Henri Poincaré

Open Court began publishingThe Monistin 1890 as a journal“devotedto the philosophy of science”that regularly included mathematics. The audiencewas understood to be“cultured people who have not a technical mathematicaltraining”but nevertheless“have a mathematical penchant.”With these constraints,the mathematical content varied from recreations to logical foundations, but every-one had something to say about non-Euclidean geometry, in debates that rangedfrom psychology to semantics. The focus in this essay is on the contested value ofmathematical expertise in legitimating what should be considered as mathematics.While some mathematicians urgedThe (...) Monistto uphold disciplinary standards ofgeometrical reasoning, authors opposed tonon-Euclidean geometry aligned theirreasoning with practical applications, universal know-how, and nonhierarchicaldemocracy. As one contributorinquired,“How isthe professional expert betterfit-ted to see more lucidly in dealing with the elements of geometry than any otherperson of good geometric faculty?”. (shrink)

Part 1 of this article exposed a tension between Poincaré’s views of arithmetic and geometry and argued that it could not be resolved by taking geometry to depend on arithmetic. Part 2 aims to resolve the tension by supposing not merely that intuition’s role is to justify induction on the natural numbers but rather that it also functions to acquaint us with the unity of orders and structures and show practices to fit or harmonize with experience. I argue that in (...) this manner, intuition serves the epistemological function of warranting generalizations and justifying practices. In particular, it justifies the application of group-theoretic notions in geometry but not the use of set-theoretic notions in arithmetic. (shrink)

This paper offers a systematic analysis of Poincaré’s understanding of beauty in science. In particular, the paper examines the epistemic significance Poincaré attributes to aesthetic judgement by reconstructing and analysing his arguments on simplicity and unity in science. I offer a consistent reconstruction of Poincaré’s account and show that for Poincaré simplicity and unity are regulative principles, linked to the aim of science—that of achieving understanding of how phenomena relate. I show how Poincaré’s account of beauty in science can be (...) incorporated within his wider philosophy of science. (shrink)

The main goal of part 1 is to challenge the widely held view that Poincaré orders the sciences in a hierarchy of dependence, such that all others presuppose arithmetic. Commentators have suggested that the intuition that grounds the use of induction in arithmetic also underlies the conception of a continuum, that the consistency of geometrical axioms must be proved through arithmetical induction, and that arithmetical induction licenses the supposition that certain operations form a group. I criticize each of these readings. (...) More fully, I argue that the justification Poincaré offers for the use of the group notion in geometry appears to extend to set-theoretic notions that would suffice to put arithmetic on a logical foundation, thus undermining his own case for the necessity of intuition in arithmetic. In part 2, I offer an interpretation of intuition’s role on which it justifies the use of group-theoretic, but not set-theoretic, notions. (shrink)

" Parmi les hommes qui ont, tous utilement, mais plus ou moins brillamment servi la science, les uns avaient reçu dans leur jeunesse une éducation classique solide, parfois raffinée, tandis que les autres n'avaient eu qu'une formation littéraire hâtive, incomplète et sommaire. On serait tenté d'en conclure que l'étude des lettres est inutile aux savants, puisque beaucoup d'entre eux ont pu s'en passer. Ce serait aller un peu vite en besogne. Est-il certain qu'on ne saurait faire de différence entre les (...) oeuvres des uns et des autres et y reconnaître une sorte de marque d'origine. C'est là une comparaison que je ne veux pas faire ici, il faudrait citer des noms propres, et je ne voudrais désobliger personne, même les morts. En pareille matière, les appréciations sont difficiles, mais quand même on aurait démontré que les uns ont été aussi bons savants que les autres, qu'est-ce que cela prouverait? Le fait s'expliquerait tout naturellement. Il y a eu de longues années où il était difficile de percer sans avoir fait ses classes, et en général de sortir de son rang. Ceux qui y sont parvenus n'ont pu le faire que grâce à une énergie exceptionnelle qui leur tenait lieu de bien d'autres avantages, et qui pouvait les mettre de pair avec des esprits plus cultivés, mais servis par des caractères moins bien trempés...". (shrink)

In the introduction to his famous book, La Science et l’hypothèse, Poincaré remarks on the necessary role and legitimacy of hypotheses. He establishes a triple classification of hypotheses, dividing them into verifiable, useful, and apparent. However, in chapter 9, entitled ‘Les hypothèses en physique’, he gives a slightly different triadic classification: natural hypotheses, indifferent hypotheses, and real generalizations. The origin of this second classification is a lecture given at the International Congress of Physics, Paris, 1900. What are the similarities and (...) differences between these two classifications? My main purpose is to provide a possible equivalence between the two classifications in order to clarify the role of hypotheses in Poincaré’s epistemology of natural science. The discussion will be based on the two fundamental texts mentioned above and on the contrast between Poincaré’s use of the notion of hypothesis and that of some of his contemporaries. (shrink)

This paper examines whether, and in what contexts, Duhem’s and Poincaré’s views can be regarded as conventionalist or structural realist. After analysing the three different contexts in which conventionalism is attributed to them – in the context of the aim of science, the underdetermination problem and the epistemological status of certain principles – I show that neither Duhem’s nor Poincaré’s arguments can be regarded as conventionalist. I argue that Duhem and Poincaré offer different solutions to the problem of theory choice, (...) differ in their stances towards scientific knowledge and the status of scientific principles, making their epistemological claims substantially different. (shrink)

Poincaré is well known for his conventionalism and structuralism. However, the relationship between these two theses and their place in Poincaré׳s epistemology of science remain puzzling. In this paper I show the scope of Poincaré׳s conventionalism and its position in Poincaré׳s hierarchical approach to scientific theories. I argue that for Poincaré scientific knowledge is relational and made possible by synthetic a priori, empirical and conventional elements, which, however, are not chosen arbitrarily. By examining his geometric conventionalism, his hierarchical account of (...) science and defence of continuity in theory change, I argue that Poincaré defends a complex structuralist position based on synthetic a priori and conventional elements, the mind-dependence of which precludes epistemic access to mind-independent structures. (shrink)

Nous reproduisons ici, avec l’aimable autorisation de la Hofbibliothek Aschaffenburg, la courte correspondance inédite entre Dingler et Henri Poincaré. Cette correspondance, conservée aux Archives Dingler dans la Hofbibliothek Aschaffenburg, consiste en trois documents : un brouillon en allemand de la main de Dingler, sa traduction française par une autre plume, mais avec des corrections ef...

Nous reproduisons ici, avec l’aimable autorisation de la Hofbibliothek Aschaffenburg, la courte correspondance inédite entre Dingler et Henri Poincaré. Cette correspondance, conservée aux Archives Dingler dans la Hofbibliothek Aschaffenburg, consiste en trois documents : un brouillon en allemand de la main de Dingler, sa traduction française par une autre plume, mais avec des corrections ef...

In this paper I argue that Poincaré’s acceptance of the atom does not indicate a shift from instrumentalism to scientific realism. I examine the implications of Poincaré’s acceptance of the existence of the atom for our current understanding of his philosophy of science. Specifically, how can we understand Poincaré’s acceptance of the atom in structural realist terms? I examine his 1912 paper carefully and suggest that it does not entail scientific realism in the sense of acceptance of the fundamental existence (...) of atoms but rather, argues against fundamental entities. I argue that Poincaré’s paper motivates a non-fundamentalist view about the world, and that this is compatible with his structuralism. (shrink)

Este texto introduz a tradução do discurso de intitulado "Sobre os números transfinitos" ("Über transfinite Zahlen"), proferido por Henri Poincaré em 27 de abril de 1909, na Universidade de Göttingen. Após uma breve apresentação do pensamento do autor acerca dos fundamentos da aritmética, procura-se citar os aspectos mais relevantes da chamada crise dos fundamentos da matemática, para então introduzir a reformulação do conceito de predicatividade aventada no referido discurso sobre números transfinitos, contribuição compreendida como um recurso teórico necessário para a (...) superação dos paradoxos relativos à teoria dos conjuntos. Com isso, pretende-se evidenciar o papel central do texto publicado nesta edição para o desenvolvimento da concepção matemática de Henri Poincaré. This article provides an introduction to the translation of the lecture entitled "On transfinite numbers" ("Über transfinite Zahlen"), given by Henri Poincaré at the University of Göttingen on April 27, 1909. Following a short presentation of Poincaré's views on the foundations of arithmetic, it identifies the aspects of the so-called crisis of the foundations of mathematics that are most relevant for understanding his reconstruction of the concept of predicativity, which is intended to be a theoretical resource that may overcome the paradoxes of set theory. In doing so, the article highlights the central role of this text in the process of maturation of Poincaré's views on the foundations of mathematics. (shrink)

Poincaré tries to tackle the question "Can science sway us into materialism?". In other words, he analyzes if science - in particular Physics - and its theories are dependent on a materialistic ontological view of the world. His final answer appears to be "no". However, in order to reach this conclusion he resorts to a brief history of Physics, providing an insightful account on the evolution of the concept of atom from Democritus to the, then, recent discoveries on the composition (...) of the atom. In his opinion, science swung and will keep swinging between two opposing approaches over the constitution of nature: the discrete and the continuous approach. (shrink)

Review of: Michael Friedman and Alfred Nordmann, eds. The Kantian Legacy in Nineteenth-Century Science. Cambridge, MA: MIT Press, 2006. Pp. vi+370. $45.00 (cloth).

Many people would like to know where scientific ideas come from and how they arise. In the case of mathematics, new ideas often come in the form of new “mathematical objects”: groups, vector spaces, sets, etc. Some people think these new objects are invented, others that they are discovered. By exploring the birth of descriptive set theory in France and Russia in the period 1890–1930 we show that the leading French mathematicians worked within a rational, secular worldview that made them (...) doubt the legitimacy of infinite sets, particularly nondenumerable ones; on the other hand, the creators of the famous Moscow school of mathematics, particularly those who subscribed to a religious doctrine known as “name‐worshipping,” believed that humans had absolute freedom to invent mathematical objects. Partly as a result of their different cultural environments, the French and the Russians took different approaches to the same problem. In the end the Russians created a new field, descriptive set theory, at a time when the French remained hesitant. (shrink)

This essay discusses attempts that have been made to explain the striking similarities between two theories propounded in 1905 by Albert Einstein and Henri Poincaré without any mutual reference.

In the late 19th century great changes in theories of light and electricity were in direct conflict with certitude, the view that scientific knowledge is infallible. What is, then, the epistemic status of scientific theory? To resolve this issue Duhem and Poincaré proposed images of fallible knowledge, Instrumentalism and Conventionalism, respectively. Only in 1919–1922, after Einstein's relativity was published, he offered arguments to support Fallibilism, the view that certainty cannot be achieved in science. Though Einstein did not consider Duhem's Instrumentalism, (...) he argued against Poincaré's Conventionalism. Hitherto, Einstein's Fallibilism, as presented at first in a rarely known essay of 1919, was left in the dark. Recently, Howard obscured its meaning. Einstein's essay was never translated into English. In my paper I provide its translation and attempt to shed light on Einstein's view and its context; I also direct attention to Einstein's images of philosophical opportunism in scientific practice. (shrink)

The central theme running throughout this outstanding new survey is the nature of the philosophical debate created by modern science's foundation in experimental and mathematical method. More recently, recognition that reasoning in science is probabilistic generated intense debate about whether and how it should be constrained so as to ensure the practical certainty of the conclusions drawn. These debates brought to light issues of a philosophical nature which form the core of many scientific controversies today. _Scientific Method: A Historical and (...) Philosophical Introduction_ presents these debates through clear and comparative discussion of key figures in the history of science. Key chapters critically discuss * Galileo's demonstrative method, Bacon's inductive method, and Newton's rules of reasoning * the rise of probabilistic `Bayesian' methods in the eighteenth century * the method of hypotheses through the work of Herschel, Mill and Whewell * the conventionalist views of Poincaré and Duhem * the inductivism of Peirce, Russell and Keynes * Popper's falsification compared with Reichenbach's enumerative induction * Carnap's scientific method as Bayesian reasoning The debates are brought up to date in the final chapters by considering the ways in which ideas about method in the physical and biological sciences have affected thinking about method in the social sciences. This debate is analyzed through the ideas of key theorists such as Kuhn, Lakatos, and Feyerabend. (shrink)

Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicist's conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no principles of (...) inference specific to a given local topic. Poincaré, a Kantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical reasoning (proof). In this essay, I try to isolate and clarify this idea and to describe the mathematical epistemology which underlies it. Central to this epistemology (which is basically Kantian in orientation, and closely similar to that advocated by Brouwer) is a principle of epistemic conservation which says that knowledge of a given type cannot be extended by means of an inference unless that inference itself constitutes knowledge belonging to the given type. (shrink)

This deeply researched, carefully constructed and very thoughtful book is fascinating in its own right as well as being indispensable background material for anyone interested in current philosophical thought about space, time, and geometry.

It is a matter of record that Henri Poincaré never responded publicly to Einstein's Special Theory of Relativity . Since almost no private papers of Poincaré are available, his attitude toward Einstein's work and his silence on that score become somewhat of a mystery. It is almost certain that Poincaré knew of Einstein's work in RT. First, he was fluent in German, having learned it as a young man when the Germans occupied his home town of Nancy in 1870. Second, (...) he often reported to the members of the Académie des Sciences on current work in electrodynamics in Germany. It is highly improbable that he would have missed the abstract of Einstein's first paper on RT or the subsequent articles by Einstein on the subject, especially those which were translated into French, since they were in areas directly related to his own interests in theoretical physics. (shrink)

THE VALUE OF SCIENCE INTRODUCTION The search for truth should be the goal of our activities; it is the sole end worthy of them. Doubtless we should first bend our efforts to assuage human suffering, but why ? Not to suffer is a negative ...

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" Vivid . . . immense clarity . . . the product of a brilliant and extremely forceful intellect." — Journal of the Royal Naval Scientific Service "Still a sheer joy to read." — Mathematical Gazette "Should be read by any student, teacher or researcher in mathematics." — Mathematics Teacher The originator of algebraic topology and of the theory of analytic functions of several complex variables, Henri Poincare (1854–1912) excelled at explaining the complexities of scientific and mathematical ideas to lay (...) readers. Science and Method, written in 1908, has been appreciated by a wide audience of nonprofessionals and translated into many languages. It defines the basic methodology and psychology of scientific discovery, particularly in regard to mathematics and mathematical physics. Drawing on examples from many fields, it explains how scientists analyze and choose their working facts, and it explores the nature of experimentation, theory, and the mind. 1914 edition. Translated by Francis Maitland. (shrink)