Mathematics as Intuition

  • Ole Ravn
  • Ole Skovsmose
Part of the History of Mathematics Education book series (HME)


This chapter explores Brouwer’s conception of mathematics. Brouwer is the principal proponent of the direction in the philosophy of mathematics referred to as intuitionism. Brouwer is also concerned about the paradoxes that troubled Frege, Russell, Whitehead and Hilbert. However, in order to eliminate the paradoxes, he suggests a radically different remedy. A basic idea of intuitionism is that the very language we use to communicate mathematics gives rise to mathematical problems and paradoxes. This also applies to formalised languages. Brouwer finds that mathematical formalisms are only imprecise and limited representations of mathematics. When we, for instance, denote the set of natural numbers as , we provide a linguistic construction within the recorded mathematics, which does not have anything to do with intuitive mathematical constructions. Brouwer’s thesis is that the linguistic representations of mathematics have departed from the intuitive mathematical constructions. Linguistic expressions bring us to believe that we express ourselves about mathematical entities, while we actually become seduced by language. According to Brouwer, linguistic formulations are the source of paradoxes and inconsistencies.

According to Brouwer, mathematics develops through a particular mathematical intuition shared by all human beings. This way, Brouwer supports the thesis that new mathematics is created. Brouwer’s constructivism opens up the possibility of interpreting mathematics as a process. The chapter summarises Heyting’s presentation of mathematical processes through an invented dialogue between an intuitionist, a formalist, a classic and other characters in mathematics.


Anti-realism Constructivism Infinity Intuitionism Principle of the excluded third 


  1. Brouwer, L. E. J. (1913). Intuitionism and formalism. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics (pp. 77–89). Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  2. Brouwer, L. E. J. (1981). In D. van Dalen (Ed.), Brouwer’s Cambridge lectures on intuitionism. Cambridge, UK: Cambridge University Press.Google Scholar
  3. Heyting, A. (1971). Intuitionism: An introduction. Amsterdam, The Netherlands: North Holland Publishing Company.Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ole Ravn
    • 1
  • Ole Skovsmose
    • 1
    • 2
  1. 1.Department of Learning and PhilosophyAalborg UniversityAalborgDenmark
  2. 2.Department of Mathematics EducationState University of São Paulo, (Universidade Estadual Paulista, Unesp)São PauloBrazil

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