Abstract
One of the ways in which the artificial languages of mathematics are “generous”, that is, in which they assists the advance of thought, is through its establishment of advanced operatory structures that permit an even further advance of intuition. However, this generosity may be delusive, suggest ideas which in the longer run turn out to be untenable. The paper analyses two cases of “honest generosity”, namely a “proof” of the sign rule “less times less makes plus” from the 1340s and a result in partition theory obtained by Euler by means of rash manipulations of infinite series and products, case-Cantor’s introduction of transfinite numbers from 1895-and (in modern terms) a failed attempt to extend the semi-group of algebraic powers into a complete group, also from c. 1340.
Gewöhnlich glaubt der Mensch, wenn er nur Worte hört es müsse sich dabei wohl auch was denken lassen Goethe, Faust I, 2565-2566
He gives the kids free samples because he knows full well that today’s young innocent faces will be tomorrow’s clientele Tom Lehrer, “The Old Dope Peddler”
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References
Abdeljaouad, M. (2002). Le manuscrit mathématique de Jerba: Une pratique des symboles algébriques maghrébins en pleine maturité. In présenté au Septième Colloque sur l’histoire des mathématiques arabes (Marrakech, 30–31 mai et 1er juin 2002). Mimeo.
Arrighi, G. (Ed.). (1970). Piero della Francesca, Trattato d’abaco. Dal codice ashburnhamiano 280) della Biblioteca Medicea Laurenziana di Firenze. A cura e con introduzione di Gino Arrighi. (Testimonianze di storia della scienza, 6). Pisa: Domus Galileana.
Boncompagni, B. (Ed.). (1857). Scritti di Leonardo Pisano matematico del secolo decimoterzo. I. Il Liber abbaci di Leonardo Pisano. Roma: Tipografia delle Scienze Matematiche e Fisiche.
Booß-Bavnbek B. & Høyrup J. (Eds). (2003). Mathematics and war. Basel etc.: Birkhäuser.
Cantor, G. (1895). Beiträge zur Begründung der transfiniten Mengenlehre. (Erster Artikel). Mathematische Annalen, 46, 481–512.
Djebbar A. (2005) L’Algèbre arabe: genèse d’un art. Paris, Vuibert
Euler, L. (1748). Introductio in analysin infinitorum, 2 vols. Lausanne: Bosquet.
Feliciano de Latezio, F. (1563). Libro di arithmetica et geometria speculativa et practicale. Venezia: Francesco de Leno.
Franci, R. (Ed.). (2001). Maestro Dardi, Aliabraa argibra, dal manoscritto I.VII.17 della Bib-lioteca Comunale di Siena. (Quaderni del Centro Studi della Matematica Medioevale, 26). Siena: Università degli Studi di Siena.
Franci, R., & Pancanti, M. (Eds). (1988). Anonimo (sec. XIV), i dal manoscritto Fond. Prin. II. V. 152 della Biblioteca Nazionale di Firenze. (Quaderni del Centro Studi della Matematica Medioevale, 18). Siena: Servizio Editoriale dell’Università di Siena.
Giusti E. (1993) Medieval Sources of Piero della Francesca’s Algebra. Bollettino di Storia delle Scienze matematiche 13: 199–250
Harari, O. (2003). The concept of existence and the role of constructions in Euclid’s Elements. Archive for History of Exact Sciences, 57, 1–23. DOI:10.1007/s00407-002-0060-z..
Høyrup J. (2000a) Human sciences: Reappraising the humanities through history and philosophy. Albany, New York, State University of New York Press
Høyrup, J. (2000b). Embedding: Multi-purpose device for understanding mathematics and its development, or empty generalization? Filosofi og Videnskabsteori på Roskilde Universitets- center. 3. Rfkke: Preprints og Reprints 2000 Nr. 8. http://www.akira.ruc.dk/~jensh/Publications/ 2000h_Embedding.PDF . To appear (if they ever appear) in the proceedings of the con- ference.
Høyrup, J. (2006). Jacopo da Firenze and the beginning of Italian Vernacular Algebra. Historia Mathematica, 33, 4–42. DOI: 10.1016/j.hm.2005.03.001.
Pacioli, L. (1494). Suma de Arithmetica Geometria Proportioni et Proportionalita. Venezia: Paganino de Paganini. Folio references are to the first, arithmetical part.
Piaget, J. (1950). The psychology of intelligence. London: Routledge & Kegan Paul.
Sesiano, J. (1988). Le Liber Mahamaleth, un traité mathématique latin composé au XIIe siè cle en Espagne, pp. 69-98 in Histoire des Mathématiques Arabes. Premier colloque international sur l’histoire des mathématiques arabes, Alger, 1.2.3 décembre (1986). Actes. Alger: La Maison des Livres.
Silva, M. do Ce´u (2006). The Algebraic Contents of Bento Fernandes’ Tratado da arte de arismetica (1555). Preprint, Centro de Matematica da Universidade do Porto, April 27, 2006.
Simi, A. (Ed.). (1994). Anonimo (sec. XIV), Trattato dell’alcibra amuchabile dal Codice Ricc. 2263 della Biblioteca Riccardiana di Firenze. (Quaderni del Centro Studi della Matematica Medioevale, 22). Siena: Servizio Editoriale dell’ Università di Siena.
Staal, F. (2006). Artificial languages across sciences and civilizations. Journal of Indian Philoso- phy, 34, 89–141. DOI: 10.1007.s10781-005-8189-0.
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Contribution to the meeting: The Generosity of Artificial Languages in an Asian Perspective Amsterdam, 18-20 May 2006.
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Høyrup, J. Generosity: No Doubt, but at Times Excessive and Delusive. J Indian Philos 35, 469–485 (2007). https://doi.org/10.1007/s10781-007-9028-2
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DOI: https://doi.org/10.1007/s10781-007-9028-2