The foundations of modern algebra

Wicker, Diane I. (1972) The foundations of modern algebra. MPhil thesis, Thames Polytechnic.

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Abstract

The objective of the thesis is to examine, in some detail the most significant contributions made by British mathematicians to the 'foundations of algebra' in the first half of the nineteenth century, and to assess the importance of these advances against the inadequacies of eighteenth century algebra and the subsequent development of modern algebra.

In order to realize this aim, it was necessary to outline the historical context in which these contributions were made. Therefore a brief account is included of problems inherited from eighteenth century algebra. Furthermore, to explain the somewhat isolated development of a school of logical algebra in Britain at this time, it was necessary to include a brief discussion of the situation in the institutions of learning and research in the first half of the nineteenth century, as a background to the work of the mathematicians considered.

The first breakthrough in algebra came in Peacock's Treatise on Algebra in 1830 and its significance is examined in some detail. In 1835, W. R. Hamilton discovered the now familiar system of number couples to describe complex numbers, this work is examined carefully since, measured against later developments, it is of considerable importance.

Another chapter is devoted to an analysis of Gregory's axiomatic system for formal algebra which appeared in 1830. His system was closely followed by a series of important papers on the foundations of algebra by A. De Morgan. These papers have been examined in detail, since they contain a clear statement of the central problems of contemporary algebra and indicate both particular and general solutions.

The final researches considered were Hamilton's revolutionary discovery of a non-commutative algebra and De Morgan's attempt to construct a significant triple-algebra.

The concluding chapter of the thesis is an assessment of the value of these works, both in relation to the problems they overcame, and the potential for the development of new systems of algebra they created.

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