  Our understanding of past evolutionary change is often based on
reconstructions based on incomplete data, raising fundamental questions about
the degree to which we can make reliable inferences about past evolutionary
processes. This was demonstrated by Louca and Pennell (2020), who showed that
each pure-birth process can be generated by an infinite number of birth-death
processes. Here, I explore what it means to reconstruct past evolutionary
change with three approaches from measure theory, group theory, and homotopy
theory to better understand structural constraint and origins of
(non)identifiability. As an example, the developed framework is applied to the
case of birth-death processes.
