This Element introduces a young field, the 'philosophy of mathematical practice'. We first offer a general characterisation of the approach to the philosophy of mathematics that takes mathematical practice seriously and contrast it with 'mathematical philosophy'. The latter is traced back to Bertrand Russell and the orientation referred to as 'scientific philosophy' that was active between 1850 and 1930. To give a better sense of the field, the Element further contains two examples of topics studied, that of mathematical structuralism and visual thinking in mathematics. These are in part presented from a methodological point of view, focussing on mathematics as an activity and questions related to how mathematics develops. In addition, the Element contains several examples from mathematics, both historical and contemporary , to illustrate and support the philosophical points.

Building on the seminal work of Kit Fine in the 1980s, Leon Horsten here develops a new theory of arbitrary entities. He connects this theory to issues and debates in metaphysics, logic, and contemporary philosophy of mathematics, investigating the relation between specific and arbitrary objects and between specific and arbitrary systems of objects. His book shows how this innovative theory is highly applicable to problems in the philosophy of arithmetic, and explores in particular how arbitrary objects can engage with the nineteenth-century concept of variable mathematical quantities, how they are relevant for debates around mathematical structuralism, and how they can help our understanding of the concept of random variables in statistics. This fully worked through theory will open up new avenues within philosophy of mathematics, bringing in the work of other philosophers such as Saul Kripke, and providing new insights into the development of the foundations of mathematics from the eighteenth century to the present day.

The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, set-theoretic and category-theoretic, arose within mathematics itself. After exposing a number of problems, the Element considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as abstract universals, modal, eliminating structures as objects in favor of freely entertained logical possibilities, and finally, modal-set-theoretic, a sort of synthesis of the set-theoretic and modal perspectives.