Abstract
In constructive mathematics, a problem is counted as solved only if an explicit solution can, in principle at least, be produced. Thus, for example, “There is an x such that P(x)” means that, in principle at least, we can explicitly produce an x such that P(x). If the solution to the problem involves parameters, we must be able to present the solution explicitly by means of some algorithm or rule when given values of the parameters. That is, “for every x there is a y such that P(x, y) means that, we possess an explicit method of determining, for any given x, a y for which P(x, y). This leads us to examine what it means for a mathematical object to be explicitly given.
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Notes
- 1.
It will be observed that in defining a constructive real number in this way we are following Cantor ’s, rather than Dedekind ’s characterization.
- 2.
In fact the converse is equivalent to Markov’s Principle, which asserts that, if, for each n, x n = 0 or 1, and if it is contradictory that x n = 0 for all n, then there exists n for which x n = 1. This thesis is accepted by some, but not all schools of constructivism.
- 3.
In Chap. 10 we shall describe a model of the real line in which the decidability of equality can be refuted.
- 4.
The stability of equality on the reals in constructive and intuitionistic analysis blocks the possibility of defining infinitesimals as real numbers ε which are “indistinguishable from 0” in the sense that ¬(ε ≠ 0). In smooth infinitesimal analysis (see Chap. 10) infinitesimals can, and indeed are, defined in this way.
- 5.
From the facts that ¬(r # s) ⇒ r = s it follows easily that r ≠ s ⇔ ¬ ¬(r # s).
- 6.
But constructive proofs of this theorem are known.
- 7.
See Heyting (1956).
- 8.
As remarked in Chap. 5, this was a central tenet of intuitionism’s founder, Brouwer .
- 9.
E.g. in intuitionistic analysis (see below) and smooth infinitesimal analysis (see Chap. 10).
- 10.
See Appendix A.
- 11.
My account here is based on Bridges (1999).
- 12.
Here and in the sequel x ≠ y is an abbreviation for ¬(x = y).
- 13.
Here and in the remainder of this chapter “nonempty” has the stronger constructive meaning of being inhabited, to wit, that an element of the set in question can actually be constructed.
- 14.
Bridges (1999), pp. 103–5.
- 15.
Dummett (1977), p. 62.
- 16.
Weyl (1949), p. 52.
- 17.
Intuitionistic analysis, nevertheless, an extension of CA .
- 18.
This may be seen to be plausible if one considers that the according to Brouwer the construction of a choice sequence is incompletable; at any given moment we can know nothing about it outside the identities of a finite number of its entries. Brouwer’s Continuity Principle amounts to the assertion that every function from ℕℕ to ℕ is continuous.
- 19.
Bridges and Richman (1987), p. 109.
- 20.
- 21.
See Appendix A.
- 22.
Van Dalen (1997).
- 23.
See Appendix A.
- 24.
More exactly, for any real number a, the complement ℝ – {a} of {a} is cohesive .
- 25.
Ibid. There the classical continuum is described as the “frozen intuitionistic continuum ”.
- 26.
Vesley (1981).
- 27.
i.e. ≠ 0, not # 0.
- 28.
At the end of the paper the author asks whether the calculus can be treated fully along these lines, and whether such an approach has advantages. The question appears to be open.
Bibliography
Bishop, E., and D. Bridges. 1985. Constructive Analysis. Berlin: Springer.
Bridges, D. 1999. Constructive mathematics: A foundation for computable analysis. Theoretical Computer Science 219: 95–109.
Bridges, D., and F. Richman. 1987. Varieties of Constructive Mathematics. Cambridge: Cambridge University Press.
Dummett, M. 1977. Elements of Intuitionism. Oxford: Clarendon Press.
Heyting, A. 1956. Intuitionism: An Introduction. Amsterdam: North-Holland.
Van Dalen, D. 1997. How connected is the intuitionistic continuum? Journal of Symbolic Logic 62: 1147–1150.
Vesley, R. 1981. An intuitionistic infinitesimal calculus. In Lecture Notes in Mathematics 873, 208–212. Berlin: Springer.
Weyl, H. 1949. Philosophy of Mathematics and Natural Science. Princeton: Princeton University Press. (An expanded Engish version of Philosophie der Mathematik und Naturwissenschaft, Leibniz Verlag, 1927.).
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Bell, J.L. (2019). The Continuum in Constructive and Intuitionistic Mathematics. In: The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics. The Western Ontario Series in Philosophy of Science, vol 82. Springer, Cham. https://doi.org/10.1007/978-3-030-18707-1_9
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