This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege's Grundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes of Grundlagen are developed: the relationship Frege envisions between arithmetic and geometry and (...) the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer. (shrink)

The ArgumentIn writing the history of science, the fluctuations between two meanings of the concept of style are of special interest: a simple or direct meaning of this concept referring to a means of expression and of presentation, and a philosophical interpretation of this term referring to “a world of objective spiritual order.” The last two chapters of this paper consider the perspective of the simple meaning of the concept, the first two chapters take the philosophical meaning as their starting (...) point.The concept of style in its general epistemological meaning emerges within a conceptual space that becomes effective as a totality at the end of the eighteenth century and which is built up of further notions such as: individual, genius, expression, symbol, education, creativity, and others.The individual and, as believed, the nevertheless infinitely creative subject has taken the place that the concept of god had occupied within rationalism. But it is not only the subject as construction and will, but also the subject who reflected in a new way about the objective foundations of his conscience and tried to bring the object and the means of knowledge into a new relation. (shrink)