Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T21:58:01.841Z Has data issue: false hasContentIssue false
Series:   Elements in the Philosophy of Mathematics

Innovation and Certainty

Published online by Cambridge University Press:  10 December 2020

Mark Wilson
Mark Wilson
Affiliation:
University of Pittsburgh

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.
Get access

Keywords

innovationcertaintyDedekindDe Morgansetting
Type
Element
Information
Series: Elements in the Philosophy of Mathematics
Online ISBN: 9781108592901
Publisher: Cambridge University Press
Print publication: 07 January 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Apelt, Ernst Friedrich (1854), Die Theorie der Induction, Wilhelm Engelmann, Leipzig.Google Scholar
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
Bell, E. T. (1937), Men of Mathematics, Simon and Schuster, New York.Google Scholar
Boole, George (1854), An Investigation of the Laws of Thought, Walton and Maberly, Cambridge.Google Scholar
Boole, George (2018), The Continued Exercise of Reason, MIT Press, Cambridge.Google Scholar
Bottazzini, Umberto (1986), The Higher Calculus, Springer-Verlag, Cham.Google Scholar
Bottazzini, Umberto, and Gray, Jeremy (2013), Hidden Harmony – Geometric Fantasies, Springer, Cham.Google Scholar
Burgess, John (2015), Rigor and Structure, Oxford University Press, Oxford.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
Cajori, Florian (1919), A History of Mathematics, MacMillan, New York.Google Scholar
Carus, Paul (1893), “Axioms,” Open Court Magazine 7 (31).Google Scholar
Cellucci, Carlo (2017), Rethinking Knowledge: The Heuristic View, Springer, Cham.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 (1872), “Continuity and Irrational Numbers,” in (Dedekind 1963).Google Scholar
Dedekind, Richard (1877), Theory of Algebraic Numbers, John Stillwell, trans., Cambridge University Press, Cambridge.Google Scholar
Dedekind, Richard (1888), “The Nature and Meaning of Numbers,” in (Dedekind 1963).Google Scholar
Dedekind, Richard (1888a), “Letter to Weber,” in (Ewald 1996).Google Scholar
Dedekind, Richard (1963), Essays on the Theory of Numbers, Dover, New York.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
Gardner, Martin (1956), Mathematics Magic and Mystery, Dover, New York.Google Scholar
Glanvill, Joseph (1661), The Vanity of Dogmatizing, Henry Eversden, London.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
Gray, Jeremy (2007), Worlds Out of Nothing, Springer-Verlag, London.Google Scholar
Hancock, Harris (1928), “The Analogy of the Theory of Kummer’s Ideal Numbers with Chemistry,” American Mathematical Monthly 35(6).Google Scholar
Hankel, Hermann (1875), Elemente der Projectivischen Geometrie, cited in (Merz 1912).Google Scholar
Hardy, G. H. (1956), Divergent Series, Oxford University Press, Oxford.Google Scholar
Hardy, G. H. (1967), A Mathematician’s Apology, Cambridge University Press, Cambridge.Google Scholar
Heaviside, Oliver (2003), Electromagnetic Theory, 3 vols., Chelsea, Providence.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
Hill, Thomas (1857), “The Imagination in Mathematics,” North American Review 85.Google Scholar
Holmes, Oliver Wendell (1897), “The Path of the Law,” Harvard Law Review 10(457).Google Scholar
Jacobi, Karl (1830), “Letter to Legendre,” cited in (Bottazzini 1986).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
Knopp, Konrad (1990), Theory and Application of Infinite Series, Dover, New York.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
Mach, Ernst (1895), Popular Scientific Lectures, Open Court, Chicago.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
Monna, A. F. (1975), Dirichlet’s Principle, Oostheok, Scheltema and Holkema, Utrecht.Google Scholar
Moritz, R. E. (1914), Memorabilia Mathematica, MacMillan, New York.Google Scholar
Mumford, David, and Tate, John (2015), “Alexander Grothendieck (1928–2014),” Nature 517.Google Scholar
Peacock, George (1830), A Treatise on Algebra, 2 vols., J. J. Deighton, Cambridge.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
Poincaré, Henri (1908), Science and Method, 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
Rayleigh, Lord (1894), Theory of Sound, MacMillan, London.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
Rowe, David E. (2018), A Richer Picture of Mathematics, Springer, Cham.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
Smith, Stephen B. (1973), The Great Mental Calculators, Columbia, New York.Google Scholar
Steiner, Jakob (1832), Systematische Entwicklung Der Abhängigkeit Geometrischer Gestalten Von Einander, F. Fincke, Berlin, cited in (Moritz 1914).Google Scholar
Sylvester, J. J. (1870), “A Plea for the Mathematician,” Nature 6(1).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
Venn, John (1894), Symbolic Logic, MacMillan and Co., London.Google Scholar
Weyl, Hermann (1939), “Invariants,” Duke Mathematical Journal, 5(3).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 (2006), Wandering Significance, Oxford University Press, Oxford.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

Save element to Kindle

To save this element to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Innovation and Certainty
  • Mark Wilson, University of Pittsburgh
  • Online ISBN: 9781108592901
Available formats
×

Save element to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Innovation and Certainty
  • Mark Wilson, University of Pittsburgh
  • Online ISBN: 9781108592901
Available formats
×

Save element to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Innovation and Certainty
  • Mark Wilson, University of Pittsburgh
  • Online ISBN: 9781108592901
Available formats
×