Thesis (M. Ed.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 56-57).

We begin with an account of the Prime Mover argument. This originated in the "laws" of Plato, where it is argued that a self-moving mover must exist as source of other motions, and that it must be a kind of soul. In Aristotle this Prime Mover is not itself moved, and elaborate proofs of its existence are offered in the "Physics": all motion requires a mover, and the series cannot go on to infinity, but must end in one or more unmoved movers. His proofs, however, were far from watertight, and later Peripatetics like Theophrastus and Strato rejected them. The argument reappeared in Proclus, but only as subsidiary to a First Cause argument; moreover,the Prime Mover is only the second member of the Neoplatonic "trinity". John Philoponus' theory of "impetus" should have undermined the argument, but in fact did not, and it continued to be used by the Arabs (despite criticism by Avicenna) and the Jews (notably Maimonides). It was taken over by the Christian scholastics like Aquinas. But criticism also continued, especially from Algaael in Islam and by Ockham and his followers in Christendom, and a detailed refutation was offered by the Jew Crescas. The arrival of non-Aristotelian physics was fatal to the argument; it is indeed still defended occasionally by neo-Scholastic philosophers, but none of their defences is adequate. Outside Scholasticism it has few supporters, though Samuel Clarke used it for the Platonic purpose of pointing out an analogy between the Prime Mover and mind, Lotze, however, advanced a quite different kind of argument, but based, like the Platonic and Aristotelian ones, on the existence of change; he argued that change ought always to be internal to that which changes, and hence that the universe must be in some sense a unity. The relationship between this unity and individual things would then be analogous to that between a mind and its states. [Continued in text ...]

Thesis (M. P. H.)--University of Hong Kong, 2007.

Thesis (M. P. H.)--University of Hong Kong, 2007. / Also available in print.

Thesis (M.Ed.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 56-57). Also available in print.

This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli ([And92]) is developed. This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin's cut- free LJT ([Her95], here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1-1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called 'permutation-free' calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding et al ([HSZ96]). It is applied to two calculi for intuitionistic logic and also to two modal logics; Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a 'permutation-free' calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to be sound and complete with respect to a proof-theoretic semantics and (weak) cut- elimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter 4.

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The subject of the study is a theory of positive and negative inquiry with emphasis in mathematics. The purposes of this study are to examine the historical development of systematic inquiry in mathematics, to identify the nature of positive and negative inquiry, to propose and develop an interrelated set of propositions regarding positive and negative inquiry, and to relate the proposition of the theory to certain basic concepts of trigonometry.

Mathematical proof is an important topic in mathematics education research. Many researchers have addressed various aspects of proof. One aspect that has not been addressed is what common traits are shared by those who are successful at creating proof. This research investigates the common traits in the thought processes of undergraduate students who are considered successful by their professors at creating mathematical proof. A successful proof is defined as a proof that successfully accomplishes at least one of DeVilliers (2003) six roles of proof and demonstrates adequate mathematical content, knowledge, deduction and logical reasoning abilities. This will typically be present in a proof that fits Weber's (2004) semantic proof category, though some syntactic proofs may also qualify. Proof creation can be considered a type of problem, and Schoenfeld's (1985) categories of resources, heuristics, control and ability are used as a framework for reporting the results. The research involved a) finding volunteers based on professorial recommendations; b) administering a proof questionnaire and conducting a video recorded interview about the results; and then c) holding a second video recorded interview where new proofs were introduced to the subjects during the interviews. The researcher used Goldin's (2000) recommendations for making task based research scientific and made interview protocols in the style of Galbraith (1981). The interviews were transcribed and analyzed using Strauss and Corbin's (1990) methods. The resulting codes corresponded with Schoenfeld's four categories, so his category names were used. Resources involved the mathematical content knowledge available to the subject. Heuristics involved strategies and techniques used by the subject in creating the proof. Control involved choices in implementing resources and heuristics, planning and using time wisely. Beliefs involved the subjects' beliefs about mathematics, proof, and their own skills. These categories are seen in other research involving proof but not all put together. The research has implications for further research possibilities in how the categories all work together and develop in successful proof creators. It also has implications for what should be taught in proofs courses to help students become successful provers.

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