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Analogy in William Rowan Hamilton's New Algebra Technical Communication Quarterly ISSN: 1057-2252 (Print) 1542-7625 (Online) Journal homepage: https://www.tandfonline.com/loi/htcq20 Analogy in William Rowan Hamilton's New Algebra Joseph Little & Maritza M. Branker To cite this article: Joseph Little & Maritza M. Branker (2012) Analogy in William Rowan Hamilton's New Algebra, Technical Communication Quarterly, 21:4, 277-289, DOI: 10.1080/10572252.2012.673955 To link to this article: https://doi.org/10.1080/10572252.2012.673955 Accepted author version posted online: 16 Mar 2012. Published online: 16 Mar 2012. Submit your article to this journal Article views: 158 Citing articles: 1 View citing articles Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=htcq20 Technical Communication Quarterly, 21: 277–289, 2012 Copyright # Association of Teachers of Technical Writing ISSN: 1057-2252 print/1542-7625 online DOI: 10.1080/10572252.2012.673955 Analogy in William Rowan Hamilton’s New Algebra Joseph Little and Maritza M. Branker Niagara University This essay offers the first analysis of analogy in research-level mathematics, taking as its case the 1837 treatise of William Rowan Hamilton. Analogy spatialized Hamilton’s key concepts—knowl- edge and time—in culturally familiar ways, creating an effective landscape for thinking about the new algebra. It also structurally aligned his theory with the real number system so his objects and operations would behave customarily, thus encompassing the old algebra while systematically bringing into existence the new. Keywords: algebra, analogy, mathematics, William Rowan Hamilton INTRODUCTION Studies of analogy in technical discourse have made important strides in the 30 years since Lakoff and Johnson (1980) ushered in the cognitive linguistic turn. Following Gross’s (1990) provocative Rhetoric of Science, Turner’s (1991) call for a cognitive rhetoric in Reading Minds, and the second edition of Ortony’s (1993) Metaphor and Thought, book-length analyses by Keller (1995), Baake (2003), Graves (2005), and Giles (2008) have taken up scientific analogy directly and extensively: Analogies, we now know, have the power to shape scientific discover- ies including their concomitant lines of reasoning. They have also played a central role in the conceptual formation of entire disciplines. Work in this area continues, recent studies by Gibson (2008) and Little (2008) having extended the collective analysis to artificial intelligence and nuclear physics, respectively. No one, however, has considered whether or how analogy func- tions in research-level mathematics. Gross (1990, p. 28) did recognize Leonhard Euler’s ‘‘daring leap by means of analogy’’ to solve for the sum of the infinite series of squares, but the details of that leap are left unexamined. Given his broader point about the different ways in which analogy functions in political, schol- arly, and scientific discourse, Gross may not have directly benefited from a more technical analysis. But the precedent he set is nonetheless significant: His influential example affirms that it is well within the norms of strong scholarship to acknowledge the imaginative leaps of analogy without addressing the specific options—in Euler’s case, mathematical options—rendered prob- able or improbable by them. In their influential argument for the primacy of conceptual blending in human cognition, Fauconnier and Turner (2002) touched on the development of complex numbers, but, like Gross, they opted for a conceptual treatment, three pages in length, leaving little room for the kind of fine-grained analysis that would have shed light on the contextualized practices at work in any given case. In the same way, Baake (2003) offered an impressive account of the ways in which metaphor and analogy underwrite the intellectual work of a group 278 LITTLE & BRANKER of Santa Fe Institute scientists. In a deft discussion of the relation of mathematics to science, he acknowledged the ‘‘deep structural similarities’’ that exist between systems of mathematics that invoke imaginary numbers (Colyvan, as cited in Baake, 2003, p. 93), but Baake stopped short of investigating those structural similarities—analogy by another name—or their effect on the mathematics at hand. In this essay, we respond to the absence of research-level mathematics in the literature by ana- lyzing the role of analogy in William Rowan Hamilton’s (1837) two-part treatise, ‘‘Theory of Con- jugate Functions, or Algebraic Couples; With a Preliminary and Elementary Essay on Algebra as the Science of Pure Time.’’ In developing his theory of algebraic couples, also known as moment couples, Hamilton was the first to place imaginary numbers on a firm, algebraic foundation, one that proceeded deductively from first principles without relying on diagrams or other forms of geo- metric interpretation, which, though conventional, were known to be potentially misleading. But before Hamilton could create his moment couples, he had to revise the basis of modern algebra, shifting its focus from the concept of quantity, with all its theoretical misgivings, to that of time. Only within such a reference system, argued Hamilton, could imaginary numbers be rescued from their conceptual inconsistencies and take on an interpretively stable form amenable to the rigors of modern mathematics. Analogy, as we will see, functioned in two distinct ways in Hamilton’s treat- ise: In his preliminary essay, it spatialized his key concepts—knowledge and time—in culturally familiar ways, creating a palpable landscape for thinking about the new algebra. Analogy was also used to extend the familiar rules of mathematics—addition, multiplication, powers, and the like— first from the real number system to his theory of moments, and then, as we show in detail, from moments to his theory of moment couples. The structural alignment afforded by analogy across these three domains—real numbers, moments, and moment couples—enabled Hamilton’s new algebra to behave in traditional ways and therefore to have the look and feel of the old algebra while nonetheless expanding the conceptual purview of mathematics far beyond its traditional reach, a purview that included a sensible account of imaginary numbers. Given the interdisciplinary context of research today, we feel the need to clarify our view of analogy. Believing, as contemporary theorists do, that central to human cognition is the seeing of relational similarity across disparate domains, we take analogy to be that seeing, that ‘‘ability to think about relational patterns’’ that underwrites so much of our intellectual work, past and present (Holyoak, Gentner, & Kokinov, 2001, p. 2). It is this relational aspect of analogy, this notion of analogy as a ‘‘resemblance of structures’’ and not of simple traits (Perelman & Olbrechts-Tyteca, 1971, p. 372), that distinguishes it from other types of pattern recognition, such as mere-appearance similarity (Gentner & Jeziorski, 1993). To take an example from the common stock of science studies, what was significant in the planetary analogies of the late-Victorian physicists were second- and third-order relational correspondences: the idea that things revolve around each other (a second-order relationship) or the idea that this revolving was caused by mutual gravitational attraction (a third-order relationship). It was higher-order relationships like these that structured the physicist’s atomic domain in patently classical and visualizable terms. Mere-appearance mappings of isolated object attributes, such as the idea that Saturn is yellow and therefore the atom must be, played no role in late-Victorian thinking. This contemporary framing of analogy, consonant with the most recent advances in cognitive science as well as rhetoric of science, brings into relief the constitutive potential of analogy by demon- strating its ability to lead to a kind of deep structural alignment between two domains, an effect we will see in the work of Hamilton (1837) in the following pages. ANALOGY IN NEW ALGEBRA 279 SITUATING HAMILTON At the time of Hamilton’s (1837) writing,p mathematiciansffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi held an uneasy relationship with imaginary numbers—those curious results, such as À7or À1=2, whose squares were known to be negative. Much of the conceptual difficulty lay in the fact that the field as a whole, still under the heavy influ- ence of ancient Greek geometry, had long been suspicious of mathematical symbols that did not lend themselves to geometric representation (Nahin, 1998). As Green (1976, p. 104) explained, ‘‘early mathematicians thought of x2 as a physical square of side x,andx3 as a physical cube of side x ....’’; higher-order symbols such as x4 and x5 were commonly interpreted as meaningless and there- fore beyond the purview of mathematics because they were geometrically inconceivable. We may find this way of thinking hard to fathom because today algebraic symbols are understood abstractly, and calculations involving squares and square roots rarely conjure, much less necessitate, mental images of actual squares with palpable sides and areas. However, it was precisely this issue that spon- sored much of the debate among Hamilton’s contemporaries over the legitimacy of such ‘‘impossible roots,’’ for what, they asked, would a square of negative area look like? For some, the debate over imaginaries simply galvanized their commitment to traditional mathematics. They, therefore, continued to insist that the symbols of algebra be considered
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