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.

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

Wilfred Sieg. Mechanical Procedures and Mathematical Experience.

Wilfred Sieg. Aspects of Mathematical Experience.

Novovjekovni ideal demokracije temelji se na apstraktnog kategoriji humaniteta, koja svoje uporište nalazi u univerzalnim vrijednostima i jednakosti ljudi kao nositelja tih vrijednosti. Prepoznatljivo obilježje takve apstraktne osobe je vezana uz racionalnost. Međutim, razumijevanje demokracije u modernosti, koje izjednačava pojedince apstrahirajući od svakog funkcionalnog/ povijesnog obilježja, predalo je svoju ulogu diskursima »razlike« i »identiteta«, što je postalo očitim naročito nakon atmosfere neizvjesnosti u 80-im godinama prošloga stoljeća. Stoga je glavna os suvremenih rasprava u političkoj filozofiji određena bipolarnošću partikularnosti i univerzalnosti. (...) S jedne strane osi nalazi se pojam univerzalnosti koji naglašava bivanje kao bivanje bestjelesnim i potpuno konceptualnim, no s druge strane se nalaze partikularizmi koji revitaliziraju ontologiju biti, koja njih same čini robovima partikularnog. U ovom radu nastojim otkriti vezu između univerzalnosti i partikularnosti na putu kritike ontologije biti. Problem s kojim se suočavam nije onaj odabira između univerzalnog i partikularnog, već je onaj prikazivanja neuništive međusobne veze između univerzalnog i partikularnog.The democracy ideal of the Modern Age is grounded on an abstract humanity category, which is based on universal values and on the equity of people who are bearer of these values. The distinctive characteristic of such an abstract person is related to rationality. However, the modernity’s understanding of democracy, which equalizes individuals through abstracting them from all of their functional/historical characteristics, left its role to the discourses of “differences” and “identity” and became obvious especially after the 1980s’ atmosphere of uncertainty. Therefore the main axis of the contemporary political philosophy debates is determined by a bipolarity of particularity and universality. On one side of the axis there is a concept of universality which emphasizes the being as being bodiless and totally conceptual, however on the other side there are particularisms that revitalize the ontology of essence which renders themselves a slave of the particular. In this study I aim to unfold the relationship between universality and particularity on the path of the critique of ontology of essence. The problem I handle here is not one of choosing between universal and particular, rather it is one of depicting the indestructible mutual connection between universal and particular.L’idéal démocratique de l’époque moderne est ancré sur une catégorie abstraite d’humanité qui trouve son fondement dans les valeurs universelles et dans l’égalité des hommes qui portent ces valeurs. Le caractère distinctif d’une telle personne abstraite est lié à la rationalité. Cependant, la compréhension moderne de la démocratie, qui égalise les individus en les extrayant de toute caractéristique fonctionnelle/historique, a laissé place aux discours de « différences » et d’ « identité », ce qui est devenu évident notamment après l’ambiance d’incertitude dans les années 80. Ainsi, l’axe principal des débats de philosophie politique contemporains est déterminé par la bipolarité de la particularité et de l’universalité. D’un côté de l’axe se trouve un concept d’universalité qui met l’accent sur l’être comme étant immatériel et complètement conceptuel, mais de l’autre côté se trouvent des particularismes qui revitalisent l’ontologie de l’essence qui les assujettit au particulier. Dans cette étude, je vise à découvrir le lien entre universalité et particularité sur la trace de la critique de l’ontologie de l’essence. Le problème dont je traite ici n’est pas celui de choisir entre l’universel et le particulier, mais plutôt de décrire la relation réciproque indestructible entre l’universel et le particulier.Das neuzeitliche Ideal der Demokratie gründet auf der abstrakten Kategorie der Humanität, die auf universalen Werten bzw. der Gleichrangigkeit der Menschen als diesbezügliche Werteträger aufbaut. Das Unterscheidungsmerkmal einer solchen abstrakten Person steht im Konnex mit der Rationalität. Das modernitätsbezogene Demokratieverständnis hingegen, welches die Individuen mittels deren Abstrahierung von sämtlichen funktionellen/geschichtlichen Charakteristika gleichstellt, übergab dessen Rolle den Diskursen der „Gegensätze“ bzw. der „Identität“ und wurde insbesondere augenfällig im Anschluss an die Atmosphäre der Ungewissheit der 1980er-Jahre. Dementsprechend wird die Hauptachse der zeitgenössischen Debatten in der politischen Philosophie durch Zweipoligkeit der Partikularität und Universalität determiniert. Auf der einen Seite der Achse ist der Universalitätsbegriff positioniert, der das Sein als körperlos und durchweg konzeptuell akzentuiert, während auf der gegenüberliegenden Seite Partikularismen ruhen, welche die Ontologie des Wesens neu beleben, die alsdann sie selbst zu Sklaven des Partikularen macht. In dieser Arbeit ziele ich darauf ab, das Verhältnis zwischen Universalität und Partikularität abzuhandeln, und zwar auf dem Wege der Kritik der Ontologie des Wesens. Das Problem, das ich hiermit behandle, figuriert nicht als jenes der Wahl zwischen Universalem und Partikularem, sondern vielmehr als jenes der Veranschaulichung des unverwüstlichen gegenseitigen Zusammenspiels zwischen Universalem und Partikularem. (shrink)

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

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 and Mark Ravaglia. David Hilbert and Paul Bernays, Grundlagen der Mathematik I and II: A Landmark.

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)

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)

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

Wilfried Sieg. Formal Systems, Church Turing Thesis, and Gödel's Theorems: Three Contributions to The MIT Encyclopedias of Cognitive Science.

Wilfred Sieg and Saverio Cittadini. Normal Natural Deduction Proof (In Non-Classical Logics.

Wilfred Sieg and Rosella Lupiccini. Computing Machines: Entry for the Second Edition of the Encyclopedia of Philsophy.

Wilfred Sieg. Effectiveness and Provability.

Wilfred Sieg and John Byrnes. Generalizing Turing's Machine and Arguments.

Wilfred Sieg and Stanley S. Wainer. Program Transformation and Proof Transformation.

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.

Wilfried Sieg and John Byrnes. Gödel, Turing, and K-Graph Machines.

Wilfred Sieg. Intercalculation Calculi for Classical Logic.

Wilfred Sieg. Mechanisms and Search: Aspects of Proof Theory.

This is a summary of developments analysed in my (1997A). A first version of that paper was presented at the workshop Modern Mathematical Thought in Pittsburgh (September 21-24, 1995).

Wilfred Sieg. “Church's Thesis”, “Consistency”, “Formalization”, “Proof Theory”: Dictionary Entries.

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

Any thorough discussion of computing machines requires the examination of rigorous concepts of computation and is facilitated by the distinction between mathematical, symbolic and physical computations. The delicate connection between the three kinds of computations and the underlying questions, "What are machines?" and "When are they computing?", motivate an extensive theoretical and historical discussion. The relevant outcome of this..

— via appropriate substitutions — syntactically identical. The method can be applied directly to quantifierfree formulae and, in this paper, will b e extended in a natural and strai ghlforward way to quantified formulae.

Wilfred Sieg. Relative Consistency and Accesible Domains.

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

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)

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)

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)

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)

Richard Scheines and Wilfred Sieg. Computer Environments for Proof Construction.