Ladislav Kvasz

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 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 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

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