Ph.D. (Mathematics) / This thesis reports on an interdisciplinary research programme: an investigation of power structures, and their applications in various fields. A power construction is an attempt to lift whatever structure may exist between the elements of a set to subsets of that set. The notions of structure considered here are algebraic, relational and topological. It is shown how power constructions are useful in a number of contexts in Mathematics, in Logic, in Computer Science and in the Philosophy of Science. The thesis is therefore an exercise in what may be called lateral research, where the aim is to look horizontally across disciplinary boundaries, identify common basic concepts, and use these to fertilise each field with results from the others. This differs from the more common vertical research method, the two manifestations of which (in Mathematics, in particular) are specialisation and generalisation. To specialise means to narrow down the field of investigation, as with a group theorist studying specifically Abelian groups. Generalisation moves in the opposite direction - one may attempt, for example, to generalise a result first proved for Abelian groups to the case of arbitrary groups. But, whether narrowing down or opening up, in vertical research mode it is the concept alone which is under investigation - its own nature, rather than its relationships to other concepts. In particular, vertical research pays little attention to the occurrence and application of the concept under investigation in other disciplines. It is sad that 'research', in Mathematics, is often unthinkingly equated with 'vertical research'. This is detrimental to scholarship in at least two ways. One is the training of new scientists - more particularly, of new PhD's. It is ironic that though the requirement for a PhD is almost universally held to be 'original research', or 'a contribution to knowledge', few things are in fact more orthodox and conformist than a PhD thesis. Here I refer not just to presentation (uniformity of which may be beneficial), but to methodology: few PhD candidates would dare to prejudice their chances with unpredictable examiners by venturing outside the paradigm of vertical research. A second effect (which is also a cause) of equating 'research' with 'vertical research' is the allocation of research funding. Project proposals and grant applications must be evaluated; this is usually done by peer review, and it seems clear that referees' reports emanating from a smallish fraternity of specialists will have a more enthusiastic ring than...
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:uj/uj:3724 |
| Date | 10 February 2014 |
| Creators | Brink, Chris |
| Source Sets | South African National ETD Portal |
| Detected Language | English |
| Type | Thesis |
| Rights | University of Johannesburg |
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