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Wandering About: Analogy, Ambiguity and Humanistic Mathematics Journal of Humanistic Mathematics Volume 3 | Issue 1 January 2013 Wandering About: Analogy, Ambiguity and Humanistic Mathematics William M. Priestley Sewanee University of the South Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Philosophy of Language Commons, and the Rhetoric Commons Recommended Citation Priestley, W. M. "Wandering About: Analogy, Ambiguity and Humanistic Mathematics," Journal of Humanistic Mathematics, Volume 3 Issue 1 (January 2013), pages 115-135. DOI: 10.5642/ jhummath.201301.10 . Available at: https://scholarship.claremont.edu/jhm/vol3/iss1/10 ©2013 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. Wandering About: Analogy, Ambiguity and Humanistic Mathematics W. M. Priestley University of the South, Sewanee TN 37383 [email protected] Synopsis This article concerns the relationship between mathematics and language, em- phasizing the role of analogy both as an expression of a mathematical property and as a source of productive ambiguity in mathematics. An historical discus- sion is given of the interplay between the notions of logos, litotes, and limit that has implications for our understanding and teaching of Dedekind cuts and, more generally, for a humanistic notion of the role of mathematics within liberal education. Mathematics and language have interacted in many ways over the ages. Examples range from the effect of prehistoric “counting tokens” upon written symbols to today’s strong presence of mathematics in linguistics and cryp- tography. We tend to overlook, however, the large role that ordinary rhetoric plays in this connection. Consider, for example, Mark Twain’s insight into the writer’s craft: A good word is to the best word as a lightning bug is to the lightning. (T) The statement (T) is, of course, an analogy, that is, an assertion of the form “A is to B as C is to D.” By succinctly indicating enlightening but unsuspected connections between pairs of things, the analogy has become a major rhetorical tool in the humanities. One side of an analogy is often more familiar to us, making the other side easier to comprehend. Is there something “mathematical” going on here? James Gregory [17, page 215] used an analogy to give us insight into the vitality of 17th-century mathematics: Journal of Humanistic Mathematics Vol 3, No 1, January 2013 116 Wandering About The power of all previous methods has the same ratio to that of series (G) as the glimmer of dawn to the splendor of the noonday sun. The appearance of the word ratio in Gregory’s analogy reminds us of an almost forgotten connection between mathematics and the humanities: an analogy is an assertion of the sameness of ratios—ratios being understood perhaps in a very general sense. Indeed, logos, the Greek word for the Latin ratio, is the main stem of analogia, from which our word analogy comes. Here the prefix ana- means “up” in a generalized sense, as used also in analysis (“loosening up”). Like analysis, a word whose meaning was extended from geometry to philosophy in ancient times, analogia originally applied to mathematical assertions and later to rhetorical analogies. Today the form of an analogy is often indicated in symbols by writing A : B :: C : D, where the four dots in the middle may remind us of the assertion of a “square” balance between the ratio A : B and the corresponding ratio C : D. Com- pletion of an analogy (giving three of its members and asking for the fourth) is sometimes featured in questions on intelligence tests. Putting the question in a multiple-choice setting lessens worry about existence and uniqueness of the answer. An open-ended variant of this is to take two “extremes” and ask whether there exists a “mean” between them. For example, does there exist an entity X serving as intermediary between the sciences and the humanities in the sense that SCIENCES : X :: X : HUMANITIES? (P) Beginning in the fourth century BCE the ancient Greeks exploited re- lations among the ideas behind logos, litotes, and limit that allowed them to initiate mathematical analysis without developing our numerical epsilon- delta approach. Recalling these efforts brings mathematics closer to the hu- manities, suggests possibilities for new approaches to teaching, and—as we move toward modern times—sheds new light on the old topic of Dedekind cuts and on the place of mathematics within the liberal arts today. Perhaps a solution to proposition (P) will suggest itself along the way. W. M. Priestley 117 Figure 1: Plato and Aristotle “wandering about” in Raphael’s School of Athens. See [15]. Bronze sculpture, 18” tall, by Jeanie Stephenson, in- stalled October, 2012, in Courtyard of Woods Laboratories, the University of the South, Sewanee TN. Photograph by Mary Priestley, reproduced by permission. 1. Logos In Greek mathematics, logos (plural, logoi) means ratio—“the size of one thing relative to another” according to Euclid V —but the word is more familiar outside mathematics. “In the beginning was the logos,” the phrase from the New Testament that opens the fourth gospel, contains perhaps its 118 Wandering About most famous usage, written at a time when the word already signified a wealth of related notions. Alternative English translations of logos include word, language, speech, and reason. In the fourth century BCE it was seen as a unifying principle— along with notions of harmony, rhythm, etc.—permeating the enkuklios paideia, as the Greeks referred to the distinctive education that they developed [21]. Logos still plays a unifying role in liberal education, as suggested by such En- glish derivatives as logic and analogy, and by the host of modern academic words, such as ecology, possessing the suffix -logy. Although a numerical ratio such as 154 : 49 would be identified today with the fraction 22/7, we should remember that the Greeks possessed no algebra of common fractions [20]. They would have had little inclination to add one ratio to another, although they saw clearly—see inequality (E4) below, for example—when one ratio might be said to be less than another. Indeed, as documented in [20] and described for the general reader in [9], our modern quantitative sense took a great many centuries to develop. The strength of this sense, however, may be seen in our quick identification of the rational number 22/7 with its large equivalence class, including such fractions as 154/49. Archytas, a late Pythagorean and a friend of Plato (427–347 BCE), is traditionally credited with a striking analogy asserting the sameness of two logoi whose four members constitute a large part of the liberal education developed by the Greeks: ARITHMETIC : MUSIC :: GEOMETRY : ASTRONOMY: (A) The quadrivium, as this fourfold collection of disciplines was much later named by Boethius (480–524), would eventually become an integral part of the classical liberal arts of medieval European universities. Archytas’s analogy is usually understood by reflecting that arithmetic is about num- bers “at rest” and geometry is about magnitudes “at rest”; whereas music and astronomy study the same things, respectively, but “in motion.” That is, the science of music concerns itself with the numbers underlying pitches produced by the vibrations of the strings of a lyre, while astronomy studies magnitudes revolving in the great celestial sphere. The use of the compact form (A) to represent Archytas’s analogy would not have appeared in ancient times when the Greeks wrote everything out “rhetorically,” rarely using abbreviations beyond allowing letters to stand for W. M. Priestley 119 numbers or for the vertices of a geometric figure. In contrast, a college in- structor today might breezily remark that if a plane cuts a cone parallel to one of the cone’s generators, a Cartesian coordinate system can be easily imposed upon the plane so that the curve of intersection has an equation cy = x2 for a certain constant c. Apollonius (260?–200? BCE), however, in making essentially this observation long ago, required a great many words to express virtually the same idea, calling attention to the equality of ar- eas of a certain square (of size x) and a certain rectangle (with sides c and y). Apollonius went on to note that the analogue of this equality of square and rectangle becomes an inequality in the case of the other types of conic sections. Accordingly, he gave them the names that became our familiar parabola, hyperbola, and ellipse, depending upon (as we would say today) whether the symbol =, >, or < applies. The connections between the three types of conic sections, these three modern symbols, and the familiar rhetor- ical terms parable, hyperbole, and ellipsis are noteworthy, both for seeing ties between mathematics and rhetoric and for apprehending how symbols can sometimes reveal analogies with surprising succinctness. See [18] for a more complete discussion of Apollonius’s ideas. While their mathematics became more involved, the Greeks remained reticent to use symbolism to break the ties joining mathematics and rhetoric. This is often seen as a hindrance. It is worth emphasizing that it took a very long time for abbreviative notations and their consequent large-scale algebraic manipulations to manifest themselves. They have proved to be so efficient, however, that we have all but forgotten about past struggles in mathematics between symbolists and rhetoricians; see [7, pages 426–431].
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