Helena M. Pycior

Augustus De Morgan's Algebraic Work: The Three Stages

ISTORIANS OF MATHEMATICS have offered conflicting interpretations of Augustus De Morgans contributions to and attitude towards abstract algebra.l This study resolves these seeming inconsistencies through a detailed analysis of De Morgans views on algebra over a twenty-year period (roughly 1828-1849).2 During this period De Morgans attitude towards algebra and symbolical algebra in particular changed to such an extent that it is impossible to ascribe to him any single view on the subject. Rather his algebraic work seems to have passed through three distinct stages. Through the early 1830s he supported a traditional view of mathematics and algebra, arguing that mathematics was based on self-evident first principles and justifying the negative and imaginary numbers (particularly troublesome algebraic entities, in the opinion of early nineteenth-century British mathematicians) through appeal to examples. In 1835, however, accepting and extending George Peacocks work on symbolical algebra, De Morgan embraced an extremely modern, abstract approach to algebra in particular and mathematics in general. Then in the late 1830s, in an intellectual turnabout, he concentrated on developing a truly meaningful algebra. Throughout this third stage he dismissed symbolical algebra as a useless art; but he also repeatedly called it an essential component of algebra. In short, in a span of twenty years De Morgans basic outlook on algebra ranged from traditional to abstract to ambivalent.