The Carnegie Foundation commissioned a collection of essays as part of the Carnegie Initiative on the Doctorate (CID). Essays and essayists represent six disciplines that are part of the CID: chemistry, education, English, history, mathematics, and neuroscience. Intended to engender conversation about the conceptual foundation of doctoral education, the essays are intended as a starting point and not the last word in disciplinary discussions. This essay, which considers doctoral programs in mathematics, discusses the importance of ensuring a source of high-quality doctoral students, ensuring that mathematics is part of science, and ensuring societal support for mathematics doctoral education. It then considers four steps to ensure that mathematics doctoral education meets its goals: make it more attractive to students and competitive with other fields, prepare doctoral students for a broad range of future careers, improve mentoring during doctoral training, and adopt some effective practices from other sciences. (Contains 16 references.) (EV)