Richard Startup

A perspective in the philosophy of mathematics is developed from a consideration of the strengths and limitations of both logicism and platonism, with an early focus on Frege’s work. Importantly, although many set-theoretic structures may be developed each of which offers limited isomorphism with the system of natural numbers, no one of them may be identified with it. Furthermore, the timeless, ever present nature of mathematical concepts and results itself offers direct access, in the face of a platonist account which generates a supposed problem of access. Crucially too, pure mathematics has its own distinctive method of confirming or validating results - mathematical proof - which supplies a higher level of confidence and objectivity than that available elsewhere. The dichotomy of invention and discovery is too jejune a framework for analysing creative mathematical activity. The Gödelian platonist perspective is evaluated and queried through scrutiny of the part played by mathematical resources and constraints in relation to human activity. It appears that there can be non-causal mathematical explanations and mathematical constraint on purely natural processes. Valuable implications of Quine’s naturalism are explored, but one must be cautious of his thesis of confirmational holism. The distinction between algebraic and non-algebraic mathematical theories usefully contributes to our understanding of the internally differentiated nature of the subject.

The extent of confusion between symbols and that which is symbolised is examined across five institutional spheres. Religion is the institution most marked by confusion of this type; indeed in some respects the symbolic mes- sage of religion may be the extent of the substantive reality. On the other hand, the very existence of the nation state may be judged to depend upon the exercise of the human imagination; hence providing a source of instability which may lead to the excesses of nationalism. In regard to social status, the main problematical element is a certain circularity: it is necessary to get peo- ple to exhibit differences in behaviour which are then used to justify or con- stitute the status differences themselves. In politics, the symbolism of left and right threatens to strangle creative thinking, while in education the tendency on all sides to orient towards public systems of measurement and grading undermines the claim that what is really important is pupil and student learn- ing. A social cost is being paid for the failure to recognise and, where possible, address the issues identified.

In a study drawing from both evolutionary biology and the social sciences, evidence and argument is assembled in support of the comprehensive appli- cation of selfish gene theory to the human population. With a focus on genes giving rise to characteristically-human cooperation (“cooperative genes”) in- volving language and theory of mind, one may situate a whole range of pat- terned behaviour—including celibacy and even slavery—otherwise seeming to present insuperable difficulties. Crucially, the behaviour which tends to propa- gate the cooperative genes may be “at cost” to the genes of some who may be party to the cooperation itself. Explanatory insights are provided by Trivers’ parent-offspring conflict theory, Lack’s principle, and Hamilton’s kin selec- tion mechanism. A primary observation is that cooperation using language and theory of mind is itself interdependent with full human conceptualization of a world of objects and of themselves as embodied beings. Human capaci- ties inhering in, or arising out of, the ability to cooperate are also responsible for a vitally important long-term process, the domestication of animals and p- lants. The approach illuminates the difference between animal and human sexual behaviour, and the emergence of kinship systems. Again, recent pat- terns of population growth become much more explicable. It is argued that the gene is the single controlling replicator; the notion of the meme as a sec- ond independent replicator is flawed.

Originally published in 1979, the aim of this work was to analyse the occupational role of the university teacher, with the help of data collected within a specific university institution. This involves examining both what is expected of university teachers and what they actually do, and accounting for the patterns which their activities exhibit. Since the university teacher's occupation is multi-faceted it is necessary to examine several areas of activity including teaching, research and 'external' professional activities, as well as a number of types of relationships in which lecturers are involved. Data is presented and interpreted from interviews with the staff of four selected departments of a British provincial university (classics, pure mathematics, civil engineering and psychology) and also from questionnaire surveys of staff and students. Sociologically the university may be regarded as an organisation or institution, and the behaviour of its members understood through the notion of 'social role'. On the educational side of the study the central concern is with the teacher-pupil relationship. We are also confronted with a basic human problem of how employed people weave elements to give meaning to their working lives.

Progress may be made in resolving the tension between free will and determinism by analysis of the necessary conditions of freedom. It is of the essence that these conditions include causal and deterministic regularities. Furthermore, the human expression of free will is informed by understanding some of those regularities, and increments in that understanding have served to enhance freedom. When the possible character of a deterministic system based on physical theory is considered, it is judged that, far from implying the elimination of human freedom, such a theory might simply set parameters for it; indeed knowledge of that system could again prove to be in some respects liberating. On the other hand, it is of the essence that the overarching biological framework is not a deterministic system and it foregrounds the behavioural flexibility of humans in being able to choose within a range of options and react to chance occurrences. Furthermore, an issue for determinism flows from the way in which randomness (e.g. using a true random number generator) and chance events could and do enter human life. Once the implications of that issue are fully understood, other elements fit comfortably together in our understanding of freely undertaken action: the contribution of reasons and causes; the fact that reasons are never sufficient to account for outcomes; the rationale for the attribution of praise and blame.

Religion emerged among early humans because both purposive and non-purposive explanations were being employed but understanding was lacking of their precise scope and limits. Given also a context of very limited human power, the resultant foregrounding of agency and purposive explanation expressed itself in religion’s marked tendency towards anthropomorphism and its key role in legitimizing behaviour. The inevitability of death also structures the religious outlook; with ancestors sometimes assigned a role in relation to the living. Subjective elements such as the experience of dreams and the internalization of moral precepts also play their part. Two important sources of variation among religions concern the adoption of a dualist or non-dualist perspective, and whether or not the religion’s early political experience is such as to generate a systematic doctrine subordinating politics to religion. The near ubiquity and endurance of religion are further illuminated by analysis of its functions and ideological role. Religion tends to be socially conservative but has the potential to be revolutionary.

Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation or theorem and its proof. An attempt is made to meet directly some of Wittgenstein’s critical comments on the mathematical treatment of infinity and irrational numbers.

A framework is developed for understanding what is “taken for granted” both in philosophy and in life generally, which may serve to orient philosophical inquiry and make it more effective. The framework takes in language and its development, as well as mathematics, logic, and the empirical sphere with particular reference to the exigencies of life. It is evaluated through consideration of seven philosophical issues concerned with such topics as solipsism, sense data as the route to knowledge, the possible reduction of geometry to logic, and the existence and status of human rights. Various dichotomies and the notion of continuity are evidently highly strategic.