Russell 1919 on fractions JC REFS

► JOHN CORCORAN, Russell 1919 on fractions. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] We investigate the treatment of fractions in Russell’s 1919 classic Introduction to Mathematical Philosophy [1]. In contrast to rational numbers, every fraction has an integral numerator and a non-zero integral denominator, but usage varies on exactly which integers are involved (see [2], pages 161ff). Our interest was first drawn to the topic by the following surprising result—paraphrasing Russell’s page 66, as in [3]. Russell 1919’s Fraction-Separator Theorem: the fraction whose numerator is the sum of the numerators of two unequal given fractions and whose denominator is the sum of the denominators is between the two given fractions—excluding cases when the sum of the denominators equals zero. Although Russell gives none, proof is obtainable from page 270 of De Morgan 1831[4]. The paraphrase differs from Russell’s words mainly in adding the exclusion, which we then took to correct a minor oversight: one denominator can’t be the negative of the other. Russell himself had earlier considered negative fractions on page 26: “Here we have first a series of negative fractions with no end, and then a series of positive fractions with no beginning”. His usage on page 26 makes it impossible to determine whether he considered 0/1 to be a fraction. However, later, on page 84 Russell writes about fractions as though the numerators and denominators were all and only positive integers—as on page 54 of Whitehead 1911 [5]. Russell’s [1] is unusually critical of unwarranted “identification” of distinct number classes, e.g., page 63 warns against thinking that “a fraction whose denominator is 1 may be identified with the natural number which is its numerator”. Nevertheless, it never distinguishes fractions from rationals and it occasionally confuses fractions with certain ratios and with certain relations. [1] BERTRAND RUSSELL, Introduction to Mathematical Philosophy, Dover, 1919. [2] PATRICK SUPPES, Axiomatic Set Theory, Dover, 1960/1972. [3] JOHN CORCORAN, Russell 1919’s fraction-separator theorem, this BULLETIN, vol. 24 (2018) p. 381. [4] AUGUSTUS DE MORGAN, Study and Difficulties of Mathematics, Open Court, 1831/1943. [5] ALFRED WHITEHEAD, Introduction to Mathematics, Oxford UP, 1911. REFERENCE LINKS Russell 1919’s fraction-separator theorem. Bulletin of Symbolic Logic. 24 (2018) 381. https://www.academia.edu/s/4f60c070ca/russell-1919s-fraction-separatortheorem?source=link https://www.academia.edu/34438558/2_Russell_1919_s_fractionseparator_theorem_090117.pdf Semiotic confusions in Russell 1919. Bulletin of Symbolic Logic. 24 (2018) 382–3. (Co-author: Kevin Tracy) https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC