Let G be a group

Traditional philosophy of mathematics has dealt extensively with the question of the nature of mathematical objects such as number, point, and line. Considerations of this question have a great interest from a historical point of view, but they have become largely outdated in light of the development of modern mathematics. The prevalent view of mathematics in the 20th (and the early part of the 21st) century has been some variant of the so-called formalistic view, according to which mathematics is a study of axiomatic systems or structures. In such structures the objects have exactly the properties set out in the axioms and nothing else can be said about their nature. The question of the nature of mathematical objects has become a non-issue. Similarly, the structure itself is completely determined by its set of axioms and one can say nothing meaningful about its nature other than that. In particular, in the formalistic understanding of mathematics a structure exists if its axioms are consistent among each other (i.e., non-contradicting). In this chapter we shall first outline this formalistic view of mathematics, then we shall follow the historical developments that led to it, and finally we shall critically analyze the formalistic view of mathematics, not from a purely philosophical point of view but from the view of the practice and history of mathematics. Instead of asking what can in principle exist and what can in principle be proved, we shall ask what is studied in practice, i.e., which mathematical structures are singled out by mathematicians and which statements in the structure are found interesting enough to be worthy of investigation. This question is not answered by the formalistic view of mathematics. In order to investigate the question one must study what drives mathematicians when they develop new mathematics. Here the history of mathematics points to intriguing problems as a particularly forceful driving force. Many (perhaps most) mathematical theories have been developed as an aid to solving problems both inside and outside of mathematics. Aristotle considered mathematical objects as abstractions of phenomena in nature. However, many other generating processes such as generalization, rigorization, and axiomatization have played an important role in the development of the mathematical structures and theories that are currently being investigated. Thus, for a historian like me, it is tempting to replace an Aristotelian ontology with a richer historical narrative taking all the driving forces and generating processes into account. Although such a narrative may not explain the ontology of mathematical structures and objects it gives an account of how actually studied mathematical structures and objects came into being and to some degree also why they were singled out among the many potentially possible structures and objects.