Abstract
It is argued that Hellman's arguments purporting to demonstrate that constructive mathematics cannot cope with unbounded operators on a Hilbert space are seriously flawed, and that there is no evidence that his thesis is correct.
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References
Bishop, Errett,Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.
Bridges, Douglas, ‘Towards a Constructive Foundation for Quantum Mechanics’, in F. Richman, ed.,Constructive Mathematics (Springer Lecture Notes in Mathematics873), 260–273.
Bridges, Douglas and Richman, Fred,Varieties of Constructive Mathematics, London Math. Soc. Lecture Notes97, Cambridge Univ. Press, London, 1987.
Calude, C.,Theories of Computational Complexity, North-Holland, Amsterdam, 1988.
Cooke, R., Keane, M. and Moran, W., ‘An Elementary Proof of Gleason's Theorem’, inThe Structure and Interpretation of Quantum Mechanics (R. G. Hughes, ed.), Harvard University Press, Cambridge, Mass., 1989.
da Costa, N. C. A. and Doria, F. A., ‘Undecidability and Incompleteness in Classical Mechanics’,Int. J. Theoret. Physics 30 (1991), 1041–1073.
van Dalen, D.(ed.),Brouwer's Cambridge Lectures on Intuitionism, Cambridge Univ. Press, London, 1981.
Dummett, M. A. E.,Elements of Intuitionism, Oxford Univ. Press, Oxford, 1977.
Gleason, A. M., ‘Measures on the Closed Subspaces of a Hilbert Space’,J. Mathematics and Mechanics 6 (1957), 885–893.
Hellman, Geoffrey, ‘Gleason's Theorem is not Constructively Provable’,J. Phil. Logic 22 (1993), 193–203.
Hellman, Geoffrey, ‘Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem’,J. Phil. Logic 22 (1993), 221–248.
Ishihara, Hajime, ‘A Note on Constructive Unbounded Operators’, to appear.
Kushner, B. A.,Lectures on Constructive Mathematical Analysis, American Mathematical Society, Providence R.I., 1985.
Penrose, R.,The Emperor's New Mind, Oxford Univ. Press, Oxford, 1989.
Pour-El, M. B. and Richards, J. I.,Computability in Analysis and Physics, Perspectives in Mathematical Logic, Springer-Verlag, Berlin-Heidelberg-New York, 1990.
Richrnan, Fred, ‘The Frog Replies’ and ‘The Last Croak’,Math. Intelligencer 9(3) (1987), 22–24, 25–26.
Specker, E., ‘Nicht konstruktiv beweisbare Sätze der Analysis’,J. Symbolic Logic 14 (1949), 145–158.
Stewart, Ian, ‘Frog and Mouse Revisited: a Review of ...Constructive Analysis by Errett Bishop and Douglas Bridges (Springer: 1985) andAn Introduction to Non-standard Real Analysis by A. E. Hurd and P. A. Loeb (Academic Press: 1985)’,Math. Intelligencer 8(4) (1986), 78–82.
Stewart, Ian, ‘Is There a Mouse in the House?’ and ‘A Final Squeak’,Math. Intelligencer 9(3) (1987), 24–25, 26.
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Bridges, D.S. Constructive mathematics and unbounded operators — A reply to Hellman. J Philos Logic 24, 549–561 (1995). https://doi.org/10.1007/BF01052602
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DOI: https://doi.org/10.1007/BF01052602