Abstract The modern mathematical theory of dynamical systems proposes a new model of mechanical motion. In this model the deterministic unstable systems can behave in a statistical manner. Both kinds of motion are inseparably connected, they depend on the point of view and researcher's approach to the system. This mathematical fact solves in a new way the old problem of statistical laws in the world which is essentially deterministic. The classical opposition: deterministic?statistical, disappears in random dynamics. The main thesis of (...) the paper is that the new theory of motion is a revolution in the research programme of classical mechanics. It is the revolution brought about by the development of mathematics. (shrink)
Abstract The first fractal constructions appeared in mathematics in the second half of the 19th century. Their history is divided into two periods. The first period lasted 100 years and is a good example of the method of proofs and refutations discovered by Lakatos. The modern history of these objects started 20 years ago, when Mandelbrot decided to create fractal geometry, a general theory concentrated on specific properties of fractals. His approach has been surprisingly effective. The aim of this paper (...) is to examine the reasons for Mandelbrot's success and for the present popularity of fractals. They are now known not only in mathematics but also in many fields of natural, social and applied sciences, and even in the arts. (shrink)
Scientists, when they want to understand properties of complex systems, they divide them into simple parts and – knowing the parts and their relations – reconstruct the whole. This approach is effective in the case of simple systems: atomic nuclei, atoms, chemical compounds, stars and so on. In biology it is supplemented by the theory of evolution. However the situation is hopeless when one wants to explain the astonishing properties of the Universe generally known as Anthropic Principle. Discussing that problem, (...) philosophers use the metaphor of God as a Mathematician. The aim of the paper is to prove the inadequacy of this metaphor. (shrink)
The purpose of this paper is to compare two conceptions of order. One, based on classical mechanics, identifies order as simplicity of components and their compositions. Within this framework it is difficult to describe and understand system which have rich, dynamical structures. In order to accomplish this, we need non-linear dynamics, and hence a new notion of order. In non-linear dynamics local processes are chaotic, but order emerges on the higher level.
This article is a polemic with the thesis held by J. Bobryk that we will never succeed in giving a unified description of human thinking and physiological processes taking place in human brain. The author rejects this view on the basis of the results achieved by modelling the brain's activities with the help of the theory of chaos. These models treat the brain as an incredibly complex dynamical system in which the order apprears as a result of mutual interactions of (...) big neuron collections. The significant empirical material was gathered, ilustrating chaotic dynamics of the sense of sight, muscles movements, concentration etc. There seems to be no unsurmountable obstacles on the way of creating the unified theory of the brain dynamics and the consciousness. (shrink)
After the rise of the theory of relativity and the quantum theory, the classical mechanics has been recognized as an approximate version of both theories - applicable only within the bounds of macro-phenomena and by relatively low velocities. The theory of chaos - unfolding nowadays, and being amplification of the classical mechanics - reverses this relation: quantum processes are treated as a certain rather simple class of non-linear and non-stable mechanics phenomena. The unexpected revival of the classical mechanics forces the (...) considerable weakening of the thesis about incommensurability of the theories separated by a scientific „revolution”. (shrink)
Elementary particle physics is an intensively developing fundamental branch of physics. It has many important results and the picture of the fundamental level of the structure of matter elaborated by this theory becomes more and more detailed and complete. Physicists call the picture the Standard Model. However, in spite of its great achievements, Standard Model has several serious problems that cannot be solved by its methods. In order to overcome these problems physicists formulate the ultimate physical theory, called the supersymmetry (...) theory. They have big expectations with respect to this theory and they build strong accelerators to perform experiments testing specific predictions of supersymmetry. Till now none of those predictions has been empirically corroborated and this fact leads to the conclusion that the idea of supersymmetry is more fantasy than the empirical scientific theory. The paper is based on the presentation of the supersymmetry theory given by G. Kane in his book Supersymmetry. Squarks, Photinos and the Unveilling of the Ultimate Laws of Nature. (shrink)
The subject of the paper is the presentation and discussion of Lee Smolin's book The Trouble with Physics . Smolin presents in his book the history and the current state of elementary particle physics. He shows that this field of research is now in a very bad situation because no progress has been achieved for last twenty five years. Since the time of Galileo and Newton, fundamental physics has been in the process of permanent progress and the recent lack of (...) success is considered by Smolin as ex-ceptional and dangerous. Superstring theory is, in his opinion, the main cause of this trouble with physics. The theory dominated elementary particle physics in USA, al-most completely eliminating other more fruitful approaches to elementary particles and their interactions. Smolin analyses the situation from sociological point of view and he proposes some preventive measures. He is completely concentrated on phys-ics in USA and he neglects the rest of the world. (shrink)
The central topic of this article is the analyse of a role which modern mathematics plays in developing empirical theories. This is done with the help of an example of how fractal theory is applied to physics.
The main objective of Micheal Sandel’s Liberalism and the Limits of Justice was to attack the concept of deontology, as formulated by John Rawls. In this article, I argue that Sandel interpreted the concept of deontology in a misleading way. There is a difference between how Rawls defined this concept and how it was interpreted by Sandel. Given this, the first part of this article will analyze the way Sandel interpreted the Rawlsian concept of deontology. The second part presents an (...) assessment of the clash between the two authors in light of the differences in the way both of them understood the meaning of deontology in political liberalism. (shrink)
A history of logic -- Patterns of reasoning -- A language and its meaning -- A symbolic language -- 1850-1950 mathematical logic -- Modern symbolic logic -- Elements of set theory -- Sets, functions, relations -- Induction -- Turning machines -- Computability and decidability -- Propositional logic -- Syntax and proof systems -- Semantics of PL -- Soundness and completeness -- First order logic -- Syntax and proof systems of FOL -- Semantics of FOL -- More semantics -- Soundness and (...) completeness -- Why is first order logic "First Order"? (shrink)
Machine generated contents note: Introduction Rudy Rucker; Part I. Perspectives on Infinity from History: 1. Infinity as a transformative concept in science and theology Wolfgang Achtner; Part II. Perspectives on Infinity from Mathematics: 2. The mathematical infinity Enrico Bombieri; 3. Warning signs of a possible collapse of contemporary mathematics Edward Nelson; Part III. Technical Perspectives on Infinity from Advanced Mathematics: 4. The realm of the infinite W. Hugh Woodin; 5. A potential subtlety concerning the distinction between determinism and nondeterminism W. (...) Hugh Woodin; 6. Concept calculus: much better than Harvey M. Friedman; Part IV. Perspectives on Infinity from Physics and Cosmology: 7. Some considerations on infinity in physics Carlo Rovelli; 8. Cosmological intimations of infinity Anthony Aguirre; 9. Infinity and the nostalgia of the stars Marco Bersanelli; 10. Infinities in cosmology Michael Heller; Part V. Perspectives on Infinity from Philosophy and Theology: 11. God and infinity: directions for future research Graham Oppy; 12. Notes on the concept of the infinite in the history of Western metaphysics David Bentley Hart; 13. God and infinity: theological insights from Cantor's mathematics Robert J. Russell; 14. A partially skeptical response to Hart and Russell Denys A. Turner. (shrink)
This paper begins by tracing interest in emergence in physics to the work of condensed matter physicist Philip Anderson. It provides a selective introduction to contemporary philosophical approaches to emergence. It surveys two exciting areas of current work that give good reason to re-evaluate our views about emergence in physics. One area focuses on physical systems wherein fundamental theories appear to break down. The other area is the quantum-to-classical transition, where some have claimed that a complete explanation of the behaviors (...) and features of the objects of classical physics entirely in quantum terms is now within our grasp. We suggest that the most useful way to approach the emergent/non-emergent distinction is in epistemic terms, and more specifically that the failure of reductive explanation is constitutive of emergence in physics. (shrink)
The issue of infinity appeared in cosmology in the form of a question on spatial and time finiteness or infinity of the universe. Recently, more and more talking is going on about “other universes” (different ones from “our”), the number of which may be infinite. Speculations on this topic emerged in effect of the discussions on the issue of the anthropic principle, and the so-called inflation scenario. In truth, this kind of speculations are hardly recognized as scientific theories, however, they (...) may be included in a sort of “scientific fringe” fulfilling a beneficial heuristic function.All of the speculations regarding numerous universes boil down to the juggling of probabilities, i.e. to the applying of the theory of probability to the universes’ set. However, without probabilistic measure being introduced onto this “set” (as it is not known whether it is a set in a technical sense of this term)—and there is no knowledge at all as to how to do this—such considerations may not go beyond a vague intuition.The producing of other universes usually results from an assumption that the disturbing of original circumstances, of values of physical constants, or of other parameters characterizing the universe is possible. On the other hand, the idea of the final theory seems to assume that the mathematical structure of this theory should be rigid, i.e. that the disturbing of its parameters leads on to the very same structure. This would have eliminated the possibility of the existence of other universes.The idea of infinite number of universes sometimes has an anti-theological undertone: there is no need for assuming purposeful acting of the Creator, since all possibilities are fulfilled. The reaction of a theologian may be as follows: Just the same, God may create just a single universe, as much as an infinite number of universes. What’s more, one may risk saying that God is not interested in nothing that may be short of infinity. (shrink)
There has long been a debate on the possible similarity between some forms of Indian and Greek idealistic monism ( Advaita and Neoplatonism ). After a basic historical introduction to the debate, the text proposes that Paramādvaita , also known as Kashmiri Shaivism , is a more suitable comparandum for Neoplatonism than any other form of Advaita , suggested in the debate. Paramādvaita ’s dynamic view of reality summarized in the terms prakāśa-vimarśa or unmeṣa-nimeṣa , corresponds quite precisely to the (...) viewpoint of Neoplatonism , summarized in the similar bipolar terms such as prohodos-epistrophe . The context of the dynamic nature of reality doctrine is also quite similar ( svataḥsiddhatva, authypostasis ). My arguments are based on the texts of Plotinus and Proclus ( Neoplatonism ) and the texts of Abhinavagupta, Utpaladeva and Kṣemarāja ( Paramādvaita ) . Several parallel doctrines of both systems are further discussed: the doctrine of creative multilevel subjectivity , the doctrine of mutual omnipresence of all in all , the doctrine of creative multilevel speech , and some corresponding doctrines on aesthetic beauty and its important role in the Soul’s return towards its ultimate source. Some implications of the high degree of correspondence between both systems are considered at the end of the paper, for instance whether some similarities of compared systems might be explained on a structural basis, since both schools ware facing similar sceptical critique ( Mādhyamika, Hellenistic scepticism ). (shrink)