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”.

Properly understood, Kant’s moral philosophy is incompatible with constitutivism. According to the constitutivist, being subject to the moral law cannot be a matter of free choice, and failure to c...

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)

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle and I (...) apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions. (shrink)

Life's Solution builds a persuasive case for the predictability of evolutionary outcomes. The case rests on a remarkable compilation of examples of convergent evolution, in which two or more lineages have independently evolved similar structures and functions. The examples range from the aerodynamics of hovering moths and hummingbirds to the use of silk by spiders and some insects to capture prey. Going against the grain of Darwinian orthodoxy, this book is a must read for anyone grappling with the meaning of (...) evolution and our place in the Universe. Simon Conway Morris is the Ad Hominen Professor in the Earth Science Department at the University of Cambridge and a Fellow of St. John's College and the Royal Society. His research focuses on the study of constraints on evolution, and the historical processes that lead to the emergence of complexity, especially with respect to the construction of the major animal body parts in the Cambrian explosion. Previous books include The Crucible of Creation (Getty Center for Education in the Arts, 1999) and co-author of Solnhofen (Cambridge, 1990). Hb ISBN (2003) 0-521-82704-3. (shrink)

This chapter explores some fundamental consequences of the correspondence between physical process and computations. Most physical questions may be answerable only through irreducible amounts of computation. Those that concern idealized limits of infinite time, volume, or numerical precision can require arbitrarily long computations, and so be considered formally undecidable. The behavior of a physical system may always be calculated by simulating explicitly each step in its evolution. Much of theoretical physics has, however, been concerned with devising shorter methods of calculation (...) that reproduce the outcome without tracing each step. Computational irreducibility is common among the systems investigated in mathematics and computation theory, but it may well be the exception rather than the rule, since most physical questions may be answerable only through irreducible amounts of computation. (shrink)

We investigate connections between the syntactic and semantic distance of programs on an abstract, recursion theoretic level. For a certain rather restrictive notion of interdependency of the two kinds of distances, there remain only few and “unnatural” numberings allowing such close relationship. Weakening the requirements leads to the discovery of universal metrics such that for an arbitrary recursively enumerable family of functions a numbering compatible with such a metric can uniformly be constructed. We conclude our considerations with some implications on (...) learning theory. (shrink)

On the one hand, means of transport can be considered as media which shape the perception of space; on the other, they can be considered as milieus which produce certain forms of social interaction. In order to relate both perspectives to each other, the present contribution outlines a topology of vehicles, drawing upon contemporary French literature set in cities. Their detailed representation of certain means of transport shows that literary texts not only decipher modes of spatial perception that are specific (...) to certain vehicles, but also devise new ways of using vehicles through their involvement with an outdated culture of transport. German Verkehrsmittel lassen sich als Medien betrachten, die auf die Wahrnehmung des Raums einwirken, aber auch als Milieus, die bestimmte Formen sozialer Interaktion erzeugen. Um beide Perspektiven aufeinander zu beziehen, umreißt der Beitrag eine Topologie der Fahrzeuge anhand von Stadttexten aus der französischen Literatur der Gegenwart. Aus deren eingehender Darstellung bestimmter Verkehrsmittel geht hervor, dass literarische Texte nicht allein fahrzeugspezifische Weisen der Raumerfahrung entziffern, sondern darüber hinaus auch in Auseinandersetzung mit einer überkommenen Transportkultur originelle Praktiken des Fahrzeuggebrauchs ersinnen. (shrink)

Recent literature describing the process and pathways of the agrarian transition in Southeast Asia suggests that the rise of agricultural intensification and the growth of commodity markets will lead to the demise of swidden agriculture. This paper offers a longitudinal overview of the conditions that drive the agrarian transition amongst indigenous swidden cultivators and migrant paddy farmers in central Palawan Island, the Philippines. In line with regional agrarian change, we describe how a history of conservation policies has criminalized and pressured (...) swidden farmers to adopt more intensive “modern” agricultural practices. We examine how indigenous swidden cultivators adjust their practice in response to recent changes in policies, security of harvests, and socio-cultural values vis-à-vis intensification. Rather than suggest that this transition will lead to the demise of swidden, results reveal that farmers instead negotiate a shifting ground in which they lean on and value swidden as a means of negotiating agrarian change. (shrink)

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle and I (...) apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions. (shrink)

***Angaben zur beteiligten Person Kann: Christoph Kann ist Professor für Philosophie an der Universität Düsseldorf.

Name der Zeitschrift: Neue Zeitschrift für Systematische Theologie und Religionsphilosophie Jahrgang: 58 Heft: 3 Seiten: 325-336.

Name der Zeitschrift: Neue Zeitschrift für Systematische Theologie und Religionsphilosophie Jahrgang: 57 Heft: 4 Seiten: 501-520.