Ladislav Kvasz

The text attempts, in a new way, to interpret the transition from Aristotelian syllogistic logic to Frege’s mathematical logic and thus builds on my previous analyses of Aristotle’s logic. Provided that the proposed interpretation shows itself to be adequate, it can help further develop formal epistemology that I presented in one of my previous contributions (Kvasz, L., Formalna epistemológia – budúca syntéza. Filosofický časopis, 64, 2016, No. 6, pp. 951–963). Formal epistemology attempts to emulate, in epistemology, Frege’s reform of logic. A new understanding of the transition from Aristotle’s to Frege’s logic will allow us to divide the formalizing epistemology into several tasks, namely the introduction of relational, compositional and deductive syntheses into the language of epistemology, i.e. tasks that correspond to the steps that Frege undertook in building his logic.

There are several interpretations of what mathesis universalis could have been. Some interpreters claim that mathesis universalis is a universal method of discovery; others understand it as coextensive with algebra as a fundamental discipline of mathematics; while still others identify it with the mathematical articulation of material reality. Each of these three interpretations is safe. We can find traces of all of them in Descartes’ work. However, each of them captures only one aspect of Descartes’ mathematical work, ignoring others. Also, they allow us to understand neither the unity of these three aspects of Descartes’ mathematical work nor the relationship between mathesis universalis and his method. The method, as Descartes presented it in the Regulae, was geometrical rather than algebraic, its aim was certainty rather than discovery, and it did not discriminate between material and non-material reality. I will show how Descartes gradually expanded and radicalized his mathesis universalis.

As Timothy Reiss pointed out, in the Discours de la Méthode Descartes includes the description of two methods. The first is given in the second part of the treatise, and consists of four rules (Descartes 1637a, p. 120; AT 6, pp. 18–19). According to Reiss, however, the Discours contains also a second method, which Descartes introduces in the sixth part of the treatise, where he speaks about his manuscript (Le Monde) and indicates the reasons why he decided not to publish it. (Descartes 1637a, pp. 143–144; AT 6, pp. 63–64). To these two expositions of method we may add a third, contained in the first pages of La Géométrie (Descartes 1937b, p. 6; AT 6, p. 372). Since La Géométrie was published as an appendix to the Discours, we can say that in 1637 Descartes published three variants of his method. The first variant (from the second part of the Discours) we will call the epistemic method; the second variant (from the sixth part of the Discours) we will call the method of reduction, and the third (from La Géométrie). My aim is to explain the relations among these three formulations of Descartes' method.

Most historians of science contrast Descartes’ verbally formulated (and in many respects mistaken) physics with Galileo’s and Newton’s mathematical physics. Garber called Descartes’ physics “metaphysical physics”. Not mathematical physics, but metaphysical physics (see Garber 1992). Schuster accepts this view, consistently labeling Descartes’ physics as “natural philosophy”. I shall show that Descartes’ physics was a mathematical physics, despite its verbal formulation. It fundamentally overcame the limitations of Galilean science and in many ways was the starting point for Newton’s innovations.

Schuster criticized Descartes’ method, which he interpreted as a unique, universal, transferable and efficacious general method of discovery and/or justification for rational disciplines. The accumulation of such absolutist attributes makes Descartes’ method untenable. However, when these attributes are understood as manifestations of the enthusiasm that naturally accompanies every discovery, they can be relativized. Thus, unique can be understood as exceptional, universal can be understood as broadly applicable, transferable can be interpreted as transferable to adjacent fields, and general can be understood as more general than the common practices of the time. These relativized attributes were shifted by Schuster’s description into the realm of ideological blindness and demagogy. Descartes had many reasons to be enthusiastic about his method. His analytic method was indeed exceptional, it opened possibilities for further discovery, and it was transferable to various disciplines, beginning with geometry, where it proved its efficacy. Therefore, rather than debunking the absolutist caricature of Descartes’ method, it is more interesting to try to understand its working core and to reconstruct its development.

Descartes is considered one of the founders of the modern theory of knowledge and the creator of a dualistic metaphysics. This is why books about Descartes are usually written by historians of philosophy. When historians of mathematics write about Descartes, they confine themselves to interpreting a particular aspect of Descartes’ work. The mathematical aspect of Descartes’ work thus becomes a kind of ornament to the work of a thinker whose life struggle occurred in the field of philosophy. Nevertheless, if we want to understand Descartes, we must remember that he was, perhaps, one of the most influential mathematicians of all time, someone like Euclid, Newton, Euler, Cantor, Frege, Hilbert or Turing. Therefore, the task of anyone who wants to deal seriously with Descartes is at least once to engage in a meditation in which everything previously learned about Descartes is discarded in an effort to come to a clear idea of the nature of his mathematical talent.

We consider the answer to this question to be the central problem of the interpretation of Descartes’ scientific work. Schuster declared Descartes’ physics, because of its verbal character, to be natural philosophy, and excluded it from the tradition of mathematical physics, which he associates with Galileo and Newton. This assessment is mistaken, and the reason for the mistake is the vagueness of the concept of mathematical physics. Galileo’s physics, which completely lacks any description of interaction, which attributes to bodies a natural tendency to accelerate in free fall, and which considers circular motion as inertial, is automatically taken to be mathematical physics. On the other hand Descartes who introduced the notion of interaction into physics, who clearly saw that the acceleration of falling bodies must be the result of interaction, and who regarded only uniform rectilinear motion to be inertial, is not included in mathematical physics. My aim is to correct this obvious error.

The language of mathematics attracted recently the attention of philosophers, historians of mathematics, and researchers in mathematics education (see Dutilh Novaes in Formal languages in logic. A philosophical and cognitive analysis. Cambridge University, Cambridge, UK, 2012, Hoyrup in The development of algebraic symbolism. College Publications, London, 2010, Kvasz in Communication in the mathematical classroom. Wydawnictwo Uniwersitetu Rzeszowskiego, Rzeszów, pp. 207–228, 2014, Lakoff and Nunez in Where mathematics comes from. Basic Books, New York, 2000, Macbeth in Realizing reason. A narrative of truth and knowing. Oxford University Press, Oxford, 2014, Serfati in La Révolution Symbolique. La Constitution De L’Ecriture Symbolique Mathematique. Editions Petra, Paris, 2005, or Sfard in Thinking as communicating. Human development, the growth of discourses, and mathematizing. Cambridge University Press, Cambridge, UK, 2008). There exist a considerable number of approaches, each of which studies one particular aspect of the language of mathematics that is relevant in the particular context. Nevertheless, what is lacking is a theoretical framework that would make it possible to integrate these different approaches to the study of the language of mathematics and use their potential for a deeper theoretical foundation of mathematics education. The aim of the present paper is to argue for the need and to outline the possible structure of an integrative theoretical framework for the study of the language of mathematics. The author is convinced that such an integrative initiative can arise from the philosophy of mathematics that is historically grounded and educationally motivated.

In this article, we propose an interpretation of Aristotle’s theory of local motion as the first stage of the mathematization of motion. We proceed from the interpretation presented by Jan Patočka in his work Aristoteles, jeho předchůdci a dědicové (Aristotle, His Predecessors and His Heirs). We try to show that Patočka approached the interpretation of Aristotle’s theory of motion as an attempt at mathematization. What prevented him from taking the final step towards such an interpretation was, on the one hand, his acceptance of Plato’s theory as the first mathematization of the cosmos, against which Aristotle’s attempt looks like a departure from mathematization, and, on the other hand, his failure to realize the difference between the mathematics of Aristotle’s time and the mathematics by which motion was later mathematized.

The invention of symbolic algebra in the sixteenth and seventeenth centuries fundamentally changed the way we do mathematics. If we want to understand this change and appreciate its importance, we must analyze it on two levels. One concerns the compositional function of algebraic symbols as tools for representing complexity; the other concerns the referential function of algebraic symbols, which enables their use as tools for describing objects (such as polynomials), properties (such as irreducibility), relations (such as divisibility), and operations (such as factorization). The reconstruction of both the compositional function and the referential function of algebraic symbols requires the use of different analytic tools and the taking of different temporal perspectives. In this chapter, we offer both: a reconstruction of the compositional function of algebraic symbols, and a reconstruction of their referential function.

This book argues that Descartes’ physics was a milestone on the road to modern mathematical physics. After Newton introduced a completely different approach to mathematical description of motion, Descartes’ physics became obsolete and even difficult to comprehend. This text follows the language of Descartes and the means of which motion can be described. It argues that Descartes achieved almost everything that later Newton was able to do—to describe the motion of interacting bodies- by different (i.e. algebraic) means. This volume completely refutes the received view according to which Descartes’ physics was merely a kind of discursive natural philosophy. To make this interpretation more plausible the book follows Descartes’ ideas from his early work in mathematics, through his invention of the analytic method towards his mature physics. It shows that Descartes followed a similar heuristic pattern. The volume appeals to students and researchers; it invites the reader equippedwith minimal understanding of college mathematics to follow Descartes on his intellectual journey through the Scientific Revolution. The reader will gain a deeper understanding of the role of mathematical language in the creation of modern physics and a glimpse into the fascinating world of Descartes’ scientific thought. Several of Descartes’ philosophical ideas can be traced back to his scientific interests and thus the book elucidates the motivation behind some of Descartes’ key positions in the area of epistemology and method. In the penultimate chapter the book presents four arguments in favor of seeing Descartes as a physicist on par with Galileo and Newton.

The aim of this paper is to connect Wittgenstein’s picture theory of meaning with the Hegelian idea of the development of the self. Combining Wittgenstein with Hegel is perhaps not so original (see Mácha/Berg 2019); nevertheless, the context by means of which they will be connected, namely the history of painting, is perhaps new. I will argue that Wittgenstein’s notion of pictorial form is an excellent tool for the analysis of the development of painting from Renaissance to modern art. The idea is to identify epochs, such as the Renaissance or Baroque, by means of the pictorial form they use. Pictorial form is closely related to the epistemic subject and thus the development of pictorial form can be used as a key to studying changes in the self. I shell discriminate eight pictorial forms in the history of western art and describe the form of selfhood characteristic of each of them.

The aim of the paper is to analyze how language affects scientific research, from planning experiments and interpreting their results, through constructing models and the testing their predictions, to building theories and justifying their principles. I try to give an overview of the potentialities of language of science. I propose to distinguish six potentialities: analytic, expressive, methodical, integrative, explanatory, and constitutive power of language. I will shortly characterize each of these potentialities and illustrate their contribution to scientific research. Although I believe that the theory is valid for a wide range of scientific disciplines, all illustrations are taken from physics.