The Epistemological Question of the Applicability of Mathematics
DOI:
https://doi.org/10.15173/jhap.v6i3.3435Abstract
The question of the applicability of mathematics is an epistemological issue that was explicitly raised by Kant, and which has played different roles in the works of neo-Kantian philosophers, before becoming an essential issue in early analytic philosophy. This paper will first distinguish three main issues that are related to the application of mathematics: (1) indispensability arguments that are aimed at justifying mathematics itself; (2) philosophical justifications of the successful application of mathematics to scientific theories; and (3) discussions on the application of real numbers to the measurement of physical magnitudes. A refinement of this tripartition is suggested and supported by a historical investigation of the differences between Kant’s position on the problem, several neo-Kantian perspectives (Helmholtz and Cassirer in particular, but also Otto Hölder), early analytic philosophy (Frege), and late 19th century mathematicians (Grassmann, Dedekind, Hankel, and Bettazzi). Finally, the debate on the cogency of an application constraint in the definition of real numbers is discussed in relation to a contemporary debate in neo-logicism (Hale, Wright and some criticism by Batitksy), in order to suggest a comparison not only with Frege’s original positions, but also with the ideas of several neo-Kantian scholars, including Hölder, Cassirer, and Helmholtz.References
Banks, Erik C., 2013. “Extension and Measurement: A Constructivist Program from Leibniz to Grassmann.” Studies in History and Philosophy of Science Part A 44: 20–31.
Batitsky, Vadim, 2002. “Some Measurement-Theoretic Concerns about Hale’s ‘Reals by Abstraction’.” Philosophia Mathematica 10: 286–303.
Bettazzi, Rodolfo, 1887. “Sul concetto di numero.” Periodico di matematica per l’insegnamento secondario 2: 97–113, 129–43.
———, 1890. Teoria delle grandezze. Pisa: Spoerri.
Biagioli, Francesca, 2016. Space, Number, and Geometry from Helmholtz to Cassirer. Cham: Springer.
Cantù, Paola, 2003. La matematica da scienza delle grandezze a teoria delle forme. L’Ausdehnungslehre di H. Grassmann. PhD thesis, Università di Genova.
Cassirer, Ernst, 1907. “Kant und die moderne Mathematik: Mit Bezug auf Bertrand Russells und Louis Couturats Werke über die Prinzipien der Mathematik.” Kant-Studien 12: 1–49.
———, 1910. Substanzbegriff und Funktionsbegriff. Berlin: Cassirer Verlag.
———, 1922. Das Erkenntnisproblem in der Philosophie und Wissenschaft der neuren Zeit, vol. 2. Berlin: Cassirer Verlag.
———, 1923a. Philosophie der symbolischen Formen, vol. 1. Berlin: Cassirer.
———, 1923b. Substance and Function, and Einstein’s Theory of Relativity. Chicago: Open Court.
Colyvan, Mark, 2001. The Indispensability of Mathematics. Oxford: Oxford University Press.
———, 2012. An Introduction to the Philosophy of Mathematics. Cambridge: Cambridge University Press.
Darrigol, Olivier, 2003. “Number and Measure: Hermann von Helmholtz at the Crossroads of Mathematics, Physics, and Psychology.” Studies in History and Philosophy of Science 34: 515–73.
Dedekind, Richard, 1872. Stetigkeit und irrationale Zahlen. Reprinted in Dedekind (1932), pp. 315–34.
———, 1888. Was sind und was sollen die Zahlen? Reprinted in Dedekind (1932), pp. 335–91.
———, 1932. Gesammelte mathematische Werke, vol. 3, edited by R. Fricke, E. Noether and Ø. Ore. Brunswick: Vieweg & Sohn.
Diez, José, 1997. “Hundred Years of Numbers: An Historical Introduction to Measurement Theory 1887–1990. Part I: The Formation Period. Two Lines of Research: Axiomatics and Real Morphisms, Scales and Invariance.” Studies in History and Philosophy of Science 28: 167–81.
Du Bois-Reymond, Paul, 1882. Die allgemeine Functionenlehre. Metaphysik und Theorie der mathematischen Grundbegriffe: Grösse, Grenze, Argument und Function. Tübingen: Laupp.
Ebert, Philip A., and Marcus Rossberg, eds., 2016. Abstractionism: Essays in Philosophy of Mathematics. New York: Oxford University Press.
Edgar, Scott, 2015a. “Intersubjectivity and Physical Laws in Post-Kantian Theory of Knowledge: Natorp and Cassirer.” In Friedman and Luft (2015), pp. 141–62.
———, 2015b. “The Physiology of the Sense Organs and Early Neo-Kantian Conceptions of Objectivity: Helmholtz, Lange, Liebmann.” In Objectivity in Science: New Perspectives from Science and Technology Studies, edited by F. Padovani, A. Richardson, and J. Y. Tsou, pp. 101–22. Dordrecht: Springer.
Ehrlich, Philip, 2006. “The Rise of non-Archimedean Mathematics and the Roots of a Misconception. I: The Emergence of non-Archimedean Systems of Magnitudes.” Archive for History of Exact Sciences 60: 1–121.
Ferreirós, José, 2006. “Riemann’s Habilitationsvortrag at the Crossroads of Mathematics, Physics, and Philosophy.” In Ferreirós and Gray (2006), pp. 67–96.
Ferreirós, José and Jeremy Gray, eds., 2006. The Architecture of Modern Mathematics. Essays in History and Philosophy. Oxford: Oxford University Press.
Flament, Dominique, 2005. “H. G. Grassmann et l’introduction d’une nouvelle discipline mathématique: l’Ausdehnungslehre.” Philosophia Scientiæ 5: 81–141.
Frege, Gottlob, 1903. Grundgesetze der Arithmetik, vol. 2. Jena: Pohle.
Friedman, J. Tyler and Sebastian Luft, eds., 2015. The Philosophy of Ernst Cassirer: A Novel Assessment. Berlin: De Gruyter.
Friedman, Michael, 1990. “Kant on Concepts and Intuitions in the Mathematical Sciences.” Synthese 84: 213–57.
Galilei, Galileo, 1623. Il Saggiatore nel quale con bilancia esquisita e giusta si ponderano le cose contenute nella libra astronomica e filosofica di Lotario Sarsi Sigensano scritto in forma di lettera all’illustrissimo e reverendissimo Monsignore D. Virginio Cesarini. Rome: Giacomo Mascardi. Translated into English as “The Assayer,” in The Controversy on the Comets of 1618, by S. Drake and C. D. O’Malley. Philadephia: University of Pennsylvania Press, 1960.
Grassmann, Hermann G., 1844. Die Wissenschaft der extensiven Grösse oder die Ausdehnungslehre, eine neue mathematische Disciplin dargestellt und durch Anwendungen erläutert, first part, Die lineale Ausdehnungslehre. Leipzig: Wigand. Reprinted in Grassmann (1894–1911), vol, I.1.
———, 1861. Lehrbuch der Arithmetik. Berlin: Enslin. Partially reprinted in Grassmann (1894–1911), vol. 2.1, pp. 295–349.
———, 1894–1911. Gesammelte mathematische und physikalische Werke, 3 vols, edited by F. Engel. Leipzig: Teubner.
Gray, Jeremy, 2007. Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century. Dordrecht: Springer.
Hale, Bob, 2000. “Reals by Abstraction.” Philosophia Mathematica 8: 100–23.
———, 2002. “Real Numbers, Quantities and Measurement.” Philosophia Mathematica 10: 304–23.
———, 2004. “Real Numbers and Set Theory–Extending the Neo-Fregean Programme Beyond Arithmetic.” Synthese 147: 21–41.
Hale, Bob and Crispin Wright, 2001. The Reason’s Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford: Clarendon Press.
Hankel, Hermann, 1867. Vorlesungen über die complexen Zahlen und ihre Functionen. Leipzig: Voss.
Heis, Jeremy, 2010. “ ‘Critical Philosophy Begins at the Very Point Where Logistic Leaves Off’: Cassirer’s Response to Frege and Russell.” Perspectives on Science 18: 383–408.
———, 2011. “Ernst Cassirer’s Neo-Kantian Philosophy of Geometry.” British Journal for the History of Philosophy 19: 759–94.
———, 2015. “Arithmetic and Number in the Philosophy of Symbolic Forms.” In Friedman and Luft (2015), pp. 123–40.
Helmholtz, Hermann von, 1887. “Zählen und Messen, erkenntnistheoretisch betrachtet.” In Philosophische Aufsätze, Eduard Zeller zu seinem fünfzigjährigen Doctorjubiläum gewidmet, pp. 11–52. Leipzig: Fues.
———, 1977. Epistemological Writings, translated by M. Lowe, edited by R. Cohen and Y. Elkana. Dordrecht: Reidel.
Hölder, Otto, 1901. “Die Axiome der Quantität und die Lehre vom Mass.” Berichte der mathematisch-physischen Classe der Königl. Sächs. Gesellschaft der Wissenschaften zu Leipzig 53: 1–64. Translated into English as “The Axioms of Quantity and the Theory of Measurement,” by J. Michell and C. Ernst. Journal of Mathematical Psychology 40 (1996): 235–52, 345–56.
Kant, Immanuel, 1900–. Kants gesammelte Schriften. Berlin: Reimer/De Gruyter.
———, 1998. Critique of Pure Reason, edited and translated by P. Guyer and A. Wood. Cambridge: Cambridge University Press.
Königsberger, Leo, 1903. Hermann von Helmholtz, vol. 2. Brunswick: Vieweg.
Lagrange, Joseph-Louis, 1797. Théorie des fonctions analytiques. Paris: Imprimerie de la République.
Lewis, Albert C., 2004. “The Unity of Logic, Pedagogy and Foundations in Grassmann’s Mathematical Work.” History and Philosophy of Logic 25: 15–36.
Linsky, Bernard and Edward N. Zalta, 2006. “What is Neologicism?” The Bulletin of Symbolic Logic 12: 60–99.
Michell, Joel, 1993. “The Origins of the Representational Theory of Measurement: Helmholtz, Hoelder and Russell.” Studies in History and Philosophy of Science 24: 185–206.
Mormann, Thomas, 2008. “Idealization in Cassirer’s Philosophy of Mathematics.” Philosophia Mathematica 16: 151–81.
———, 2018. “Zur mathematischen Wissenschaftsphilosophie des Marburger Neukantianismus.” In Philosophie und Wissenschaft bei Hermann Cohen, edited by C. Damböck. Dordrecht: Springer.
Otte, Michael, 1989. “The Ideas of Hermann Grassmann in the Context of the Mathematical and Philosophical Tradition since Leibniz.” Historia Mathematica 16: 1–35.
Panza, Marco, 2015. “From Lagrange to Frege: Functions and Expressions.” In Functions and Generality of Logic: Reflections on Dedekind’s and Frege’s Logicisms, pp. 59–95. Cham: Springer.
Panza, Marco and Andrea Sereni, 2013. Plato’s Problem: An Introduction to Mathematical Platonism. Basingstoke: Palgrave Macmillan.
Patton, Lydia, 2009. “Signs, Toy Models, and the A Priori: From Helmholtz to Wittgenstein.” Studies in History and Philosophy of Science Part A 40: 281–89.
Petsche, Hans-Joachim, Albert C. Lewis, Jörg Liesen, and Steve Russ, eds., 2011. From Past to Future: Graßmann’s Work in Context. Graßmann Bicentennial Conference, September 2009. Berlin and Heidelberg: Springer-Verlag.
Reck, Erich H., 2013. “Frege or Dedekind? Towards a Reevaluation of Their Legacies.” In The Historical Turn in Analytic Philosophy, edited by E. Reck, pp. 139–70. Basingstoke: Palgrave Macmillan.
Riemann, Bernhard, 1868. “Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (1854).” Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13: 133–42. Reprinted in Riemann (1876), pp. 254–69. Translated into English as Riemann (2016).
———, 1876. Bernhard Riemann’s Gesammelte mathematische Werke und Wissenschaftlicher Nachlass, edited by R. Dedekind and H. Weber. Leipzig: Tuebner.
———, 2016. On the Hypotheses Which Lie at the Bases of Geometry, translated by W. K. Clifford, edited by J. Jost. Cham: Springer.
Schubring, Gert, ed., 1996. Hermann Günther Grassmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar. Papers from a Sesquicentennial Conference. Dordrecht: Kluwer.
Shapiro, Stewart, 2000. “Frege Meets Dedekind: A Neologicist Treatment of Real Analysis.” Notre Dame Journal of Formal Logic 41: 335–64.
———, 2003. “Prolegomenon to Any Future Neo-Logicist Set Theory: Abstraction and Indefinite Extensibility.” The British Journal for the Philosophy of Science 54: 59–91.
Steiner, Mark, 1995. “The Applicabilities of Mathematics.” Philosophia Mathematica 3: 129–56.
Stolz, Otto, 1885–1886. Vorlesungen über allgemeine Arithmetik, 2 vols. Leipzig: Tuebner.
Suppes, Patrick, 1951. “A Set of Independent Axioms for Extensive Quantities.” Portugaliae Mathematica 10: 163–72.
Suppes, Patrick and Joseph L. Zinnes, 1963. “Basic Measurement Theory.” In Handbook of Mathematical Psychology, edited by R. D. Luce, R. R. Bush, and E. Galanter, pp. 1–76. New York: Wiley & Sons.
Sutherland, Daniel, 2006. “Kant on Arithmetic, Algebra, and the Theory of Proportions.” Journal of the History of Philosophy 44: 533–58.
Tappenden, Jamie, 2006. “The Riemannian Background to Frege’s Philosophy.” In Ferreirós and Gray (2006), pp. 97–132.
Torretti, Roberto, 1978. Philosophy of Geometry From Riemann to Poincaré. Dordrecht: Reidel.
Veronese, Giuseppe, 1891. I fondamenti della geometria. Padoa: Tipografia del Seminario.
Wright, Crispin, 1997. “On the Philosophical Significance of Frege’s Theorem.” In Language, Thought and Logic: Essays in Honour of Michael Dummett, edited by R. Heck, pp. 201–44. Oxford: Clarendon Press. Reprinted in Hale and Wright (2001), pp. 272–306.
———, 2000. “Neo-Fregean Foundations for Real Analysis: Some Reflections On Frege’s Constraint.” Notre Dame Journal of Formal Logic 41: 317–34.
Yap, Audrey, 2017. “Dedekind and Cassirer on Mathematical Concept Formation.” Philosophia Mathematica 25: 369–89.
Batitsky, Vadim, 2002. “Some Measurement-Theoretic Concerns about Hale’s ‘Reals by Abstraction’.” Philosophia Mathematica 10: 286–303.
Bettazzi, Rodolfo, 1887. “Sul concetto di numero.” Periodico di matematica per l’insegnamento secondario 2: 97–113, 129–43.
———, 1890. Teoria delle grandezze. Pisa: Spoerri.
Biagioli, Francesca, 2016. Space, Number, and Geometry from Helmholtz to Cassirer. Cham: Springer.
Cantù, Paola, 2003. La matematica da scienza delle grandezze a teoria delle forme. L’Ausdehnungslehre di H. Grassmann. PhD thesis, Università di Genova.
Cassirer, Ernst, 1907. “Kant und die moderne Mathematik: Mit Bezug auf Bertrand Russells und Louis Couturats Werke über die Prinzipien der Mathematik.” Kant-Studien 12: 1–49.
———, 1910. Substanzbegriff und Funktionsbegriff. Berlin: Cassirer Verlag.
———, 1922. Das Erkenntnisproblem in der Philosophie und Wissenschaft der neuren Zeit, vol. 2. Berlin: Cassirer Verlag.
———, 1923a. Philosophie der symbolischen Formen, vol. 1. Berlin: Cassirer.
———, 1923b. Substance and Function, and Einstein’s Theory of Relativity. Chicago: Open Court.
Colyvan, Mark, 2001. The Indispensability of Mathematics. Oxford: Oxford University Press.
———, 2012. An Introduction to the Philosophy of Mathematics. Cambridge: Cambridge University Press.
Darrigol, Olivier, 2003. “Number and Measure: Hermann von Helmholtz at the Crossroads of Mathematics, Physics, and Psychology.” Studies in History and Philosophy of Science 34: 515–73.
Dedekind, Richard, 1872. Stetigkeit und irrationale Zahlen. Reprinted in Dedekind (1932), pp. 315–34.
———, 1888. Was sind und was sollen die Zahlen? Reprinted in Dedekind (1932), pp. 335–91.
———, 1932. Gesammelte mathematische Werke, vol. 3, edited by R. Fricke, E. Noether and Ø. Ore. Brunswick: Vieweg & Sohn.
Diez, José, 1997. “Hundred Years of Numbers: An Historical Introduction to Measurement Theory 1887–1990. Part I: The Formation Period. Two Lines of Research: Axiomatics and Real Morphisms, Scales and Invariance.” Studies in History and Philosophy of Science 28: 167–81.
Du Bois-Reymond, Paul, 1882. Die allgemeine Functionenlehre. Metaphysik und Theorie der mathematischen Grundbegriffe: Grösse, Grenze, Argument und Function. Tübingen: Laupp.
Ebert, Philip A., and Marcus Rossberg, eds., 2016. Abstractionism: Essays in Philosophy of Mathematics. New York: Oxford University Press.
Edgar, Scott, 2015a. “Intersubjectivity and Physical Laws in Post-Kantian Theory of Knowledge: Natorp and Cassirer.” In Friedman and Luft (2015), pp. 141–62.
———, 2015b. “The Physiology of the Sense Organs and Early Neo-Kantian Conceptions of Objectivity: Helmholtz, Lange, Liebmann.” In Objectivity in Science: New Perspectives from Science and Technology Studies, edited by F. Padovani, A. Richardson, and J. Y. Tsou, pp. 101–22. Dordrecht: Springer.
Ehrlich, Philip, 2006. “The Rise of non-Archimedean Mathematics and the Roots of a Misconception. I: The Emergence of non-Archimedean Systems of Magnitudes.” Archive for History of Exact Sciences 60: 1–121.
Ferreirós, José, 2006. “Riemann’s Habilitationsvortrag at the Crossroads of Mathematics, Physics, and Philosophy.” In Ferreirós and Gray (2006), pp. 67–96.
Ferreirós, José and Jeremy Gray, eds., 2006. The Architecture of Modern Mathematics. Essays in History and Philosophy. Oxford: Oxford University Press.
Flament, Dominique, 2005. “H. G. Grassmann et l’introduction d’une nouvelle discipline mathématique: l’Ausdehnungslehre.” Philosophia Scientiæ 5: 81–141.
Frege, Gottlob, 1903. Grundgesetze der Arithmetik, vol. 2. Jena: Pohle.
Friedman, J. Tyler and Sebastian Luft, eds., 2015. The Philosophy of Ernst Cassirer: A Novel Assessment. Berlin: De Gruyter.
Friedman, Michael, 1990. “Kant on Concepts and Intuitions in the Mathematical Sciences.” Synthese 84: 213–57.
Galilei, Galileo, 1623. Il Saggiatore nel quale con bilancia esquisita e giusta si ponderano le cose contenute nella libra astronomica e filosofica di Lotario Sarsi Sigensano scritto in forma di lettera all’illustrissimo e reverendissimo Monsignore D. Virginio Cesarini. Rome: Giacomo Mascardi. Translated into English as “The Assayer,” in The Controversy on the Comets of 1618, by S. Drake and C. D. O’Malley. Philadephia: University of Pennsylvania Press, 1960.
Grassmann, Hermann G., 1844. Die Wissenschaft der extensiven Grösse oder die Ausdehnungslehre, eine neue mathematische Disciplin dargestellt und durch Anwendungen erläutert, first part, Die lineale Ausdehnungslehre. Leipzig: Wigand. Reprinted in Grassmann (1894–1911), vol, I.1.
———, 1861. Lehrbuch der Arithmetik. Berlin: Enslin. Partially reprinted in Grassmann (1894–1911), vol. 2.1, pp. 295–349.
———, 1894–1911. Gesammelte mathematische und physikalische Werke, 3 vols, edited by F. Engel. Leipzig: Teubner.
Gray, Jeremy, 2007. Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century. Dordrecht: Springer.
Hale, Bob, 2000. “Reals by Abstraction.” Philosophia Mathematica 8: 100–23.
———, 2002. “Real Numbers, Quantities and Measurement.” Philosophia Mathematica 10: 304–23.
———, 2004. “Real Numbers and Set Theory–Extending the Neo-Fregean Programme Beyond Arithmetic.” Synthese 147: 21–41.
Hale, Bob and Crispin Wright, 2001. The Reason’s Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford: Clarendon Press.
Hankel, Hermann, 1867. Vorlesungen über die complexen Zahlen und ihre Functionen. Leipzig: Voss.
Heis, Jeremy, 2010. “ ‘Critical Philosophy Begins at the Very Point Where Logistic Leaves Off’: Cassirer’s Response to Frege and Russell.” Perspectives on Science 18: 383–408.
———, 2011. “Ernst Cassirer’s Neo-Kantian Philosophy of Geometry.” British Journal for the History of Philosophy 19: 759–94.
———, 2015. “Arithmetic and Number in the Philosophy of Symbolic Forms.” In Friedman and Luft (2015), pp. 123–40.
Helmholtz, Hermann von, 1887. “Zählen und Messen, erkenntnistheoretisch betrachtet.” In Philosophische Aufsätze, Eduard Zeller zu seinem fünfzigjährigen Doctorjubiläum gewidmet, pp. 11–52. Leipzig: Fues.
———, 1977. Epistemological Writings, translated by M. Lowe, edited by R. Cohen and Y. Elkana. Dordrecht: Reidel.
Hölder, Otto, 1901. “Die Axiome der Quantität und die Lehre vom Mass.” Berichte der mathematisch-physischen Classe der Königl. Sächs. Gesellschaft der Wissenschaften zu Leipzig 53: 1–64. Translated into English as “The Axioms of Quantity and the Theory of Measurement,” by J. Michell and C. Ernst. Journal of Mathematical Psychology 40 (1996): 235–52, 345–56.
Kant, Immanuel, 1900–. Kants gesammelte Schriften. Berlin: Reimer/De Gruyter.
———, 1998. Critique of Pure Reason, edited and translated by P. Guyer and A. Wood. Cambridge: Cambridge University Press.
Königsberger, Leo, 1903. Hermann von Helmholtz, vol. 2. Brunswick: Vieweg.
Lagrange, Joseph-Louis, 1797. Théorie des fonctions analytiques. Paris: Imprimerie de la République.
Lewis, Albert C., 2004. “The Unity of Logic, Pedagogy and Foundations in Grassmann’s Mathematical Work.” History and Philosophy of Logic 25: 15–36.
Linsky, Bernard and Edward N. Zalta, 2006. “What is Neologicism?” The Bulletin of Symbolic Logic 12: 60–99.
Michell, Joel, 1993. “The Origins of the Representational Theory of Measurement: Helmholtz, Hoelder and Russell.” Studies in History and Philosophy of Science 24: 185–206.
Mormann, Thomas, 2008. “Idealization in Cassirer’s Philosophy of Mathematics.” Philosophia Mathematica 16: 151–81.
———, 2018. “Zur mathematischen Wissenschaftsphilosophie des Marburger Neukantianismus.” In Philosophie und Wissenschaft bei Hermann Cohen, edited by C. Damböck. Dordrecht: Springer.
Otte, Michael, 1989. “The Ideas of Hermann Grassmann in the Context of the Mathematical and Philosophical Tradition since Leibniz.” Historia Mathematica 16: 1–35.
Panza, Marco, 2015. “From Lagrange to Frege: Functions and Expressions.” In Functions and Generality of Logic: Reflections on Dedekind’s and Frege’s Logicisms, pp. 59–95. Cham: Springer.
Panza, Marco and Andrea Sereni, 2013. Plato’s Problem: An Introduction to Mathematical Platonism. Basingstoke: Palgrave Macmillan.
Patton, Lydia, 2009. “Signs, Toy Models, and the A Priori: From Helmholtz to Wittgenstein.” Studies in History and Philosophy of Science Part A 40: 281–89.
Petsche, Hans-Joachim, Albert C. Lewis, Jörg Liesen, and Steve Russ, eds., 2011. From Past to Future: Graßmann’s Work in Context. Graßmann Bicentennial Conference, September 2009. Berlin and Heidelberg: Springer-Verlag.
Reck, Erich H., 2013. “Frege or Dedekind? Towards a Reevaluation of Their Legacies.” In The Historical Turn in Analytic Philosophy, edited by E. Reck, pp. 139–70. Basingstoke: Palgrave Macmillan.
Riemann, Bernhard, 1868. “Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (1854).” Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13: 133–42. Reprinted in Riemann (1876), pp. 254–69. Translated into English as Riemann (2016).
———, 1876. Bernhard Riemann’s Gesammelte mathematische Werke und Wissenschaftlicher Nachlass, edited by R. Dedekind and H. Weber. Leipzig: Tuebner.
———, 2016. On the Hypotheses Which Lie at the Bases of Geometry, translated by W. K. Clifford, edited by J. Jost. Cham: Springer.
Schubring, Gert, ed., 1996. Hermann Günther Grassmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar. Papers from a Sesquicentennial Conference. Dordrecht: Kluwer.
Shapiro, Stewart, 2000. “Frege Meets Dedekind: A Neologicist Treatment of Real Analysis.” Notre Dame Journal of Formal Logic 41: 335–64.
———, 2003. “Prolegomenon to Any Future Neo-Logicist Set Theory: Abstraction and Indefinite Extensibility.” The British Journal for the Philosophy of Science 54: 59–91.
Steiner, Mark, 1995. “The Applicabilities of Mathematics.” Philosophia Mathematica 3: 129–56.
Stolz, Otto, 1885–1886. Vorlesungen über allgemeine Arithmetik, 2 vols. Leipzig: Tuebner.
Suppes, Patrick, 1951. “A Set of Independent Axioms for Extensive Quantities.” Portugaliae Mathematica 10: 163–72.
Suppes, Patrick and Joseph L. Zinnes, 1963. “Basic Measurement Theory.” In Handbook of Mathematical Psychology, edited by R. D. Luce, R. R. Bush, and E. Galanter, pp. 1–76. New York: Wiley & Sons.
Sutherland, Daniel, 2006. “Kant on Arithmetic, Algebra, and the Theory of Proportions.” Journal of the History of Philosophy 44: 533–58.
Tappenden, Jamie, 2006. “The Riemannian Background to Frege’s Philosophy.” In Ferreirós and Gray (2006), pp. 97–132.
Torretti, Roberto, 1978. Philosophy of Geometry From Riemann to Poincaré. Dordrecht: Reidel.
Veronese, Giuseppe, 1891. I fondamenti della geometria. Padoa: Tipografia del Seminario.
Wright, Crispin, 1997. “On the Philosophical Significance of Frege’s Theorem.” In Language, Thought and Logic: Essays in Honour of Michael Dummett, edited by R. Heck, pp. 201–44. Oxford: Clarendon Press. Reprinted in Hale and Wright (2001), pp. 272–306.
———, 2000. “Neo-Fregean Foundations for Real Analysis: Some Reflections On Frege’s Constraint.” Notre Dame Journal of Formal Logic 41: 317–34.
Yap, Audrey, 2017. “Dedekind and Cassirer on Mathematical Concept Formation.” Philosophia Mathematica 25: 369–89.
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