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)

We give a survey on truth theories for applicative theories. It comprises Frege structures, universes for Frege structures, and a theory of supervaluation. We present the proof-theoretic results for these theories and show their syntactical expressive power. In particular, we present as a novelty a syntactical interpretation of ID1 in a applicative truth theory based on supervaluation.

In this paper, we study a concept of universe for a truth predicate over applicative theories. A proof-theoretic analysis is given by use of transfinitely iterated fixed point theories . The lower bound is obtained by a syntactical interpretation of these theories. Thus, universes over Frege structures represent a syntactically expressive framework of metapredicative theories in the context of applicative theories.

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.

In this paper we summarize some results about sets in Frege structures. The resulting set theory is discussed with respect to its historical and philosophical significance. This includes the treatment of diagonalization in the presence of a universal set.

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”.

We study the logical relationship of various forms of induction, as well as quantification operators in applicative theories. In both cases the introduced notion of $\hbox{\sf N}$ -strictness allows us to obtain the appropriate results.

In this book, Andy Mueller examines the ways in which epistemic and practical rationality are intertwined. In the first part, he presents an overview of the contemporary debates about epistemic norms for practical reasoning, and defends the thesis that epistemic rationality can make one practically irrational. Mueller proposes a contextualist account of epistemic norms for practical reasoning and introduces novel epistemic norms pertaining to ends and hope. In the second part Mueller considers current approaches to pragmatic encroachment in epistemology, ultimately (...) arguing in favor of a new principle-based argument for pragmatic encroachment. While the book defends tenets of the knowledge-first programme, one of its main conclusions is thoroughly pragmatist: in an important sense, the practical has primacy over the epistemic. (shrink)

We give a reading of binary necessity statements of the form “ϕ is necessary for ψ” in terms of proofs. This reading is based on the idea of interpreting such statements as “Every proof of ψ uses ϕ”.

In this paper we introduce applicative theories which characterize the polynomial hierarchy of time and its levels. These theories are based on a characterization of the functions in the polynomial hierarchy using monotonicity constraints, introduced by Ben-Amram, Loff, and Oitavem.

Mario Bunge forcefully argues for Dual Axiomatics, i.e., an axiomatic method applied to natural sciences which explicitly takes into account semantic aspects of the concepts involved in an axiomatization. In this paper we will discuss how dual axiomatics is equally important in mathematics; both historically in Hilbert and Bernays’s conception as well as today in a set-theoretical environment.

This source publication of one of the earliest Arabic pharmacopoeias is framed by a philological analysis of the unique manuscript, a biographical sketch of its author, an etymological survey of terms, a glossary, and indices.

In seiner Zeit am Institute of Advanced Study wurde Kurt Gödel ein enger Freund von Albert Einstein und hat sich insbesondere mit der Relativitätstheorie beschäftigt. Ein Ergebnis dieser Untersuchung war die Entdeckung des sogenannten Gödel-Universums (Genau genommen handelt es sich um eine ganze Klasse von Universen; die Singularform steht hier als Kollektivum.), einem Modell der Einsteinschen Feldgleichungen der allgemeinen Relativitätstheorie, in dem sich keine absolute Zeit definieren lässt – theoretisch wären in einem solchen Universum Zeitreisen möglich.

In this paper, we discuss the question how Peano’s Arithmetic reached the place it occupies today in Mathematics. We compare Peano’s approach with Dedekind’s account of the subject. Then we highlight the role of Hilbert and Bernays in subsequent developments.

Gerhard Gentzen has been described as logic’s lost genius, whom Gödel called a better logician than himself. This work comprises articles by leading proof theorists, attesting to Gentzen’s enduring legacy to mathematical logic and beyond. The contributions range from philosophical reflections and re-evaluations of Gentzen’s original consistency proofs to the most recent developments in proof theory. Gentzen founded modern proof theory. His sequent calculus and natural deduction system beautifully explain the deep symmetries of logic. They underlie modern developments in computer (...) science such as automated theorem proving and type theory. (shrink)

Es ist bekannt, dass Kurt Gödel im „mathematischen Realismus“ seine philosophische Grundposition wiedergegeben sah. Zielsetzung dieses Essays ist es zu zeigen, dass sich auch umgekehrt ein mathematischer Realismus praktisch zwangsläufig als eine Konsequenz aus dem Unvollständigkeitssatz ergibt. Im Weiteren zeigt sich damit eine unüberwindliche Hürde für eine reduktionistische Mathematik.

“[T]here is something rotten at the heart of medicine” —this is one of the central statements of Jeffrey Paul Bishop in his book The Anticipatory Corpse. Medicine, Power and the Care of the Dying. The obvious, if somewhat morbid, thought that “rotten” would refer to the decaying body as the central subject of investigation is, however, misleading. Instead, Bishop aims to demonstrate that the modern trend of medicalizing dying and death is the wrong way.The book explores contemporary medicine’s practices, their (...) historical evolvement, and their underpinnings with regard to the care of the dying. Informed by Foucault’s genealogy of medicine, the book argues that the dead body has become the epistemologically normative body for medicine: medical knowledge of the living body is derived from investigating the dead body. With the help of autopsies, medicine has learned to view life as “matter in motion” and people as moving machines with interchangeable parts. Furthermore, medicine has .. (shrink)

We discuss Lorenzen’s consistency proof for ramified type theory without reducibility, published in 1951, in its historical context and highlight Lorenzen’s contribution to the development of modern proof theory, notably by the introduction of the ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-rule.

Los ciclos de la naturaleza han marcado profundamente tanto la cosmovisión reflejada en los himnos védicos como el desarrollo del ritualismo brahmánico. Siguiendo un progresivo proceso de abstracción, el hombre asocia cada vez más su destino a estos ciclos cósmicos lo que conduce a la doctrina de la transmigración de las almas.

This paper provides a discussion to which extent the Mathematician David Hilbert could or should be considered as a Philosopher, too. In the first part, we discuss some aspects of the relation of Mathematicians and Philosophers. In the second part we give an analysis of David Hilbert as Philosopher.

We give an overview of recent results in ordinal analysis. Therefore, we discuss the different frameworks used in mathematical proof-theory, namely "subsystem of analysis" including "reverse mathematics", "Kripke-Platek set theory", "explicit mathematics", "theories of inductive definitions", "constructive set theory", and "Martin-Löf's type theory".