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University of Birmingham Brouwer Versus Wittgenstein on the Infinite University of Birmingham Brouwer versus Wittgenstein on the Infinite and the Law of Excluded Middle Rumfitt, Ian DOI: 10.1163/9789401211949_008 License: None: All rights reserved Document Version Peer reviewed version Citation for published version (Harvard): Rumfitt, I 2014, 'Brouwer versus Wittgenstein on the Infinite and the Law of Excluded Middle', Grazer Philosophische Studien, vol. 89, no. 1, pp. 93-108. https://doi.org/10.1163/9789401211949_008 Link to publication on Research at Birmingham portal Publisher Rights Statement: Final published version available at: http://dx.doi.org/10.1163/9789401211949_008 Checked October 2015 General rights Unless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or the copyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposes permitted by law. •Users may freely distribute the URL that is used to identify this publication. •Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of private study or non-commercial research. •User may use extracts from the document in line with the concept of ‘fair dealing’ under the Copyright, Designs and Patents Act 1988 (?) •Users may not further distribute the material nor use it for the purposes of commercial gain. Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive. If you believe that this is the case for this document, please contact [email protected] providing details and we will remove access to the work immediately and investigate. Download date: 25. Sep. 2021 BROUWER VERSUS WITTGENSTEIN ON THE INFINITE AND THE LAW OF EXCLUDED MIDDLE* Ian Rumfitt In Grazer Philosophische Studien 89 (2014): 93-108 Abstract: Wittgenstein and Brouwer were agreed that some of the higher mathematics of their day rested upon a projection into the infinite of methods that legitimately apply only within finite domains. In this paper I compare and assess the different treatments the two philosophers give of problematic cases involving infinity. For Brouwer, certain claims about infinite sequences provide exceptions to the law of excluded middle; while Wittgenstein argues that the same claims are without sense, since for him the law of excluded middle is a criterion of being a proposition. I end the paper by outlining how the intuitionist might respond to Wittgenstein’s arguments. According to Herbert Feigl, who was with him on the day, Wittgenstein was provoked into returning to philosophy by hearing L.E.J. Brouwer’s lecture, ‘Mathematik, Wissenschaft und Sprache’, in Vienna on 10 March 1928 (see the quotation from Feigl in Pitcher 1964, 8n). While Wittgenstein’s later writings reject several central Brouwerian theses, a comparison between these thinkers is instructive. As I hope this paper will show, Wittgenstein accepts one of Brouwer’s key negative contentions— namely, that some of the higher mathematics of their day rests upon an illegitimate projection into the infinite of methods that properly apply only within finite domains. While they differ over the remedy, agreement on that negative point and Wittgenstein’s close engagement with Brouwer’s positive theory * I am much indebted to Lucy Baines, Hanjo Glock, Joachim Schulte, and Göran Sundholm for their comments on drafts of this paper. 1 belie the widespread view—inspired by a notorious obiter dictum in the transcript of a 1939 lecture— that, for Wittgenstein, ‘Intuitionism is all bosh—entirely’ (LFM, 237).\1/ 1. The intuitionists on infinity Nowadays, under the influence of the late Sir Michael Dummett, we are apt to associate the intuitionist critique of classical mathematics and logic with the adoption of verificationist semantic theories, in which the meaning of a declarative sentence (henceforth, a statement) is given by specifying the conditions in which a speaker would be entitled to assert it, rather than by specifying the conditions under which it would be true. It is important to set these associations aside in reading the early intuitionists, for the founding fathers of the school were not verificationists. In a paper of 1923, Brouwer wrote that ‘a complete empirical corroboration of the inferences drawn [about the “world of perception”] is usually materially excluded a priori and there cannot be any question of even a partial corroboration in the case of (juridical and other) inferences about the past’ (Brouwer 1923, 336). A verificationist would conclude from that claim that talk about the past is meaningless; Brouwer, though, expressly holds that it is meaningful. Indeed, he allows that the laws of classical logic, including Excluded Middle, may validly be applied in reasoning about the world of perception, so long as we are able to think of the ‘objects and mechanisms of [that] world…as (possibly partly unknown) finite discrete systems’ (ibid., emphasis in the original). More exactly, it is the possibility of projecting ‘a finite discrete system upon the objects in question’ that is the ‘condition of the applicability’ of Excluded Middle to judgements concerning those objects. We see here a fundamental difference between Brouwer and Dummett. For Dummett, the basic mistake of the classical mathematicians is that they apply a realist or truth-conditional semantic theory to the language of mathematics. For Brouwer, by contrast, their error was to apply distinctively classical logical rules ‘even in the mathematics of infinite systems’, where the rules’ condition of applicability does not obtain. A.N. Kolmogorov, another pioneer of intuitionism, agreed with Brouwer. He understood Brouwer’s 1 The account of intuitionism that directly precedes this dictum in the lecture notes (which were taken down by some students) is in any case eccentric. 2 writings to have ‘revealed that it is illegitimate to use the principle of excluded middle in the domain of transfinite argument’ (Kolmogorov 1925, 416). As Brouwer’s reference to ‘infinite systems’ implies, the early intuitionists did not impugn as unintelligible expressions, such as ‘the sequence of natural numbers’, that purport to designate infinite mathematical structures. They did, however, claim that talk about such structures, if it makes sense at all, is disguised talk about the mathematical principles that characterize them. Thus, to say that the natural number sequence has a property is to say that the property in question is entailed by the laws of Heyting Arithmetic, these laws (the intuitionistic analogue of the Peano Postulates) being the principles that characterize that structure. This marks a fundamental contrast with the finite case. A finite structure might be characterized by certain mathematical principles but, even when it is so characterized, it may still have properties that are not entailed by the principles. As one might put it, in the finite case the extension of certain mathematical principles will have mathematical properties over and above those consequent on the principles themselves. According to the intuitionist, this is conceptually impossible in the infinite case. A finite initial segment of an infinite sequence may have properties over and above those entailed by the principles that generate the sequence. But if we speak of the infinite sequence as a whole, we must be referring (perhaps elliptically) to the generating principles themselves. For the intuitionist, one might say, infinite structures cannot be conceived purely extensionally. So to conceive them is illegitimately to project into the infinite a notion that only makes sense in the finite case. Wittgenstein understood and heeded Brouwer’s warning not to treat infinite collections as though they were large finite ones. In §19 of Part V of the Remarks on the Foundations of Mathematics (RFM), which its editors date to between 1942 and 1944, he asks: Isn’t it like this? The concepts of infinite decimals in mathematical propositions are not concepts of series, but of the unlimited technique of expansion of series. We learn an endless technique: that is to say, something is done for us first, and then we do it; we are told rules and we do exercises in following them; perhaps some expression like ‘and so on ad inf.’ is also used, but what is in question is not some gigantic extension (278-9). A little later, in §36, he says: 3 Our difficulty really already begins with the infinite straight line; although we learn even as children that a straight line has no end, and I do not know that this idea has ever given anyone any difficulty…But the straight line is a law for producing further (290). Remarks such as these—which are typical of Part V of RFM—nicely express Brouwer’s basic objection to the conception of the infinite that prevailed in his day and still prevails in ours. While Wittgenstein and Brouwer differ over the best prophylactic against this popular misconception, they are at one in perceiving a deep problem in the standard view of the infinite, and as such they are allies against the majority of mathematicians. 2. Brouwer against the Law of Excluded Middle According to classical logic, we are entitled to assert A A no matter what meaningful statement A might be. Brouwer argues, though, that there are meaningful mathematical statements A for which an assertion of A A conflicts with a correct view of the infinite. Accordingly, a correct view of the infinite forces us to revise classical logic. In particular, it forces us to restrict the Law of Excluded Middle.\2/ Since this revisionist claim is one that Wittgenstein rejects, it will be worth setting out Brouwer’s grounds for it carefully.
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