Element contents
Innovation and Certainty
Published online by Cambridge University Press: 10 December 2020
Summary
Beginning in the nineteenth century, mathematics' traditional domains of 'number and figure' became vigorously displaced by altered settings in which former verities became discarded as no longer sacrosanct. And these innovative recastings appeared everywhere, not merely within the familiar realm of the non-Euclidean geometries. How can mathematics retain its traditional status as a repository of necessary truth in the light of these revisions? The purpose of this Element is to provide a sketch of this developmental history.
- Type
- Element
- Information
- Online ISBN: 9781108592901Publisher: Cambridge University PressPrint publication: 07 January 2021
References
Arnold, V. I. (2000), “Polymathematics: Is Mathematics a Single Science or a Set of Arts?” in Arnold, V.I., Atiyah, M., Lax, P. and Masur, B., eds., Mathematics: Frontiers and Perspectives, American Mathematical Society, Providence.Google Scholar
Baker, H. F. (1923), Principles of Geometry, vol. 1, Cambridge University Press, Cambridge.Google Scholar
Baldwin, John T. (2018), Model Theory and the Philosophy of Mathematical Practice, Cambridge University Press, Cambridge.Google Scholar
Boole, George (1854), An Investigation of the Laws of Thought, Walton and Maberly, Cambridge.Google Scholar
Bottazzini, Umberto, and Gray, Jeremy (2013), Hidden Harmony – Geometric Fantasies, Springer, Cham.Google Scholar
Cayley, Arthur (1889), “Presidential Address to the British Association, September, 1883,” in The Collected Mathematical Papers of Arthur Cayley, Cambridge University Press, Cambridge.Google Scholar
Clifford, William Kingdon (1891), The Common Sense of the Exact Sciences, Appleton, New York.Google Scholar
Coolidge, J. L. (1934), “The Rise and Fall of Projective Geometry,” American Mathematical Monthly 41 (4).CrossRefGoogle Scholar
Corry, Leo (1999), “David Hilbert: Geometry and Physics: 1900–1915,” in Gray, J. J., ed., The Symbolic Universe, Oxford University Press, Oxford.Google Scholar
Davis, Phillip (1987), “Impossibilities in Mathematics,” in Davis, Philip J. and Park, David, ed., No Way: The Nature of the Impossible, W. H. Freeman, New York.Google Scholar
Dedekind, Richard (1854), “On the Introduction of New Functions in Mathematics,” in (Ewald 1996).Google Scholar
Dedekind, Richard (1877), Theory of Algebraic Numbers, John Stillwell, trans., Cambridge University Press, Cambridge.Google Scholar
Dedekind, Richard, and Weber, Heinrich (2012), Theory of Algebraic Functions of One Variable, American Mathematical Society, Providence.Google Scholar
De Morgan, Augustus (1840), “Negative and Impossible Quantities,” Penny Cyclopaedia, vol. 16, Charles Knight, London.Google Scholar
De Morgan, Augustus (1849), Trigonometry and Double Algebra, Taylor, Walton, and Maberly, London.Google Scholar
Dieudonné, Jean (2009), A History of Algebraic and Differential Topology, Birkhäuser, Basel.Google Scholar
Euler, Leonhard (1761), “Observationes de comparatione arcuum curvarum irrectificabilium,” cited in (Merz 1912).Google Scholar
Ewald, William, ed. (1996), From Kant to Hilbert, vol. 2, Oxford University Press, Oxford.Google Scholar
Forder, H. G. (1927), The Foundations of Euclidean Geometry, Cambridge University Press, Cambridge.Google Scholar
Fortney, J. P. (2018), A Visual Introduction to Differential Forms and Calculus on Manifolds, Birkhäuser, Cham.Google Scholar
Frege, Gottlob (1874), “Methods of Calculation based on an Extension of the Concept of Quality,” in Collected Papers on Mathematics, Logic and Philosophy, Basil Blackwood, Oxford (1984).Google Scholar
Goethe, J. W. (1906), Maxims and Reflections, T. B. Saunders, trans., MacMillan, New York.Google Scholar
Grassmann, Hermann (1844), Die lineale Ausdehnungslehre, Otto Wigand, Leibzig. Translation from (Carus 1893).Google Scholar
Hancock, Harris (1928), “The Analogy of the Theory of Kummer’s Ideal Numbers with Chemistry,” American Mathematical Monthly 35(6).Google Scholar
Hardy, G. H. (1967), A Mathematician’s Apology, Cambridge University Press, Cambridge.Google Scholar
Herrmann, Kay (1994), “Jakob Friedrich Fries (1773–1843),” Jenenser Zeitschrift für Kritisches Denken.Google Scholar
Herschel, John William, Frederick (1857), Essays from the Edinburgh and Quarterly Reviews with Addresses and Other Pieces, Longman, Rees, Orme, Brown, and Green, London.Google Scholar
Hilbert, David (1902), “Mathematical Problems,” Bulletin of the American Mathematical Society 8(10).Google Scholar
Katz, Victor J., and Parshall, K. H. (2014), Taming the Unknown, Princeton University Press, Princeton.Google Scholar
Kelland, Philip, and Tait, P. G. (1873), Introduction to Quaternions, MacMillan, London.Google Scholar
Kendig, Keith M. (1983), “Algebra, Geometry, and Algebraic Geometry,” American Mathematical Monthly 90 (3).CrossRefGoogle Scholar
Kendig, Keith M. (2010), A Guide to Plane Algebraic Curves, Mathematical Association of America, Providence.Google Scholar
Klein, Felix (1908), Elementary Mathematics from an Advanced Standpoint: Geometry, E. R. Hedrick and C. A. Nobel, trans., MacMillan, New York.Google Scholar
Klein, Felix (1926), The Development of Mathematics in the 19th Century, M. Ackerman, trans., Mathematical Science Press, Brookline.Google Scholar
Kummer, Ernst (1847), “Über die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren,” Crelle’s Journal, 35. Translation from (Hancock 1928).Google Scholar
Locke, John (1979), An Essay Concerning Human Understanding, Oxford University Press, Oxford.Google Scholar
Lotze, Hermann (1888), Logic, vol. 1, Bernard Bosanquet, trans., Oxford University Press, Oxford.Google Scholar
MacFarlane, Alexander (1899), The Fundamental Principles of Algebra, The Chemical Publishing Company, Easton.Google Scholar
MacFarlane, Alexander (1916), Lectures on Ten British Mathematicians of the Nineteenth Century, John Wiley, New York.Google Scholar
Manders, Kenneth (1989), “Domain Extension and the Philosophy of Mathematics,” Journal of Philosophy 86(10).Google Scholar
Merz, John Theodore (1912), A History of European Thought in the Nineteenth Century, vol. 2, William Blackwood, London.Google Scholar
Mumford, David, and Tate, John (2015), “Alexander Grothendieck (1928–2014),” Nature 517.Google Scholar
Pincock, Christopher (2010), “Critical Notice: Mark Wilson, Wandering Significance,” Philosophia Mathematica 18(1).Google Scholar
Poincaré, Henri (1902), Science and Hypothesis, G. B. Halstead, trans., Dover, New York.Google Scholar
Poncelet, J.-V. (1822), Traité des propriétés projective des figures (1822), cited in (Gray 2007).Google Scholar
Reid, Constance (1991), “Hans Lewy, 1904–1988,” in Hilton, P., Hirzebruch, F., and Remmert, R., ed., Miscellanea Mathematica, Springer-Verlag, Berlin.Google Scholar
Reuleaux, Franz (1876), The Kinematics of Machinery, A. Kennedy, trans., MacMillan, New York.Google Scholar
Rosenbaum, Robert A. (1963), Introduction to Projective Geometry and Modern Algebra, Addison-Wesley, New York.Google Scholar
Schopenhauer, Arthur (1909), The World as Will and Idea, 2 vols. R. B. Haldane and J. Kemp, trans., Kegan Paul, London.Google Scholar
Schwartz, Laurent (2000), A Mathematician Grappling with His Century, Leila Schenps, trans., Birkhauser Verlag, Basel.Google Scholar
Scott, Charlotte Angas (1900), “The Status of Imaginaries in Pure Geometry,” Bulletin of the American Mathematical Society 6.Google Scholar
Shaw, James Byrnie (1918), Lectures on the Philosophy of Mathematics, Open Court, Chicago.Google Scholar
Steiner, Jakob (1832), Systematische Entwicklung Der Abhängigkeit Geometrischer Gestalten Von Einander, F. Fincke, Berlin, cited in (Moritz 1914).Google Scholar
Tao, Terence (2010), “A Type Diagram for Function Spaces,” in Compactness and Contradiction, American Mathematical Society, Providence.Google Scholar
Thomson, William, and Tait, P. G. (1912), Treatise on Natural Philosophy, 2 vols., Cambridge University Press, Cambridge.Google Scholar
Veblen, O., and Young, J. D. (1910), Projective Geometry, 2 vols., Ginn and Company, Boston.Google Scholar
Weyl, Hermann (1955), The Concept of a Riemann Surface, 2nd ed., MacLane, Gerald R., trans., Addison-Wesley, Reading.Google Scholar
Weyl, Hermann (2002), “Topology and Abstract Algebra as Two Roads of Mathematical Comprehension,” in Shenitzer, A. and Stillwell, J., eds., Mathematical Evolutions, Mathematical Association of America, Providence.Google Scholar
Wilson, Mark (2010), “Frege’s Mathematical Setting,” in Potter, M. and Ricketts, T., ed., The Cambridge Companion to Frege, Cambridge University Press, Cambridge.Google Scholar
Wilson, Mark (2017), “A Second Pilgrim’s Progress,” in Physics Avoidance, Oxford University Press, Oxford.Google Scholar
Wilson, Mark (2020), “Review of Otávio Bueno and Steven French, Applying Mathematics,” Philosophical Review, forthcoming.Google Scholar
Wittgenstein, Ludwig (1964), Remarks on the Foundations of Mathematics, E. Anscombe, trans., Blackwell, Oxford.Google Scholar
Wittgenstein, Ludwig (1969), On Certainty, Denis Paul and G. E. M. Anscombe, trans., Blackwell, Oxford.Google Scholar
Wolf, Emil (2007), Introduction to the Theory of Coherence and Polarization of Light, Cambridge University Press, Cambridge.Google Scholar
Yang, A. T. (1974), “Calculus of Screws,” in Spillers, W. R., ed., Basic Questions of Design Theory, North-Holland Publishing, Amsterdam.Google Scholar
- 16
- Cited by