Introduction to mathematical logic. Part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.

An attempt to revise the special relativity genesis at the expense of comprehending all Einstein’s 1905 papers as a whole is provided. It is argued that light quanta hypothesis and special relativity turn out to be mere stages of implementation of the programme of maxwellian electrodynamics, statistical mechanics and thermodynamics reconciliation. The conception of luminiferous ether was an insurmountable stumbling block for Einstein’s statistical thermodynamics programme in which the leading role was played by the light quanta paper . Einstein’s 1905 (...) unificationist modus operandi was close to Mach’s principle of economy of thought in ingenious conjunction with bashful inclinations of Kantian epistemology. Keywords: intertheoretic context, special relativity, light quanta , Unity of Nature, Mach, Kant. (shrink)

Timelines is an inquiry into the nature of time, both as an apparent feature of the external physical world and as a fundamental feature of our experience of ourselves in the world. The principal argument of Timelines is that our coventional ideas about time are largely mistaken and that what we think of as independent physical time is actually our calibration of a certain relation between events. Namely, the relation between time-keeping events and the causal sequential differences of physical processes (...) which are real and independent of us. (shrink)

Observational and experimental discoveries of new factual entities such as objects, systems, or processes, are major contributors to some advances in the life sciences. Yet, whereas discovery of theories was extensively deliberated by philosophers of science, very little philosophical attention was paid to the discovery of factual entities. This paper examines historical and philosophical aspects of the experimental discovery by Carl Woese of archaea, prokaryotes that comprise one of the three principal domains of the phylogenetic tree. Borrowing Kuhn’s terminology, this (...) discovery of a major biological entity was made during a ‘normal science’ project of building molecular taxonomy for prokaryotes. Unexpectedly, however, an observed anomaly instigated the discovery of archaea. Substantiation of the existence of the new archaeal entity and consequent reconstruction of the phylogenetic tree prompted replacement of a long-held model of a prokarya and eukarya bipartite tree of life by a new model of a tripartite tree comprising of bacteria, archaea, and eukarya. This paper explores the history and philosophical implications of the progression of Woese’s project from normal science to anomaly-instigated model-changing discovery. It is also shown that the consequential discoveries of RNA splicing and of ribozymes were similarly prompted by unexpected irregularities during normal science activities. It is thus submitted that some discoveries of factual biological entities are triggered by unforeseen observational or experimental anomalies. (shrink)

A famous essay by Wigner is reexamined in view of more recent developments around its topic, together with some remarks on the metaphysical character of its main question about mathematics and natural sciences.

Science teaching always engages a philosophy of science. This article introduces a modern philosophy of science and indicates its implications for science education. The hermeneutic philosophy of science is the tradition of Kant, Heidegger, and Heelan. Essential to this tradition are two concepts of truth, truth as correspondence and truth as disclosure. It is these concepts that enable access to science in and of itself. Modern science forces aspects of reality to reveal themselves to human beings in events of disclosure. (...) The achievement of each event of disclosure requires the precise manipulation of equipment, which is an activity that depends on truth as correspondence. The implications of the hermeneutic philosophy of science for science education are profound. The article refers to Newton’s early work on optics to explore what the theory implies for teaching. Modern science—as the event of truth—is a relationship between an individual student, equipment, and reality. Science teachers provide for their students’ access to truth and they may show how their discipline holds a special relationship to reality. If the aim of science teaching is to enable students to disclose reality, the science curriculum will challenge some of the current practices of schooling. If teachers base science teaching upon the hermeneutic philosophy of science, science will assert itself as the intellectual discipline that derives from nature, and not from the inclinations of human beings. Science teachers teach nature’s own science. (shrink)