Introduction
Published online by Cambridge University Press: 26 January 2021
Summary
Abstract mathematics, from its earliest times in ancient Greece right up to the present, has always presented a major challenge for philosophical understanding. On the one hand, mathematics is widely considered a paradigm of providing genuine knowledge, achieving a degree of certainty and security as great as or greater than knowledge in any other domain. A part of this, no doubt, is that it proceeds by means of deductive proofs, thereby inheriting the security of necessary truth preservation of deductive logical inference. But proofs have to start somewhere: ultimately there need to be axioms, and these are the starting points, not end points, of logical inference. But what then grounds or justifies axioms?
- Type
- Chapter
- Information
- Mathematics and Its LogicsPhilosophical Essays, pp. 1 - 16Publisher: Cambridge University PressPrint publication year: 2021
References
Boolos, G. [1971] “The iterative conception of set,” reprinted in Logic, Logic, and Logic, Jeffrey, R., ed. (Cambridge, MA: Harvard University Press, 1998), pp. 13–29.Google Scholar
Boolos, G. [1985] “Nominalist Platonism,” Philosophical Review 94: 327–344, reprinted in Logic, Logic, and Logic, Jeffrey, R., ed. (Cambridge, MA: Harvard University Press, 1998), pp. 73–87.Google Scholar
Boolos, G. [1998] “Must we believe in set theory?,” reprinted in Logic, Logic, and Logic, Jeffrey, R., ed. (Cambridge, MA: Harvard University Press, 1998), pp. 120–132.Google Scholar
Burgess, J. and Rosen, G. [1997] A Subject with No Object: Strategies for Nominalist Interpretation of Mathematics (Oxford: Oxford University Press).Google Scholar
Burgess, J., Hazen, A. and Lewis, D. [1991] “Appendix,” in Lewis, D., Parts of Classes (Oxford: Blackwell), pp. 121–149.Google Scholar
Feferman, S. [1977] “Categorical foundations and foundations of category theory,” in Butts, R. E. and Hintikka, J., eds., Logic, Foundations of Mathematics and Computability Theory (Dordrecht: Reidel), pp. 149–169.CrossRefGoogle Scholar
Gödel, K. [1947] “What is Cantor’s continuum problem?,” reprinted in Benacerraf, P. and Putnam, H., eds., Philosophy of Mathematics: Selected Readings, 2nd edn. (Cambridge: Cambridge University Press,1983), pp. 470–485.Google Scholar
Hellman, G. [1989] Mathematics without Numbers: Towards a Modal-Structural Interpretation (Oxford: Oxford University Press).Google Scholar
Hellman, G. [1996] “Structuralism without structures,” Philosophia Mathematica 4(2): 100–123.CrossRefGoogle Scholar
Hellman, G. [2003] “Does category theory provide a framework for mathematical structuralism?” Philosophia Mathematica 11: 129–157.Google Scholar
Hellman, G. [2005] “Structuralism,” in Shapiro, S., ed., Oxford Handbook of Philosophy of Mathematics and Logic (Oxford: Oxford University Press), pp. 536–562.CrossRefGoogle Scholar
Hellman, G. and Shapiro, S. [2018] Varieties of Continua: From Regions to Points and Back (Oxford: Oxford University Press).Google Scholar
Hellman, G. and Shapiro, S. [2019] Mathematical Structuralism (Cambridge: Cambridge University Press).Google Scholar
Hempel, C. G. [1945] “On the nature of mathematical truth,” reprinted in Benacerraf, P. and Putnam, H., eds., Philosophy of Mathematics: Selected Readings, 2nd edn. (Cambridge: Cambridge University Press, 1983), pp. 377–393.Google Scholar
Isaacson, D. [1987] “Arithmetical truth and hidden higher-order concepts,” reprinted in Hart, W. D., ed., Oxford Readings in Philosophy of Mathematics (Oxford: Oxford University Press, 1996), pp. 203–224.Google Scholar
Maddy, P. [2011] Defending the Axioms: On the Philosophical Foundations of Set Theory (Oxford: Oxford University Press).Google Scholar
Maddy, P. [to appear] “Enhanced if-thenism,” in A Plea for Natural Philosophy and Other Essays (Oxford: Oxford University Press).Google Scholar
Putnam, H. [1967] “Mathematics without foundations,” reprinted in Benacerraf, P. and Putnam, H., eds. Philosophy of Mathematics: Selected Readings, 2nd edn. (Cambridge: Cambridge University Press, 1983), pp. 295–311.Google Scholar
Quine, W.V. [1936] “Truth by convention,” reprinted in Benacerraf, P. and Putnam, H., eds., Philosophy of Mathematics: Selected Readings, 2nd edn. (Cambridge: Cambridge University Press, 1983), pp. 329–354.Google Scholar
Zermelo, E. [1930] “Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre,” Fundamenta Mathematicae, 16: 29–47; translated as “On boundary numbers and domains of sets: new investigations in the foundations of set theory,” in Ewald, W., ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume 2 (Oxford: Oxford University Press, 1996), pp. 1219–1233.Google Scholar