Truth and Probability

Contains two other essays as well: Further Considerations & Last Papers: Probability and Partial Belief.

逆数学と可算代数系

Tohoku University山崎武課

Analysis of methods for extraction of programs from non-constructive proofs

The present thesis compares two computational interpretations of non-constructive proofs: refined A-translation and Gödel's functional "Dialectica" interpretation. The behaviour of the extraction methods is evaluated in the light of several case studies, where the resulting programs are analysed and compared. It is argued that the two interpretations correspond to specific backtracking implementations and that programs obtained via the refined A-translation tend to be simpler, faster and more readable than programs obtained via Gödel's interpretation. Three layers of optimisation are suggested in order to produce faster and more readable programs. First, it is shown that syntactic repetition of subterms can be reduced by using let-constructions instead of meta substitutions abd thus obtaining a near linear size bound of extracted terms. The second improvement allows declaring syntactically computational parts of the proof as irrelevant and that this can be used to remove redundant parameters, possibly improving the efficiency of the program. Finally, a special case of induction is identified, for which a more efficient recursive extracted term can be defined. It is shown the outcome of case distinctions can be memoised, which can result in exponential improvement of the average time complexity of the extracted program

Philosophy of mathematics education

PHILOSOPHY OF MATHEMATICS EDUCATION\ud This thesis supports the view that mathematics teachers should be aware of differing views of the nature of mathematics and of a range of teaching perspectives. The first part of the thesis discusses differing ways in which the subject 'mathematics' can be identified, by relying on existing philosophy of mathematics. The thesis describes three traditionally recognised philosophies of mathematics: logicism, formalism and intuitionism. A fourth philosophy is constructed, the hypothetical, bringing together the ideas of Peirce and of Lakatos, in particular. The second part of the thesis introduces differing ways of teaching mathematics, and identifies the logical and sometimes contingent connections that exist between the philosophies of mathematics discussed in part 1, and the philosophies of mathematics teaching that arise in part 2. Four teaching perspectives are outlined: the teaching of mathematics as aestheticallyorientated, the teaching of mathematics as a game, the teaching of mathematics as a member of the natural sciences, and the teaching of mathematics as technology-orientated. It is argued that a possible fifth perspective, the teaching of mathematics as a language, is not a distinctive approach. A further approach, the Inter-disciplinary perspective, is recognised as a valid alternative within previously identified philosophical constraints. Thus parts 1 and 2 clarify the range of interpretations found in both the philosophy of mathematics and of mathematics teaching and show that they present realistic choices for the mathematics teacher. The foundations are thereby laid for the arguments generated in part 3, that any mathematics teacher ought to appreciate the full range of teaching 4 perspectives which may be chosen and how these link to views of the nature of mathematics. This would hopefully reverse 'the trend at the moment... towards excessively narrow interpretation of the subject' as reported by Her Majesty's Inspectorate (Aspects of Secondary Education in England, 7.6.20, H. M. S. O., 1979). While the thesis does not contain infallible prescriptions it is concluded that the technology-orientated perspective supported by the hypothetical philosophy of mathematics facilitates the aims of those educators who show concern for the recognition of mathematics in the curriculum, both for its intrinsic and extrinsic value. But the main thrust of the thesis is that the training of future mathematics educators must include opportunities for gaining awareness of the diversity of teaching perspectives and the influence on them of philosophies of mathematics

The External Tape Hypothesis: a Turing machine based approach to cognitive computation

The symbol processing or "classical cognitivist" approach to mental computation suggests that the cognitive architecture operates rather like a digital computer. The components of the architecture are input, output and central systems. The input and output systems communicate with both the internal and external environments of the cognizer and transmit codes to and from the rule governed, central processing system which operates on structured representational expressions in the internal environment. The connectionist approach, by contrast, suggests that the cognitive architecture should be thought of as a network of interconnected neuron-like processing elements (nodes) which operates rather like a brain. Connectionism distinguishes input, output and central or "hidden" layers of nodes. Connectionists claim that internal processing consists not of the rule governed manipulation of structured symbolic expressions, but of the excitation and inhibition of activity and the alteration of connection strengths via message passing within and between layers of nodes in the network. A central claim of the thesis is that neither symbol processing nor connectionism provides an adequate characterization of the role of the external environment in cognitive computation. An alternative approach, called the External Tape Hypothesis (ETH), is developed which claims, on the basis of Turing's analysis of routine computation, that the Turing machine model can be used as the basis for a theory which includes the environment as an essential part of the cognitive architecture. The environment is thought of as the tape, and the brain as the control of a Turing machine. Finite state automata, Turing machines, and universal Turing machines are described, including details of Turing's original universal machine construction. A short account of relevant aspects of the history of digital computation is followed by a critique of the symbol processing approach as it is construed by influential proponents such as Allen Newell and Zenon Pylyshyn among others. The External Tape Hypothesis is then developed as an alternative theoretical basis. In the final chapter, the ETH is combined with the notion of a self-describing Turing machine to provide the basis for an account of thinking and the development of internal representations

Spectroscopy at Microwave and Radio Frequencies

Part A introduces the origin of proton chemical shifts and coupling constants, and discusses the significance of these quantities for studies of conformation and configuration in organic molecules. The general theories of chemical shifts and coupling constants are then applied to pyranosides and in particular to the high resolution proton magnetic resonance spectra of saturated deuterochloroform solutions of the four compounds: A: methyl 2-acetoxymercuri-2-deoxy-beta-D-glucopyranoside triacetate. B: methyl 2-chIoromercuri-2-deoxy-beta-D-glucopyranoside triacetate. C: methyl 2-chloromereuri-2-deoxy-alpha-D-mannopyranoside triacetate. D: methyl 2-chloromercuri-2-deoxy-alpha-D-talopyranoside triacetate. Compounds containing pyranoside rings exist usually in one of the two possible chair conformations, designated by Reeves as Cl and 1C: other things being equal, the Cl conformation is preferred for most D-hexoses and their derivatives. The spectra of the four compounds have been analysed, fully for the ring proton absorptions, using programs written for the DEUCE and KDF 9 computers in Glasgow University. The chemical shifts and coupling constants resulting from the analyses are given. Interpretation of these chemical shifts and coupling constants has confirmed that the compounds do have the structures and configurations described by A, B, C and D above. This is especially important for compound C, about whose configuration there has been some controversy. The ring proton coupling constants in particular show that compounds A, B, and C in saturated deuterochloroform solution have essentially C1 chair forms, with some distortion, and that compound D exists as a very distorted Cl chair conformer, almost in a half-boat conformation: some of these conclusions are supported by X-ray analyses. Apart from the analysis of the ring proton absorption peaks, the methoxy proton chemical shifts agree well with previous findings concerning this substituent, and the acetoxy proton chemical shifts are not inconsistent with the results of other workers. No spin-spin coupling between mercury isotopes and the protons H(1), H(2) or H(3) has been explicitly observed. The peaks in the spectrum of compound D are broadened slightly relative to the peaks in the spectra of the other compounds. This may be because of kinetic effects involving the presence in low concentration of another conformer in the solution, or to unresolved couplings, perhaps long-range proton-proton couplings. In Part B, nuclear quadrupole resonance (NQR) spoctroaccpy is introduced, and quadrupole resonance is treated theoretically, with special reference to the 14N nucleus. A description is given of the two main types of instrument used to detect NQR: the marginal oscillator and the super-regenerative oscillator. Modulation is discussed. Two complete spectrometer systems for detection of NQR have been designed and constructed, and details of these systems are given. The first system uses a marginal oscillator; the second uses a super-regenerative oscillator which is externally quenched. In Part C, the factors which contribute to the electric field gradient tensor and so to the NQR frequencies in an isolated molecule are analysed. Methods of finding various molecular and atomic parameters which are needed for estimating the contributions of these factors are reviewed and discussed, and a few possible extensions or modifications of some of those methods are suggested. Mathematical techniques for evaluating the integrals which come out of these methods are also briefly reviewed, Intermolecular effects on the electric field gradient tensor are discussed briefly. The results are then given of the application to some simple molecules of the methods described and suggested earlier, with some further discussion of points of important detail. These results seem to be promising enough to justify a proper, more detailed, study of the possibilities of semi-empirical calculations of NQR frequencies