Ladislav Kvasz

The aim of this paper is to compare two approaches to semantics, namely the standard Tarskian theory and Wittgenstein’s picture theory of meaning. I will compare them with respect to an unusual subject matter, namely to geometrical pictures. The choice of geometry rather than arithmetic or set theory as the basis, on which this comparison will be made has two reasons. One reason is related to Wittgenstein’s picture theory of meaning. This theory was developed more or less as a metaphor, comparing the language to a picture. Nevertheless, if we take pictures themselves in the role of the language to which we apply Wittgenstein’s picture theory of meaning, this theory stops being a metaphor and starts to work in a technical way. I believe that in this way we can create a new approach to semantics in geometry. The other reason for taking geometry as the basis for our investigation is that the Tarskian approach to semantics does not work in geometry as we would wish

The aim of the paper is to analyse the geometrical aspects of a series of modern paintings and to show the parallel between them and the development of modern geometry. It starts with El Greco, offering a geometrical explanation of his painting the figures in a prolonged manner. Further the analogy between the impressionist way of creating space and the geometrical idea of Cayley to use projective space as a basis for non-Euclidean geometry is reconstructed. Next the paper describes the parallel between the creation of space in the paintings of Cézanne and Picasso and the concept of space in algebraic topology. In conclusion the paper deals with the analogy between Kandinski's abstract paintings and the set-theoretical foundations of geometry

The aim of this paper is a philosophical generalisation of the results, which we obtained through the analysis of the development of synthetic geometry. I our papers Náčrt analytickej teórie subjektu and Topológia versus teória množín we proposed a method of analysis of the development of geometry based on Wittgensteinś Picture theory of meaning from the Tractatus. It turned out, that the concept of the form of language can be effectively used to characterise the changes, which occured in the course of development of geometry. In the present paper we would like to generalize our approach and outline a universal approach to epistemology, which we call formal epistemology

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

The aim of the paper is to present a theoretical framework which would make it possible to embed the conflict between the natural and human sciences into a broader historical context. With the help of categories such as paradigmatic disciplines, mixed discipline of the paradigm, metaphorical realm of the paradigm, and elusive realm of the paradigm we describe the dynamics of changes in the classification of scientific disciplines which accompany a scientific revolution. Our approach is thus an alternative to T. S. Kuhn’s sociological theory of scientific revolutions. Instead by a social conflict between the proponents and opponents of the new paradigm we interpret the scientific revolution as a cognitive conflict between the mixed disciplines and the metaphorical realm of the old paradigm

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 aim of this paper is to introduce Wittgenstein’s concept of the form of a language into geometry and to show how it can be used to achieve a better understanding of the development of geometry, from Desargues, Lobachevsky and Beltrami to Cayley, Klein and Poincaré. Thus this essay can be seen as an attempt to rehabilitate the Picture Theory of Meaning, from the Tractatus. Its basic idea is to use Picture Theory to understand the pictures of geometry. I will try to show, that the historical evolution of geometry can be interpreted as the development of the form of its language. This confrontation of the Picture Theory with history of geometry sheds new light also on the ideas of Wittgenstein.

The aim of the paper is to describe the main epistemolo­gi­cal ruptures in the history of modern physics. Our ap­proach is based on the re­construction of the formal language of physical theories. We examine how particular aspects of the formal language, such as its analytical, expressive, or explanatory power, as well as its analytical and expressive boundaries, have changed in the course of the historical development of physics. In the closing part of the paper we discuss the results of our historical reconstruc­tion from several viewpoints

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 … "

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 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 question whether Kuhn's theory of scientific revolutions could be applied to mathematics caused many interesting problems to arise. The aim of this paper is to discuss whether there are different kinds of scientific revolution, and if so, how many. The basic idea of the paper is to discriminate between the formal and the social aspects of the development of science and to compare them. The paper has four parts. In the first introductory part we discuss some of the questions which arose during the debate of the historians of mathematics. In the second part, we introduce the concept of the epistemic framework of a theory. We propose to discriminate three parts of this framework, from which the one called formal frame will be of considerable importance for our approach, as its development is conservative and gradual. In the third part of the paper we define the concept of epistemic rupture as a discontinuity in the formal frame. The conservative and gradual nature of the changes of the formal frame open the possibility to compare different epistemic ruptures. We try to show that there are four different kinds of epistemic rupture, which we call idealisation, re-presentation, objectivisation and re-formulation. In the last part of the paper we derive from the classification of the epistemic ruptures a classification of scientific revolutions. As only the first three kinds of rupture are revolutionary (the re-formulations are rather cumulative), we obtain three kinds of scientific revolution: idealisation, re-presentation, and objectivisation. We discuss the relation of our classification of scientific revolutions to the views of Kuhn, Lakatos, Crowe, and Dauben.

The aim of the present paper is to offer a new analysis of the multifarious relations between mathematics and reality. We believe that the relation of mathematics to reality is, just like in the case of the natural sciences, mediated by instruments . Therefore the kind of realism we aim to develop for mathematics can be called instrumental realism. It is a kind of realism, because it is based on the thesis, that mathematics describes certain patterns of reality. And it is instrumental realism, because it pays atten-tion to the role of instruments by means of which mathematics identifies these patterns. The article concludes by offering solutions to some famous semantic paradoxes based on the diagonal construction as corroboration for this claim

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