In this programmatic paper we renew the well-known question “What is a proof?”. Starting from the challenge of the mathematical community by computer assisted theorem provers we discuss in the first part how the experiences from examinations of proofs can help to sharpen the question. In the second part we have a look to the new challenge given by “big proofs”.

In “The Book of the Crown (_Kitāb al-Iklīl_) of pseudo-Rhazes” Oliver Kahl and Henrietta Sharp Cockrell offer a facsimile edition, with annotated English translation and introductory study, of a unique and highly unusual medieval Arabic medical text.

A close analysis of truth and freedom in Heidegger’s Beiträge zur Philosophie is offered, demonstrating that an engagement with Friedrich Wilhelm Joseph vonSchelling is decisive in bringing Heidegger to an understanding of Dasein in terms of freedom, community, culture, and history. The controversial claim that a reconsideration of German Idealism can provide a new way of accessing Heidegger’s later work is strongly supported in this essay, demonstrating that the payoff of this approach is vast as well as highly coherent with (...) many recent developments in Heidegger scholarship which focus on practical philosophy. (shrink)

A close analysis of truth and freedom in Heidegger’s Beiträge zur Philosophie is offered, demonstrating that an engagement with Friedrich Wilhelm Joseph vonSchelling is decisive in bringing Heidegger to an understanding of Dasein in terms of freedom, community, culture, and history. The controversial claim that a reconsideration of German Idealism can provide a new way of accessing Heidegger’s later work is strongly supported in this essay, demonstrating that the payoff of this approach is vast as well as highly coherent with (...) many recent developments in Heidegger scholarship which focus on practical philosophy. (shrink)

In this study, we use a mixed methods research design to investigate how national cultural forces may impede or enhance the positive impact of females’ economic and political empowerment on increasing gender diversity of corporate boards. Using both a longitudinal correlation-based methodology and a configurational approach with fuzzy-set qualitative comparative analysis, we integrate theoretical mechanisms from gender schema and institutional theories to develop a mid-range theory about how female empowerment and national culture shape gender diversity on corporate boards around the (...) world. With our configurational approach, we conceptually and empirically model the complexity that is associated with the simultaneous interdependencies, both complementary and substitutive ones, between female empowerment processes and various cultural dimensions. Our findings contribute unique insights to research focused on board gender diversity as well as provide information for firm decision makers and policymakers about possible solutions for addressing the continuing issue of the underrepresentation of women on corporate boards. (shrink)

This article provides the proof-theoretic analysis of the transfinitely iterated fixed point theories $\widehat{ID}_\alpha and \widehat{ID}_{ the exact proof-theoretic ordinals of these systems are presented.

We give recursion-theoretic characterizations of the counting class \(\textsf {\#P} \), the class of those functions which count the number of accepting computations of non-deterministic Turing machines working in polynomial time. Moreover, we characterize in a recursion-theoretic manner all the levels \(\{\textsf {\#P} _k\}_{k\in {\mathbb {N}}}\) of the counting hierarchy of functions \(\textsf {FCH} \), which result from allowing queries to functions of the previous level, and \(\textsf {FCH} \) itself as a whole. This is done in the style of (...) Bellantoni and Cook’s safe recursion, and it places \(\textsf {\#P} \) in the context of implicit computational complexity. Namely, it relates \(\textsf {\#P} \) with the implicit characterizations of \(\textsf {FPTIME} \) (Bellantoni and Cook, Comput Complex 2:97–110, 1992) and \(\textsf {FPSPACE} \) (Oitavem, Math Log Q 54(3):317–323, 2008), by exploiting the features of the tree-recursion scheme of \(\textsf {FPSPACE} \). (shrink)

This article provides the proof-theoretic analysis of the transfinitely iterated fixed point theories $\widehat{ID}_\alpha and \widehat{ID}_{<\alpha};$ the exact proof-theoretic ordinals of these systems are presented.

In 2000, a draft note of David Hilbert was found in his Nachlass concerning a 24th problem he had consider to include in the his famous problem list of the talk at the International Congress of Mathematicians in 1900 in Paris. This problem concerns simplicity of proofs. In this paper we review the traces of this problem which one can find in the work of Hilbert and his school, as well as modern research started on it after its publication. We (...) stress, in particular, the mathematical nature of the problem.1. (shrink)

Falk Wagner was the leading Hegelian in late 20th century German Protestant theology. Wagner who had studied philosophy with Theodor W. Adorno and Wolfgang Cramer in Frankfurt am Main and Systematic Theology with Wolfhart Pannenberg at the University of Mainz taught Systematic Theology at the Universities of Munich and, since 1988, Vienna. He published several influential books. In his lectures and seminars he frequently handed out short theory papers to his students which should serve as the basis for debate. The (...) five texts presented here for the first time deal with the problems of individual in society and of the modern history of theology, with the doctrine of God, with the significance of the Bible for the systematic theology and finally with the concept of „Subjektivität“ in G. W. F. Hegel’s philosophy. These texts are an essential source for the development of Wagner’s thought when teaching at the university of Munich until 1988. (shrink)

This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Feferman's.

English summary: In this volume, the Munich and Vienna-based theologian Falk Wagner's work in the fields of philosophical theology, sociology and lived religion is taken up for the first time against the backdrop of current theological and philosophical controversies. The essays integrate Wagner's thinking into the history of theology and philosophy in the twentieth century, reconstruct fundamental elements of his philosophical theology between the poles of speculative theology and lived religion, while also highlighting the constellations and context wherein his theology (...) is situated. The authors explore from a work- and problem-historical perspective the contribution of the systematic theologian to the relevant twenty first century debates. German description: Der vorliegende Band thematisiert erstmals vor dem Hintergrund der gegenwartigen theologischen und philosophischen Kontroversen das Gesamtwerk des Munchener und Wiener Theologen Falk Wagner im Spannungsfeld von philosophischer Theologie, Sozialwissenschaften und gelebter Religion. Die einzelnen Beitrage ordnen dessen Denken in die theologie- und philosophiegeschichtliche Entwicklung des 20. Jahrhunderts ein, rekonstruieren grundlegende Aufbauelemente seiner philosophischen Theologie und beleuchten die Konstellationen, in denen seine Theologie steht. Dabei loten die Autoren sowohl in werkgeschichtlicher als auch in problemgeschichtlicher Perspektive den Beitrag des systematischen Theologen fur die theologischen Debatten im 21. Jahrhundert aus. Mit Beitragen von: Christine Axt-Piscalar, Ulrich Barth, Christian Danz, Wilhelm Grab, Kathrin Mette, Michael Murrmann-Kahl, Jan Rohls, Ewald Stubinger, Folkart Wittekind, Kurt Walter Zeidler, Manuel Zelger. (shrink)

We draw from resource dependence and institutional theories to explore how board characteristics associated with directors’ capacities to provide resources and legitimacy (i.e., board size, the number of non-executive, interlocking, and female directors) along with regulative, normative, and cultural-cognitive institutional conditions combine to shape firm environmental, social, and governance (ESG) performance. Using a process of configurational theorizing with fuzzy set qualitative comparative analysis and data from firms in 32 countries, we identify multiple equifinal configurations that are associated with high and (...) low ESG performance. We find that high and low ESG performance have different drivers due to complementarities among the presence and absence of board characteristics. Our results also show that the effectiveness (or not) of the bundles of boards’ characteristics for ESG performance varies across institutional contexts. By leveraging these findings to construct a typology of board archetypes that lead to high and low ESG performance, we offer novel theoretical and empirical insights to scholars as well as implications for practice. (shrink)

This anthology of articles provides contemporary international feminist perspectives on issues of identity, agency, and difference as they pertain to both feminist politics in particular, and contemporary western politics more generally.

Solidarity is one of the central concepts in social theory and has gained much attention due to the multiple challenges that the EU has been facing the last decade and due to the most recent COVID-19 pandemic. Although the debate on the nature and conditions of solidarity has been revitalized, there remains a large variety in how to conceptualize solidarity. In contrast to other approaches, we do not conceive solidarity as normative concept, but as descriptive–analytical one. Therefore, we provide a (...) theory-based definition that is prone to capture the empirical dimensions of solidarity. Accounting for the dynamic and interactive character of solidarity as subject to permanent societal and political renegotiation, we conceptualize solidarity from a discourse perspective and follow a multilevel design breaking down the understandings of solidarity on different levels. This approach contributes to the research of solidarity that is interested to capturing the ‘real world’ dimensions of solidarity. (shrink)

In 2000, Rüdiger Thiele [1] found in a notebook of David Hilbert, kept in Hilbert's Nachlass at the University of Göttingen, a small note concerning a 24th problem. As Hilbert wrote, he had considered including this problem in his famous problem list for the International Congress of Mathematicians in Paris in 1900.

The paper lists all the predictions and observations of eclipses of Jupiter's satellites 1668–1678 and compares them with modern computations of the these eclipses by J. H. Lieske. We discuss Roemer's method that led to his discovery of the retardment of light and finally we shall interpret Roemer's calculations.

In this note, we review paradoxes like Russell’s, the Liar, and Curry’s in the context of intuitionistic logic. One may observe that one cannot blame the underlying logic for the paradoxes, but has to take into account the particular concept formations. For proof-theoretic semantics, however, this comes with the challenge to block some forms of direct axiomatizations of the Liar. A proper answer to this challenge might be given by Schroeder-Heister’s definitional freedom.

In his introductory paper to first-order logic, Jon Barwise writes in the Handbook of Mathematical Logic :[T]he informal notion of provable used in mathematics is made precise by the formal notion provable in first-order logic. Following a sug[g]estion of Martin Davis, we refer to this view as Hilbert’s Thesis.This paper reviews the discussion of Hilbert’s Thesis in the literature. In addition to the question whether it is justifiable to use Hilbert’s name here, the arguments for this thesis are compared with (...) those for Church’s Thesis concerning computability. This leads to the question whether one could provide an analogue for proofs of the concept of partial recursive function. (shrink)

In 2000, Rüdiger Thiele [1] found in a notebook of David Hilbert, kept in Hilbert's Nachlass at the University of Göttingen, a small note concerning a 24th problem. As Hilbert wrote, he had considered including this problem in his famous problem list for the International Congress of Mathematicians in Paris in 1900.

Diagonalization is a transversal theme in Logic. In this work, it is shown that there exists a common origin of several diagonalization phenomena — paradoxes and Löb's Theorem. That common origin comprises a common reasoning and a common logical structure. We analyse the common structure from a philosophical point-of-view and we draw some conclusions.

We study the decidability of k-provability in \—the relation ‘being provable in \ with at most k steps’—and the decidability of the proof-skeleton problem—the problem of deciding if a given formula has a proof that has a given skeleton. The decidability of k-provability for the usual Hilbert-style formalisation of \ is still an open problem, but it is known that the proof-skeleton problem is undecidable for that theory. Using new methods, we present a characterisation of some numbers k for which (...) k-provability is decidable, and we present a characterisation of some proof-skeletons for which one can decide whether a formula has a proof whose skeleton is the considered one. These characterisations are natural and parameterised by unification algorithms. (shrink)

Kreisel’s conjecture is the statement: if, for all$n\in \mathbb {N}$,$\mathop {\text {PA}} \nolimits \vdash _{k \text { steps}} \varphi (\overline {n})$, then$\mathop {\text {PA}} \nolimits \vdash \forall x.\varphi (x)$. For a theory of arithmeticT, given a recursive functionh,$T \vdash _{\leq h} \varphi $holds if there is a proof of$\varphi $inTwhose code is at most$h(\#\varphi )$. This notion depends on the underlying coding.${P}^h_T(x)$is a predicate for$\vdash _{\leq h}$inT. It is shown that there exist a sentence$\varphi $and a total recursive functionhsuch that$T\vdash (...) _{\leq h}\mathop {\text {Pr}} \nolimits _T(\ulcorner \mathop {\text {Pr}} \nolimits _T(\ulcorner \varphi \urcorner )\rightarrow \varphi \urcorner )$, but, where$\mathop {\text {Pr}} \nolimits _T$stands for the standard provability predicate inT. This statement is related to a conjecture by Montagna. Also variants and weakenings of Kreisel’s conjecture are studied. By the use of reflexion principles, one can obtain a theory$T^h_\Gamma $that extendsTsuch that a version of Kreisel’s conjecture holds: given a recursive functionhand$\varphi (x)$a$\Gamma $-formula (where$\Gamma $is an arbitrarily fixed class of formulas) such that, for all$n\in \mathbb {N}$,$T\vdash _{\leq h} \varphi (\overline {n})$, then$T^h_\Gamma \vdash \forall x.\varphi (x)$. Derivability conditions are studied for a theory to satisfy the following implication: if, then$T\vdash \forall x.\varphi (x)$. This corresponds to an arithmetization of Kreisel’s conjecture. It is shown that, for certain theories, there exists a functionhsuch that$\vdash _{k \text { steps}}\ \subseteq\ \vdash _{\leq h}$. (shrink)