Abstract
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...