Abstract
In the philosophy of mathematics, the realist vs. anti-realist debate continues today with differing positions on the status of mathematical objects. For realists, objects sit in “Plato’s heaven”, immovable, objective, eternal, and we contemplate them, whereas anti-realists (or Constructionists) are the opposite, and emphasize epistemology over ontology, saying that we construct mathematical objects. There are numerous results in mathematics which can be arrived at both from a realist and an anti-realist viewpoint. In other words, they can be contemplated (proved) via methods deemed unsuitable by anti-realists- or simply arrived at it through methods (or construction) as the anti-realist would say. In this chapter, we argue that realism and anti-realism can be seen as two sides of the same coin, or different ways of knowing the same thing, and therefore the so called dichotomy between these positions is reconcilable for particular mathematical objects.
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Notes
- 1.
We deliberately rule out formalism for the primary reason that in keeping with Heyting’s (1974) observation: “There is no conflict between intuitionism and formalism when each keeps to its own subject, intuitionism to mental constructions, formalism to the construction of a formal system, motivated by its internal beauty or by its utility for science and industry. They clash when formalists contend that their systems express mathematical thought. Intuitionists make two objections against this contention. In the first place, …[m]ental constructions cannot be rendered exactly by means of language; secondly the usual interpretation of the formal system is untenable as a mental construction.” (p. 89).
- 2.
In this chapter we use the terms Realism and Constructionism for these two positions.
- 3.
The proposal originally made by Georg Cantor that there is no infinite set with a cardinal number between that of the infinite set of integers x 0 and the infinite set of real numbers (the “continuum”).
- 4.
See for instance a classic and constructive proofs for the fundamental theorem of Algebra.
- 5.
A rigorous proof ζ(−1) = −1/12 can be found in: Stopple, J. (2003). A primer of analytic number theory: from Pythagoras to Riemann. Cambridge University Press.
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Sriraman, B., Haavold, P. (2018). Reconciling the Realist/Anti Realist Dichotomy in the Philosophy of Mathematics. In: Wuppuluri, S., Doria, F. (eds) The Map and the Territory. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-319-72478-2_20
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