Abstract
The situation in constructive mathematics in the nineties is so vastly different from that in the thirties, that it is worthwhile to pause a moment to survey the development in the intermediate years. In doing so, I follow the example of Heyting, who at certain intervals took stock of intuitionistic mathematics, which for a long time was the only variety of constructive mathematics. Heyting entered the foundational debate in 1930 at the occasion of the famous Königsberg meeting.
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References
Mi. Beeson, Foundations of Constructive Mathematics. Springer Verlag, Berlin.
D. Bridges, F. Richman, Varieties of Constructive Mathematics. Cambridge University Press, Cambridge.
E. Bishop, Foundations of Constructive Analysis. McGraw-Hill, New York.
E. Bishop, D. Bridges, Constructive Analysis. Springer Verlag, Berlin.
L.E.J. Brouwer, On the Foundations of Mathematics (Dutch) Diss. Maas en Van Suchtelen, Amsterdam. English transl. in [10].
L.E.J. Brouwer, Besitzt jede reelle Zahl eine Dezimalbruchentwickelung?Math. Annalen 83, 201–210.
L.E.J. Brouwer, Mathematik, Wissenschaft und Sprache. Monatshf Math.-Phys. 36, 153–164.
L.EJ. Brouwer, Volition, Knowledge, Speech. (Dutch) Euclides 9, 177–193.
L.E.J. Brouwer, Consciousness, Philosophy and Mathematics. In: E.W. Beth, H.J. Pos, H.J.A. Holak (eds.) Proc. 10th Intern. Congress Philosophy. North-Holland Publ. Co., Amsterdam, 1235–1249.
L.E.J. Brouwer, Collected Works. I (ed. A. Heyting) North-Holland Publ. Co., Amsterdam.
D. Bridges, F. Richman, Varieties of Constructive Mathematics. Cambridge University Press, Cambridge.
Th. Coquand and G. Huet, The Calculus of Constructions. Information and Computation 76, 95–120.
D. van Dalen, The use of Kripke’s schema as a reduction principle. Journ. Symb. Logic. 42, 238–240.
D. van Dalen, An interpretation of intuitionistic analysis. Ann. Math. Logic 13, 1–43.
D. van Dalen, The continuum and first-order intuitionistic logic. Journ. Symb. Logic 57, 1417–1424.
M.A.E. Dummett, Elements of Intuitionism Oxford University Press, Oxford.
M. Dummett, The Philosophical basis of Intuitionism. In `H.E. Rose, J.C. Shepherson (eds.) Logic Colloquum-1973 Bristol’. North-Holland, Amsterdam, 5–40.
A.S. Esenin-Volpin, Le programme ultra-intuitioniste des fondements des mathématiques. In: Infinitistic Methods,Warsaw, 201–223.
J.-Y. Girard, Y. Lafont, P. Taylor, Proofs and Types. Cambridge University Press, Cambridge.
G.F.C. Griss, Negationless Mathematics. NederlandseAk. van Wetenschappen. Verh. TweedeAfd. Nat. 53, 261–268.
L.A. Harrington, M.D. Morley, A. Scedrov, S.G. Simpson (eds.), Harvey Friedman’s Research on the Foundations of Mathematics. North-Holland Publ. Co., Amsterdam.
W.A. Howard, The formulae as types notion of construction. In: J.P.Seldin, J.R.Hindley (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda calculus and Formalism Academic Press, New York, 480–490.
A. Heyting, Mathematische Grundlagen forschung, Intuitionismus, Beweistheorie. Springer Verlag, Berlin.
A. Heyting, Blick von der intuitionistische Waite. Dialectica 12, 332–345.
A. Heyting, After thirty years. In [42], 194–197.
J.M.E. Hyland, The effective topos. In [45], 165–216.
J.M.E. Hyland, A small complete category. Ann. of Pure and Appl. Logic,40, 135–166.
S.C. Kleene, R. Vesley, The Foundations of Intuitionistic Mathematics. North-Holland Pub. Co., Amsterdam.
S.C. Kleene, Realizability: a retrospective survey. In [38], 95–112.
G. Kreisel, On the interpretation of non-finitist proofs II. Journ. Symb. Logic 17, 43–48.
G. Kreisel, A remark on free choice sequences and the topological completeness proofs. Journ. Symb. Logic 23, 369–388.
G. Kreisel, Foundations of intuitionistic logic. In [42], 198–210.
G. Kreisel, A. Maclntyre, Constructive Logic versus Algebraization I. In [45], 217–260.
B.A. Kushner, Lectures on constructive mathematical analysis. AMS, Providence.
J. Lambek, Ph. Scott, Introduction to Higher-Order Categorical Logic. Cambridge University Press, Cambridge.
H. Luckhardt, Herbrand-Analysenzweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken.Journ. Symb. Logic 54, 234–263.
P. Martin-Löf, Intuitionistic Type Theory. Bibliopolis, Napoli.
A. Mathias, H. Rogers (eds.), Cambridge Summer School in Mathematical Logic. Springer Verlag, Berlin.
D. McCarty, Realizability and Recursive Mathematics. Diss., Pittsburgh.
R. Mines, F, Richman, W. Ruitenburg,A Course in Constructive Algebra. Springer Verlag, Berlin.
J. Moschovakis, A topological interpretation of second-order intuitionistic arithmetic. Comp. Math., 26, 261–275.
E.Nagel, P. Suppes, A. Tarski (eds.), Logic, Methodology and Philosophy of Science: Proceedings of the 1960 International Congress. Stanford University Press, Stanford.
D. Prawitz, Meanings and Proofs: On the conflict between classical and intuitionistic logic. Theoria, 43, 2–40.
A.S. Troelstra, Notes on second-order arithmetic. In [38], 171–205.
A.S. Troelstra, D. van Dalen, The L.EJ. Brouwer Centenary Symposium. North-Holland Publ. Co., Amsterdam.
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van Dalen, D. (1995). Why Constructive Mathematics?. In: Depauli-Schimanovich, W., Köhler, E., Stadler, F. (eds) The Foundational Debate. Vienna Circle Institute Yearbook [1995], vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3327-4_11
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