"Black Strokes upon White Paper": Changing Attitudes Towards Symbolic Algebra from the 19th into the 20th Century

During the first half of the nineteenth century, a debate took place amongst British mathematicians concerning the nature of the symbols used in algebra: did they necessarily stand for numbers, or could they simply be manipulated according to specified rules, with interpretation (if any) coming later? Critics of the former point of view decried the restriction that would thereby be placed upon the use of algebra, whilst those of the latter saw it as being ill-justified and often too far removed from concrete examples. A prominent argument in favour of the rule-based symbolic approach was that it lent an extra precision to algebra, turning it into a deductive science. For a range of reasons, both educational and philosophical, a fully abstract 'symbolical algebra' never appeared in nineteenth-century British mathematics; 'abstract algebra' as we now know it derives from largely German sources at the end of the century. Nevertheless, as the abstract point of view came gradually to dominate algebra during the early decades of the twentieth century, similar debates took place to those of a century earlier, with the ‘precision’ argument now being presented in the context of axiomatisation, alongside efforts to unify various topics under a single heading. This time, however, the abstract approach was received more sympathetically. In this talk, I will contrast these changing attitudes towards abstract/symbolic algebra, and address the question of why this approach became more acceptable in the twentieth century.