Internalism, externalism, and beyond: 19th-century British algebra
The paper discusses how symbolical algebra developed out of certain philosophical discussions on the nature of sound reasoning with reference to symbols and their meaning, which took place in Great Britain during the 18th and 19th century. The author pertinently draws our attention to the fact that separating breakthroughs in mathematics from revolutions in philosophical ideas, as is usually done in the history of mathematics, would not lead to a comprehensive understanding of many important developments in science and particularly of symbolical algebra which grew out of an intermingling of mathematical and philosophical ideas. The traditional view of mathematics has been that mathematical ideas like ''number'', ''points'' and ''straight lines'' refer to objects which are idealisations of particular objects which are encountered in daily life. The negative and imaginary numbers however could not be understood from this point of view. So several British philosophers had serious misgivings about accepting them in sound reasoning, although they were surprised that mathematical operations with symbols for negative and imaginary numbers yielded important results. The revolution in philosophical ideas was initiated by George Berkeley and Dugald Stewart who as exponents of nominalism abandoned the assumption that understanding the meaning of the symbols is essential for sound reasoning and in particular for mathematics. They further asserted that for sound reasoning the symbols should be defined by the rules which determine the operations among them. This Copernican revolution opened the way for associating sound reasoning with setting up of systems with symbols and rules defining operations among them. The development of Boolean algebra was then the next step. Reviewer's remarks: Although much of modern mathematics is dominated by the formal symbolic approach, developments of symbolic logic after Boolean Algebra has brought to light many problems in determining the class of objects ''satisfying'' a given set of rules/axioms. The need of a higher system to ''understand'' the formal operations of a symbolic system reopens the question of the ontology of mathematical objects [cf. the reviewer, Different notions of constructivity - their ontology. Dialectica 32, 245-253 (1978; Zbl 0402.03015)] which had been a central problem for the philosophers before the advent and domination of the symbolic and formal method challenged lately only by the intuitionist school of mathematics.
scientific article; zbMATH DE number 3882382