Skip to main content
SpringerLink
Log in

Mathematics as Problem Solving

  • Chapter
  • First Online:
Book cover Rethinking Knowledge

Part of the book series: European Studies in Philosophy of Science ((ESPS,volume 4))

  • 524 Accesses

Abstract

Contrary to the belief of several mathematicians that philosophy is irrelevant to mathematics, this chapter argues that philosophy is relevant to it, because it may expose the inadequacy of some basic mathematical concepts, it may provide an analysis of some basic mathematical concepts, and may help to formulate new rules of discovery. What is relevant to mathematics, however, is not classical philosophy of mathematics, which has basic limitations, but an alternative approach to the philosophy of mathematics according to which, like all other knowledge, mathematics is problem solving by the analytic method. In this perspective, the chapter argues that the aim of mathematics is not truth but plausibility, and that intuition has no role in mathematics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Barrow, John David. 2000. Between inner space and outer space: Essays on science, art, and philosophy. Oxford: Oxford University Press.

    Google Scholar 

  • Berkeley, George. 1948–1957. Works. London: Nelson.

    Google Scholar 

  • Bostock, David. 2009. Philosophies of mathematics: An introduction. Oxford: Wiley-Blackwell.

    Google Scholar 

  • Bourbaki, Nicholas. 1950. The architecture of mathematics. The American Mathematical Monthly 57 : 221–232.

    Article  Google Scholar 

  • Byers, William. 2007. How mathematicians think. Princeton: Princeton University Press.

    Google Scholar 

  • Church, Alonzo. 2011. Letter to Stephen Kleene, 29 November 1935. In Appendix B to Rod Adams, An early history of recursive functions and computability: From Gödel to Turing, 223–227. Boston: Docent Press.

    Google Scholar 

  • Cooper, William. 2001. The evolution of reason: Logic as a branch of biology. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Davis, Philip J., and Reuben Hersh. 1981. The mathematical experience. Boston: Birkhäuser.

    Google Scholar 

  • Dehaene, Stanislas. 2009. Origins of mathematical intuitions: The case of arithmetic. Annals of the New York Academy of Sciences 1156 : 232–259.

    Article  Google Scholar 

  • Devlin, Keith. 2005. The math instinct: Why you’re a mathematical genius (along with lobsters, birds, cats, and dogs). New York: Thunder’s Mouth Press.

    Google Scholar 

  • Dieudonné, Jean. 2013. Mathematics – the music of reason. Berlin: Springer.

    Google Scholar 

  • Feferman, Solomon. 1998. In the light of logic. Oxford: Oxford University Press.

    Google Scholar 

  • Frege, Gottlob. 1960. The foundations of arithmetic: A logico-mathematical enquiry into the concept of number. New York: Harper.

    Google Scholar 

  • ———. 1967. Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. In From Frege to Gödel: A source book in mathematical logic, 1879–1931, ed. Jean van Heijenoort, 5–82. Cambridge: Harvard University Press.

    Google Scholar 

  • George, Alexander, and Daniel J. Velleman. 2002. Philosophies of mathematics. Oxford: Blackwell.

    Google Scholar 

  • Gödel, Kurt. 1986–2002. Collected works. Oxford: Oxford University Press.

    Google Scholar 

  • Goodman, Nicolas D. 1991. Modernizing the philosophy of mathematics. Synthese 88 : 119–126.

    Article  Google Scholar 

  • Gowers, Timothy. 2006. Does mathematics need a philosophy? In 18 unconventional essays on the nature of mathematics, ed. Reuben Hersh, 182–200. New York: Springer.

    Chapter  Google Scholar 

  • Grabiner, Judith. 2005. The origins of Cauchy’s rigorous calculus. Mineola: Dover.

    Google Scholar 

  • Grosholz, Emily. 2007. Representation and productive ambiguity in mathematics and the sciences. Oxford: Oxford University Press.

    Google Scholar 

  • ———. 2015. Review of Carlo Cellucci, Rethinking logic: Logic in relation to mathematics, evolution and method. Philosophia Mathematica 23 : 136–140.

    Google Scholar 

  • Hart, Wilbur Dyre. 1996. Introduction. In The philosophy of mathematics, ed. Wilbur Dyre Hart, 1–13. Oxford: Oxford University Press.

    Google Scholar 

  • Hersh, Reuben. 1979. Some proposals for reviving the philosophy of mathematics. Advances in Mathematics 31 : 31–50.

    Article  Google Scholar 

  • ———. 2014. Experiencing mathematics: What do we do, when we do mathematics? Providence: American Mathematical Society.

    Google Scholar 

  • Kitcher, Philip. 1983. The nature of mathematical knowledge. Oxford: Oxford University.

    Google Scholar 

  • Kline, Morris. 1964. Mathematics in Western culture. Oxford: Oxford University Press.

    Google Scholar 

  • Knorr, Wilbur Richard. 1993. The ancient tradition of geometric problems. Mineola: Dover.

    Google Scholar 

  • Körner, Stephan. 1986. The philosophy of mathematics: An introductory essay. Mineola: Dover.

    Google Scholar 

  • Lehman, Hugh. 1979. Introduction to the philosophy of mathematics. Oxford: Blackwell.

    Google Scholar 

  • Locke, John. 1975. An essay concerning human understanding. Oxford: Oxford University Press.

    Google Scholar 

  • Ó Cairbre, Fiacre. 2009. The importance of being beautiful in mathematics. IMTA Newsletter 109: 29–45.

    Google Scholar 

  • Poincaré, Henri. 2013. The foundations of science: Science and hypothesis – The value of science – Science and method. Cambridge: Cambridge University Press.

    Google Scholar 

  • Rodin, Andrei. 2014. Axiomatic method and category theory. Berlin: Springer.

    Book  Google Scholar 

  • Spelke, Elizabeth. 2011. Natural number and natural geometry. In Space, time and number in the brain, ed. Stanislas Dehaene and Elizabeth Merrit Brannon, 287–317. London: Elsevier.

    Chapter  Google Scholar 

  • Thomas, Robert. 2014. Reflections on the objectivity of mathematics. In From a heuristic point of view, ed. Cesare Cozzo, and Emiliano Ippoliti, 241–256. Newcastle upon Tyne: Cambridge Scholars Publishing.

    Google Scholar 

  • Turing, Alan Mathison. 2001. Collected works: Mathematical logic. Amsterdam: North-Holland.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, Sapienza University of Rome, Rome, Italy

    Carlo Cellucci

Authors
  1. Carlo Cellucci
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Cellucci, C. (2017). Mathematics as Problem Solving. In: Rethinking Knowledge. European Studies in Philosophy of Science, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-53237-0_18

Download citation

Publish with us

Policies and ethics

Search

Navigation