L. E. J. Brouwer’s ‘Unreliability of the logical principles’: a new translation, with an introduction. (English) Zbl 1372.01108
This article takes on the task of translating Brouwer’s “The unreliability of the logical principles” (De onbetrouwbaarheid der logische principes, 1908), from Dutch to English as literally as possible, in order to preserve the essentials of Brouwer’s original manuscript, which can sometimes get lost in other translations. An analysis of Brouwer’s key elements are presented, followed by both the Dutch and English versions of Brouwer’s paper.
Brouwer’s signature contention with the place of language inside mathematics is discussed; that is, mathematical ideas that emerge out of logic cannot be seen as “reliable” since logic is a part of language. Since language exists within the world of time (of “repetition in time of interidentifiable succession-sequences of qualitative differentiation through time”, p. 15), these linguistic mathematical results cannot be “reliable” either (“Least of all can a mathematical system of posits, separated from the observations it made intelligible, when continued indefinitely, remain reliable when directing along those observations”, p. 15).
The authors make a step-by-step dissection of Brouwer’s work, delineating certain elements in order to make Brouwer’s intention clear. For example, the section entitled “Unreliability in the natural sciences and wisdom” disparages logic’s use in the natural sciences because it is an unreliable tool of induction used to get from premise to conclusion; the use of logic in the realm of wisdom (“religious truth”), however, is confronted with a different dilemma, “as the appearance of time is no longer accepted [in wisdom], even less thus the reliability of logic” (p. 17).
The discussion of the unreliability of the law of the excluded middle shines brightly. In his “Unreliability”, Brouwer corrects a position held in his dissertation [JFM 38.0081.04] about the law of the excluded middle, specifying that \(\neg A\) means that from the hypothesis of a demonstration of \(A\) one can construct “the blockage by impossibility” (p. 19). It is only through this correction that the principle of the excluded middle becomes unreliable.
Aspects of truth and meaning, held by Brouwer, are discussed in addition to Brouwer’s correspondences and the direct influences of this paper, in particular on Griss’ negationless intuitionistic mathematics.
For a French translation by the authors et al. see [Zbl 1333.01018].
Brouwer’s signature contention with the place of language inside mathematics is discussed; that is, mathematical ideas that emerge out of logic cannot be seen as “reliable” since logic is a part of language. Since language exists within the world of time (of “repetition in time of interidentifiable succession-sequences of qualitative differentiation through time”, p. 15), these linguistic mathematical results cannot be “reliable” either (“Least of all can a mathematical system of posits, separated from the observations it made intelligible, when continued indefinitely, remain reliable when directing along those observations”, p. 15).
The authors make a step-by-step dissection of Brouwer’s work, delineating certain elements in order to make Brouwer’s intention clear. For example, the section entitled “Unreliability in the natural sciences and wisdom” disparages logic’s use in the natural sciences because it is an unreliable tool of induction used to get from premise to conclusion; the use of logic in the realm of wisdom (“religious truth”), however, is confronted with a different dilemma, “as the appearance of time is no longer accepted [in wisdom], even less thus the reliability of logic” (p. 17).
The discussion of the unreliability of the law of the excluded middle shines brightly. In his “Unreliability”, Brouwer corrects a position held in his dissertation [JFM 38.0081.04] about the law of the excluded middle, specifying that \(\neg A\) means that from the hypothesis of a demonstration of \(A\) one can construct “the blockage by impossibility” (p. 19). It is only through this correction that the principle of the excluded middle becomes unreliable.
Aspects of truth and meaning, held by Brouwer, are discussed in addition to Brouwer’s correspondences and the direct influences of this paper, in particular on Griss’ negationless intuitionistic mathematics.
For a French translation by the authors et al. see [Zbl 1333.01018].
Reviewer: Celia M. Schacht (Glendale)
MSC:
| 01A75 | Collected or selected works; reprintings or translations of classics |
| 03-03 | History of mathematical logic and foundations |
| 03A05 | Philosophical and critical aspects of logic and foundations |
| 01A60 | History of mathematics in the 20th century |
| 01A70 | Biographies, obituaries, personalia, bibliographies |
Keywords:
intuitionismBiographic References:
Brouwer, Luitzen Egbertus JanReferences:
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