Contributions to a science of contemporary mathematics, preprint; current draft at http:// www.math.vt.edu/people/quinn

Abstract. This essay provides a description of modern mathematical prac-tice, with emphasis on differences between this and practices in the nineteenth century, and in other sciences. Roughly, modern practice is well adapted to the structure of the subject and, within this constraint, much better adapted to the strengths and weaknesses of human cognition. These adaptations greatly increased the effectiveness of mathematical methods and enabled sweeping de-velopments in the twentieth century. The subject is approached in a bottom-up ‘scientific ’ way, finding patterns in concrete micro-level observations and being eventually lead by these to un-derstanding at macro levels. The complex and intensely-disciplined technical details of modern practice are fully represented. Finding accurate commonal-ities that transcend technical detail is certainly a challenge, but any account that shies away from this cannot be complete. As in all sciences, the final result is complex, highly nuanced, and has many surprises. A particular objective is to provide a resource for mathematics education. Elementary education remains modeled on the mathematics of the nineteenth century and before, and outcomes have not changed much either. Modern methodologies might lead to educational gains similar to those seen in profes-sional practice. This draft is about 90 % complete, and comments are welcome. 1