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Nonsense Version of This Geometry, Independently Derived Nonsense version of this geometry, independently derived. Carl SEE ALSO CONTINUUM;EXTENSION;INFINITY;MATHEMATICS, Friedrich Gauss (1777–1855), often called the “princeps PHILOSOPHY OF;MEASUREMENT. mathematicorum” (prince of mathematicians), simulta- neously developed this geometry, calling it “non- BIBLIOGRAPHY Coffa, Alberto. “From Geometry to Tolerance: Sources of Euclidean,” but he kept his results private, fearing that Conventionalism in Nineteenth-century Geometry.” In From the learned public was unprepared for this development. Quarks to Quasars: Philosophical Problems of Modern Physics, This type of geometry is now called hyperbolic geometry. edited by Robert Colodny, 3–70. Pittsburgh, PA: University In 1766, Johann Heinrich Lambert (1728–1777) of Pittsburgh Press, 1986. showed that hyperbolic geometry could be envisioned as Greenberg, Marvin Jay. Euclidean and Non-Euclidean the geometry that is valid on a sphere of imaginary Geometries: Development and History. 4th ed. New York: radius. But being imaginary, this reasoning did not Freeman, 2008. convince Lambert of the consistency of hyperbolic Stillwell, John. Sources of Hyperbolic Geometry. Providence, RI: geometry. Lambert also recognized that the great circles American Mathematical Society, 1996. Includes a critical on a sphere behave like lines without parallels, since introduction and English translations of key articles by Bel- trami, Felix Klein, and Henri Poincaré. every two “great circles” (paths on the sphere dividing it into two equal hemispheres) meet in two points. This idea gave rise to another type of non-Euclidean geometry Andrew Arana Associate Professor of Philosophy and Mathematics called elliptic geometry, which was investigated more University of Illinois at Urbana-Champaign (2013) systematically by Bernhard Riemann (1826–1866). Ri- emann studied geometries on curved surfaces, including spheres and ellipsoids, and on curved spaces more generally. Riemann’s work was instrumental in convincing NONSENSE mathematicians of the value of non-Euclidean geometries. However, the question of their legitimacy Nonsense, in general, means simply “lacking sense.” As remained. One of the chief nineteenth-century ac- a term of philosophical criticism, it is often associated complishments in this arena was to show that if there with the work of the Austrian philosopher Ludwig WIT- were a contradiction in hyperbolic geometry, then there TGENSTEIN (1889–1951). For Wittgenstein, both in his would be a contradiction in Euclidean geometry as well. early work (principally the Tractatus Logico-Philosophicus It follows that if Euclidean geometry is consistent, then (TLP) and his later work (principally the Philosophical there is neither a proof nor a disproof of the parallel Investigations (PI), philosophy as traditionally conceived postulate available from the rest of Euclid’s postulates; it is, for the most part, nonsense. Philosophical activity as is independent of the other postulates. Thus, attempts to Wittgenstein conceived it, by contrast, is a kind of prove Euclid’s fifth postulate from the others were in therapy for the illusion involved in the claims of vain, assuming the consistency of Euclidean geometry. traditional philosophy to make sense where, in Wittgen- The work of Eugenio Beltrami (1835–1900) was stein’s view, no sense is being made. In the Investigations, critical in establishing the relative consistency of he reports that his aim is “to teach you to pass from a hyperbolic geometry with Euclidean geometry. He piece of disguised nonsense to something that is patent showed that the lines on a surface in Euclidean space nonsense” (PI §464). Philosophical activity is an activity called the pseudo-sphere behaved like straight lines in of clarification: clarification both of the many things hyperbolic geometry. He also showed that the geometry that individuals could mean by their words and of the on this surface could be mapped to a disk in the ways in which their attempts to use words depart from Euclidean plane. This provided a way to visualize or conflate those things. In a sense, then, the philoso- hyperbolic geometry in Euclidean space, whose legiti- pher’s treatment of a question is “like the treatment of macy was not in question. From this, geometers came to an illness” (PI §255). believe that Beltrami’s disk constituted a model of hyperbolic geometry, thereby providing a Euclidean Wittgenstein’s Tractatus Propositions. In the Tracta- interpretation of the terms of hyperbolic statements. tus, Wittgenstein’s propositions serve a double purpose Geometers concluded from this reinterpretability that as representative of both the disease and its cure. The the meanings of geometrical terms are irrelevant to the activity of clarification there is carried out by means of information they convey. Thus the truth of geometrical propositions that Wittgenstein describes as themselves statements was no longer at issue, but simply their nonsensical. He writes: “my propositions serve as consistency relative to an already-accepted theory. This elucidations in the following way: he who understands remains the canon of admissibility for new mathematical me eventually recognizes them as nonsensical, when he theories today. has used them—as steps—to climb up beyond them” NEW CATHOLIC ENCYCLOPEDIA SUPPLEMENT 2012−13: ETHICS AND PHILOSOPHY, VOLUME 3 1081 (c) 2013 Cengage Learning. All Rights Reserved. Nonsense (TLP §6.54). Commentators have interpreted that philosophy of LOGICAL POSITIVISM and the verification remark, even at a broad level, in vastly different ways. theory of meaning, according to which, in broad terms, A central issue that is at stake in that disagreement a sentence is meaningful if and only if it is either true is the nature of nonsense and what it can or cannot help by definition alone or is empirically verifiable (either in one to achieve. For many commentators (Gertrude principle or in practice). However, repeated attempts Elizabeth Margaret ANSCOMBE [1919–2001], Peter (most notably by A. J. Ayer in his Language, Truth, and Geach [1919–], and P. M. S. Hacker [1939–] foremost Logic) to articulate criteria of verifiability in such a way among them), the nonsense-sentences of the Tractatus as to exclude as nonsense traditional philosophy, but not achieve their elucidatory and clarificatory purposes not scientific generalities, met with failure, and the principle merely by revealing the emptiness of the forms of the itself, being neither true by definition nor empirically words to which individuals are attracted in philosophy, verifiable, is widely held to imply its own nonsensicality. but by bringing them to appreciate certain fundamental Moreover, and as with other putative criteria of and inexpressible features of the logical structure of the meaningfulness, application of the verification principle world and of language, features that underwrite the very always seems to come too late: to tell whether or not a possibility of making sense but that cannot themselves given sentence is indeed verifiable one must first be captured in propositions that do make sense. They understand what the sentence says, but having done that do this by deliberately flouting the rules of logical syntax, it would be absurd then to say that the sentence in fact rules that determine what it is to make sense, and says nothing, and if one fails to understand it despite combining words in ways that violate the logical one’s best efforts, then one does not have anything of structure of language (and are therefore nonsense) the right category to which to apply the principle, and precisely to draw attention to that structure. In this way, nothing more would be learned in applying it. Hence, they gesture at what in Wittgenstein’s terminology can the use for a theory of meaning imagined by the positiv- only be shown and cannot be said (TLP §4.1212). Thus, ists, to identify criteria for distinguishing sense from they are nonsense of a quite special kind, a kind that is nonsense, seems to be a chimera. philosophically illuminating in a unique way. A final question connected to the use of nonsense A New Wittgenstein? A series of articles by James Con- as a term of philosophical criticism is whether it is ant and Cora Diamond (Conant 1998 and 2002; plausible that one could ever suffer from illusions of Diamond 1995) focusing on the philosophical viability meaning something in the manner suggested. Here, Wit- tgenstein credits Bertrand RUSSELL (1872–1970) with of that notion of nonsense and the plausibility of its showing that the logical form of a sentence may differ attribution to Wittgenstein has led increasing numbers utterly from its surface appearance and so with showing of commentators to reject that picture of Wittgenstein’s how, deceived by the surface appearance of a sentence, aims. Conant and Diamond argue that such a view one may be led unwittingly into talking nonsense. involves adherence to what Conant calls a substantial view of nonsense, according to which there are (at least) SEE ALSO EMPIRICISM,LOGICAL;VERIFIABILITY AND VERIFICATIONISM. two logically distinct kinds of nonsense: mere nonsense, consisting of words to which no meaning has been given BIBLIOGRAPHY in that context, and substantial nonsense, consisting of Ayer, A. J. Language, Truth, and Logic. 2nd ed. London: Victor words whose meanings clash, thereby creating a Gollancz, 1946. determinate but illogical whole. Putative examples of the Conant, James. “The Search for Logically Alien Thought: Des- latter often discussed in the
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