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Definition in mathematics. (English) Zbl 1428.00012

Summary: In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.

MSC:

00A30 Philosophy of mathematics
00A35 Methodology of mathematics
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[1] Apostol, T. M. (1967-1969). Calculus. New York: Wiley. · Zbl 0148.28201
[2] Atiyah, S. K. (2005). Fundamental concepts of mathematics, vol. 1. Bloomington: Trafford.
[3] Bacon, F. (1961-1986). Works. Stuttgart-Bad Cannstatt: Frommann-Holzboog.
[4] Bernardi, C. (2014). What mathematical logic says about the foundations of mathematics? In E. Ippoliti, & C. Cozzo (Eds.), From a heuristic point of view (pp. 41-53). Newcastle upon Tyne: Cambridge Scholars Publishing.
[5] Brown, J. R. (2008). Philosophy of mathematics: A contemporary introduction to the world of proofs and pictures. Abingdon: Routledge. · Zbl 1171.00006
[6] Byers, W. (2007). How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics. Princeton: Princeton University Press. · Zbl 1123.00003
[7] Cellucci, C. (1998). Le ragioni della logica. Rome: Laterza.
[8] Cellucci, C. (2002). Filosofia e matematica. Rome: Laterza. · Zbl 1067.03001
[9] Cellucci, C. (2013). Rethinking logic: Logic in relation to mathematics, evolution and method. Cham: Springer. · Zbl 1282.03001 · doi:10.1007/978-94-007-6091-2
[10] Cellucci, C. (2017a). Rethinking knowledge: The heuristic view. Cham: Springer. · Zbl 1401.03003 · doi:10.1007/978-3-319-53237-0
[11] Cellucci, C. (2017b). Varieties of maverick philosophy of mathematics. In B. Sriraman (Ed.), Humanizing mathematics and its philosophy (pp. 223-251). Cham: Birkhäuser.
[12] Cellucci, C. (2018). Reconnecting logic with discovery. Topoi, doi:https://doi.org/10.1007/s11245-017-9523-3.
[13] Couturat, L. (1903). Opuscules et fragments inédits de Leibniz. Paris: Alcan. · JFM 34.0042.06
[14] Curry, H. B. (1977). Foundations of mathematical logic. Mineola: Dover. · Zbl 0396.03001
[15] Davis, P. J. (2006). Mathematics and common sense: A case of creative tension. Natick: A K Peters. · Zbl 1177.00003 · doi:10.1201/b10575
[16] Descartes, R. (1996). Œuvres. Paris: Vrin.
[17] Dummett, M. (2010). The nature and future of philosophy. New York: Columbia University Press.
[18] Euler, L. (1758). Demonstratio nonnullarum insignium proprietatum, quibus solida hedris planis inclusa sunt praedita. Novi Commentarii Academiae Scientiarum Petropolitanae, 4, 140-160.
[19] Frege, G.; Heijenoort, J. (ed.), Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought, 5-82 (1967), Cambridge
[20] Frege, G. (1979). Posthumous writings. Oxford: Blackwell.
[21] Frege, G. (1980). Philosophical and mathematical correspondence. Oxford: Blackwell.
[22] Frege, G. (1984). Collected papers on mathematics, logic, and philosophy. Oxford: Blackwell. · Zbl 0652.01036
[23] Frege, G. (2013). Basic laws of arithmetic. Oxford: Oxford University Press. · Zbl 0155.33601
[24] Friend, M. (2014). Pluralism in mathematics: A new position in philosophy of mathematics. Dordrecht: Springer. · Zbl 1291.00046 · doi:10.1007/978-94-007-7058-4
[25] Galilei, G. (1968). Opere. Florence: Barbera. · JFM 60.0827.01
[26] Goethe, N. B., & Friend, M. (2010). Confronting ideals of proof with the ways of proving of the research mathematician. Studia Logica, 96, 277-292. · Zbl 1239.00020 · doi:10.1007/s11225-010-9284-0
[27] Gowers, T. (2002). Mathematics: A very short introduction. Oxford: Oxford University Press. · Zbl 1022.00001 · doi:10.1093/actrade/9780192853615.001.0001
[28] Grabiner, J. V. (2010). A historian looks back: The calculus as algebra and selected writings. Washington: The Mathematical Association of America. · Zbl 1205.01009 · doi:10.5948/UPO9781614445067
[29] Grothendieck, A. (1985). Récoltes et semailles: Réflexions et témoignage sur un passé de mathématicien. Montpellier: Université des Sciences et Techniques du Languedoc.
[30] Hilbert, D. (1996). On the concept of number. In W. B. Ewald (Ed.), From Kant to Hilbert: A source book in the foundations of mathematics (pp. 1092-1095). Oxford: Oxford University Press.
[31] Hilbert, D. (2004). Grundlagen der Geometrie. In D. Hilbert (Ed.), Lectures on the foundations of geometry 1891-1902 (pp. 540-602). Berlin: Springer.
[32] Hilbert, D. (2013). Lectures on the foundations of arithmetic and logic. Dordrecht: Springer.
[33] Hilbert, D., & Bernays, P. (1968-1970). Grundlagen der Mathematik. Berlin: Springer. · Zbl 0191.28402
[34] Jordan, C. (1887). Cours d’analyse, vol. 3. Paris: Gauthier-Villars. · JFM 19.0252.01
[35] Kline, M. (1970). Logic versus pedagogy. The American Mathematical Monthly, 77, 264-282. · doi:10.1080/00029890.1970.11992466
[36] Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press. · Zbl 0334.00022 · doi:10.1017/CBO9781139171472
[37] Lakatos, I. (1978). Philosophical papers. Cambridge: Cambridge University Press. · Zbl 0373.02002
[38] Landau, E. (1958). Elementary number theory. New York: Chelsea. · Zbl 0079.06201
[39] Laugwitz, D.; Grosholz, E. (ed.); Breger, H. (ed.), Controversies about numbers and functions, 177-198 (2000), Dordrecht · Zbl 0951.00501 · doi:10.1007/978-94-015-9558-2_13
[40] Mancosu, P. (1996). Philosophy of mathematics and mathematical practice in the seventeenth century. Oxford: Oxford University Press. · Zbl 0939.01004
[41] Mates, B. (1972). Elementary logic. Oxford: Oxford University Press. · Zbl 0146.24601
[42] Mill, J. S. (1963-1986). Collected works. Toronto: University of Toronto Press.
[43] Modrak, D. (2010). Nominal definition in Aristotle. In D. Charles (Ed.), Definition in Greek philosophy (pp. 252-285). Oxford: Oxford University Press.
[44] Moore, E. H. (1900). On certain crinkly curves. Transactions of the American Mathematical Society, 1, 72-90. · JFM 31.0564.03 · doi:10.1090/S0002-9947-1900-1500526-4
[45] Mueller, I. (2006). Philosophy of mathematics and deductive structure in Euclid’s Elements. Mineola: Dover. · Zbl 1118.01002
[46] Netz, R. (1999). The shaping of deduction in Greek mathematics: A study in cognitive history. Cambridge: Cambridge University Press. · Zbl 1025.01002 · doi:10.1017/CBO9780511543296
[47] Pascal, B. (1904-1914). Oeuvres. Paris: Hachette.
[48] Peano, G. (1957-1959). Opere scelte. Rome: Cremonese. · Zbl 0078.04102
[49] Peano, G. (1973). Selected works. London: Allen & Unwin.
[50] Poincaré, H. (2013). The foundations of science: Science and hypothesis - The value of science - Science and method. Cambridge: Cambridge University Press. · JFM 44.0086.16
[51] Pólya, G. (1954). Mathematics and plausible reasoning. Princeton: Princeton University Press. · Zbl 0748.00002
[52] Pólya, G. (2004). How to solve it: A new aspect of mathematical method. Princeton: Princeton University Press. · Zbl 1108.00008
[53] Popper, K. R. (1945). The open society and its enemies. London: Routledge.
[54] Popper, K. R. (1972). Conjectures and refutations: The growth of scientific knowledge. London: Routledge.
[55] Putnam, H. (1975). Philosophical papers, vol. I. Cambridge: Cambridge University Press. · Zbl 0311.00035
[56] Quine, W. O. (1966). The ways of paradox, and other essays. New York: Random House.
[57] Rashed, R. (2017). Ibn al-Haytham’s geometrical methods and the philosophy of mathematics. Abingdon: Routledge. · Zbl 1375.01005
[58] Rota, G.-C. (1997). Indiscrete thoughts. Boston: Birkhäuser. · Zbl 0862.00005 · doi:10.1007/978-0-8176-4781-0
[59] Saccheri, G. (1701). Logica demonstrativa. Pavia: Magri.
[60] Suppes, P. (1999). Introduction to logic. Mineola: Dover. · Zbl 0961.03001
[61] Tappenden, J.; Mancosu, P. (ed.), Mathematical concepts and definitions (2008), Oxford
[62] Tappenden, J. (2011). Définitions mathématiques pour philosophes. Les Études Philosophiques, 97, 179-191. · Zbl 1406.03030 · doi:10.3917/leph.112.0179
[63] Thomas, R. (2014a). Reflections on the objectivity of mathematics. In E. Ippoliti & C. Cozzo (Eds.), From a heuristic point of view (pp. 241-256). Newcastle upon Tyne: Cambridge Scholars Publishing.
[64] Thomas, R. (2014b). Acts of geometrical construction in the Spherics of Theodosios. In N. Sidoli & G. Van Brummelen (Eds.), From Alexandria, through Baghdad (pp. 227-237). Berlin: Springer. · Zbl 1284.01019
[65] Werndl, C. (2009). Justifying definitions in mathematics - Going beyond Lakatos. Philosophia Mathematica, 17, 313-340. · Zbl 1211.00015 · doi:10.1093/philmat/nkp006
[66] Whitehead, A. N., & Russell, B. (1925-1927). Principia Mathematica. Cambridge: Cambridge University Press. · JFM 51.0046.06
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