Ladislav Kvasz

This paper offers an epistemological reconstruction of the historical development of algebra from al-Khwrizm, Cardano, and Descartes to Euler, Lagrange, and Galois. In the reconstruction it interprets the algebraic formulas as a symbolic language and analyzes the changes of this language in the course of history. It turns out that the most fundamental epistemological changes in the development of algebra can be interpreted as changes of the pictorial form of the symbolic language of algebra. Thus the paper develops further the method of reconstruction which the author introduced for the analysis of the development of geometry

The aim of the paper is to examine the changes, which occurred in the epistemological structure of classical mechanics during its development from Newton to Poincaré. The analysis is based on the reconstruction of the language form. Attention is paid to such aspects of the language of classical mechanics as the notion of pace or the description of action . Even though these notions do not have direct denotation, they, nevertheless, constitute the general framework, on which the relation between the denotative descriptions of the language of classical mechanics and the reality is based

The book is a detailed and penetrating study of the relations between Bolzano’s philosophical views and his mathematical achievements. It opens with a Biographical sketch, written with a good understanding of the political situation in Central Europe at Bolzano’s times. The words of the Emperor Francis II: “I do not need scholars but obedient citizens” , and of an English visitor of Austria at those times: “These school-books are the most barren and stupid extracts which ever left the printing press.” express the problems of the intellectuals in this part of the world

Mathematics is traditionally considered being an apriori discipline consisting of purely analytic propositions. The aim of the present paper is to offer arguments against this entrenched view and to draw attention to the experiential dimension of mathematical knowledge. Following Husserl’s interpretation of physical knowledge as knowledge constituted by the use of instruments, I am trying to interpret mathematical knowledge also as acknowledge based on instrumental experience. This interpretation opens a new view on the role of the logicist program, both in philosophy of mathematics and in philosophy of science

Cieľom článku je osvetliť niektoré historické okolnosti vzniku analytickej filozofie. Prvou z nich je zrod novovekej fyziky, ktorý viedol k rozšíreniu nového, empirického prístupu ku skutočnosti. Snažíme sa ukázať úlohu, ktorú v procese vzniku novovekej fyziky zohrala scholastická filozofia spolu s kartezianizmom. Ďalšia časť state objasňuje niektoré aspekty subjektivizmu, ktorý zohral významnú úlohu v novovekej filozofii. V jadre subjektivizmu leží podľa nášho názoru dezinterpretácia Descartovej filozofie. Prekonanie subjektivizmu analytickou filozofiou možno potom interpretovať ako návrat na miesto, na ktorom Descartes pôvodne stál

Coming from a mathematical background, I was always puzzled by Popper’s view, according to which, after the falsification of a scientific theory its degree of corroboration becomes zero. Most of the scientific theories taught in the physics departments have already been falsified, and what is the point of teaching theories, whose degree of corroboration is zero? The first important observation to make is that not all cases of falsification are the same. In some cases, as for instance in the case of the theory of phlogiston, the falsification happened in agreement with the Popperian picture. Scientists discarded the falsified theory and opted for its alternative. Therefore, nowadays nobody tries to make a scientific contribution to the theory of phlogiston. Nevertheless, there are cases, and Newtonian mechanics is surely the most important among them, when the behaviour of the scientists after the falsification of the theory is from the Popperian point of view incomprehensible. Many scientists were not ready to discard the falsified theory, and not for irrational reasons. Some of the deepest discoveries in Newtonian mechanics as for instance the famous Kolmogorov, Arnold, Moser theorem were made many years after this theory was falsified. I think that Andrej Kolmogorov, Vladimir Arnold or Jürgen Moser cannot be compared to an Aristotelian philosopher, who adheres to his pet theory after its falsification. What these three mathematicians did was a fundamental contribution to modern science.

The latest book of Reviel Netz presents a highly erudite analysis of the style of Hellenistic mathematics. Besides the Introduction and Conclusion the book is composed of four chapters. Before turning to more general remarks I would like first to outline the contents of the book.The Introduction starts with the presentation of Archimedes’ Spiral lines. In a condensed form, the author outlines his approach and calls attention to the stylistic peculiarities of that work. Most of the themes discussed in more detail in the following chapters, such as the analysis of the narrative structure of the treatise, its working with surprise, the exposition of the rhetorical devices, the discussion of technical tools, and the breaking of genre-boundaries emerge here side by side in a single treatise.The first chapter, called The carnival of calculation, starts with another work of Archimedes, the Stomachion, for which Netz offers a new interpretation as a treatise in geometrical combinatorics. Then he turns to Archimedes’ On the measurement of the circle, where, in a detailed analysis of the calculations made by Archimedes, Netz presents a pattern that he identifies in several other works of Hellenistic mathematics, a pattern he calls ‘asymptotic calculation’. It consists in a combination of strict geometric arguments with calculations of proportions, some precise, some only approximate, which together with rather arbitrary simplifications lead to a result that is achieved after an abrupt and unexpected halt. Netz shows that, according to the surviving testimonies, a similar pattern can be found in many Hellenistic authors, such as Aristarchus, Hipparchus, and Apollonius of Perga. "[I]n many Hellenistic mathematical works we see calculations that give rise to values that are big and unwieldy, or else are only asymptotically found. Such calculations end up not with the satisfaction of having simplified … "

Mathematics was considered by the members of the Vienna Circle an a priori discipline consisting of analytic propositions. The aim of the present paper is to draw attention to the experiential dimension of mathematical knowledge. Following the interpretation of physical knowledge as knowledge constituted by the use of instruments, I am trying to interpret also mathematical knowledge as knowledge based on instrumental experience. This interpretation opens a new perspective on the place of the logicist program in philosophy of mathematics. The aim of the present paper is to argue that mathematics is based on experience and that mathematical experience is instrumental. The difference between mathematics and physics is that in the case of mathematics the instruments are not measuring devices but tools of symbolic representation. In order to distinguish the mathematical experience from the physical one, I will call the instrumental experience in mathematics as symbolic experience. In analogy with the measuring devices of physics I will interpret the tools of symbolic representation as tools constituting symbolic experience.

The nature of changes in mathematics was discussed recently in Revolutions in Mathematics. The discussion was dominated by historical and sociological arguments. An obstacle to a philosophical analysis of this question lies in a discrepancy between our approach to formulas and to pictures. While formulas are understood as constituents of mathematical theories, pictures are viewed only as heuristic tools. Our idea is to consider the pictures contained in mathematical text, as expressions of a specific language. Thus we get formulas and pictures into one linguistic framework, and so we are able to analyse their interplay in the course of history.

The paper attempts to summarize the debate on Kant’s philosophy of geometry and to offer a restricted area of mathematical practice for which Kant’s philosophy would be a reasonable account. Geometrical theories can be characterized using Wittgenstein’s notion of pictorial form . Kant’s philosophy of geometry can be interpreted as a reconstruction of geometry based on one of these forms — the projective form . If this is correct, Kant’s philosophy is a reasonable reconstruction of such theories as projective geometry; and not only as they were practiced in Kant’s time, but also as architects use them today

There are many interpretations of the birth of modern science. Most of them are, nevertheless, confined to the analysis of certain historical episodes or technical details, while leaving the very notion of mathematization unanalyzed. In my opinion this is due to a lack of a proper philosophical framework which would show the process of mathematization as something radically new. Most historians assume that the world is just like it is depicted by science. Thus they are not aware of the radical novelty of the mathematization of nature and focus their attention on the details of this process. Phenomenology by its radical questioning of the traditional interpretations of reality provides an ideal means for the reconstruction of the process of mathematization of nature and the birth of mathematical natural science. In a series of papers I tried to show the power of Husserl 's concept of mathematization by filling in the historical details into his interpretation. In the present article I want to compare Husserl 's approach with the approach of Heidegger. I believe that by comparing these two phenomenological theories of mathematization the advantages of Husserl 's approach will come to the fore. _German_ Es gibt mehrere Interpretationen der Entstehung der modernen Wissenschaft. Die meisten von ihnen beschränken sich allerdings auf die Analyse bestimmter historischer Episoden oder technischer Details, während der Begriff der Mathematisierung unanalysiert bleibt. Meiner Meinung nach ist dies auf das Fehlen eines richtigen philosophischen Rahmens zurückzuführen, der den Prozess der Mathematisierung als etwas radikal Neues zeigen würde. Die meisten Historiker gehen davon aus, dass die Welt so ist, wie sie von der Wissenschaft dargestellt wird. So verkennen sie die Radikalität der Mathematisierung der Natur und konzentrieren ihre Aufmerksamkeit auf die Details dieses Prozesses. Die Phänomenologie bietet dank ihrer radikalen Kritik der traditionellen Interpretationen der Realität ein ideales Mittel für die Rekonstruktion des Prozesses der Mathematisierung der Natur und der Geburt der mathematischen Naturwissenschaft. In einer Reihe von Arbeiten habe ich versucht, die Stärke der Husserlschen Interpretation der Mathematisierung zu zeigen, indem ich die historischen Details in seine Interpretation einfügte. Im vorliegenden Artikel möchte ich Husserls Ansatz mit dem Ansatz von Heidegger vergleichen. Durch diesen Vergleich der beiden phänomenologischen Theorien der Mathematisierung der Natur werden die Vorteile von Husserls Ansatz in den Vordergrund treten