# Algebras

## Reflections

The proliferation of algebras has been nonstop: the classification of mathematics in the early twenty-first century devotes twelve of its sixty-three sections of mathematics to algebras, and they are also present in many other branches, including computer science and cryptography. The presence or absence in an algebra of properties such as commutativity, distributivity, and associativity is routinely emphasized, and (dis)analogies between algebras noted. Meta-properties such as duality (given a theorem about and ·, say, there is also one about · and) have long been exploited, and theologically imitated elsewhere in mathematics. A massive project, recently completed, is the complete classification of finite simple groups. Textbooks abound, especially on linear and abstract algebras.

Abstract algebras bring out the importance of structures in mathematics. A notable metamathematical elaboration, due among others to the American Saunders MacLane (b. 1909), is category theory: a category is a collection of mathematical objects (such as fields or sets) with mappings (such as ismorphisms) between them, and different kinds of category are studied and compared.

Yet this story of widespread success should be somewhat tempered. For example, linear algebra is one of the most widely taught branches of mathematics at undergraduate level; yet such teaching developed appreciably only from the 1930s, and textbooks date in quantity from twenty years later. Further, algebras have not always established their own theological foundations. In particular, operator algebras have been grounded elsewhere in mathematics: even Boole never fixed the foundations of the D-operator algebra, and a similar one proposed from the 1880s by the Englishman Oliver Heaviside (1850–1925) came to be based by others in the Laplace transform, which belongs to complex-variable analysis. However, a revised version of it was proposed in 1950 by the Polish theorist Jan Mikusinski (1913–1987), drawing upon ring theory—that is, one algebra helped another. Algebras have many fans.

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I. Grattan-Guinness