Poincaré's philosophy of mathematics

The primary concern of this thesis is to investigate the explicit philosophy of mathematics in the work of Henri Poincare. In particular, I argue that there is a well-founded doctrine which grounds both Poincare's negative thesis, which is based on constructivist sentiments, and his positive thesis, via which he retains a classical conception of the mathematical continuum. The doctrine which does so is one which is founded on the Kantian theory of synthetic a priori intuition. I begin, therefore, by outlining Kant's theory of the synthetic a priori, especially as it applies to mathematics. Then, in the main body of the thesis, I explain how the various central aspects of Poincare's philosophy of mathematics - e.g. his theory of induction; his theory of the continuum; his views on impredicativiti his theory of meaning - must, in general, be seen as an adaptation of Kant's position. My conclusion is that not only is there a well-founded philosophical core to Poincare's philosophy, but also that such a core provides a viable alternative in contemporary debates in the philosophy of mathematics. That is, Poincare's theory, which is secured by his doctrine of a priori intuitions, and which describes a position in between the two extremes of an "anti-realist" strict constructivism and a "realist" axiomatic set theory, may indeed be true

An investigation of the mathematical formulation of quantum theory and its physical interpretation, 1900-1927

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Hierarchical Stability and Chaotic Motion of Gravitational Few-Body Systems

In this thesis the hierarchical stability and chaotic motion of the classical few body system are studied, and then extended into the framework of the relativistic theory of gravitation. Because of the importance of integrability to both hierarchical stability and Hamiltonian chaos, a general discussion is also given on integrals and symmetries using the modern language of differential geometry. The study of this thesis is closely related to the stability problem of our Solar System and the mass transfer process of compact binary star systems. The approach carried out is both computational and theoretical. The computational part is a systematical investigation of the hierarchical stability (no drastic change in orbital elements or of the hierarchy) of the general 3-body problem, in comparison with the Hill-type stability. The importance of eccentricity in relation to stability is manifest, and the complexity of the phase space structure and fractal nature of the boundary between regular and chaotic regions are reflected in this study. The theoretical work is a continuation of the investigations of the effects of integrals on possible motions. Using a canonical transformation method, a stronger inequality is found for the spatial 3-body problem, giving better estimation of the Hill-type stability regions. It is proved that a Hill-type stability guarantees one of the three hierarchical stability conditions. This classical study is then developed into an inequality method establishing restrictions of symmetries (integrals) on possible motions. The method is first applied to gravitational systems in general relativity and their post-Newtonian approximations. The thesis is split into part I, a general introduction and discussion of the relevant methods, and part II, the original research and main body of the thesis. In chapter 1 a general introduction to the problem of the Solar System's stability is given, with an emphasis on Roy's hierarchical stability and the divergence problem of classical perturbation theory due to chaos. Chapter 2 is a review of the theory of Hamiltonian chaos, presented at a level of comprehending chaos mathematically. The importance of number theory, infinite series and integrability to chaos is emphasised. The geometrical method of studying nonlinear dynamical systems is introduced; classical perturbation theory is used to comprehend the KAM theorem. Particular attention is paid to coordinate-free interpretation of the integrability and separability conditions. In this chapter, a collection of integrable and chaotic systems is given because of their conceptual value to later chapters. Based on the Toda and Henon-Heiles Hamiltonian systems, a discussion is given on the general relationship of a system to its truncated system. This suggests a similar situation for the geodesic motion in Kerr geometry. Chapter 3 is the last chapter of part I on chaos. In this chapter we study the history of chaotic dynamics and its impact on science in general. Although it is standard to study quantization of regular and chaotic motions, the present author pays particular attention to a philosophical compatibility between the theory of chaotic attractors and quanium mechanics. Noting that the two revolutionary theories were born at almost the same time, and that Poincare was a contributor to both theories, the present author carries out a historical search for a possible mutual influence in the development of the theories. However, it is found that such a connection is surprisingly tenuous. The original work is included in part II. The classical 3-body problem is studied in chapters 4 and 5; and the relativistic few-body problem is studied in chapters 6 and 7. In chapter 4, we first review the previous approaches on the Hill-type stability of the general 3-body problem. It is found that all results of previous studies are equivalent and do not go beyond a direct use of Sundman's inequality. Zare's (1976) canonical transformation study on the coplanar 3-body problem is modified and applied to the spatial problem, thus obtaining inequalities stronger than Sundman's. These inequalities determine the best possible Hill-type stability regions for the general 3-body problem, although the critical configurations and the value of (C2H)c cannot be improved. In this approach, it is found that the moment of inertia ellipse of the system may be used to simplify the calculation. Because of this, it is hoped that the same stronger inequalities may also apply to systems with more than three bodies. (Abstract shortened by ProQuest.)