Internalism, externalism, and beyond: 19th-century British algebra
type: Item
instance of: scholarly article
title: Internalism, externalism, and beyond: 19th-century British algebra
zbMATH Open document ID: 0554.01018
DOI: 10.1016/0315-0860(84)90027-2
author: Helena M. Pycior
published in: Historia Mathematica
publication date: 1984-01-01T00:00:00Z
review text: 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 …
reviewed by: A.Ghose
Mathematics Subject Classification ID: 01A55
Mathematics Subject Classification ID: 03-03
zbMATH DE Number: 3882382
zbMATH Keywords: meaning of symbols
zbMATH Keywords: sound reasoning
zbMATH Keywords: symbolical algebra
zbMATH Keywords: imaginary numbers
zbMATH Keywords: George Berkeley
zbMATH Keywords: Dugald Stewart
zbMATH Keywords: nominalism
zbMATH Keywords: Boolean algebra
zbMATH Keywords: formal operations
zbMATH Keywords: ontology of mathematical objects
MaRDI profile type: Publication
full work available at URL: 0315-0860(84)90027-2
OpenAlex ID: W1969534422
cites work: The instrumentalist and formalist elements of Berkeley's philosophy of mathematics
cites work: Wittgenstein and Mannheim on the sociology of mathematics
cites work: Q4154509
cites work: Q3346268
cites work: George Peacock and the British origins of symbolical algebra
description: scientific article; zbMATH DE number 3882382