Logic as the Universal Science : Bertrand Russell's Early Conception of Logic and Its Philosophical Context

Bertrand Russell (1872 1970) introduced the English-speaking philosophical world to modern, mathematical logic and foundational study of mathematics. The present study concerns the conception of logic that underlies his early logicist philosophy of mathematics, formulated in The Principles of Mathematics (1903). In 1967, Jean van Heijenoort published a paper, Logic as Language and Logic as Calculus, in which he argued that the early development of modern logic (roughly the period 1879 1930) can be understood, when considered in the light of a distinction between two essentially different perspectives on logic. According to the view of logic as language, logic constitutes the general framework for all rational discourse, or meaningful use of language, whereas the conception of logic as calculus regards logic more as a symbolism which is subject to reinterpretation. The calculus-view paves the way for systematic metatheory, where logic itself becomes a subject of mathematical study (model-theory). Several scholars have interpreted Russell s views on logic with the help of the interpretative tool introduced by van Heijenoort,. They have commonly argued that Russell s is a clear-cut case of the view of logic as language. In the present study a detailed reconstruction of the view and its implications is provided, and it is argued that the interpretation is seriously misleading as to what he really thought about logic. I argue that Russell s conception is best understood by setting it in its proper philosophical context. This is constituted by Immanuel Kant s theory of mathematics. Kant had argued that purely conceptual thought basically, the logical forms recognised in Aristotelian logic cannot capture the content of mathematical judgments and reasonings. Mathematical cognition is not grounded in logic but in space and time as the pure forms of intuition. As against this view, Russell argued that once logic is developed into a proper tool which can be applied to mathematical theories, Kant s views turn out to be completely wrong. In the present work the view is defended that Russell s logicist philosophy of mathematics, or the view that mathematics is really only logic, is based on what I term the Bolzanian account of logic . According to this conception, (i) the distinction between form and content is not explanatory in logic; (ii) the propositions of logic have genuine content; (iii) this content is conferred upon them by special entities, logical constants . The Bolzanian account, it is argued, is both historically important and throws genuine light on Russell s conception of logic.

Opettajan uskomukset opetustyön lähtökohtana : näkökulmia pedagogiseen ajatteluun

The purpose of this research was to examine teacher’s pedagogical thinking based on beliefs. It aimed to investigate and identify beliefs from teachers’ speech when they were reflecting their own teaching. Placement of beliefs in levels of pedagogical thinking was also examined. The second starting point for a study was the Instrumental Enrichment -intervention, which aims to enhance learning potential and cognitive functioning of students. The goal of this research was to investigate how five main principles of the intervention come forward in teachers’ thinking. Specifying research question was: how similar teachers’ beliefs are to the main principles of intervention. The teacher-thinking paradigm provided the framework for this study. The essential concepts of this study are determined exactly in the theoretical framework. Model of pedagogical thinking was important in the examination of teachers’ thinking. Beliefs were approached through the referencing of varied different theories. Feuerstein theory of Structural cognitive modifiability and Mediated learning experience completed the theory of teacher thinking. The research material was gathered in two parts. In the first part two mathematics lessons of three class teachers were videotaped. In second part the teachers were interviewed by using a stimulated recall method. Interviews were recorded and analysed by qualitative content analysis. Teachers’ beliefs were divided in themes and contents of these themes were described. This part of analysis was inductive. Second part was deductive and it was based on theories of pedagogical thinking levels and Instrumental Enrichment -intervention. According to the research results, three subcategories of teachers’ beliefs were found: beliefs about learning, beliefs about teaching and beliefs about students. When the teachers discussed learning, they emphasized the importance of understanding. In teaching related beliefs student-centrality was highlighted. The teachers also brought out some demands for good education. They were: clarity, diversity and planning. Beliefs about students were divided into two groups. The teachers believed that there are learning differences between students and that students have improved over the years. Because most of the beliefs were close to practice and related to concrete classroom situation, they were situated in Action level of pedagogical thinking. Some teaching and learning related beliefs of individual teachers were situated in Object theory level. Metatheory level beliefs were not found. Occurrence of main principles of intervention differed between teachers. They were much more consistent and transparent in the beliefs of one teacher than of the other two teachers. Differences also occurred between principles. For example reciprocity came up in every teacher’s beliefs, but modifiability was only found in the beliefs of one teacher. Results of this research were consistent with other research made in the field. Teachers’ beliefs about teaching were individual. Even though shared themes were found, the teachers emphasized different aspects of their work. Occurrence of beliefs that were in accordance with the intervention were teacher-specific. Inconsistencies were also found within teachers and their individual beliefs.

On the fundamentals of coherence theory

In the thesis I study various quantum coherence phenomena and create some of the foundations for a systematic coherence theory. So far, the approach to quantum coherence in science has been purely phenomenological. In my thesis I try to answer the question what quantum coherence is and how it should be approached within the framework of physics, the metatheory of physics and the terminology related to them. It is worth noticing that quantum coherence is a conserved quantity that can be exactly defined. I propose a way to define quantum coherence mathematically from the density matrix of the system. Degenerate quantum gases, i.e., Bose condensates and ultracold Fermi systems, form a good laboratory to study coherence, since their entropy is small and coherence is large, and thus they possess strong coherence phenomena. Concerning coherence phenomena in degenerate quantum gases, I concentrate in my thesis mainly on collective association from atoms to molecules, Rabi oscillations and decoherence. It appears that collective association and oscillations do not depend on the spin-statistics of particles. Moreover, I study the logical features of decoherence in closed systems via a simple spin-model. I argue that decoherence is a valid concept also in systems with a possibility to experience recoherence, i.e., Poincaré recurrences. Metatheoretically this is a remarkable result, since it justifies quantum cosmology: to study the whole universe (i.e., physical reality) purely quantum physically is meaningful and valid science, in which decoherence explains why the quantum physical universe appears to cosmologists and other scientists very classical-like. The study of the logical structure of closed systems also reveals that complex enough closed (physical) systems obey a principle that is similar to Gödel's incompleteness theorem of logic. According to the theorem it is impossible to describe completely a closed system within the system, and the inside and outside descriptions of the system can be remarkably different. Via understanding this feature it may be possible to comprehend coarse-graining better and to define uniquely the mutual entanglement of quantum systems.