AstrAlgo cWeb Vol. 2013 (1031) issue
GOL X #1190

### The formal arrow of physics. An introduction to
applicable category theory

__Michael Heather__*, Nick Rossiter*

England

Session: PL

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.