AstrAlgo cWeb Vol. 2013 (1031) issue GOL X #1190
Michael Heather, Nick Rossiter
The simplicity of nature according to Alfred North Whitehead is a
simplicity that comes from the far side of complexity. The
formalisation of nature might be expected to follow the same path and
the test for simplicity is a good test to validate any theory of
physics. Pure mathematics on the other hand is not confined to
reality. It is not subject to the simplifying power of nature and can
therefore not be expected to escape from its own self-generated
complications without a lead from the science of the real world.
However there is a sharp contrast between ‘category land’
as applied to categorise the predicative mathematics of classical set
theory and with the impredicative category theory applicable to
physics and metaphysics.
Impredication is top-down with the starting point as the World consisting of the physical Universe with all its internal interactions. Through our senses we perceive the World to have an ordered existence that without recourse to any assumptions may be called ‘process’. This process as an ordering may be represented as an arrow that enables us to proceed down as natural reasoning still without the need for assumptions. By what we may subsequently recognize as Occam’s razor the World is syntactically one big arrow and the details are just segments of that arrow. Whitehead in his 1929 Process & Reality provides a very full informal elaboration of the details as speculative categories. It is now possible to provide the details formally in applicable category theory as the semantics of the big arrow. In contrast to Hilbert’s ‘finitary mathematics’ this is ‘finitary physics’. For our own better understanding it is convenient to distinguish different interpretations with the following possible types of arrow in rough top-down order: the topos, cartesian closed, pre-order, dolittle diagram, pullback, pushout, monad, unit of adjunction, limits, co-limits, natural transformation, adjointness, implicative, functor, covariant, contravariant, dual, category, identity functor, terminal, initial, Identity, object, composition and partial order.
The object of this lecture is to explore this semantics of the formal arrow of physics.