Abstract

What characterizes twentieth-century mathematics? Or twenty-first-century mathematics? As opposed to mathematical practices? The former is too broad and subject to different answers depending on the individual or the group being addressed. Hilbert’s 23 problems published in 1900 would probably be mentioned because they influenced generations of mathematicians in the ensuing 123 years. Many names could also be mentioned such as Hardy, Ramanujan, Kolmogorov, Arnold, Poincaré, Grothendieck, the Bourbaki, the Langlands program, and others as shaping the agenda of twentieth- and twenty-first-century mathematics. Mathematical practice, on the other hand, is a broader arena than mathematics per se. It is broader because it includes not only mathematicians making original contributions to the field but also others that use mathematics in other contexts. These can vary from applications of mathematics in the real world, revisiting historical sources, (auto) biographical and metamathematical accounts, to making mathematics more accessible through textbooks. Others could simply view mathematics as a recreational activity (e.g., Bhaskara’s Lilavati, Victorian Mathematics, etc.). In this section of the Handbook, a selection of chapters are found that serve as examples of mathematical practice in a broader sense. The perspective conveyed in this section provides a balance between Western and non-Western contributions to mathematical practice.