“That Which a Minority Construct”: Abelian Mathematics in Abel, Galois, Noether, and Grothendieck

This chapter considers the nature of radical transformations of mathematics, enabled by minority mathematics. It will be particularly concerned with modern mathematics, which emerged roughly around 1800, as abstract mathematics – abstracted from mathematics’ relations to the natural world and physics, relations that previously dominated mathematics. As, however, defined here (transferring Gilles Deleuze and Félix Guattari’s concept of a minor(ity) literature, as exemplified by F. Kafka’s work), a minority mathematics is not something that exists entirely outside a major mathematics, to be distinguished here from a majority mathematics. Instead, it is a mathematics that, while still exterior to the major mathematics to which it juxtaposes, constructs itself within and even at the very core of this major mathematics. This may be seen as merely a special form of revolutionary vis-à-vis normal mathematical practice in T. Kuhn’s sense. I shall argue, however, by using the work of N. H. Abel, É. Galois, E. Noether, and A. Grothendieck, as my main cases, that this “special” type of revolutionary practice is the primary and even the only form of revolutionary practice possible in mathematics. I designate this mathematics Abelian mathematics, the term commonly associated with formal mathematical properties (such as commutative group or abelian categories), because Abel’s work was, arguably, the first manifested case of a minority mathematics in this sense in modern mathematics.