The Russellian Influence on Hilbert and his School

The aim of the paper is to discuss the influence exercised by Russell's thought inGöttingen in the period leading to the formulation of Hilbert's program in theearly twenties. I show that after a period of intense foundational work, culminatingwith the departure from Göttingen of Zermelo and Grelling in 1910 we witnessa reemergence of interest in foundations of mathematics towards the end of 1914. Itis this second period of foundational work that is my specific interest. Through theuse of unpublished archival sources I will describe how Hilbert, Behmann, and Bernays,among others, were influenced by and reacted to the technical and philosophical thesespresented in Principia Mathematica. I also argue that there are some elements of continuity between Russell's approach and Hilbert's program as it was presented inthe early twenties.     

PRAGMATISM,REALISM, AND RELIGION

Pragmatism is often thought to be incompatible with realism, the view that there are knowable mind‐independent facts, objects, or properties. In this article, I show that there are, in fact, realist versions of pragmatism and argue that a realist pragmatism of the right sort can make important contributions to such fields as religious ethics and philosophy of religion. Using William James's pragmatism as my primary example, I show (1) that James defended realist and pluralist views in metaphysics, epistemology, ethics, and philosophy of religion, and (2) that these views not only cohere with his pragmatism but indeed are basic to it. After arguing that James's pragmatism provides a credible and useful approach to a number of basic philosophical and religious issues, I conclude by reflecting on some ways in which we can apply and potentially improve James's views in the study of religion.     

丹尼特的“异类现象学”——新实用主义谱系中一个被忽略的环节

一般认为,"新实用主义"乃经典实用主义的一个后现代主义的变种,它无法在知识论的角度接受心灵表征的客观实在性。本文则认为,反实在论并非是我们发展实用主义的心灵哲学时必然陷入的理论窠臼。如果我们采纳美国哲学家丹尼特主张的"异类现象学"方法,我们就有可能将科学方法和工具主义立场相调和,或在对他人的心灵状态做出本体论承诺时,不必向反实在论投降。     

一种实用主义的数学哲学观

作者在这篇文论中提出了一种实用主义的数学哲学观。其中的想法是,通过分析各种数学哲学思想对于数学实践的实际影响来比较它们的价值。为解释这种想法的可行性,作者将论证,正是数学的某些特性使得实用主义的方法能够被自然地运用于对数学哲学诸观点的分析。在之后的一节中,作者会给出一些案例分析以展示某些哲学观点是如何具体地影响数学研究的。文章的写作假设读者具备了一些关于经典逻辑和集合论的基础知识。     

On the Very Idea of Brandom’s Pragmatism

Although Brandom is critical of some features of narrowly conceived classical pragmatism, at the same time he explicitly embraces a version of pragmatism, both in his overall philosophical outlook, and in his philosophy of language. Brandom’s distinctive theoretical approach is based on what he calls rationalist pragmatism, which is a version of fundamental pragmatism. Within the philosophy of language it takes the form of semantic pragmatism. The paper briefly discusses Brandomian version of fundamental pragmatism and its semantic underpinning, and subsequently formulates a basic dilemma it encounters there.     

Hertz and Wittgenstein's Philosophy of Science

The German physicist Heinrich Hertz played a decisive role for Wittgenstein's use of a unique philosophical method. Wittgenstein applied this method successfully to critical problems in logic and mathematics throughout his life. Logical paradoxes and foundational problems including those of mathematics were seen as pseudo-problems requiring clarity instead of solution. In effect, Wittgenstein's controversial response to David Hilbert and Kurt Gödel was deeply influenced by Hertz and can only be fully understood when seen in this context. To comprehend the arguments against the metamathematical programme, and to appreciate how profoundly the philosophical method employed actually shaped the content of Wittgenstein's philosophy, it is necessary to make an intellectual biographical reconstruction of their philosophical framework, tracing the Hertzian elements in the early as well as in the later writings. In order to write Wittgenstein's biography, we have to take seriously the coherence of his thought throughout his life, and not let convenient philosophical ideologies be our guidance in drawing up a “Wittgensteinian philosophy”. To do so, we have to take a second look upon what he actually wrote, not only in the already published material, but in the entire Nachlass. Clearly, this is not easily done, but it is a necessary task in the historical reconstruction of Wittgenstein's life and work.     

The Public Has to Define Itself

In the article texts by John Dewey, JürgenHabermas and Richard Rorty are discussed in thelight of different meanings of the Public. Thisis done by discussing foundational andnon-foundational claims on a philosophy ofpragmatism and democracy, and by looking atdifferent meanings of intersubjectivity. Onecrucial difference I am pointing at, is thatwhile Dewey's intersubjectivity is stemmingfrom philosophical arguments as well aspolitical, Habermas's intersubjectivity isrestricted to the level of (an almostscientific) philosophical abstractargumentation without any concrete language ofpolitics. When it comes to Rorty I stress thathe is far closer than Habermas to Dewey'spragmatism, especially his ambition to includethe specific individuals and to stressindividuality in a philosophy of communicationand public conduct.     

Learning Logical Tolerance: Hans Hahn on the Foundations of Mathematics

Hans Hahn's long-neglected philosophy of mathematics is reconstructed here with an eye to his anticipation of the doctrine of logical pluralism. After establishing that Hahn pioneered a post-Tractarian conception of tautologies and attempted to overcome the traditional foundational dispute in mathematics, Hahn's and Carnap's work is briefly compared with Karl Menger's, and several significant agreements or differences between Hahn's and Carnap's work are specified and discussed.     

FREGE IN CONTEXT

Although the Cambridge Professor of Mental Philosophy and Logic James Ward was once one of Britain's most highly regarded Psychologists and Philosophers, today his work is unjustly neglected. This is because his philosophy is frequently misrepresented as a reactionary anti-naturalistic idealist theism. In this article, I argue, first, that this reading is false, and that by viewing Ward through the lens of pragmatism we obtain a fresh interpretation of his work that highlights the scientific nature of his philosophy and his original and promising theory of ‘evolutionary Kantianism’, with its applications to the philosophy of mind, epistemology, and metaphysics. Second, I show that reading Ward as a pragmatist provides us with (1) a more complex history of the reception of pragmatism at Cambridge at the turn of the twentieth century than the straightforwardly hostile one traditionally told; and (2) a more detailed understanding of the wide range of philosophical problems to which pragmatism was deemed at this time to have an appropriate application.     

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.     

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.     

Gödel,Kant, and the Path of a Science

Gödel's philosophical views were to a significant extent influenced by the study not only of Leibniz and Husserl, but also of Kant. Both Gödel and Kant aimed at the secure foundation of philosophy, the certainty of knowledge and the solvability of all meaningful problems in philosophy. In this paper, parallels between the foundational crisis of metaphysics in Kant's view and the foundational crisis of mathematics in Gödel's view are elaborated, especially regarding the problem of finding the “secure path of a science” for both mathematics and philosophy. Gödel's temporal subjectivism and metaphysical conceptual objectivism are presented as positively or negatively motivated by Kant's viewpoints. A remark on Gödel's collapse of modalities (in accordance with the collapse of objective time) is added.     

Between Pragmatism and Critical Theory: Social Philosophy Today

This paper aims at renovating the prospects for social philosophy through a confrontation between pragmatism and critical theory. In particular, it contends that the resources of pragmatism for advancing a project of emancipatory social philosophy have so far been neglected. After contrasting the two major traditions in social philosophy—the analytical and the critical—I proceed to outline the main traits of a pragmatist social philosophy. By inscribing pragmatism within the tradition of social philosophy, my aim is to promote a new understanding of pragmatism as one of the central Euro-American traditions in social and political philosophy, deserving to be on an equal footing with critical theory and political liberalism. And, furthermore, one whose critical and radical force may be of great help in the wake of the dismissal of the metaphysical certainties upon which the critical program of social philosophy had once set its hopes of social emancipation.     

Did Rorty’s Pragmatism Have Foundations?

Abstract

There is an overt tension between Rorty’s pragmatist critique of philosophy and his apparent epistemological and metaphysical commitments, which it is instructive to examine in order to assess not only Rorty’s overall position, but also renewed contemporary interest in pragmatism and its metaphilosophical implications. After showing why Rorty’s attempts to limit the scope of his critique failed to resolve this tension, I try reading him as a constructive metaphysician who was attempting to balance a causal account of the language / world relation with panrelationism. However, Rorty intended these commitments to be interpreted in light of his pragmatism about vocabularies, and relied upon a ‘social standpoint strategy’ to render his overall position consistent. I conclude that to the extent that this strategy succeeds, it removes almost all of the argumentative force from Rorty’s pragmatism.     

Kant's Conception of Inner Value

Abstract: This article addresses a foundational issue in Kant's moral philosophy, the question of the relation of the Categorical Imperative to value. There is an important movement in current Kant scholarship that argues that there is a value underlying the Categorical Imperative. However, some scholars have raised doubts as to whether Kant has a conception of value that could ground the Categorical Imperative. In this paper I seek to add to these doubts by arguing, first, that value would have to be of a particular kind in order to be the foundation of Kant's moral philosophy. Second, I argue that Kant does not have such a conception of value, and that his arguments rule out that value could ground his moral philosophy. I then outline an alternative reading of how Kant uses ‘inner value’. My conclusion will be that Kant does not derive the Categorical Imperative from an underlying value. While some of his passages could also be read as if value were foundational for Kant, a close look at these passages and his arguments point away from this conclusion.     

The Meaning of Category Theory for 21st Century Philosophy

Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific issues, in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy. Text of a talk given at the Workshop and Symposium on the Ramifications of Category Theory, Florence, November 18–22, 2003. For further documentation on the conference, see     

“Metaphysics Without Ethics is Blind”: The Legacy of Hilary Putnam

Two convictions underlie the following article. The first is that Hilary Putnam has been one of the greatest thinkers of our time, a philosopher who was able to propose groundbreaking ideas in virtually every area of philosophy. As the reader will see, the topics he tackled in his writings included questions of philosophy of science, philosophy of language, philosophy of mathematics and logic, philosophy of mind, metaethics, the fact-value dichotomy, the interpretation of Wittgenstein’s later thought, the question of relativism, the analysis of rationality, the analysis of religious experience, the character of Jewish philosophy, the interpretation of pragmatism, the elucidation of the concept of truth, the question of realism, the relationship between mind and the world. The second is that the changes some of his positions underwent, far from being a point of weakness—as some critics have sometimes felt compelled to claim—reveal the freshness and genuineness of Putnam’s way of philosophising and at the same time the essence of philosophical discussion itself.     

Descartes's Regulae, mathematics, and modern psychology: "the Noblest example of all" in Light of Turing's (1936) On computable Numbers

There are surprisingly strong connections between the philosophy of mind and the philosophy of mathematics. One particular important example can be seen in the Regulae (1628) of Descartes. In "the noblest example of all," he used his new abstract understanding of numbers to demonstrate how the brain can be considered as a symbol machine and how the intellect's algebraic reasoning can be mirrored as operations on this machine. Even though his attempt failed, it is illuminating to explore it because Descartes launched 2 traditions--mechanistic philosophy of mind and abstract mathematics--that would diverge until A. Turing (1936) approached symbolic reasoning in a similar "symbol machine-existence proof" way. Descrates's and Turing's thought experiments, which mark the beginning of modern psychology and cognitive science, respectively, indicate how important the development of mathematics has been for the constitution of the science of mind.     

How Gödel Relates Platonism to Mathematics

It is well known that Gödel takes his realistic world view as closely related to mathematics, especially to his own work in the foundations of mathematics. He reports, publicly as well as privately, that Platonism is fundamental to his major work in logic and set theory, and suggests that this philosophical position can be seen as a product of reflections on mathematics. These views of Gödel, however, are often regarded as being insufficiently formulated or argued for. In this article, the author tries to consider some points which are related to the understanding of the Gödelian mode of the interaction between mathematics and philosophy.     

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.