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

Demokratische Systeme sind auch dann, wenn sie den grundrechtlichen Forderungen politischer Gerechtigkeit genügen, kritikwürdig und reformbedürftig, solange es ihnen nicht gelingt, allen Bürgern gerechte Anteile an den gesellschaftlich produzierten Reichtümern zu sichern. Bei den Auseinandersetzungen um Arbeitsmarktpolitik, Steuerpolitik, Erziehungspolitik, Meinungs- und Willensfreiheit steht auch Grundsätzlicheres zur Debatte: Was verstehen wir unter sozialer Gerechtigkeit? In Gerechtfertigte Ungleichheiten werden die Grundzüge einer Theorie sozialer Gerechtigkeit entwickelt. Es handelt sich um eine egalitäre Theorie, die soziale Ungleichheit ausdrücklich zulässt, diese aber an das Vorliegen (...) bestimmter öffentlicher Rechtfertigungsgründe bindet. (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)

Wilfred Sieg. Relative Consistency and Accesible Domains.

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

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)

Zuerst im 2CV, dann im umgebauten VW-Bus: Das Zürcher Fotografenpaar Margrit und Ernst Baumann, 1929 bzw. 1928 geboren, begann in den 1950er-Jahren rund um den Erdball zu reisen. Ihre Fotografien publizierten sie in Zeitschriften und Zeitungen wie Stern, Neue Zürcher Zeitung oder Das gelbe Heft und brachten so die Welt in die Wohnzimmer. Kosmopolitan und neugierig kamen sie zu Motiven mit Seltenheitswert: Farbporträts von Che Guevara gehören ebenso dazu wie Reportagen über die letzten Kopfjäger im ecuadorianischen Urwald. Ein Schwerpunkt dieser (...) reich bebilderten Doppelmonografie ist ihre Reise entlang der “Panamericana” 1957-1959, die Süd- und Nordamerika verbindet. Sie fotografierten Reportagen, Porträts und Landschaftsbilder und drehten den wohl ersten Farbfilm über den legendären Verkehrsweg. Dieser Film und ein Filporträt über die beiden Fotografen sind dem Buch auf DVD beigefügt. Das Buch präsentiert Funde aus ihrem Fotoarchiv - viele Bilder erstmals im Originalausschnitt - sowie Publikationen in Zeitschriften. Der Historiker und Schriftsteller Wilfried Meichtry erzählt vom ereignisreichen Leben des Paars, das sich zeitlebens als journalistische Handwerker verstanden hat. Der Fotohistoriker Markus Schürpf ordnet ihr Werk in die Foto- und Pressegeschichte ein, Nadine Olonetzky verfasst Begleittexte zu den Bildern. Eine Entdeckung in der Schweizer Fotogeschichte. (shrink)

Dieser Artikel widerspricht der Deutung der sog. Schriftkritik am Ende des Phaidros, wie sie von den Vertretern der esoterischen Platoninterpretation vorgetragen wird. In esoterischer Sicht handelt es sich um den einzigen Abschnitt in den Dialogen, der ausdrücklich auf eine mündliche Lehre Platons hinweist. Der Hinweis soll in Sokrates' These liegen, im Unterschied zu anderen Autoren übertreffe der Philosoph seine Schriften mit wertvolleren Inhalten, die er nur mündlich lehre.Der Einwand gegen diese Interpretation lautet, daß Platon den inhaltlichen Wertunterschied nicht zwischen Schriften (...) und mündlichen Reden von Philosophen sieht, sondern zwischen kunstlosen Reden und Schriften konventioneller Autoren einerseits und kunstgerechten, d.h. dialektischen Reden und Schriften von Philosophen andererseits. Philosophische Reden unterscheiden sich von philosophischen Schriften in formaler Hinsicht: nur Reden belehren, Schriften sind Erinnerungshilfe. (shrink)

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

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)

This commentary focuses on Hommel et al.'s inferences on action planning. It discusses the relevance of anticipated extrinsic movement effects for action control, the problems of a feature-based representation of actions, and the necessity of the acquisition of conditional movement-effect associations.

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)

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. Mechanical Procedures and Mathematical Experience.

By combining the methods of two former papers of ours we develop a finitary ordinal analysis of the axiom system KPi of Kripke-P atek set theory with an inaccessible universe. As a main result we obtain an upper bound for the provably recursive functions of KPi.

In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one.

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

Etwa seit Beginn der 90er Jahre des letzten Jahrhunderts hat die innerdeutsche wie die internationale Wahrheitsdebatte einen ausserordentlichen und so nicht vorhersehbaren Aufschwung genommen. Dies lasst sich parallel auf philosophischem wie auf theologischem Gebiet beobachten - ein geistesgeschichtlich durchaus bemerkenswerter, vielleicht einzigartiger Umstand. Differenzierte Reflexe dieser erneuten Aufmerksamkeit fur Eigenart und Bedeutung des Wahrheitsproblems bieten auch die vier Beitrage des vorliegenden Bandes. Aus religionsgeschichtlicher, exegetischer und fundamentaltheologischer Perspektive nahern sie sich je auf ihre Weise einer Antwort auf diejenigen Fragen, die (...) in diesem Zusammenhang den Kern des Problems ausmachen: Was ist Wahrheit, christlich verstanden? Was ist Christentum in Wahrheit? Ist das Christentum wahr? (shrink)