatized only what was relevant for their proof-theoretic investigations. Semantically, this left the ε-operator underspecified. After briefly reviewing the literature on semantics of Hilbert's epsilon operator, we propose a new semantics with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the ε simplifies proof search and is natural in the sense that it mirrors some cases of referential interpretation of indefinite articles in natural language.     

The Mother Relationship and Artistic Inhibition in the Lives of Leonardo da Vinci and Erik H. Erikson

In four earlier articles, I focused on the theme of the relationship of melancholia and the mother, and suggested that the melancholic self may experience humor (Capps, 2007a), play (Capps, 2008a), dreams (Capps, 2007c), and art (Capps, 2008b) as restorative resources. I argued that Erik H. Erikson found these resources to be valuable remedies for his own melancholic condition, which had its origins in the fact that he was illegitimate and was raised solely by his mother until he was three years old, when she remarried. In this article, I focus on two themes in Freud’s Leonardo da Vinci and a memory of his childhood (1964): Leonardo’s relationship with his mother in early childhood and his inhibitions as an artist. I relate these two themes to Erikson’s own early childhood and his failure to achieve his goal as an aspiring artist in his early twenties. The article concludes with a discussion of Erikson’s frustrated aspirations to become an artist and his emphasis, in his psychoanalytic work, on children’s play. Donald Capps is Professor of Pastoral Psychology at Princeton Theological Seminary. His books include Men, Religion, and Melancholia (1997), Freud and Freudians on Religion (2001), and Men and Their Religion: Honor, Hope, and Humor     

A note on russell's paradox in locally cartesian closed categories

Working in the fragment of Martin-Löfs extensional type theory [12] which has products (but not sums) of dependent types, we consider two additional assumptions: firstly, that there are (strong) equality types; and secondly, that there is a type which is universal in the sense that terms of that type name all types, up to isomorphism. For such a type theory, we give a version of Russell's paradox showing that each type possesses a closed term and (hence) that all terms of each type are provably equal. We consider the kind of category theoretic structure which corresponds to this kind of type theory and obtain a categorical version of the paradox. A special case of this result is the degeneracy of a locally cartesian closed category with a morphism which is generic in the sense that every other morphism in the category can be obtained from it via pullback.The financial support of the Royal Society in London is gratefully acknowledged.     

What is The Red Book for analytical psychology?

Abstract: The Red Book enters the field of analytical psychology belatedly, yet presents itself as a foundational text. In this paper, I try to estimate how it will affect the field and to reflect on how it can be used by contemporary readers and clinicians. Jung's own works remain the baseline for theorizing and teaching the fundamental perspectives of analytical psychology, and yet the field has grown far beyond his own contributions. Is this late inheritance of central value or only of peripheral interest for the field? It can be used in various ways and certainly shows many interesting features that link it to Jung's later works, but does it detract or does it add? And if it adds, what does it add? In The Red Book, Jung is speaking to us from the grave. What is he saying? In this paper, I explore how it can be used as an inspirational text for keeping alive the spirit of the ancestors in the field and for linking the present to the deep past.     

Realization, Reduction And Psychological Autonomy

It is often thought that the computational paradigm provides a supporting case for the theoretical autonomy of the science of mind. However, I argue that computation is in fact incompatible with this alleged aspect of intentional explanation, and hence the foundational assumptions of orthodox cognitive science are mutually unstable. The most plausible way to relieve these foundational tensions is to relinquish the idea that the psychological level enjoys some special form of theoretical sovereignty. So, in contrast to well known antireductionist views based on multiple realizability, I argue that the primary goal of a computational approach to the mind should be to facilitate a translation of the psychological to the neurophysiological.     

Słupecki's Generalized Mereology and Its Flaws

One of the streams in the early development of set theory was an attempt to use mereology, a formal theory of parthood, as a foundational tool. The first such attempt is due to a Polish logician, Stanis?aw Le?niewski (1886–1939). The attempt failed, but there is another, prima facie more promising attempt by Jerzy S?upecki (1904–1987), who employed his generalized mereology to build mereological foundations for type theory. In this paper I (1) situate Le?niewski's attempt in the development of set theory, (2) describe and evaluate Le?niewski's approach, (3) describe S?upecki's strategy without unnecessary technical details, and (4) evaluate it with a rather negative outcome. The issues discussed go beyond merely historical interests due to the current popularity of mereology and because they are related to nominalistic attempts to understand mathematics in general. The introduction describes very briefly the situation in which mereology entered the scene of foundations of mathematics — it can be safely skipped by anyone familiar with the early development of set theory. Section 2 describes and evaluates Le?niewski's attempt to use mereology as a foundational tool. In Section 3, I describe an attempt by S?upecki to improve on Le?niewski's work, which resulted in a system called generalized mereology. In Section 4, I point out the reasons why this attempt is still not successful. Section 5 contains an explanation of why Le?niewski's use of Ontology in developing arithmetic also is not nominalistically satisfactory.