The aim of this paper is to examine the idea of metamathematical deduction in Hilbert’s program showing its dependence of epistemological notions, specially the notion of intuitive knowledge. It will be argued that two levels of foundations of deduction can be found in the last stages (in the 1920s) of Hilbert’s Program. The first level is related to the reduction – in a particular sense – of mathematics to formal systems, which are ‘metamathematically’ justified in terms of symbolic manipulation. The (...) second level of foundation consists in warranting epistemologically the validity of the combinatory processes underlying the symbolic manipulation in metamathematics. In this level the justification was carried out with the aid of notions from modern epistemology, particularly the notion of intuition. Finally, some problems concerning Hilbert’s use of this notion will be shown and it will be compared with Brouwer’s. (shrink)

This article examines how the theory of proportions was explained during the second half of the seventeenth century in the works of Andreas Tacquet, Antoine Arnauld, Ignace Gaston Pardies, Bernard Lamy, and Jacques Rohault. These five authors had very different conceptions of this subject, and on one hand, they show that this question was not forgotten, even after the Geometry of Descartes, and on the other hand, their work displays the progressive transformation of mathematical objects. While Tacquet deepened Euclidean thought, (...) the others stopped taking the Euclidean model as paradigmatic and tried to change the order of the Elements and to establish book V of Euclid in new ways. We shall see that this multiplicity of the approaches highlights both the vitality of the reflections and the difficulty in developing a new ontology of mathematics. Some of them nevertheless opened new perspectives that were to bloom only much later. We shall also see the increasingly important place of the algebra as time went by. (shrink)

Computation and formalization are not modalities of pure abstractive operations. The essay tries to revise the assumption of the constitutive nonsensuality of the formal. The argument is that formalization is a kind of linear spatialization, which has significant visual dimensions. Thus, a connection can be discovered between visualization by figurative graphism and formalization by symbolic calculations: Both use spatial relations not only to represent but also to operate on epistemic, nonspatial, nonvisual entities. Descartes was one of the pioneers of using (...) this kind of two-dimensional spatiality as a cognitive instrument. (shrink)

It is not just rationalist metaphysics, but also Kant’s transcendental philosophy that is teeming with a priori concepts. According to Kant, some of these a priori concepts are “ideas,” and similar to the categories, some of these ideas in turn belong to the nature of human reason, while others can be derived from them. It is therefore part of Kant’s claim in the “Transcendental Dialectic” to be able to explain not only the leading ideas of rational psychology, cosmology, and theology (...) as natural concepts of reason, but the origin of the entire a priori vocabulary of the metaphysica specialis as well. This article outlines how Kant’s derivations of these concepts work. After explaining Kant’s classification of a priori concepts and the derivation of the transcendental ideas, his derivations are explored in more detail using the example of the concepts of the pure doctrine of the soul. This contributes to a better understanding of Kant’s theory of philosophical concepts while also shedding light on the rather rarely treated topic of the “Transcendental Dialectic.”. (shrink)

This paper explicates Descartes’ theory of intuition (intuitus). Departing from certain commentators, I argue that intuition, for Descartes, is a form of clear and distinct intellectual perception. Because it is clear and distinct, it is indubitable, infallible, and provides a grade of certain knowledge he calls ‘cognitio’. I pay special attention to why he treats intuition as a form of perception, and what he means when he says it is ‘clear and distinct’. Finally, I situate his view in relation to (...) those of his Scholastic predecessors on one hand and current theories on the other. His view anticipates the contemporary ‘perceptual model of intuition’, though it is much bolder with its promise of certainty. (shrink)

In recent years, the growing scientific interest in design has led to great advances in our knowledge of authentic design processes. However, as these findings go counter to the existing theories in both design research and cognitive science, they pose a serious challenge for both disciplines: there is a wide gap between what the existing theories predict and what designers actually do. At the same time, there is a growing movement of research on authentic cognitive activities, which has among other (...) things documented the central roles of action and the physical environment in these activities, something that existing cognitive theories have overlooked and cannot properly account for. This creates an explanatory gap analogous to the one found in design. This book aims to fill both of these gaps with a cognitive theory of how designers work. It revolves around the roles of physical activities and working materials in design, and the theory explains at length how these have functions that are essential to cognition. The two threads of design and cognition run in parallel throughout the book: the cognitive theory is applied to design, but is also consistently related to cognition in general. The result is, in back cover text parlance, a 'provocative' account of cognition and human performance, which should be of interest to cognitive scientists as well as to design researchers. (shrink)

O artigo tem como objetivo investigar as razões que levaram Descartes a não concluir as Regras para a direção do Espírito, estabelecendo uma relação entre a interrupção da obra e a teoria de 1630 sobre a natureza criada das verdades eternas. Com a doutrina da livre criação das verdades, Descartes apresenta uma proposta metafísica que exigia uma revisitação dos pressupostos da sua própria epistemologia. Se nas Regras a matemática e a geometria eram consideradas isentas de toda incerteza, com a entrada (...) em cena da teoria de 1630 essas disciplinas encontram-se submetidas à causalidade arbitrária da onipotência divina. Essa consideração metafísica requer uma reflexão ulterior para avaliar as suas consequências. O projeto das Regras – interrompido, mas não abandonado – será assim reformulado à luz da novidade metafísica de 1630. As Meditações se encarregarão dessa tarefa, demonstrando de que modo a natureza criada das verdades eternas não implica colocar em questão o conhecimento humano. (shrink)

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

Vít Gvoždiak published a reconciliatory analysis of Searle’s social ontology with semiotics in Gvoždiak (2012). Without prior knowledge of his paper, an analysis of the same subject appeared in Msimang (2014). Even though Searle’s social ontology is a common point of reference in the formulation of semiotics in these papers, it also serves as a point of departure in their understanding of semiotics and its development. The semiotic theory expressed in Gvoždiak (2012) is an inherently linguistic (speech act centred) theory, (...) whereas the semiotic theory presented in Msimang (2014) tends more towards a general theory of communicative systems in which social ontology, which follows from speech act theory, is an interesting part. It is my purpose in this note to contrast the two positions of semiotic theory as they appear in the aforementioned papers in reference to their appropriation of Searle’s social ontology. (shrink)

In moral epistemology, the method of reflective equilibrium is often characterized in terms of intuitions or understood as a method for justifying intuitions. An analysis of reflective equilibrium and current theories of moral intuitions reveals that this picture is problematic. Reflective equilibrium cannot be adequately characterized in terms of intuitions. Although the method presupposes that we have initially credible commitments, it does not presuppose that they are intuitions. Nonetheless, intuitions can enter the process of developing a reflective equilibrium and, if (...) the process is successful, be justified. Since the method of reflective equilibrium does not essentially involve intuitions, it does not constitute a form of intuitionism in any substantial sense. It may be classified as intuitionist only in the minimal sense of not reducing justification to a matter of inference relations alone. (shrink)

Le débat qui se déroule dans les années 1720 à l'Académie royale des inscriptions et belles-lettres, sur la possibilité ou non de connaître l'histoire des premiers siècles de Rome, est généralement interprété comme une mise au point épistémologique par l'un des protagonistes. Une mise en contexte qui prend en considération les enjeux politiques de la dispute permet de comprendre que ce débat est l'amorce d'un processus d'autonomisation du champ des études historiques savantes, non seulement du point de vue de la (...) division des savoirs, mais aussi par rapport au pouvoir monarchique. (shrink)

Exchanges in the nineteenth century between Sir William Hamilton, James Frederick Ferrier and the French philosopher Victor Cousin are crucial to understanding contemporary efforts to preserve the continuity of the Scottish philosophical tradition on the part of those alive to new themes emanating from Kant and philosophy in Germany. Ferrier's strategy aimed at re-invigorating Descartes and Berkeley by drawing on elements in Adam Smith's social philosophy. But the promising steps taken in this direction in Ferrier's essays on consciousness were seriously (...) undermined, in Cousin's view, by the aprioristic character of his Institutes, in which he abandoned the careful empirical method characteristic of Scottish philosophy as practiced by Reid. (shrink)

The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in Posterior (...) analytics is used to distinguish between conceptions that share the same name but are substantively different: for example the search for a broader genus including all mathematical objects; the search for a common character of different species of mathematical objects; and the effort to treat magnitudes as numbers. (shrink)

In this paper we discuss the difference between (...) proliferations of logic systems, the questions of relativity and universality of logic and the position and interaction of logic with regards to other sciences such as physics, biology, mathematics and computer science. (shrink)

Descartes’ Rules for the direction of the mind presents us with a theory of knowledge in which imagination, considered as an “aid” for the intellect, plays a key role. This function of schematization, which strongly resembles key features of Proclus’ philosophy of mathematics, is in full accordance with Descartes’ mathematical practice in later works such as La Géométrie from 1637. Although due to its reliance on a form of geometric intuition, it may sound obsolete, I would like to show that (...) this has strong echoes in contemporary philosophy of mathematics, in particular in the trend of the so called “philosophy of mathematical practice”. Indeed Ken Manders’ study on the Euclidean practice, along with Reviel Netz’s historical studies on ancient Greek Geometry, indicate that mathematical imagination can play a central role in mathematical knowledge as bearing specific forms of inference. Moreover, this role can be formalized into sound logical systems. One question of general epistemology is thus to understand this mysterious role of the imagination in reasoning and to assess its relevance for other mathematical practices. Drawing from Edwin Hutchins’ study of “material anchors” in human reasoning, I would like to show that Descartes’ epistemology of mathematics may prove to be a helpful resource in the analysis of mathematical knowledge. (shrink)

After recalling the distinction between logic as reasoning and logic as theory of reasoning, we first examine the question of relativity of logic arguing that the theory of reasoning as any other science is relative. In a second part we discuss the emergence of universal logic as a general theory of logical systems, making comparison with universal algebra and the project of mathesis universalis. In a third part we critically present three lines of research connected to universal logic: logical pluralism, (...) non-classical logics and cognitive science. (shrink)

El presente artículo trata de poner de manifiesto, tomando como hilo conductor la polémica Foucault-Derrida, las tensiones que recorren el proceso de la duda en Descartes. En dichas tensiones (entre un orden deductivo y otro demostrativo, así como entre un entendimiento racional y una voluntad razonable) hay un elemento que transita desde la locura hasta la divinidad; dicho elemento opera como un resto que, una y otra vez, reaparece para amenazar todo intento de fundamentación.

Starting from a suggestion of Stephen Toulmin and through an interpretation of the criticism to which Neurath, one of the founders of the Vienna Circle, submits Descartes’ views on science, the paper attempts to outline a pattern of modernity opposed to the Cartesian one, that has been obtaining over the last four centuries. In particular, it is argued that a new alliance has to be established between science and education, overcoming Descartes’ banishment against education. In a Neurathian perspective education is (...) a key-moment of the scientific enterprise without which science itself is in danger of going astray and no scientific outlook is promoted in the society at large. Such an anti-Cartesian attitude is a leitmotiv of the whole Neurath’s production and characterizes his fundamental approach to the sense of modernity. For this reasons, despite all its shortcomings, Neurath’s proposal represents a very promising option for a new agenda of the modernity away from Descartes’ spell. By elaborating on Neurath’s (and Dewey’s) insights, the paper puts forward the idea that philosophy of science (such as it was originated by neopositivism in its Reichenbachian version) should give way to an educational philosophy of science which could allow us “to bring the genuine modern into existence”. (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)

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.

Review of Garrod, R., & Marr, A. (Eds.). (2021). Descartes and the ingenium: the embodied soul in Cartesianism. Leiden: Brill.

This paper aims to reassess the philosophical puzzle of the “applicability of mathematics to physical sciences” as a misunderstood disciplinary interplay. If the border isolating mathematics from the empirical world is based on appropriate criteria, how does one explain the fruitfulness of its systematic crossings in recent centuries? An analysis of the evolution of the criteria used to separate mathematics from experimental sciences will shed some light on this question. In this respect, we will highlight the historical influence of three (...) major disciplinary paradigms. According to the Aristotelian classification of the sciences, the separation of mathematics from physics is based on their respective objects of study. The Baconian system distinguishes these sciences by the type of knowledge involved in each field. Finally, the Whewellian disciplinary layout categorises these disciplines by their respective methods. In this paper, we argue that the cascading effect of such disciplinary categorisations—based successively on ontological, epistemological, and heuristic criteria—resulted in a profound redefinition of the border between mathematics and physics, with the consequence of obscuring the foundations of the interplay between these fields. Our approach intends to put forward such a mechanism as a constitutive piece of the puzzle of the applicability of mathematics. (shrink)

From both physical and epistemological viewpoints, the following theses, which nowadays are often discussed in the literature, are examined: Nonlinear thermodynamics renders it possible to grasp evolutionary physical processes; for thermodynamics it introduces, instead of idealized reversible time, a directed time into physics; thus a science is established that is nearer to reality than classical physics. To analyze these theses, the relation of thermodynamics to dynamical physics is considered. In particular, it is demonstrated that, in classical as well as in (...) modern thermodynamics, irreversibility is introduced via conditions which must be formulated in addition to the dynamical laws. To show the reason for this, the epistemological status of the physical time conception is analyzed, and its character as a physical measurement quantity is established. (shrink)