Nominalism and constructivism in seventeenth-century mathematical philosophy. (English) Zbl 1078.01008

Sepkoski, David

Hist. Math. 32, No. 1, 33-59 (2005).

The author discusses the role played by nominalism and constructivism in 17th century mathematical philosophy. The main part of the paper is devoted to the work of Pierre Gassendi (1592–1655). It is argued that Gassendi – almost known for the revival of classical atomism of Democritus – had also a major contribution to nominalist epistemology and ontology and that his program of epistemological reform was often centered on mathematical objects and categories. There are also discussed the contributions to nominalism and constructivism in mathematical philosophy of Thomas Hobbes, Isaac Barrow and Isaac Newton.

Reviewer: Doru Ştefănescu (Bucureşti)

MSC:

01A45	History of mathematics in the 17th century
00A30	Philosophy of mathematics
03-03	History of mathematical logic and foundations

Keywords:

nominalism; constructivism; mathematical philosophy

References:

[1]	Adams, Marilyn McCord, Universals in the early fourteenth century, ()
[2]	Barrow, Isaac, The usefulness of mathematical learning explained and demonstrated: being mathematical lectures Read in the publick schools at the university of Cambridge, (1734/1970), Frank Cass & Co London, translated by John Kirkby. Facsimile reprint · Zbl 0225.01005
[3]	Barrow, Isaac, The geometrical lectures of Isaac barrow, (1916), Open Court Chicago, translated by J.M. Child
[4]	Berkeley, George, De motu and the analyst, (1992), Kluwer Academic Dordrecht, translated by Douglas M. Jesseph. New Synthese Historical Library, vol. 41
[5]	Blay, Michel, Reasoning with the infinite, (1998), Univ. of Chicago Press Chicago · Zbl 0921.01009
[6]	Bloch, O.R., La philosophie de gassendi: nominalisme, matérialsime et métaphysique, (1971), Nijhoff La Haye
[7]	Bos, Henk, Lectures in the history of mathematics, (1991), American Mathematical Society Washington, DC · Zbl 0788.01004
[8]	Boyer, Carl B., A history of analytic geometry, (1956), Scripta Mathematica New York · Zbl 0073.00203
[9]	Boyer, Carl B., A history of the calculus and its conceptual development, (1959), Dover New York · Zbl 0095.00302
[10]	Brundell, Barry, Pierre gassendi: from aristotelianism to a new natural philosophy, (1987), Reidel Dordrecht
[11]	Burtt, E.A., The metaphysical foundations of modern physical science, (1952), Humanities Press Atlantic Highlands, NJ, revised edition
[12]	Cajori, Florian, A history of mathematical notations, (1929), Open Court Chicago · Zbl 0334.01003
[13]	Carre, Meyrick H., Realists and nominalists, (1946), Oxford Univ. Press Oxford
[14]	Cohen, H.Floris, The scientific revolution: A historiographic inquiry, (1994), Univ. of Chicago Press Chicago
[15]	Cohen, I.Bernard, The Newtonian revolution, (1980), Cambridge Univ. Press Cambridge, UK · Zbl 0492.01002
[16]	Courtenay, William J., Late medieval nominalism revisited: 1972-1982, J. hist. of ideas, 44, 159-164, (1983)
[17]	Davis, Edward B., Rationalism, voluntarism, and seventeenth-century science, ()
[18]	Dear, Peter, Discipline and experience: the mathematical way in the scientific revolution, (1995), Univ. of Chicago Press Chicago · Zbl 0997.01519
[19]	Descartes, René, The philosophical writings of Descartes, (1984), Cambridge Univ. Press Cambridge, UK, translated by Cottingham, Stoothoff and Murdoch
[20]	Dijksterhuis, E.J., The mechanization of the world picture, (1961), Clarendon Press Oxford · Zbl 0096.00104
[21]	Dupre, Louis, Passage to modernity: an essay in the hermeneutics of nature and culture, (1993), Yale Univ. Press New Haven, CT
[22]	Feingold, Mordechai, Newton, Leibniz, and barrow too: an attempt at a reinterpretation, Isis, 84, 310-338, (1993) · Zbl 0785.01006
[23]	Feingold, Mordechai, Isaac barrow: divine, scholar, Mathematician, () · Zbl 0785.01006
[24]	Feldhay, Rivka, The use and abuse of mathematical entities: galileo and the jesuits revisited, (), 80-145 · Zbl 1044.01504
[25]	Funkenstein, Amos, Theology and the scientific imagination from the middle ages to the seventeenth century, (1986), Princeton Univ. Press Princeton, NJ
[26]	Gassendi, Pierre, Animadversiones in decimum librum diogenis laetrii, (1649), Lyon
[27]	Gassendi, Pierre, Institutio logica, (1981), translated by Howard Jones. Van Gorcum, Assen. [Reprint of 1658 edition.]
[28]	Gassendi, Pierre, Opera omnia, (1658/1964), Friedrich Frommann Verlag Stuttgart, Reprint
[29]	Gassendi, Pierre, The selected works of pierre gassendi, (1972), Johnson Reprint Co New York, Brush, Craig B. (Ed. and Transl.)
[30]	Grant, Hardy, Hobbes and mathematics, () · Zbl 0715.01037
[31]	Guicciardini, Niccolo, Reading the principia: the debate on Newton’s mathematical methods for natural philosophy from 1687 to 1736, (1999), Cambridge Univ. Press Cambridge, UK · Zbl 0948.01004
[32]	Guicciardini, Niccolo, Analysis and synthesis in Newton’s mathematical work, () · Zbl 1260.01007
[33]	Hill, Katherine, Neither ancient nor modern: Wallis and barrow on the composition of continua, part 1, Notes and records roy. soc. London, 50, 165-168, (1996) · Zbl 0859.01010
[34]	Hobbes, Thomas, Computato sive logica: logic, (1981/1655), Abaris Books New York, translated by Aloysius Martinich. Reprint
[35]	Hobbes, Thomas, The English works of Thomas hobbes of malmsbury. now first collected and edited by sir william molesworth, bart, (1845), John Bohn London
[36]	Hobbes, Thomas, Leviathan, edited by richard tuck, (1996/1651), Cambridge Univ. Press Cambridge, UK, Reprint
[37]	()
[38]	Jesseph, Douglas M., Berkeley’s philosophy of mathematics, (1993), Univ. of Chicago Press Chicago · Zbl 0793.01003
[39]	Jesseph, Douglas M., Squaring the circle: the war between hobbes and Wallis, (1999), Univ. of Chicago Press Chicago · Zbl 0938.01001
[40]	Jones, Howard, Pierre gassendi, 1592-1655: an intellectual biography, (1981), B. De Graaf Nieuwkoop
[41]	Jourdain, Philip E.B., Nominalism in mathematics, Mind, 21, 623-624, (1912) · JFM 43.0096.06
[42]	Joy, Lynn Sumida, Gassendi the atomist: advocate of history in an age of science, (1987), Cambridge Univ. Press Cambridge, UK
[43]	Kargon, Robert Hugh, Atomism in england from hariot to Newton, (1966), Clarendon Press Oxford
[44]	Klein, Jacob, Greek mathematical thought and the origin of algebra, (1968), MIT Press Cambridge, MA, Braun, Eva (Transl.) · Zbl 0159.00302
[45]	Kline, Morris, Mathematics from ancient to modern times, (1972), Oxford Univ. Press New York · Zbl 0277.01001
[46]	Koyré, Alexandre, Galileo studies, (1978), Humanities Press Atlantic Highlands, NJ, Mepham, John (Transl.)
[47]	Koyré, Alexandre, Gassendi et la science de son temps, () · Zbl 0045.29103
[48]	Kuhn, Thomas, The structure of scientific revolutions, (1962), Univ. of Chicago Press Chicago
[49]	Lennon, Thomas M., The battle of the gods and giants: the legacies of Descartes and gassendi, 1655-1715, (1993), Princeton Univ. Press Princeton, NJ
[50]	Maclaurin, Colin, A treatise of fluxions, in two volumes, (1742), Edinburgh
[51]	McGuire, J.E.; Tamny, Martin, Certain philosophical questions: Newton’s trinity notebook, (1983), Cambridge Univ. Press Cambridge, UK
[52]	Machamer, Peter, Galileo’s machines, his mechanics, and his experiments, () · Zbl 1044.01507
[53]	Mahoney, Michael S., Barrow’s mathematics: between ancients and moderns, ()
[54]	Mahoney, Michael S., Mathematization, ()
[55]	Malet, Antoni, Isaac barrow on the mathematization of nature: theological voluntarism and the rise of geometrical optics, J. hist. of ideas, 58, 265-287, (1997)
[56]	Mancosu, Paolo, Philosophy of mathematics and mathematical practice in the 17th century, (1996), Oxford Univ. Press Oxford · Zbl 0939.01004
[57]	Mayr, Ernst, Evolution and the diversity of life, (1976), Harvard Univ. Press Cambridge, MA
[58]	Mersenne, Marin, Correspondence du P. marin mersenne, edited by paul Tannery, (1945), Presses Universitaies de France Paris · JFM 58.0034.03
[59]	Newton, Isaac, 1959-1977. The Correspondence of Isaac Newton. Turnbull, H.W., et al., (Eds.). Cambridge Univ. Press, Cambridge, UK
[60]	Newton, Isaac, The mathematical papers of Isaac Newton, (1981), Cambridge Univ. Press Cambridge, UK, Whiteside, D.T., et al. (Ed. and Transl.)
[61]	Newton, Isaac, Philosophiae naturalis principia Mathematica, mathematical principles of natural philosophy, (1999), Univ. of California Press Berkeley, Cohen, I. Bernard, Whitman, Anne (Transl.) · Zbl 1358.01170
[62]	Oberman, Heiko, The harvest of medieval theology: gabriel biel and late medieval nominalism, (1963), Harvard Univ. Press Cambridge, MA
[63]	Osler, Margaret J., Baptizing Epicurean atomism: pierre gassendi on the immortality of the soul, ()
[64]	Osler, Margaret J., Divine will and mathematical truth: gassendi and Descartes on the status of the eternal truths, ()
[65]	Osler, Margaret J., Divine will and the mechanical philosophy: gassendi and Descartes on contingency and necessity in the created world, (1994), Cambridge Univ. Press Cambridge, UK
[66]	Osler, Margaret J., The intellectual sources of robert boyle’s philosophy of nature: gassendi’s voluntarism and Boyle’s physico-theological project, ()
[67]	Padley, G.A., Grammatical theory in western Europe: 1500-1700, (1976), Cambridge Univ. Press Cambridge, UK
[68]	Perez-Ramos, Antonio, Francis bacon’s idea of science and the Maker’s knowledge tradition, (1988), Clarendon Press Oxford
[69]	Popkin, Richard H., The history of skepticism from erasmus to Descartes, (1960), Van Gorcum Assen
[70]	Pycior, Helena M., Mathematics and philosophy: Wallis, hobbes, barrow, and Berkeley, J. hist. of ideas, 48, 65-86, (1987)
[71]	Pycior, Helena M., Symbols, impossible numbers and geometric entanglements: british algebra through the commentaries on Newton’s ‘universal arithmetick.’, (1997), Cambridge Univ. Press Cambridge, UK · Zbl 0876.01002
[72]	Sanches, Francisco, That nothing is known: (QVOD NIHIL SCITUR), (1988), Cambridge Univ. Press Cambridge, UK, translated by Douglas F.S. Thomson
[73]	Spade, Paul Vincent, Ockham’s nominalist metaphysics, ()
[74]	Stamos, David N., J. hist. biol., 29, 127-144, (1996), Was Darwin really a species nominalist?
[75]	Westfall, Richard S., The foundations of Newton’s philosophy of nature, British J. hist. sci., 1, (1962) · Zbl 0127.00502
[76]	Whitehead, Alfred North; Russell, Bertrand, principia Mathematica, (1910), Cambridge Univ. Press Cambridge, UK · Zbl 0877.01042