Andrea Reichenberger

The philosophical work of Grete Hermann on the foundations of quantum mechanics has recently received increased attention and appreciation. This includes her critique of John von Neumann’s no-hidden-variables proof and her Neo-Kantian considerations on causality. I argue that Hermann, who was a German mathematician, physicist, and philosopher, took a critical view of Leonard Nelson’s interpretation of Kant. Her aim was to challenge Nelson’s dogmatism through her interpretation of quantum physics. Support for this thesis can be found in her interpretation of Bohr’s complementarity principle and her criticism of Heisenberg’s cut and von Neumann’s proof of the impossibility of hidden variables. Over the years, Hermann progressively distanced herself from Nelson, adopting a more liberal and democratic interpretation of Kant, and became a committed advocate of research ethics and responsibility in science and politics.

Der vorliegende Beitrag beschäftigt sich mit Grete Hermanns neukantianischer Interpretation der Quantenmechanik. Im Mittelpunkt stehen ihre Kritik an Johann von Neumanns Beweis der Unmöglichkeit verborgener Parameter sowie ihre Diskussionen mit dem Münsteraner Physiker Adolf Kratzer und dem Mathematiker Paul Bernays über das Messproblem. In der klassischen Mechanik beschreibt der Phasenraum die Menge aller möglichen Zustände eines dynamischen Systems. Wegen der Eindeutigkeit der Lösung der Bewegungsgleichungen kann durch jeden Punkt des Phasenraums nur eine Trajektorie führen. In der Quantenphysik werden Zustände nicht im Phasenraum, sondern durch Vektoren im komplexen Hilbertraum beschrieben. Kanonisch konjugierte Größen sind aufgrund der Unschärferelation nicht gleichzeitig exakt bestimmbar. Grete Hermann griff auf Immanuel Kants transzendentale Erkenntnislehre als Werkzeug und Mittel zurück, um diesen Perspektivenwechsel in der Physik zu interpretieren. Nach Kant richten sich die Gegenstände, sofern wir sie erkennen können, nach der Erkenntnis, d. h. auf ein hypothetisch konstruiertes Objekt. Wir können nur deswegen auf Objekte referieren und diese lokalisieren, weil wir selbst eine Position in Raum und Zeit haben und unter Bezugnahme auf sie Ereignisse bestimmen. Folglich sind beides, die Forderungen nach Wert-Definitheit und Nicht-Kontextualität, ergo Lokalisierbarkeit und Kausalität, nicht gleichzeitig erfüllbar und realisierbar.

Finite model theory is currently not one of the hot topics in the philosophy and history of mathematics, not even in the philosophy and history of mathematical logic. The philosophy of mathematics and mathematical logic has concentrated on infinite structures, closely related to foundational issues. In that context, finite models deserved only marginal attention because it was taken for granted that the study of finite structures is trivial compared to the study of infinite structures. In retrospect, research on finite structures turned out to be neither trivial nor irrelevant but had an important impact on theoretical computer science. A closer look at the early history of the spectrum problem shows that the birth of finite model theory has roots in formal logic, at a time when computer science did not yet exist.

Finite model theory is currently not one of the hot topics in the philosophy and history of mathematics, not even in the philosophy and history of mathematical logic. The philosophy of mathematics and mathematical logic has concentrated on infinite structures, closely related to foundational issues. In that context, finite models deserved only marginal attention because it was taken for granted that the study of finite structures is trivial compared to the study of infinite structures. In retrospect, research on finite structures turned out to be neither trivial nor irrelevant but had an important impact on theoretical computer science. A closer look at the early history of the spectrum problem shows that the birth of finite model theory has roots in formal logic, at a time when computer science did not yet exist.

In the vein of a renewed interest in diagrammatic reasoning, this paper challenges an opposition between logic diagrams and formal languages that has traditionally been the common view in philosophy of logic and linguistics. We examine, from a philosophical point of view, what we call five dogmas of logic diagrams. These are as follows: (1) diagrams are non-linguistic; (2) diagrams are visual representations; (3) diagrams are iconic, and not symbolic; (4) diagrams are non-linear; (5) diagrams are heterogenous, and not homogenous. Using historical examples, we argue that none of these dogmas is an adequate criterion to distinguish logic diagrams from formal languages. Instead, we advocate that there is a common core between linguistic and diagrammatic representation and reasoning.

In den letzten Jahren sind Ernst Cassirers lange Zeit marginalisierten und ignorierten Beiträge zu einer Philosophie der Mathematik zunehmend in den Fokus insbesondere der anglophonen Fachwelt gerückt. In diesem Zuge etablierte sich schnell das Narrativ, Ernst Cassirer sei ein früher Ver­treter, ja Pionier des mathematischen Strukturalismus gewesen. In diesem Beitrag geht es um eine andere, bisher kaum in Erwägung gezogene Lesart, die Cassirers Philosophie der Mathematik mit dessen Kulturphilosophie zusammenführt und Cassirers Transformation der Kantischen Kritik der Vernunft zu einer Kulturkritik auch für die Mathematik geltend macht. Ernst Cassirers Kritik an Gottlob Frege s Grundlegung der natürlichen Zahlen, Richard Dedekind s Konstruktion der reellen Zahlen und Georg Cantors Beschreibung der Mächtigkeit (Kardinalität) von Mengen sowie dem Beweis der Überabzählbarkeit des Zahlenkontinuums sind Beispiele, die nicht nur die viel diskutierte Abkehr assirers vom Substanzbegriff hin zum Funktionsbegriff explizieren. Vielmehr konkretisieren diese Bei­spiele auch Cassirers transzendentalidealistische Grundüberzeugung von der Freiheit mathematischer Begriffsbildung und deren Verankerung in einem humanistischen Liberalismus.

Hilbert’s famous list of problems helped set the research agenda for mathematics in the 20th century. Some of his problems have been solved, among them Cantor’s continuum hypothesis or the solvability of any Diophantine equation, and some are still open, among them the Riemann hypothesis [Gray, 2000]. However, Hilbert did not include the 24th problem concerning the simplicity of a proof in his renowned list of open problems given at the 1900 International Congress of Mathematicians in Paris nor in his published texts. Rüdiger Thiele, a German historian of mathematics, discovered the 24th problem in an unpublished notebook among Hilbert’s Nachlass in Göttingen in 2000 [Thiele, 2003]. This discovery provoked a series of discussions and speculations. Why did Hilbert originally want to include the simplicity of proofs as a mathematical problem in his list and why did he decide against it? The question of simplicity seems to have never been taken up by the Hilbert school. Why? Is it at all justified to regard the simplicity of proofs as a mathematical problem? Criteria for simplicity depend on context, purpose, and acceptability, and are embodied in social practices. The same holds true for the concept of proof. If one reduces mathematics to a formalized logical axiom system, and determines the criterion for the simplicity of a proof in terms of the length of the proof, one certainly does not grasp the diversity of what is considered a proof in mathematical practice. We will discuss how the simplicity of proofs can be understood differently by focusing on the example of diagram chasing as a proof technique in category theory.

Gestures are usually regarded as a casual element of communication processes between logicians. By contrast, we aim to show that gestures have played a significant role in logic. We argue that the development of communication techniques and their standardization have led to the rise of formal notation systems commonly used in logic today. In order to substantiate this claim, the historical development of the use of gestures in (early) modern logic is investigated. This investigation uncovers exemplary communication and proof techniques that illustrate the efficacy of gestures as representational media in formal logic. Revisiting the tradition of gestures in logic offers promising paths and insights for today’s logic as a communication practice in the digital age.

Im Mittelpunkt der vorliegenden Studie steht die Frage nach der Tragweite und Anwendungsrelevanz der Methodenlehre Émilie du Châtelets für die Physik im 18. Jahrhundert, mit der sich die Französin an der Diskussion um Energie- und Impulserhaltung und um das Prinzip der kleinsten Wirkung beteiligte. Andrea Reichenberger zeigt, dass Prinzipien und Hypothesen für Émilie du Châtelet als Fundament und Gerüst wissenschaftlicher Erkenntnis gelten. Im Zusammenspiel beider Komponenten erweisen sich das Prinzip des Widerspruchs und das Prinzip des zureichenden Grundes als regulative Leitlinien und Handlungsmaxime für die auf Hypothesen gestützte Theoriebildung und -begründung. Die sich daraus ergebenden Konsequenzen für den Status und Inhalt der Newtonschen Axiome werden exemplarisch aufgezeigt.

The topic of this article is Hilbert’s axiom of solvability, that is, his conviction of the solvability of every mathematical problem by means of a finite number of operations. The question of solvability is commonly identified with the decision problem. Given this identification, there is not the slightest doubt that Hilbert’s conviction was falsified by Gödel’s proof and by the negative results for the decision problem. On the other hand, Gödel’s theorems do offer a solution, albeit a negative one, in the form of an impossibility proof. In this sense, Hilbert’s optimism may still be justified. Here I argue that Gödel’s theorems opened the door to proof theory and to the remarkably successful development of generalized as well as relativized realizations of Hilbert’s program. Thus, the fall of absolute certainty came hand in hand with the rise of partially secure and reliable foundations of mathematical knowledge. Not all was lost and much was gained.

Today, there is a large consensus in science, politics and society about the relevance and necessity for advancing gender equality. Despite increased measures and initiatives for gender-appropriate research and teaching and for funding programs, women are still strongly underrepresented in science. The number of women in philosophy of science is conspicuously low. While gender and diversity issues are at the top of the agenda in other sciences, disciplines, and scientific cultures, and gender research has long since found its way into practice, there is a considerable research deficit in philosophy of science vis-à-vis such approaches, apart from a few critical voices from feminist science studies. The aim of this chapter is to provide guidance on how to make women’s participation in research visible from a historical, constructive, and systematic point of view by using selected examples from philosophy and history of physics.

Émilie Du Châtelet’s Foundations of Physics (Institutions de physique, 1740/42) has recently been attracting increasing interest from analytical philosophy in the anglophone world. Du Châtelet’s conception of space and time constitutes a controversial issue. I argue that the current debate underestimates the modal approach and epistemological turn in Du Châtelet’s view on space and time. A historical perspective on Abraham Gotthelf Kästner’s criticism and Jean Henry Samuel Formey’s plagiarism of Du Châtelet underlines the significance of this turning point. Against this background, I revisit Leonhard Euler’s space-time realism and its influence on Immanuel Kant. Les Institutions de physique (1740/1742) d’Émilie Du Châtelet suscitent depuis peu un intérêt croissant dans le monde anglophone, chez les philosophes analytiques. Une question controversée concerne les concepts d’espace et de temps de Du Châtelet. Je soutiens que le débat actuel sous-estime l’approche modale et la tournure épistémologique des considérations de Du Châtelet sur l’espace et le temps. Un regard historique sur la critique d’Abraham Gotthelf Kästner et le plagiat de Du Châtelet par Jean Henry Samuel Formey souligne l’importance de ce tournant. Dans ce contexte, je revisite le réalisme spatio-temporel de Leonhard Euler et son influence sur Emmanuel Kant.

The following article describes a pilot study on the possible integration of digital historiography into teaching practice. It focuses on Émilie Du Châtelet’s considerations of space and time against the background of Leibniz’s program of analysis situs. Historians have characterized philosophical controversies on space and time as a dichotomy between the absolute and relational concepts of space and time. In response to this, the present case study pursues two aims: First, it shows that the common portrayal simplifies the complex pattern of change and the semantic shift from absolute-relational concepts of space and time to invariance and conservation principles. Second, against this background, I present the Online Reading Guide on Émilie Du Châtelet’s Foundations of Physics, a teaching and research project designed to help navigate Du Châtelet’s Institutions physiques. This project makes Du Châtelet’s important text visible to a broad audience and allows for a more critical and deeper view on classical topics of the history of philosophy and science in a more accessible way than traditional introductions.

Freges Werk gilt heute als Klassiker der Philosophie und Philosophiegeschichte. Dessen Einfluss auf unterschiedlichste Bereiche der Philosophie, von der Logik und Mathematik bis hin zur Ontologie, Epistemologie und Sprachphilosophie, ist unbestritten. Vor diesem Hintergrund scheint die Annahme naheliegend, dass Freges Wirkungsgeschichte umfassend erforscht ist. Tatsächlich gilt dies nicht für die komplexe Rezeptionsgeschichte Freges, auch wenn es dazu durchaus eine Reihe von neuen, lehrreichen und detaillierten Studien gibt. 1 Der vorliegende Beitrag befasst sich mit zwei bisher kaum bekannten und unbeachtet gebliebenen Frege-Rezeptionen aus den 30er und 40er Jahren des 20. Jahrhunderts. Es handelt sich dabei zum einen um Wilma Papsts Studie Gottlob Frege als Philosoph (1932), zum anderen um Editha Krenzs Doktorarbeit Der Zahlbegriff bei Frege (1942). In der Literatur werden die Namen der Autorinnen zwar wiederholt in Fußnoten oder im Literaturverzeichnis genannt, doch bis auf wenige Ausnahmen nicht diskutiert. Im Folgenden soll anhand einer problemgeschichtlichen Kontextualisierung gezeigt werden, dass beide Arbeiten interessante Aspekte zu Frege und seiner Rezeption beleuchten.

This broad and insightful book presents current scholarship in important subfields of philosophy of science and addresses an interdisciplinary and multidisciplinary readership. It groups carefully selected contributions into the four fields of I) philosophy of physics, II) philosophy of life sciences, III) philosophy of social sciences and values in science, and IV) philosophy of mathematics and formal modeling. Readers will discover research papers by Paul Hoyningen-Huene, Keizo Matsubara, Kian Salimkhani, Andrea Reichenberger, Anne Sophie Meincke, Javier Suárez, Roger Deulofeu, Ludger Jansen, Peter Hucklenbroich, Martin Carrier, Elizaveta Kostrova, Lara Huber, Jens Harbecke, Antonio Piccolomini d'Aragona and Axel Gelfert. This collection fosters dialogue between philosophers of science working in different subfields, and brings readers the finest and latest work across the breadth of the field,illustrating that contemporary philosophy of science has successfully broadened its scope of reflection. It will interest and inspire a wide audience of philosophers as well as scholars of the natural sciences, social sciences and the humanities. The volume shares selected contributions from the prestigious second triennial conference of the German Society for Philosophy of Science/ Gesellschaft für Wissenschaftsphilosophie (GWP.2016, March 8, 2016 - March 11, 2016).