Abstract
In the present paper, we will discuss some of the known reconstructions of Brouwer’s theorem of negative continuity and contend that the theory of the creative subject is the proper frame in which to understand Brouwer’s argument. We will also point out a sense in which negative continuity is a cogent consequence of the general intuitionistic tenets while positive continuity is not.
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Notes
- 1.
To derive 1.1 from NCP, as well as 1.3 from CP, one has to assume that any rational approximation f of a real function is extensional on number generators. Though this fact is often accepted as unproblematic (Veldman tacitly assumes nothing less than the extensionality of every function on choice sequences!), it is far from being evident. All that follows from the definition of real function is only that \(\alpha = \beta \rightarrow f(\alpha ) \approx f(\beta )\) where \(\approx \) is the relation “m touches n” between rational segments (coded by natural numbers) (see Veldman 1982, p. 14). However, replacing extensionality with the latter condition, the proof of NCP, mutatis mutandis, holds again. So the above assumption of extensionality is not needed.
- 2.
Perhaps it is worth noticing that a proof, at stage 0, that f is defined on \(\gamma \) does not necessarily provide, at the same stage 0, the value of \(f(\gamma )\) because this may depend on some future information (see Martino 1982, p. 316). Such a proof gives only a method for calculating, in a finite number of stages, \(f(\gamma )\). And this is enough for our purposes.
- 3.
Our reconstruction is similar to the second one of (Posy 1976, p. 113) but the latter is contaminated by misleading problems. In fact, Posy rightly recognises the possibility of getting an outright absurdity by opposing \(\lnot \vdash _{0} \exists n f(\gamma ) = n\) to \(\vdash _{0} \exists n f(\gamma ) = n\). Notwithstanding, he does not realise that \(\vdash _{0} \exists n f(\gamma ) = n\) straightforwardly follows from the a priori knowledge of the fullness of f. So he tries to justify it by conjecturing certain hypotheses of determinateness on f which Brouwer might have tacitly understood. By this way he goes off the rails and incurs Troelstra’s criticism (see Troelstra 1982, note 10).
- 4.
The crucial importance in Brouwer’s argument of the fullness hypothesis was not realised by Veldman. In fact, he “teases” Brouwer by remarking that, following the line of his reasoning, one could construct a real number at which the characteristic function \(c_{\mathbb {Q}}\) of the set of rational numbers is not defined (Veldman 1982, p. 13). But Brouwer could well reply: “according to my reasoning, one could construct it if \(c_{\mathbb {Q}}\) were full! This leads only to the right conclusion that \(c_{\mathbb {Q}}\) fails to be full”.
- 5.
Such a justification of 6.1, based on the mere time element, is also upheld by Dummett (1977, p. 349).
- 6.
In Troelstra (1983), Troelstra is aware that the code of the initial segment specified in advance occurs among the data of a lawless sequence (p. 212). As to individuality, he seems to disregard it mostly. He is forced, however, to take it into account when formulating the general form of the axiom of open data (Troelstra 1969, p. 36). In this context, he seems to believe that one can exploit individuality only by means of the relation of equality. Perhaps he intends to restrict himself to properties and functions for which this is the case.
References
Brouwer, L. (1927). Über Definitionsbereiche von Funktionen. Mathematische Annalen, 97, 60–75. English translation in From Frege to Gödel, Cambridge MA, 1967, pp. 446–463.
Dummett, M. (1977). Elements of intuitionism. Oxford: Clarendon Press.
Martino, E. (1982). Creative subject and bar theorem. In D. V. Dalen & A. Troelstra (Eds.), The L. E. J. Brouwer Centenary Symposium (pp. 311–318). North Holland: Amsterdam. Reprinted here as chapter 2.
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Troelstra, A. (1969). Principles of intuitionism. Berlin: Springer.
Troelstra, A. (1982). On the origin and development of Brouwer’s concept of choice sequence. In D.V. Dalen & A. Troelstra (Eds.), The L. E. J. Brouwer Centenary Symposium (pp. 465–486). North-Holland: Amsterdam.
Troelstra, A. (1983). Analysing choice sequences. Journal of Philosophical Logic, 12, 197–259.
Veldman, W. (1982). On the continuity of functions in intuitionistic real analysis. Technical report, Nijmegen.
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Martino, E. (2018). On the Brouwerian Concept of Negative Continuity. In: Intuitionistic Proof Versus Classical Truth. Logic, Epistemology, and the Unity of Science, vol 42. Springer, Cham. https://doi.org/10.1007/978-3-319-74357-8_5
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