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)

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)

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

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.

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)

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

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

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)

Several tendencies in the process of political decision-making in Belgium tend to put under strain the official model of representative democracy. One of these evolutions is the set up of many advisory councils, and especially the dynamism of some of these advisory councils to move up to participation in the decision-making itself.The process starts at the functional basis of the advisory councils: an information function, an advice function, a function to make proposals, an investigation function and a selection function. It (...) is possible to get to the care of the decision for reason of a liability to advice, the quality as regards content of the advice, the composition of the councils and the appeal to the public opinion. The dysfunctions are the obstacles that need to be advoided: the alibi function, a basis for non-decision, the delay and concealing of the decision-making itself, a certain obliqueness in the composition and the undermining of the representative function of parliament. (shrink)

The Belgian polyarchy has developed into a «particracy» in which the political parties have a preponderant impact on the political decisionmaking.As in every polyarchy, the political parties fulfill a set of vital functions in the Belgian system. But that does not explain its «particratie» particularity. The particracy arises as a consequence of four factors. The parties are fixed by the «zuilen» into an «internal-imperialistic» role. The power of the parties increased considerably, an increase of which the politization of the instruments (...) for, and the agencies of, policy is one of the most important, together with the capability of making and unmaking issues. The impact and power of the opponents of the parties in the decision-making process have been diminished to a considerable extent. This weakening involves the king, the parliament, theadministration, the electorate, and, ultimately, the cabinet because of the growing impact of the parties on policy design and policy output as well as on the appointment of ministers. Finally, the choice of the consociational pattern of politics of accomodation by the elite indicates the instrumental function of particracy in Belgian politics. (shrink)

Wilfred Sieg. Effectiveness and Provability.

English summary: Does pastoral care require ethical competence in order to fulfil its task properly? And, conversely, to what extent does ethical reflection require feedback from pastoral experience in order to remain true to life and of practical relevance? Wilfried Sturm examines the way hospital pastors deal with ethical challenges and conflict situations in neonatal clinics and enquires into the insights that can be gained from them for the relationship of pastoral care and ethics. German description: Braucht Seelsorge ethische (...) Kompetenz, um ihrem Auftrag gerecht zu werden? Inwiefern bedarf umgekehrt die ethische Reflexion der Ruckkopplung an seelsorgerliche Erfahrung, um lebensnah und praxisrelevant zu bleiben? Wilfried Sturm untersucht den Umgang von Klinikseelsorgerinnen und -seelsorgern mit ethischen Herausforderungen und Konfliktsituationen in der Fruh- und Neugeborenenmedizin und fragt, welche Einsichten sich daraus fur das Verhaltnis von Seelsorge und Ethik gewinnen lassen. (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.

Using the concept of notations for infinitary derivations we give an explanation of Takeuti's reduction steps on finite derivations (used in his consistency proof for Π1 1-CA) in terms of the more perspicious infinitary approach from [BS88].

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)

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)

Wilfred Sieg. Mechanical Procedures and Mathematical Experience.

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)

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)

In this article we prove that a certain intuitionistic version of the well-known fixed point theory \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\widehat{\rm ID}_1$\end{document} is conservative over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mbox{\sf HA}$\end{document} for almost negative formulas.

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

La dissolution assez inattendue des Chambres, le 24 septembre 1971 impliquait une promesse d'élection directe du gouvernement. Cette promesse s'est cependant vite volatilisée. Des élections calmes et traditionnelles ont pris le relais, le gouvernement ayant choisi uniquement le moment le moins défavorable pour les partis de la majorité.La composition des listes s'effectue dans un plus grand calme que de coutume et se déroule davantage par le biais de petits comités ou d'assemblées peu nombreuses. Cependant un grand changement de personnel se (...) réalise par le retrait plus au mains forcé de 65 parlementaires sortants; ce chiffre est, d'après la tradition établie depuis plusieurs élections, très élevé.En dehors de Bruxelles la campagne est calme. Une personnification se manifeste de plus en plus, aussi bien en Wallonie qu'en Flandre.En ce qui concerne la personnification, Bruxelles a une longueur d'avance. Le penchant pour l'«argumentation» symbolique est une autre caractéristique de la campagne. On ne s'oppose pas sur les points litigieux, on ne les expose même pas.Les résultats électoraux en W allonie et à Bruxelles diffèrent fondamentalement de ceux de la Flandre. En Wallonie et à Bruxelles de grands changements se font jour : le P.L.P. est le grand perdant tandis que le F.D.F./R.W. sort des élections en grand vainqueur. En Flandre, par contre, il y a peu de changements dans les chiffres globaux, mais de région à région de petits gains et de petites pertes sant enregistrés.Ceux-ci cependant vont dans des sens divers et même souvent radicalement opposés. L'explication de ces changements reste encore à fournir. Une fois de plus, le nombre des votes blanc et non valables à augmenté. La composition de la Chambre des Représentants et du Sénat issue de ces élections ne permet qu'une bipartite entre C.V.P. -P.S.C. et P.S.B. Les trois partis traditionnels possèdent encore numériquement plus des 2/3 des sièges. Certaines complications dans la réorganisation de l'Etat établie par la révision constitutionnelle, se font déjà entrevoir. Et cela n'est qu'un des problèmes que ces élections ont portés à l'attention de l'opinion publique. (shrink)

Abstract:The shutdown due to the COVID-19 pandemic has accelerated the implementation of digital technology to advance many new applications. Digital applications were believed to be indispensable for changes in learning environments and strategies that would enhance the capacity and quality of learning through focused motivation, communicative interaction, and stronger self-determination. This text will discuss prominent arguments for digital learning and digital technologies that might initiate a digital turn. To this end, this paper reflects on the psychological and mental conditions of (...) human learning, evaluates the potential opportunities of digital tools within the context of teaching and learning, and concludes with consequences for music learning in public schools. (shrink)

Was ist Religion in der Religionsphilosophie von Heinrich Scholz? / Theodor Mahlmann -- Husserls phänomenologischer Zugangsweg zu den Problemen der Religion / Ludwig Landgrebe -- Religion, Religionsphilosophie und Theologie bei Geradus van der Leeuw / Hubertus G. Hubbeling -- Anders Nygrens Religionsbegriff / Hampus Lyttkens -- Charles S. Peirce's Religionsphilosophie als Metaphysik der Evolution / Hermann Deuser -- Die theoretische Theologie der Prozessphilosophie Whiteheads / Ingolf U. Dalferth -- Metaphysik und Religion im Spätwerk Knud Logstrups / Svend Andersen.

Keine ausführliche Beschreibung für "Sein und Gnade" verfügbar.

Das Werk ist die erste komplett an allen Gebieten der Physik systematisch ausgerichtete Darstellung der Geschichte der Physik. Geschrieben von einem erfahrenen Didaktiker seines Faches und als Ergebnis jahrzehntelanger Recherche soll dieses Buch ein tieferes Verständnis seines Faches vermitteln, als dies traditionelle Lehrbücher vermögen. Der rote Faden sind die Konzepte und Theoriebildungen, wie sie sich durch die Jahrhunderte nachzeichnen lassen. Das Ergebnis ist ein lebendig geschriebenes Fachbuch, das sich sowohl zum Lernen und Nachschlagen systematisch nutzen wie auch als packende Physikgeschichte (...) von der ersten bis letzten Seite spannend lesen lässt. (shrink)

Thema der vorliegenden Arbeit ist der dialektische Gehalt der wichtigsten bildungstheoretischen Abhandlungen des padagogischen Theoretikers Josef Derbolav. Ergebnis der zum grossten Teil immanent durchgefuhrten Analyse ist der Aufweis, dass Derbolav die dialektische Fassung didaktischer, anthropologischer und handlungstheoretischer Prinzipien aufgrund metaphysischer Denkstrukturen nicht oder nur teilweise gelingt. Inwieweit einige dieser Prinzipien dialektisches Potential enthalten konnten, zeigt der Verfasser anhand der Entwicklung eines eigenen, an der hermeneutischen Philosophie orientierten dialektischen Standpunktes.".