Never Say “Never”!

Geoffrey Hellman

doi:10.1017/9781108657419.013

12 - Never Say “Never”!

On the Communication Problem between Intuitionism and Classicism

from Part III - Logics of Mathematics

Published online by Cambridge University Press: 26 January 2021

Geoffrey Hellman

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Geoffrey Hellman: Affiliation:
University of Minnesota

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Summary

It is commonplace that opposing philosophical schools – whatever their subject – have difficulty communicating with one another, and, indeed frequently talk past one another rather than engage in rational debate. At times, this seems to have been Brouwer’s own view of the relation between intuitionistic and classical approaches to foundations of mathematics. In such circumstances, it is frequently obscure just wherein genuine disagreement – as opposed to merely apparent or verbal disagreement – actually resides, or even whether there really is at bottom any genuine disagreement at all.

Type: Chapter
Information: Mathematics and Its Logics
Philosophical Essays
, pp. 191 - 211

DOI: https://doi.org/10.1017/9781108657419.013 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2021

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References

Bishop, E. [1967] Foundations of Constructive Analysis (New York: McGraw Hill).Google Scholar

Brouwer, L. E. J. [1975] Collected Works, Volume I (Amsterdam: North Holland).Google Scholar

Dummet, M. [1977] Elements of Intuitionism (Oxford: Oxford University Press).Google Scholar

Gödel, K. [1932] “Eine Interpretation des lntuitionistichen Aussagenkalkuls,” Ergebnisse eines Mathematischen Kolloquiums 4: 39–40.Google Scholar

Goodman, N. [1970] “A theory of constructions equivalent to arithmetic,” in Kino, A., Myhill, J., and Vesley, R. E., eds., Intuitionism and Proof Theory (Amsterdam: North Holland), pp. 101–120.Google Scholar

Hellman, G. [1989] Mathematics without Numbers: Towards a Modal-Structural Interpretation (Oxford: Oxford University Press).Google Scholar

Heyting, A. [1966] Intuitionism: An Introduction (Amsterdam: North Holland).Google Scholar

Kreisel, G. [1962] “Foundations of intuitionistic logic,” in Nagel, E., Suppes, P., and Tarski, A., eds., Logic, Methodology, and Philosophy of Science (Stanford, CA: Stanford University Press), pp. 198–210.Google Scholar

Kripke, S. A. [1965] “Semantical analysis of intuitionistic logic I,” in Crossley, J. N. and Dummett, M. A. E., eds., Formal Systems and Recursive Functions (Amsterdam: North Holland), pp. 92–130.CrossRef Google Scholar

Lifschitz, V. [1985] “Calculable natural numbers,” in Shapiro, S., ed., Intensional Mathematics (Amsterdam: Elsevier), pp. 173–190.Google Scholar

Moore, G. H. [1982] Zermelo’s Axiom of Choice (New York: Springer).CrossRef Google Scholar

Putnam, H. [1975] “What is mathematical truth?,” reprinted in Mathematics, Matter, and Method: Philosophical Papers, Volume I (Cambridge: Cambridge University Press, 1979), pp. 60–78.Google Scholar

Shapiro, S. [1985] “Epistemic and intuitionistic arithmetic,” in Shapiro, S., ed., Intensional Mathematics (Amsterdam: Elsevier), pp. 11–46.CrossRef Google Scholar

Tait, W. W. [1983] “Against intuitionism: constructive mathematics is part of classical mathematics,” Journal of Philosophical Logic 12: 173–195.Google Scholar

Troelstra, S. A. [1969] Principles of Intuitionism (Berlin: Springer).CrossRef Google Scholar

Troelstra, S. A. [1977] Choice Sequences (Oxford: Oxford University Press).Google Scholar

Weinstein, S. [1977] “Some remarks on the philosophical foundations of intuitionistic mathematics,” delivered to the American Philosophical Association, Eastern Division, December 29.Google Scholar

Weinstein, S. [1983] “The intended interpretation of intuitionist logic,” Journal of Philosophical Logic 12: 261–270.CrossRef Google Scholar