This paper presents an in-depth analysis of the anatomy of both thermodynamics and statistical mechanics, together
with the relationships between their constituent parts. Based on this analysis, using the renormalization group
and finite-size scaling, we give a definition of a large but finite system and argue that phase transitions are
represented correctly, as incipient singularities in such systems. We describe the role of the thermodynamic limit.
And we explore the implications of this picture of critical phenomena for the questions of reduction and emergence.
This paper builds on an earlier paper on the constitutive role of large numbers for theory development, but considerably
extends its scope. We analyse the role that large numbers, and large (anthropomorohic) scales, i.e. large compared to atomic
distances, and large compared to typical time-scales of atomistic processes are playing in creating the perception of
stable macroscopic properties of matter. We discuss the relation to limit theorems of mathematical statistics, and their
rationalization within a renormalization group approach (as pioneered by Jona-Lasinio). We discuss corresponding large scale
phenomenology in the context of phase transitions and critical phenomena. We note in particular that "trivial" high- or
low-temperature fixed points of the renomalization group which describe situations with vanishing correlation lenghts are
primarily responsible for the emergence of stable macroscopic phenomena (exept, that is, in the immediate vicinity of phase
transitions). We also discuss the implications of these findings for the debates around reduction and emergence. Finally
we explain in which sense the renormalization group approach is able to provide a post-facto justification for the succsess
of "simple" models in describing collective phenomena.
This paper describes the power but also the limitations of images concerning their role in generating, coding and
communicating insights in physics.
The present contribution deals with the central role that large numbers have played in the process of developing theories about macroscopic systems. We begin by analysing the empirical foundations of this observation, the emergence of stability and regularity through averaging in large systems, and describe their formalisation via limit theorems of mathematical statistics. We choose to adopt a formalisation which emphasises properties of descriptions on scales which are much larger than the atomic or molecular scale at which, according to current understanding, the phenomena being described have their origin. By going on to consider the consequences that the empirical foundation of our observation has for our neural information processing apparatus, we are forced to conclude that large numbers play a crucial role already in allowing stable perception and representation of external and internal reality, and thus appear to be constitutive for all theorising about the world.