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: Conceptual Analysis.

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.

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)

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

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

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 Hume’s Of Miracles the person movecl by faith is put in a dilemma between faith and reason. Can one resolve this dilemma as a compleat Humean? The answer is yes. Within the Humean context different approaches can be developed ta overcome Hume’s dilemma. One uses Hume’s theory of utility to defend the belief in the afterlife. The other requires Hume to place faith on a par with beauty and therefore among the passions to which reason must be a slave. (...) Historically there was at least one compleat Humean who was also moved by faith --- Johann Georg Hamann [1730- 1788]. (shrink)

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

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)

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)

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)

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)

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)

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.

The paper discusses the problem of global distributive justice. It proposes to distinguish between principles for the domestic and for the global or intersocietal distribution of wealth. It is argued that there may be a plurality of partly diverging domestic conceptions of distributive justice, not all of which need to be liberal egalitarian conceptions. It is maintained, however, that principles regulating the intersocietal distribution of wealth have to be egalitarian principles. This claim is defended against Rawls's argument in The Law (...) of Peoples that egalitarian principles of distributive justice should not be applied globally. Moreover, it is explained in detail, why Rawls's “duty of assistance to burdened societies” cannot be an appropriate substitute for a global principle of distributive justice. (shrink)

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)

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)

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)

Wilfred Sieg. Mechanical Procedures and Mathematical Experience.

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)

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)

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

Wilfred Sieg. Relative Consistency and Accesible Domains.

We still lack an operational theory for a complete analysis of early socialization processes. Bourdieu has stressed their bodily dimension but has done so at the expense of more symbolic aspects. This theoretical option corresponds to a very general goal of the Bourdieusian theory of practice: analysing sociality without suffering an intellectualist bias. However, symbolic activity and socializing language in particular can be approached as a practical phenomenon (i.e. habitual, informal, unconscious, etc.). From this viewpoint, the sociology of embodiment may (...) have much to learn from cultural psychology, which has focused on early language as it is naturally performed – when people speak to children, and when children speak. The work of cultural psychologists on the role of everyday words and narratives in the making of the self can be extended through a sociological ethnography of language. The features and interests of such an ethnography are discussed in the final section of the article, on the basis of fieldwork conducted in a nursery. This final development illustrates and analyses the concrete appropriation of caregiver injunctions by 2- to 3-year-old children. (shrink)