Papers by Gerhard Heinzmann
After Godel’s results the limitations of the three principal “foundational schools” became more a... more After Godel’s results the limitations of the three principal “foundational schools” became more and more evident, while the “working scientists” continued their activity caring more for the acquisition of “results” than for logical rigor. This “pragmatic turn” was perceivable also in philosophy of science due to an influence of pragmatism that replaced the previous influence of logical empiricism and analytic philosophy.
Selon Beth, le climat philosophique de l'epoque d'apres-guerre est caracterise par une te... more Selon Beth, le climat philosophique de l'epoque d'apres-guerre est caracterise par une tension entre scientisme et relativisme subjectif. Quelle consequence faut-il en tirer pour la philosophie des mathematiques? Refusant de lier la rationalite a une evidence absolue, la synthese envisagee par Beth est assez proche des solutions proposees par Gonseth et Bernays. Dans sa partie centrale, cet article examine le point de vue de Beth a partir de deux exemples qui concernent l'engagement ontologique: 1° par rapport a la position de Carnap dans les annees trente; 2° par rapport aux consequences a tirer du theoreme de Lowenheim-Skolem.
Paul Lorenzen -- Mathematician and Logician, 2021
In this article we give an overview, from a philosophical point of view, of Lorenzen’s constructi... more In this article we give an overview, from a philosophical point of view, of Lorenzen’s construction of the natural and the real numbers. Particular emphasis is placed on Lorenzen’s classification in the tradition of predicative approaches that stretches from Poincaré to Feferman.
Paul Lorenzen und die konstruktive Philosophie, 2016
Revue de métaphysique et de morale, 2020
Varieties of Scientific Realism, 2017
This paper proposes a reconsideration of mathematical structuralism, inaugurated by Bourbaki, by ... more This paper proposes a reconsideration of mathematical structuralism, inaugurated by Bourbaki, by adopting the “practical turn” that owes much to Henri Poincare. By reconstructing his group theoretic approach of geometry, it seems possible to explain the main difficulty of modern philosophical eliminative and non-eliminative structuralism: the unclear ontological status of ‘structures’ and ‘places’. The formation of the group concept—a ‘universal’—is suggested by a specific system of stipulated sensations and, read as a relational set, the general group concept constitutes a model of the group axioms, which are exemplified (in the Goodmanian sense) by the sensation system. In other words, the shape created in the mind leads to a particular type of platonistic universals, which is a model (in the model theoretical sense) of the mathematical axiom system of the displacement group. The elements of the displacement group are independent and complete entities with respect to the axiom system of the group. But, by analyzing the subgroups of the displacement group (common to geometries with constant curvature) one transforms the variables of the axiom system in ‘places’ whose ‘objects’ lack any ontological commitment except with respect to the specified axioms. In general, a structure R is interpreted as a second order relation, which is exemplified by a system of axioms according to the pragmatic maxim of Peirce.
Freedom in Mathematics, 2016
Sylvestre Huet: We are now going to explore three major points concerning aspects of “mathematics... more Sylvestre Huet: We are now going to explore three major points concerning aspects of “mathematics and society”: education and training, mathematics and industry, and finally the use of mathematics in political debates and generally speaking in the humanities.
Freedom in Mathematics, 2016
The origins of mathematics accompanied the evolution of social systems. Many, many social needs r... more The origins of mathematics accompanied the evolution of social systems. Many, many social needs require calculation and numbers. Conversely, the calculation of numbers enables more complex relations and interactions between peoples. Numbers and calculations with them require a well organized operational system. Such systems were perhaps the earliest models of complex rigorous systems. As we will see, not just one but several number systems come to us from antiquity. However, as interesting as the basic notions of counting may be, the origins of mathematics include more than just enumeration, counting, and arithmetic. Thus, also considered are other issues of mathematics to be considered. Number provides a common link between societies and a basis for communication and trade. This chapter illustrates various entries into our ancient knowledge of how our number system began. Examples range from prehistory to contemporary. The ancient evidence is easy to accept, but the modern examples yield more conclusions. It is remarkable that even though mathematics achieved its first zenith twenty three hundred years ago, more than a hundred generations, there are peoples today that still count with their fingers or with stones, that have no real language for numbers, and moreover have not the general concept of number beyond specific examples. There is every reason to believe that in the future, near or distant, there will exist a far more credible theory for the origin of counting than there may ever for the development of geometric proof, the PythagoreanTheorem in particular.
The main thesis of this paper is that Poincaré conventionalism must be ranked among these sources... more The main thesis of this paper is that Poincaré conventionalism must be ranked among these sources of the logical empiricists, that seem to be at the same time a source of Quine's criticism of the two dogmas of the logical empiricism and survives consequently Quine's criticism.
The Significance of the Hypothetical in the Natural Sciences, 2009
To all these institutions we express our warm gratitude. We are also grateful to the members of t... more To all these institutions we express our warm gratitude. We are also grateful to the members of the PILM scientific committee for their invaluable help in preparing the program and reporting on so many lectures, as well as to the staff of the Poincaré Archives for their help in the preparation of the conference and their expert assistance in various ways. We particularly express our gratitude to Dr. Prosper Doh (Poincaré Archives) who has taken on the job of technical editor for this book and who has realized the index and the camera-ready copy. Finally, the editors are indebted to the editorial board of "Logic, Epistemology, and the Unity of Science" for accepting this volume in their series. They would also like to thank Springer Publishers and, in particular, Floor Oosting.
The Western Ontario Series in Philosophy of Science, 2014
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.
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Papers by Gerhard Heinzmann