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.

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)

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

Die Logik, einst Grundbestand der Philosophie, hat sich in unserem J ahrhundert zu einer selbstandigen Wissenschaft ent 1 wickelt. Sie wird zur Abgrenzung gegeniiber friiheren Gestalten "neue Logik" oder "moderne Logik" genannt; auch sind Be zeichnungen wie "theoretische Logik", "mathematische Logik", "symbolische Logik", "formale Logik", "Logistik" gebrauch 2 lich, urn die Eigenart der grundlegenden Methoden anzudeuten. Die genannten Benennungen werden in dieser Arbeit gleichbe rechtigt gebraucht. Mit 1. M. BOCHENSKI und A. MENNE kann man unter Logistik in weiterem Sinne "die (...) Lehre vom Logikkalkiil, seinen Voraus 3 setzungen und Anwendungen" verstehen. Die Absicht dieser Arbeit ist eine philosophische; sie fragt nach dem, was die Aussage als Aussage zum Element positiver Logik werden laBt. In Frage ist daher die Aussagen- und (ele mentare) Pradikatenlogik, die in formaler Hinsicht "durch den Vollstandigkeitssatz (Go DEL I930) und den Unentscheidbarkeits- 1 Texte zur Geschichte der modernen Logik: I.M. BOCHENSKI, Formale Logik, Freiburg-Miinchen I956, S. 3II-477. 2 s. z.B.: B. JUH6s, Elemente der neuen Logik, Wien I954. G. HASEN]AEGER, Einfuhrung in die Grundbegritfe und Probleme der modernen Logik, Freiburg-Miinchen I962. D. HILBERT und W. ACKERMANN, Grundzi~ge der theoretischm Logik, 3. Aufl., B- lin-Gi:ittingen-Heidelberg I949. H. SCHOLZ und G. HASEN]AEGER, Grundzuge der mathematischen Logik, Berlin 1961. H. HERMES, Einfuhrung in die mathematische Logik, Stuttgart 1963. R. CARNAP, Symbolische Logik, Wien 1954. P. LORENZEN, Formale Logik. I.M. BOCHENSKI, GrundrifJ der Logistik, aus dem Franzi:isischen iibers., neubearb. und erweitert von A. MENNE, 2. erweiterte Auf I., Paderborn 1962. (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)

There is a tension between the idea of popular sovereignty and our understanding that basic constitutional rights and liberties have a normative authority which is independent from the results of democratic decision‐making procedures. On the one hand there is the claim that the content of political justice, at least as far as the basic liberties are concerned, is to be fixed solely by substantive moral and political argument, while on the other there is the claim that it is the people (...) who determine the specific scheme of basic liberties that they live under. The apparent tension between the two claims can be resolved by introducing, firstly, a distinction between the concept of justice and the concept of legitimacy and, secondly, between proto‐rights and constitutional rights. Unlike constitutional rights, proto‐rights are moral, not legal, rights, and they are less specific than constitutional rights: one set of proto‐rights allows for various schemes of legal constitutional rights. Proto‐rights are part of the principles of justice and can be established only by substantive argument while constitutional rights have to be established by the procedures of democratic legitimization, and, therefore, they are a result of the exercise of popular sovereignty. The correlated distinctions between justice and legitimacy, and between proto‐rights and constitutional rights, allow us to specify the scope of the two claims and, hence, to affirm both simultaneously. The philosophical and legal relevance of the distinctions made is elucidated by a brief discussion of the constitutional rights of freedom of speech in the US and Germany. (shrink)

In his classic paper On Computable Numbers Turing analyzed what can be done by a human computor in a routine, “mechanical” way. He argued that mechanical op-erations obey locality conditions and are carried out on configurations satisfying boundedness conditions. Processes meeting these restrictive conditions can be shown to be computable by a Turing machine. Turing viewed memory limitations of computors as the ultimate reason for the restrictive conditions. In contrast, Gandy analyzed in his paper Church’s Thesis and Principles for Mechanisms (...) what can be done by “discrete deterministic mechanical devices” and appealed to physical considerations to motivate restrictive principles. These devices include, crucially, ones that carry out parallel processes, e.g., cellular automata. Following Turing’s methodological ways and guided by Conway’s “Game of Life” as a paradigm, Gandy formulated four restrictive principles and proved that devices satisfying them are computationally equivalent to Turing machines. (shrink)

[1] Textteil : Grundlagen, Prozeduren, Evaluation -- [2] Anhang : Interviewleitfaden, Tonbandtranskript und Auswertungsbeispiele.

§30. Significance of Desargues's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 CHAPTER VI. PASCAL'S THEOREM. §31. ...

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)

The major theoretical framework relative to the perception of illusory figures is reviewed and discussed in the attempt to provide a unifying explanatory account for these phenomena.

Following the pioneering studies of the receptive field (RF), the concept gained further significance for visual perception by the discovery of input effects from beyond the classical RF. These studies demonstrated that neuronal responses could be modulated by stimuli outside their RFs, consistent with the perception of induced brightness, color, orientation, and motion. Lesion scotomata are similarly modulated perceptually from the surround by RFs that have migrated from the interior to the outer edge of the scotoma and in this way (...) provide filling-in of the void. Large RFs are advantageous to this task. In higher visual areas, such as the middle temporal and inferotemporal lobe, RFs increase in size and lose most of their retinotopic organization while encoding increasingly complex features. Whereas lowerlevel RFs mediate perceptual filling-in, contour integration, and figure–ground segregation, RFs at higher levels serve the perception of grouping by common fate, biological motion, and other biologically relevant stimuli, such as faces. Studies in alert monkeys while freely viewing natural scenes showed that classical and nonclassical RFs cooperate in forming representations of the visual world. Today, our understanding of the mechanisms underlying the RF is undergoing a quantum leap. What had started out as a hierarchical feedforward concept for simple stimuli, such as spots, lines, and bars, now refers to mechanisms involving ascending, descending, and lateral signal flow. By extension of the bottom-up paradigm, RFs are nowadays understood as adaptive processors, enabling the predictive coding of complex scenes. Top-down effects guiding attention and tuned to task-relevant information complement the bottom-up analysis. (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.".

In this rapid commentary, a mini-review is given of the present state-of-knowledge regarding the etiology and epidemiology of the new coronavirus 2019-nCoV and the risks for developing Acute respiratory distress syndrome. The available knowledge on the viral genomics, molecular biology and pathogenicity of viruses of the Coronaviridae family and other Nidovirales, forms a helpful template for understanding the present pandemic outbreak. However, important questions remain unanswered about the underlying mechanism causing the very high case fatality ratios and mechanisms regarding severe (...) reactions like ARDS, fatal cardiac and renal failures, associated with a number of important comorbidity factors. Immunological reactions to lung alveoles in particular in late phase ARDS in SARS-like CoV diseases, so far may not have received enough attention. Finally a shortlist of questions for high priority further research is suggested. (shrink)

Die theoretische Logik, auch mathematische oder symbolische Logik genannt, ist eine Ausdehnung der fonnalen Methode der Mathematik auf das Gebiet der Logik. Sie wendet fUr die Logik eine ahnliche Fonnel­ sprache an, wie sie zum Ausdruck mathematischer Beziehungen schon seit langem gebrauchlich ist. In der Mathematik wurde es heute als eine Utopie gelten, wollte man beim Aufbau einer mathematischen Disziplin sich nur der gewohnlichen Sprache bedienen. Die groBen Fortschritte, die in der Mathematik seit der Antike gemacht worden sind, sind zum (...) wesentlichen Teil mit dadurch bedingt, daB es gelang, einen brauchbaren und leistungsfahigen Fonnalismus zu finden. - Was durch die Formel­ sprache in der Mathematik erreicht wird, das solI auch in der theoretischen Logik durch diese erzielt werden, namlich eine exakte, wissenschaftliche Behandlung ihres Gegenstandes. Die logischen Sachverhalte, die zwischen Urteilen, Begriffen usw. bestehen, finden ihre Darstellung durch Formeln, deren Interpretation frei ist von den Unklarheiten, die beim sprachlichen Ausdruck leicht auftreten konnen. Der Dbergang zu logischen Folgerungen, wie er durch das SchlieBen geschieht, wird in seine letzten Elemente zerlegt und erscheint als fonnale Umgestaltung der Ausgangsfonneln nach gewissen Regeln, die den Rechenregeln in der Algebra analog sind; das logische Denken findet sein Abbild in einem LogikkalkUl. Dieser Kalkiil macht die erfolgreiche Inangriffnahme von Problemen moglich, bei denen das rein inhaltliche Denken prinzipiell versagt. Zu diesen gehort z. B. (shrink)

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.