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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.

BIBLIOGRAPHY

Corry, Leo. Modern Algebra and the Rise of Mathematical Structures. Basel, Switzerland, and Boston: Birkhäuser, 1996.

Cournot, Antoine Augustin. De l'origine et des limites da la correspondance entre l'algèbre et la géométrie. Paris and Algiers: Hachette, 1847.

Crowe, Michael J. A History of Vector Analysis: The Evolution of the Idea of a Vertical System. Notre Dame and London: Notre Dame University Press, 1967. Reprint, New York: Dover, 1987.

Dieudonné, Jean. A History of Algebraic and Differential Topology 1900–1960. Basel, Switzerland, Birkhäuser, 1989.

Grattan-Guinness, I., ed. Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. 2 vols. London: Routledge, 1994. Reprint, Baltimore: Johns Hopkins University Press, 2003. See especially Part 6.

Hawkins, Thomas W. Emergence of Lie's Theory of Groups: An Essay in the History of Mathematics, 1869–1926. Berlin: Springer, 2002.

Høyrup, Jens. Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin. Berlin and New York: Springer, 2002.

Hutton, Charles. "A History of Algebra." In vol. 2 of his Tracts on Mathematical and Philosophical Subjects. London: F. C. and R. Rivington, 1812.

Klein, J. Greek Mathematical Thought and the Origins of Algebra. Cambridge, Mass.: MIT Press, 1968.

Lam, Lay Yong, and T. S. Ang. Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China. Singapore: World Scientific Publishing, 1992.

Mehrtens, Herbert. Die Entstehung der Verbandstheorie. Hildesheim, Germany: Gerstenberg, 1979.

Nesselmann, G. H. F. Versuch einer kritischen Geschichte der Algebra. Reprint, Frankfurt, Germany: Minerva, 1969.

Novy, Lubos. The Origins of Modern Algebra. Translated by Jaroslav Taner. Prague: Academia, 1973.

Pycior, Helena M. Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra through the Commentaries on Newton's Universal Arithmetick. Cambridge, U.K., and New York: Cambridge University Press, 1997.

Scholz, Erhard, ed. Geschichte der Algebra: Eine Ausführung. Mannheim: Wissenschaftsverlag, 1990. Up to Noether, but no algebraic logic, differential operators, or functional equations.

Sinaceur, Hourya. Corps et modèles: Essai sur l'histoire de l'algèbra réele. Paris: Vrin, 1991. English translation Basel, Switzerland: Birkh.

Stedall, Jacqueline A. A Discourse Concerning Algebra: English Algebra to 1685. Oxford, and New York: Oxford University Press, 2002.

van der Waerden, B. L. A History of Algebra: From al-Khwarizmi to Emmy Noether. Berlin: Springer, 1985. See the remark above on Scholz.

Vercelloni, Luca. Filosofia delle strutture. Florence, Italy: La Nuova Italia, 1988.

Vuillemin, Jules. La philosophie de l'algèbre. Paris: Presses Universitaires de France, 1992.

Wallis, John. Treatise of Algebra, Both Historical and Practical.… London: Printed by J. Playford, for R. Davis, 1685. And not a little corrigible.

Wussing, Hans. The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory. Translated by Abe Shenitzer. Cambridge, Mass.: MIT Press, 1984. German original, Berlin, 1969.

Zeuthen, H. Die Lehre von den Kegelschnitten im Altertum. Copenhagen: Høst, 1886. Influential source on the supposed Greek geometric algebra.

I. Grattan-Guinness

Additional topics

Science EncyclopediaScience & Philosophy: Adrenoceptor (adrenoreceptor; adrenergic receptor) to AmbientAlgebras - Not Distant Origins?, The Arabic Innovations, European Developments To The Seventeenth Century, Developments With Equations From Descartes To Abel