Peter Jelavich's new study pursues a double agenda: while it examines the role of radio and film in the broader context of cultural politics in Weimar Germany, it at the same time explores the transformation of Alfred Döblin's novel Berlin Alexanderplatz (1929) into a radio play (1930) and then a film (1931). The detailed and intriguing intermedial comparison serves to demonstrate Jelavich's main thesis that the death of the innovative and critical culture of the first German Republic predates the end (...) of the democracy. Challenging the widely shared assumption that only Hitler's rise to power put an end to the…. (shrink)

In 12th-Century France, political Augustinianism inherited from Gregory and Isidore was based on the absolute supremacy of spiritual power over temporal power in the name of the absolute primacy of grace over fallen nature. This could easily solve the question of what should true dominium be in a society based on Christian values. Just rule was connected with a ruler who was sanctioned by the Church: the king of France, defender of the Pax Dei. The Siege of Germigny shows (...) how cynical realpolitik finds philosophical justification in political Augustinianism. (shrink)

C. F. Gellert's 1748 novel "Leben der schwedischen Gräfin von G***" portrays the moral actions of Jews as a result of good Christian conduct. In reaction, G. E. Lessing disputes this depiction in his one-act-play "Die Juden" from 1749. The recognition that a Jew could fulfill the ideals of the Enlightenment helped overcome the prejudices of Christian stage characters and of the audience, but it failed in the social circumstances of the time. Christian reception understood a "noble (...) Jew" to be a violation of the poetic principle of probability - and even for Lessing such a figure was the exception. Moses Mendelssohn, however, believed that Judaism was especially well-suited for fostering the development of certain virtues. In light of far-reaching interreligious concepts on the one hand and religiously motivated conflicts on the other, Lessing's early works reveal today the persistent value of self-critical examination concerning hidden prejudices towards religious and cultural minorities. (shrink)

It is a remarkable fact that Hilbert's programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic from the late 1890s is often understood from that vantage point. My essay pursues one main goal, namely, to contrast Hilbert's formal axiomatic method from the early 1920s with his existential axiomatic approach from the 1890s. Such a contrast illuminates (...) the circuitous beginnings of the finitist consistency program and connects the complex emergence of existential axiomatics with transformations in mathematics and philosophy during the 19th century; the sheer complexity and methodological difficulties of the latter development are partially reflected in the well known, but not well understood correspondence between Frege and Hilbert. Taking seriously the goal of formalizing mathematics in an effective logical framework leads also to contemporary tasks, not just historical and systematic insights; those are briefly described as “one direction” for fascinating work. (shrink)

The identification of an informal concept of ‘effective calculability’ with a rigorous mathematical notion like ‘recursiveness’ or ‘Turing computability’ is still viewed as problematic, and I think rightly so. I analyze three different and conflicting perspectives Gödel articulated in the three decades from 1934 to 1964. The significant shifts in Gödel's position underline the difficulties of the methodological issues surrounding the Church-Turing Thesis.

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the fifteenth publication in the Lecture Notes in Logic series, collects papers presented at the symposium 'Reflections on the Foundations of Mathematics' held in celebration of Solomon Feferman's 70th birthday (The 'Feferfest') at Stanford University, California in 1988. (...) Feferman has shaped the field of foundational research for nearly half a century. These papers reflect his broad interests as well as his approach to foundational research, which emphasizes the solution of mathematical and philosophical problems. There are four sections, covering proof theoretic analysis, logic and computation, applicative and self-applicative theories, and philosophy of modern mathematical and logic thought. (shrink)

Wilfred Sieg. Mechanical Procedures and Mathematical Experience.

Decision-making under normative uncertainty requires an agent to aggregate the assessments of options given by rival normative theories into a single assessment that tells her what to do in light of her uncertainty. But what if the assessments of rival theories differ not just in their content but in their structure -- e.g., some are merely ordinal while others are cardinal? This paper describes and evaluates three general approaches to this "problem of structural diversity": structural enrichment, structural depletion, and multi-stage (...) aggregation. All three approaches have notable drawbacks, but I tentatively defend multi-stage aggregation as least bad of the three. (shrink)

Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.

We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Minc [14]; each has been shown to be of considerable interest for both mathematical practice and metamathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems. The results are (...) generalized to relate a hierarchy of subsystems, all contained in the theory of arithmetic properties, to a corresponding hierarchy of fragments of arithmetic. The proof theoretic tools employed there are used to re-establish in a uniform, elementary way relationships between various fragments of arithmetic due to Parsons, Paris and Kirby, and Friedman. (shrink)

Kants Musikästhetik wird weithin unterschätzt. Dabei bietet sie die entscheidenden Ansätze zur Befreiung der Musik aus den Fängen der Nachahmungsästhetik, wie sie vor allem E.T.A.Hoffman kongenial umgesetzt hat.

Hilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has (...) to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert's and Bernays's contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert's Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work. (shrink)

David Hilbert was one of the great mathematicians who expounded the centrality of their subject in human thought. In this collection of essays, Wilfried Sieg frames Hilbert's foundational work, from 1890 to 1939, in a comprehensive way and integrates it with modern proof theoretic investigations.

The principle that rational agents should maximize expected utility or choiceworthiness is intuitively plausible in many ordinary cases of decision-making under uncertainty. But it is less plausible in cases of extreme, low-probability risk (like Pascal's Mugging), and intolerably paradoxical in cases like the St. Petersburg and Pasadena games. In this paper I show that, under certain conditions, stochastic dominance reasoning can capture most of the plausible implications of expectational reasoning while avoiding most of its pitfalls. Specifically, given sufficient background uncertainty (...) about the choiceworthiness of one's options, many expectation-maximizing gambles that do not stochastically dominate their alternatives "in a vacuum" become stochastically dominant in virtue of that background uncertainty. But, even under these conditions, stochastic dominance will not require agents to accept options whose expectational superiority depends on sufficiently small probabilities of extreme payoffs. The sort of background uncertainty on which these results depend looks unavoidable for any agent who measures the choiceworthiness of her options in part by the total amount of value in the resulting world. At least for such agents, then, stochastic dominance offers a plausible general principle of choice under uncertainty that can explain more of the apparent rational constraints on such choices than has previously been recognized. (shrink)

Les disciplines historiques, littéraires et philosophiques font un emploi abondant des catégories historiographiques. Parmi celles-ci, les termes en ismes sont très fréquents pour référer à une doctrine, un courant artistique, une idéologie ou des événements spécifiques. On fait cependant remarquer que ces désignations posent de nombreux problèmes d’interprétation. En particulier, que l’origine exacte d’une catégorie est souvent méconnue et que sa signification est plus équivoque qu’on ne le croit habituellement. La formation d’un terme en isme s’explique souvent dans un contexte (...) polémique et sert à condamner ou critiquer plutôt qu’à comprendre une pensée, un mouvement ou des faits historiques. Les catégories d’athéisme, de classicisme ou de féminisme en sont de bons exemples. Le présent volume vise à contribuer à l’étude critique des catégories historiographiques et des notions en isme en s’attardant à l’époque moderne (XVI e XVIII e siècles). Il s’agit d’un moment clef de l’histoire européenne pendant laquelle nombre de ces notions apparaissent ou pour laquelle l’historiographie en a fait un emploi abondant. Les chapitres en examinent la valeur théorique par rapport à une origine ou des usages, et déterminent de quelles manières elles contribuent à l’interprétation de l’histoire, de la littérature ou de la philosophie, de la Renaissance jusqu’à la Révolution française. (shrink)

Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ (...) -definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of effective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper, Why Church needed Gödel's recursiveness for his Thesis. In Section 5, I contrast Church's analysis with that of Alan Turing and explore, in the very last section, an analogy with Dedekind's investigation of continuity. (shrink)

How should you decide what to do when you’re uncertain about basic normative principles? A natural suggestion is to follow some "second-order:" norm: e.g., obey the most probable norm or maximize expected choiceworthiness. But what if you’re uncertain about second-order norms too—must you then invoke some third-order norm? If so, any norm-guided response to normative uncertainty appears doomed to a vicious regress. This paper aims to rescue second-order norms from the threat of regress. I first elaborate and defend the claim (...) some philosophers have made that the regress problem forces us to accept normative externalism, the view that at least one norm is incumbent on all agents regardless of their normative beliefs. But, I then argue, we need not accept externalism about first-order norms, thus closing off any question of how agents should respond to normative uncertainty. Rather, we can head off the threat of regress by ascribing external force to a single second-order norm: the enkratic principle. (shrink)

Wilfried Sieg. Calculations by Man and Machine: Conceptual Analysis.

Dedekind’s structuralism is a crucial source for the structuralism of mathematical practice—with its focus on abstract concepts like groups and fields. It plays an equally central role for the structuralism of philosophical analysis—with its focus on particular mathematical objects like natural and real numbers. Tensions between these structuralisms are palpable in Dedekind’s work, but are resolved in his essay Was sind und was sollen die Zahlen? In a radical shift, Dedekind extends his mathematical approach to “the” natural numbers. He creates (...) the abstract concept of a simply infinite system, proves the existence of a “model”, insists on the stepwise derivation of theorems, and defines structure-preserving mappings between different systems that fall under the abstract concept. Crucial parts of these considerations were added, however, only to the penultimate manuscript, for example, the very concept of a simply infinite system. The methodological consequences of this radical shift are elucidated by an analysis of Dedekind’s metamathematics. Our analysis provides a deeper understanding of the essay and, in addition, illuminates its impact on the evolution of the axiomatic method and of “semantics” before Tarski. This understanding allows us to make connections to contemporary issues in the philosophy of mathematics and science. (shrink)

Herbrand's Theorem, in the form of $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\exists } $$ -inversion lemmata for finitary and infinitary sequent calculi, is the crucial tool for the determination of the provably total function(al)s of a variety of theories. The theories are (second order extensions of) fragments of classical arithmetic; the classes of provably total functions include the elements of the Polynomial Hierarchy, the Grzegorczyk Hierarchy, and the extended Grzegorczyk Hierarchy $\mathfrak{E}^\alpha $ , α < ε0. A subsidiary aim of the paper is to show (...) that the proof theoretic methods used here are distinguished by technical elegance, conceptual clarity, and wide-ranging applicability. (shrink)

Wilfred Sieg. Relative Consistency and Accesible Domains.

Wilfried Sieg and John Byrnes. AnModel for Parallel Computation: Gandy's Thesis.

Church's and Turing's theses dogmatically assert that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computability. I present an analysis of calculability that is embedded in a rich historical and philosophical context, leads to precise concepts, but dispenses with theses. To investigate effective calculability is to analyze symbolic processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting work (...) of Gandy, I formulate boundedness and locality conditions for two types of calculators, namely, human computing agents and mechanical computing devices (discrete machines). The distinctive feature of the latter is that they can carry out parallel computations. The analysis leads to axioms for discrete dynamical systems (representing human and machine computations) and allows the reduction of models of these axioms to Turing machines. Cellular automata and a variety of artificial neural nets can be shown to satisfy the axioms for machine computations. (shrink)

What kind of animals are human beings? And how do our visions of the human shape our theories of social action and institutions? In Moral, Believing Animals>, Christian Smith advances a creative theory of human persons and culture that offers innovative, challenging answers to these and other fundamental questions in sociological, cultural, and religious theory. Smith suggests that human beings have a peculiar set of capacities and proclivities that distinguishes them significantly from other animals on this planet. Despite the (...) vast differences in humanity between cultures and across history, no matter how differently people narrate their lives and histories, there remains an underlying structure of human personhood that helps to order human culture, history, and narration. Drawing on important recent insights in moral philosophy, epistemology, and narrative studies, Smith argues that humans are animals who have an inescapable moral and spiritual dimension. They cannot avoid a fundamental moral orientation in life and this, says Smith, has profound consequences for how sociology must study human beings. (shrink)

On June 4, 1925, Hilbert delivered an address to the Westphalian Mathematical Society in Miinster; that was, as a quick calculation will convince you, almost exactly sixty years ago. The address was published in 1926 under the title Über dasUnendlicheand is perhaps Hilbert's most comprehensive presentation of his ideas concerning the finitist justification of classical mathematics and the role his proof theory was to play in it. But what has become of the ambitious program for securing all of mathematics, once (...) and for all? What of proof theory, the very subject Hilbert invented to carry out his program? The Hilbertian ambition reached out too far: in its original form, the program was refuted by Gödel's Incompleteness Theorems. And even allowing more than finitist means in metamathematics, the Hilbertian expectations for proof theory have not been realized: a constructive consistency proof for second-order arithmetic is still out of reach. Nevertheless, remarkable progress has been made. Two separate, but complementary directions of research have led to surprising insights: classical analysis can be formally developed in conservative extensions of elementary number theory; relative consistency proofs can be given by constructive means for impredicative parts of second order arithmetic. The mathematical and metamathematical developments have been accompanied by sustained philosophical reflections on the foundations of mathematics. This indicates briefly the main themes of the contributions to the symposium; in my introductory remarks I want to give a very schematic perspective, that is partly historical and partly systematic. (shrink)

The incompleteness theorems constitute the mathematical core of Gödel’s philosophical challenge. They are given in their “most satisfactory form”, as Gödel saw it, when the formality of theories to which they apply is characterized via Turing machines. These machines codify human mechanical procedures that can be carried out without appealing to higher cognitive capacities. The question naturally arises, whether the theorems justify the claim that the human mind has mathematical abilities that are not shared by any machine. Turing admits that (...) non-mechanical steps of intuition are needed to transcend particular formal theories. Thus, there is a substantive point in comparing Turing’s views with Gödel’s that is expressed by the assertion, “The human mind infinitely surpasses any finite machine”. The parallelisms and tensions between their views are taken as an inspiration for beginning to explore, computationally, the capacities of the human mathematical mind. (shrink)

Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame (...) is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas. To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at (see link below). (shrink)

Wilfried Sieg. Calculations by Man and Machine: Mathematical Presentation.

Wilfred Sieg and John Byrnes. K-Graph Machines: Generalizing Turing's Machines and Arguments.

Dedekind's mathematical work is integral to the transformation of mathematics in the nineteenth century and crucial for the emergence of structuralist mathematics in the twentieth century. We investigate the essential components of what Emmy Noether called, his ‘axiomatic standpoint’: abstract concepts, models, and mappings.

Nudge is a popular public policy tool that harnesses well-known biases in human judgement to subtly guide people’s decisions, often to improve their choices or to achieve some socially desirable outcome. Thanks to recent developments in artificial intelligence (AI) methods new possibilities emerge of how and when our decisions can be nudged. On the one hand, algorithmically personalized nudges have the potential to vastly improve human daily lives. On the other hand, blindly outsourcing the development and implementation of nudges to (...) “black box” AI systems means that the ultimate reasons for why such nudges work, that is, the underlying human cognitive processes that they harness, will often be unknown. In this paper, we unpack this concern by considering a series of examples and case studies that demonstrate how AI systems can learn to harness biases in human judgment to reach a specified goal. Drawing on an analogy in a philosophical debate concerning the methodology of economics, we call for the need of an interdisciplinary oversight of AI systems that are tasked and deployed to nudge human behaviours. (shrink)

Two young logicians, whose work had a dramatic impact on the direction of logic, exchanged two letters in early 1931. Jacques Herbrand initiated the correspondence on 7 April and Kurt Gödel responded on 25 July, just two days before Herbrand died in a mountaineering accident at La Bérarde (Isère). Herbrand's letter played a significant role in the development of computability theory. Gödel asserted in his 1934 Princeton Lectures and on later occasions that it suggested to him a crucial part of (...) the definition of a general recursive function. Understanding this role in detail is of great interest as the notion is absolutely central. The full text of the letter had not been available until recently, and its content (as reported by Gödel) was not in accord with Herbrand's contemporaneous published work. Together, the letters reflect broader intellectual currents of the time: they are intimately linked to the discussion of the incompleteness theorems and their potential impact on Hilbert's Program. (shrink)

The Carnegie Mellon Proof Tutor project was motivated by pedagogical concerns: we wanted to use a "mechanical" (i.e. computerized) tutor for teaching students..

Longtermists claim that what we ought to do is mainly determined by how our actions might affect the very long-run future. A natural objection to longtermism is that these effects may be nearly impossible to predict — perhaps so close to impossible that, despite the astronomical importance of the far future, the expected value of our present actions is mainly determined by near-term considerations. This paper aims to precisify and evaluate one version of this epistemic objection to longtermism. To that (...) end, I develop two simple models for comparing ‘longtermist’ and ‘neartermist’ interventions, incorporating the idea that it is harder to make a predictable difference to the further future. These models yield mixed conclusions: if we simply aim to maximize expected value, and don’t mind premising our choices on minuscule probabilities of astronomical payoffs, the case for longtermism looks robust. But on some prima facie plausible empirical worldviews, the expectational superiority of longtermist interventions depends heavily on these ‘Pascalian’ probabilities. So the case for longtermism may depend either on plausible but non-obvious empirical claims or on a tolerance for Pascalian fanaticism. (shrink)

If I am aware that p, say, that it is raining, is it the case that I must be aware that I am aware that p? Does introspective or object-awareness entail the apprehension of mental states as being of some kind or another: self-monitoring or intentional? That is, are cognitive events implicitly self-aware or is “self-awareness” just another term for metacognition? Not surprisingly, intuitions on the matter vary widely. This paper proposes a novel solution to this classical debate by reframing (...) the problem of self-knowledge in terms of the relation between phenomenal concepts and phenomenal knowledge. Concepts of consciousness such as “introspective awareness” and “reflexive self-awareness” are grounded in phenomenal experiences rather than physical events and processes. As such they yield a different kind of self-knowledge than what can be gained by applying externalist conceptual schemas to understanding the mind. I argue that Dharmakīrti’s theory of content can be seen as endorsing the efficacy of phenomenal experience as a vehicle for self-knowledge. (shrink)

Natural deduction (for short: nd-) calculi have not been used systematically as a basis for automated theorem proving in classical logic. To remove objective obstacles to their use we describe (1) a method that allows to give semantic proofs of normal form theorems for nd-calculi and (2) a framework that allows to search directly for normal nd-proofs. Thus, one can try to answer the question: How do we bridge the gap between claims and assumptions in heuristically motivated ways? This informal (...) question motivates the formulation of intercalation calculi. Ic-calculi are the technical underpinnings for (1) and (2), and our paper focuses on their detailed presentation and meta-mathematical investigation in the case of classical predicate logic. As a central theme emerges the connection between restricted forms of nd-proofs and (strategies for) proof search: normal forms are not obtained by removing local "detours", but rather by constructing proofs that directly reflect proof-strategic considerations. That theme warrants further investigation. (shrink)

The paper discusses tools for teaching logic used in Logic & Proofs, a web-based introduction to modern logic that has been taken by more than 1,300 students since the fall of 2003. The tools include a wide array of interactive learning environments or cognitive mini-tutors; most important among them is the Carnegie Proof Lab. The Proof Lab is a sophisticated interface for constructing natural deduction proofs and is central, as strategically guided discovery of proofs is the distinctive focus of the (...) course. My discussion makes explicit the broader intellectual context, but also the pursuit of pedagogical goals and their experimental examination. The intellectual context includes i) the theoretical background for the proof search algorithm AProS and its use for a dynamic Proof Tutor, and ii) the programmatic expansion of the course to Computational Logic. 1. (shrink)