Book contents
- Mathematics and Its Logics
- Mathematics and Its Logics
- Copyright page
- Contents
- Acknowledgements
- Introduction
- Part I Structuralism, Extendability, and Nominalism
- Part II Predicative Mathematics and Beyond
- Part III Logics of Mathematics
- 11 Logical Truth by Linguistic Convention
- 12 Never Say “Never”!
- 13 Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem
- 14 If “If-Then” Then What?
- 15 Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis
- Index
- References
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
Book contents
- Mathematics and Its Logics
- Mathematics and Its Logics
- Copyright page
- Contents
- Acknowledgements
- Introduction
- Part I Structuralism, Extendability, and Nominalism
- Part II Predicative Mathematics and Beyond
- Part III Logics of Mathematics
- 11 Logical Truth by Linguistic Convention
- 12 Never Say “Never”!
- 13 Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem
- 14 If “If-Then” Then What?
- 15 Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis
- Index
- References
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 LogicsPhilosophical Essays, pp. 191 - 211Publisher: Cambridge University PressPrint publication year: 2021
References
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
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.CrossRefGoogle Scholar
Lifschitz, V. [1985] “Calculable natural numbers,” in Shapiro, S., ed., Intensional Mathematics (Amsterdam: Elsevier), pp. 173–190.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.CrossRefGoogle Scholar
Tait, W. W. [1983] “Against intuitionism: constructive mathematics is part of classical mathematics,” Journal of Philosophical Logic 12: 173–195.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.CrossRefGoogle Scholar