In what follows, I examine three main points which may help us to understand the deep nature of Einstein's objections to quantum mechanics. After having played a fundamental pioneer role in the birth of quantum physics, Einstein was, as is well known, far less enthusiastic about its constitution as a quantum mechanics and, since 1927, he constantly argued against the pretention of its founders and proponents to have settled a definitive and complete theory. I emphasize first the importance of the (...) philosophical climate, which was dominated by the Copenhagen orthodoxy and Bohr's idea of complementarity: What Einstein was primarily reluctant to was to accept the fundamental character of quantum mechanics as such, and to modify for it the basic principles of knowledge. I thus stress the main lines of Einstein's own programme in respect to quantum physics, which is to be considered in relation to his other contemporary attempts and achievements. Finally, I show how Einstein's arguments, when dealing with his objections, have been fruitful and some of them still worthy, with regard to recent developments concerning local nonseparability as well as concerning the problems of completeness and accomplishment of quantum theory. (shrink)

Le débat sur l'interprétation de la mécanique quantique est, aujourd'hui, sensiblement différent de ce qu'il était dans la période de «fondation» de cette théorie. Cette modification tient à deux causes : l'ancrage des conceptions quantiques dans la pensée des physiciens, favorisé par l'utilisation systématique et fructueuse de la théorie quantique en physique atomique et subatomique, d'une part et, d'autre part, les développements théoriques et expérimentaux survenus au cours des vingt dernières années, qui ont amené à considérer comme des faits physiques (...) des énoncés qui paraissaient naguère relever davantage de l'interprétation, avec liberté d'options, voire de la spéculation. Quelle est la signification des énoncés et des concepts théoriques de la physique quantique ? et comment celle-ci s'est-elle modifiée ? Qu'entend-on par «interprétation» ? Telles sont les questions qui servent de guide à ce travail. Nous nous proposons de clarifier ce qui, dans les questions de signification, relève respectivement de la physique (essentiellement les contenus conceptuels et théoriques), et de perspectives philosophiques. Cette distinction est plus nette aujourd'hui qu'elle ne l'était au moment de l'établissement de la mécanique quantique, ce qui permet d'approfondir notre compréhension de chacun de ces deux aspects, qui s'enrichissent mutuellement de cette clarification. Nous nous interrogeons, pour terminer, sur ce que peut être une conception réaliste en termes d'un «monde d'objets quantiques». (shrink)

We consider the problem of therelationship between the quantum and theclassical domains from the point of view that itis possible to speak of a direct physicaldescription of quantum systems havingphysical properties. We put emphasis, inevidencing it, on the specific quantum conceptof indistinguishability of identical in aconceptual way (and not in a logical way in thevein of ``da Costa's school''). In essence, thesubsequent argumentation deals with therelationship between the classical and thequantum, with the problem of the quantum theoryof measurement. Even in (...) the absence of adefinitive response to this problem, the bestattitude for the time being, as we cannot reducethe classical and the quantum one to the other,seems to be to accept their pacific coexistence,and this is possible with the toleranceprinciple of the ``pragmatic truth'' developedfrom a logical point of view by Newton daCosta.RESUMO. As áreas quântica eclássica enquanto provisórioscoexistentes parallelos. Abordamos oproblema da relação entre as áreasdo quântico e do clássico considerandoque é possível falar de umadescrição física direta de sistemas quânticos tendo propriedades.Insistimos, para isto, sobre o conceitoespecificamente quântico daindicernabilidade dos idênticos de um ponto devista conceptual (não de um ponto de vistalógico à maneira da ``escola da Costa'')como evidenciando isto. O essencial daargumentação a seguir tem como enfoquea relação clássico-quântico, como problema da teoria quântica damedição. Mesmo não tendo umaresposta definitiva para este, a melhor atitudepor enquanto, já que não se podemreduzir um ao outro o clássico e oquântico, nos parece ser esta de aceitar suacoexistência pacífica, o que épossível com o princípio detolerância da ``verdade pragmática''desenvolvida logicamente por Newton da Costa. (shrink)

La quatrième de couverture indique : "Fruit d'une expérience intellectuelle vécue au carrefour de la science et de la philosophie, ce livre se présente comme la synthèse de préoccupations méthodologiques concernant l'étude critique des sciences. Il veut rendre manifeste le lien de nécessité et de constitution qui existe entre les sciences, l'épistémologie (comme examen critique de leurs concepts et propositions), l'histoire de leur développement, et la philosophie de la connaissance. Les six premiers chapitres portent sur des problèmes de philosophie et (...) d'histoire des sciences. Les huit suivants sont consacrés à des problèmes d'élaboration et d'interprétation de la physique contemporaine. Tout l'ouvrage est placé sous le signe de l'appropriation et de la communication des connaissances, en relation au problème de la place prise par la science dans le monde contemporain. Ce livre ne fait aucun appel à des connaissances spécialisées, et devrait être accessible à tout lecteur raisonnablement cultivé.". (shrink)

Il semble que la science, au sens de l'activité des scientifiques, ne soit pas vraiment la même que celles dont s'occupent, chacune de leur coté, la réflexion philosophique et la recherche historique. Les scientifiques eux-mêmes ne sont pas toujours sensibles au lien que leur travail entretient avec la philosophie et avec l'histoire des sciences, bien qu'il concerne des significations et soit en situation d'état provisoire, entre une connaissance passée et des développements ou des interprétations futures. Cependant, les leçons de la (...) découverte et du travail scientifique, telles que ceux-ci se révèlent à travers la formulation par Einstein de la théorie de la relativité restreinte, cas étudié ici à titre d'exemple, montrent les rapports réciproques et nécessaires entre ces diverses dimensions de ce qu'on appelle la science. Cette mise en relation permet d'aborder d'une manière renouvelée certains problèmes ou certaines apories traditionnelles de la pensée de la connaissance scientifique. (shrink)

A idéia da “universalidade da ciência” é objeto, nos debates atuais, de posições muito opostas, que dependem de o ponto de vista ser o de uma “ciência ideal” ou aquele da “produção social das ciências”. No primeiro caso, a ciência é concebida como o “núcleo duro” de suas proposições e de seus resultados na época considerada, e sua suposta universalidade igno­ra os fatores que tornam relativos seus conteúdos de conhecimento e que podem ser tanto de natureza conceitual como social. Inversamente, (...) uma atenção exclusiva com os aspectos so­ciais da produção dos conhecimentos científicos ignora o caráter objetivo daqueles conteúdos de conhecimento, que tanto tratam dos objetos do pensamento, como aqueles da matemática, quanto dos fenômenos do mundo real, sejam psicobiológicos, sejam humanos e sociais. Essas duas posições extremas, caricaturais e, entretanto, freqüentemente encontradas ilustram a ausência ou o desconhecimento de análises interdisciplinares entre a filosofia, a ciência e a história da ciência. Consideraremos inicialmente alguns elementos das críticas que são feitas à universalidade da ciência, tal como a entendemos hoje, e que provêm da filosofia do conhecimento, sociologia do conhecimento, história da ciência, história e antropologia. Tentaremos, então, estabelecer o problema da universalidade da ciência como uma idéia filosófica intimamente vinculada à ciência e à filosofia, desde sua gênese. Através das diferentes etapas da história do pensamento, confrontaremos o enunciado filosófico da universalidade da ciência com a realidade histórica da produção, difusão e assimilação ou apropriação do conhecimento científico concebido segundo suas diferentes dimensões, incluindo suas aplicações e suas ligações com a técnica e a tecnologia. (shrink)

Em sua concepção da dinâmica como ciência das mudanças dos movimentos dos corpos, D'Alambert propõe-se a exprimir estas mudanças somente em função das grandezas do movimento. Ele estabelece assim a possibilidade de concebê-las fisicamente, em relação às suas causas efetivas, sem recorrer a conceitos externos como o de força, afirmado a co-naturalidade da causa física e de seus efeitos, traduzida pelaç noção de aceleração instantânea, cuja significação física vincula-se à forma diferencial da grandeza tempo e ao conceito de movimento virtual.

We want to consider anew the question, which is recurrent along the history of philosophy, of the relationship between rationality and mathematics, by inquiring to which extent the structuration of rationality, which ensures the unity of its function under a variety of forms (and even according to an evolution of these forms), could be considered as homeomorphic with that of mathematical thought, taken in its movement and made concrete in its theories. This idea, which is as old as philosophy itself, (...) although it has not been dominant, has still been present to some degree in the thought of modern science, in Descartes as well as in Kant, Poincaré or Einstein (and a few other scientists and philosophers). It has been often harshly questioned, notably in the contemporaneous period, due to the failure of the logistic programme, as well as to the variety of “empirical” knowledges, and, in a general way, to the character of knowledges that show them as transitory, evolutive and mind-built. However, the analysis of scientific thought through its inventive and creative processes leads to characterize this thought as a type of rational form whose configurations can be detailed rather precisely. In this work we shall propose, first, a quick sketch of some philosophical requirements for such a research programme, among which the need for an harmonization, and even a conciliation, between the notions of rational (or rationality), of intuitive grasp and of creative thought. Then we shall examine some processes of creative scientific thought bearing on the knowledge and the understanding of the world, distinct from mathematics although keeping tight relations with them. Contemporary physical theories are privileged witnesses in this respect, for in them the rational thought of phenomena makes an intrinsic use of mathematical thought, which contributes to the structuration of the formers and to the expression of their concepts (which entails the physical contents of the latter). The General Theory of Relativity and the Quantum Theory are exemplar to this, as they directly reveal what can be called the “drag of physical thought par the mathematical form”, which makes possible to overcome the limitations of the physical knowledge previously adquired. This process is tightly related to the modalities and to the stucture of the rational thought underlying it. This is what we would like to show. DOI:10.5007/1808-1711.2011v15n2p303. (shrink)

No artigo "Einstein y el efecto Compton", publicado neste número de SCIENTIÆ UDIA: , os autores estranham o fato de Einstein não ter declarado mais claramente o quanto esse efeito comprovava definitivamente o carácter corpuscular da radiação. A presente nota crítica pretende fornecer elementos adicionais de apreciação que permitam acompanhar o método de exploração do domínio dos quanta elaborado por Einstein, na ausência de uma teoria adequada, e praticado por ele de 1905 à 1925, evidenciando por esse meio propriedades inéditas (...) e inusuais dos fenômenos quânticos; e também explicitar o alvo que ele queria almejar com sua investigação e entender melhor seu pensamento acerca dos quanta. Três argumentos principais são delineados. (1) O problema dos quanta de luz está estreitamente ligado com aquele das propriedades quânticas da matéria atômica em geral, do qual não pode ser separado. (2) Outro experimento, realizado pouco depois do experimento de Compton, trouxe um elemento adicional, necessário para que se possa atribuir sem nenhuma reticiência uma quantidade de movimento à radiação, isto é, o caráter individual da interação da qual a radiação toma parte. (3) Enfim, os trabalhos de Einstein sobre a estatística quântica, realizados no mesmo período, apontavam na direção da generalização do duplo caráter onda-corpúsculo à radiação, como também à matéria, evidenciando a indiscernibilidade dos estados idênticos e revelando, assim, propriedades fundamentais radicalmente novas que toda teoria quântica futura haveria de incluir. In the article "Einstein y el efecto Compton", published in this issue of SCIENTIÆ UDIA: , the authors wonder why Einstein did not claim unambiguously that this phenomenon was a clear and definitive proof of the corpuscular character of radiation. The aims of the present critical note are to provide additional comments that serve to clarify the method that Einstein used for exploring the quantum domain, in the absence of a satisfactory theory - a method he practiced from 1905 to 1925, and from which he obtained evidence of new and unusual properties of quantum phenomena; and to try to formulate what he wanted to get at in his investigations, so as to understand better his thinking about quanta. Three main arguments are sketched. (1) The problem of light quanta cannot be separated from that of the quantum properties of atomic matter in general. (2) Another experiment, performed shortly after Compton's yielded a further element that was necessary to obtain a definitive answer to the question of whether a quantity of motion can be attributed to radiation. (3) Finally, Einstein's works on quantum statistics during that period pointed towards a generalization of the double wave-corpuscle behavior of radiation, and also of matter, providing evidence for the indistinguishability of identical states, thus revealing radically new fundamental properties that should be accounted for by any further quantum theory. (shrink)

In this paper, we investigate the constitutive problems and other several aspects of what a research entitled 'formal epistemology' should be. The interest in this subject has to do with the possibility of reaching a privileged point of view or axis of research - i.e., the 'formal' one - that would allow, a better grasp of the richness and variety of the facts and problems tackled by precise (local) epistemology of theories (for example, in physics). This approach is likely to (...) enable one to hold the main structural lines that organize those theories according to a more com prehensive, unifying and synthetic intelligibilit. By the same token, it would eventually allow a better handling of the changes required in the organization of knowledge, putting emphasis on its main directions, drawing up a rational inventory of this knowledge, and perhaps anticipating others. At first, we deal with the `thought of changes' that no approach of the 'form' can afford to leave aside, since the meaning of this concept is inseparable from the contents that come with constructions and modifications. We examine then the notion of 'epistemic operation' as an instrument to create new forms on the theoretical as well as on the meta-theoretical levels. ln the wake of it, we analyze the characteristics of the form and of the formal, as well as their relationship with the contents of knowledge. We also take the notion of object into account, since it depends upon the decision of a subject and upon conventional choices. We finally inquire about the link between `epistemic operations' as specified above and algorithmic functions for knowledge statements, and emphasize the risk of reductionism that might follow from a naturalistic conception of representation. (shrink)

On propose quelques réflexions sur le concept de temps, tout d'abord rappelant la diversité des expériences et des consciences du temps, et montrant comment le temps des sciences et de la physique est relié à cette expérience et à cette conscience qui en est prise, notamment en ce qui concerne le rapport entre l'instant et la durée. On s'efforce ensuite de tirer deux leçons des développements sur le concept de temps tel qu'il se présente en physique. La première est que (...) le concept de temps est construitpar la pensée, et ne nous est pas directement donné, soit par la nature, soit par notre «sens interne». Cela apparait, notamment, avec l'introduction du temps comme paramètre de la dynamique par Galilée, puis avec la définition du temps «absolu et mathématique » par Newton en vue de constituer le «temps instantané» de la loi fondamentale de la mécanique, et enfin avec la critique des limites de cette conception qui aboutit à la formulation de la relativité du temps et de son lien de constitution à l'espace (construction de l'espace-temps). La seconde leçon est que le temps se définit et se détermine par les phénomènes(et les objets) de la nature (qui se ramènent aux propriétés de la matière), alors qu'on a longtemps cru l'inverse. Cela apparait notamment avec la soumission du temps, dans la reconstruction du concept effectuée par Einstein, d'abord à des principes physiques(avec la relativité restreinte), ensuite avec sa détermination, conjointement à l'espace (dans l'espace-temps), par les champs de gravitation, qui sont des propriétés de la matière. Ce caractère est également confirmé par la deuxième loi de la thermodynamique et par les développements récents de la cosmologie. (shrink)

Forward foundations. On the rationality of mathematics and of formalized sciences. — The failure of logicism for the question of foundations of mathematics invites us to consider this question under an epistemological point of view, in terms of rationality and not only of logic, and to extend it from mathematics to formalized sciences bearing on nature, such as physics, and even to scientific thought in general. One must take into account, in all cases, changes which correspond to conceptual constructions : (...) they secure at the same time, afterwards, the well-foundedness of the theories which have prepared them ; consequently, we should admit as a general rule that rational foundations can only be obtained “forward”. These changes ask also the question of their conditions of possibility. We get the conclusion, for such changes to be possible and together with them enhancement of knowledge, that one must admit correlated transformations in the very forms of rationality, mathematical ones, physical ones and of scientific rationality in general.RésuméL’insuffisance du logicisme pour la question des fondements des mathématiques nous invite à la poser sous un point de vue épistémologique, en termes de rationalité et non plus seulement de logique, et à l’étendre aux sciences formalisées portant sur la nature, comme la physique, voire à la pensée scientifique en général. On doit tenir compte, dans tous les cas, des changements qui correspondent à des constructions conceptuelles : ce sont eux qui assurent en même temps, après coup, le bien-fondé des théories qui les ont préparés, en sorte que si des fondements rationnels peuvent être obtenus, ce n’est, en règle générale, que « vers l’avant ». Ces changements posent également la question de leurs conditions de possibilités. La conclusion est qu’il faut admettre, pour que ces changements, et avec eux un accroissement de la connaissance, soient possibles, des transformations corrélatives dans les formes mêmes de la rationalité, mathématique, physique et, d’une manière générale, de la rationalité scientifique. (shrink)

Novelty and emergence in the quest for foundations. The search for firm foundations for a given knowledge, notably in the case of a formalized one, can be seen as a particular case of the search for deeper intelligibility. It generally brings to modifying the structural elements of the received knowl-edge (this having to do with questions of ontological relativity and epistemological holism), letting appear new elements of thought and of ‘reality’, emergent conceptual structures on the ground of the preceding ones. (...) We propose to show the link between the emergence of new kinds of knowledge, non thinkable previously, which enlarge the field of possible knowledge, and the effective “forward” motion of the search for foundations. (shrink)

En contrepoint à son œuvre mathématique et physique — et en relation avec elle — d'Alembert a développé une théorie de la connaissance influencée par Locke et le sensualisme de Condillac, mais centrée avant tout sur une épistémologie de la physique newtonienne. Réaliste, prônant le recours à l'expérience, il est en même temps profondément rationaliste, et même précisément, quoiqu'il s'en défende plutôt, dans la lignée de Descartes, Mais, bien que la Raison soit sa référence fondamentale, à tel point qu'il voudrait (...) fonder sur ses principes les plus évidents toute la science physico-mathématique — c'est-à-dire par excellence la Mécanique —, son programme ne peut être dit cartésien : non seulement il rejette les idées innées, mais il accepte la critique d'une rationalité apparente requise par la considération de faits irréductibles (l'attraction par exemple). Son épistémologie est un réalisme rationnel référé à l'être même de la Nature (la Raison et la Nature se rejoignent en profondeur). C'est en fonction de ces conceptions qu'il accepte ou rejette certaines notions physiques soit ambiguës, soit incertaines. Son rejet du concept de force comme celui de la considération d'une texture intime des corps semblent en faire, par le refus de ce qui ne serait pas directement mesurable, l'annonciateur du positivisme de Laplace et de Comte : une telle interprétation serait inexacte, et d'Alembert considère que la pensée peut parvenir à la connaissance du réel. Si ses conceptions s'élaborent en contrepoint de son activité scientifique, c'est en même temps en lutte et polémique contre la métaphysique, au sein de son engagement philosophique — il est tête de file, avec Voltaire, du « parti philosophique » — et notamment dans l'Encyclopédie. Bien qu'il se réfère, comme Newton, à une Intelligence suprême à l'œuvre dans l'Univers, il n'est pas déiste et affiche bientôt une position sceptique. Sa recherche et son affirmation de l'autonomie des lois de la Nature sont a-thées au sens privatif qui annonce Laplace. Il s'oppose au matérialisme de d'Holbach ou Helvétius, mais il se rapproche peu à peu — et notamment vers 1765, sans doute sous l'influence de Diderot qui rédigea vers cette époque Le rêve de d'Alembert et avec qui il venait de renouer — d'un matérialisme dynamique : mais, ce matérialisme, il ne le professe qu'en privé (essentiellement dans sa correspondance avec Frédéric de Prusse). L'âme est matérielle, et si « le plus simple raisonnement prouve qu'il y a un être éternel » , ce Dieu est matériel, il « n'est que la matière en tant qu'intelligente » , ce qui rejoint la définition du matérialisme donnée par Diderot dans l'Encyclopédie. Toutefois, refusant de se prononcer sur l'en-soi des choses, et sur la nature de la matière elle-même, comme de se définir métaphysiquement, son matérialisme tardif est encore marqué de scepticisme. (shrink)

We want to consider anew the question, which is recurrent along the history of philosophy, of the relationship between rationality and mathematics, by inquiring to which extent the structuration of rationality, which ensures the unity of its function under a variety of forms, could be considered as homeomorphic with that of mathematical thought, taken in its movement and made concrete in its theories. This idea, which is as old as philosophy itself, although it has not been dominant, has still been (...) present to some degree in the thought of modern science, in Descartes as well as in Kant, Poincaré or Einstein. It has been often harshly questioned, notably in the contemporaneous period, due to the failure of the logistic programme, as well as to the variety of “empirical” knowledges, and, in a general way, to the character of knowledges that show them as transitory, evolutive and mind-built. However, the analysis of scientific thought through its inventive and creative processes leads to characterize this thought as a type of rational form whose configurations can be detailed rather precisely. In this work we shall propose, first, a quick sketch of some philosophical requirements for such a research programme, among which the need for an harmonization, and even a conciliation, between the notions of rational, of intuitive grasp and of creative thought. Then we shall examine some processes of creative scientific thought bearing on the knowledge and the understanding of the world, distinct from mathematics although keeping tight relations with them. Contemporary physical theories are privileged witnesses in this respect, for in them the rational thought of phenomena makes an intrinsic use of mathematical thought, which contributes to the structuration of the formers and to the expression of their concepts. The General Theory of Relativity and the Quantum Theory are exemplar to this, as they directly reveal what can be called the “drag of physical thought par the mathematical form”, which makes possible to overcome the limitations of the physical knowledge previously adquired. This process is tightly related to the modalities and to the stucture of the rational thought underlying it. This is what we would like to show. (shrink)

As I am not at all a specialist on Gottlob Frege’s work, my comments intended initially to be focused on an aspect that emerges in the last part of Peter Clark’s paper “Frege, neo-logicism and applied mathematics” 1, where he treats the question of “applied mathematics” — an aspect that appealed to me and that was triggered by Frege’s relationship between numbers and concepts, and reasoning. Starting with this concern, I have been led by my subject to propose some considerations (...) about foundations and rationality which will go — briefly — in two directions. The first direction is that of a distinction between logical and rational foundations, whilst the second direction is that of taking into account as a fact the historical development of mathematics among the sciences, which modifies the terms of any foundational program. In delineating these considerations, I found that what I had in mind could apply to mathematics itself as well as to “applied mathematics”, thus deviating somewhat from my first explicit intention. In conclusion I shall consider the possibility of a rational foundation programme for mathematical and physico-mathematical sciences which would take into account the changes in the scientific contents and the widenings of the forms of rationality that, in my view, make these changes possible. Such foundations for knowledge would not be any more static, but dynamical and would be possibly considered only retrospectively: they would be “forward foundations”, in a sense that will be discussed in detail elsewhere. (shrink)

In this paper, we investigate the constitutive problems and other several aspects of what a research entitled 'formal epistemology' should be. The interest in this subject has to do with the possibility of reaching a privileged point of view or axis of research – i.e., the 'formal' one – that would allow a better grasp of the richness and variety of the facts and problems tackled by precise (local) epistemology of theories (for example, in physics). This approach is likely to (...) enable one to hold the main structural lines that organize those theories according to a more comprehensive, unifying and synthetic intelligibility. By the same token, it would eventually allow a better handling of the changes required in the organization of knowledge, putting emphasis on its main directions, drawing up a rational inventory of this knowledge, and perhaps anticipating others. At first, we deal with the 'thought of changes' that no approach of the 'form' can afford to leave aside, since the meaning of this concept is inseparable from the contents that come with constructions and modifications. We examine then the notion of 'epistemic operation' as an instrument to create new forms on the theoretical as well as on the meta-theoretical levels. In the wake of it, we analyze the characteristics of the form and of the formal, as well as their relationship with the contents of knowledge. We also take the notion of object into account, since it depends upon the decision of a subject and upon conventional choices. We finally inquire about the link between 'epistemic operations' as specified above and algorithmic functions for knowledge statements, and emphasize the risk of reductionism that might follow from a naturalistic conception of representation. (shrink)

De l'élimination des causes finales par la mathématisation de la physique à l'hypothèse de Kant-Laplace sur la formation du système solaire, puis au développement des sciences de la vie qui déterminent un nouveau champ de rationalité, le thème de la création au 18ème siècle est particulièrement apte à manifester l'évolution des rapports entre sciences, philosophie et métaphysique, jusqu'à son progressif effacement dans des philosophies aussi différentes que celles de Diderot, Hume et Kant.

As I am not at all a specialist on Gottlob Frege’s work, my comments intended initially to be focused on an aspect that emerges in the last part of Peter Clark’s paper “Frege, neo-logicism and applied mathematics” 1, where he treats the question of “applied mathematics” — an aspect that appealed to me and that was triggered by Frege’s relationship between numbers and concepts, and reasoning. Starting with this concern, I have been led by my subject to propose some considerations (...) about foundations and rationality which will go — briefly — in two directions. The first direction is that of a distinction between logical and rational foundations, whilst the second direction is that of taking into account as a fact the historical development of mathematics among the sciences, which modifies the terms of any foundational program. In delineating these considerations, I found that what I had in mind could apply to mathematics itself as well as to “applied mathematics”, thus deviating somewhat from my first explicit intention. In conclusion I shall consider the possibility of a rational foundation programme for mathematical and physico-mathematical sciences which would take into account the changes in the scientific contents and the widenings of the forms of rationality that, in my view, make these changes possible. Such foundations for knowledge would not be any more static, but dynamical and would be possibly considered only retrospectively: they would be “forward foundations”, in a sense that will be discussed in detail elsewhere. (shrink)