Analogy in William Rowan Hamilton's New Algebra

Technical Communication Quarterly

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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

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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—knowledge 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 functions 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 geometric 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 treatise: 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 7or 1=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 influence 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 therefore 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 spe- cious unless securely tethered to geometry. And they had many influential predecessors: The early modern Italian mathematician Girolamo Cardano, for example, whose influence was felt well into the 19th century, reluctantly experimented with imaginaries in his famous treatise, Ars Magna (cited in Green, 1976). Green contends that Cardano was the first mathematician to use an imaginary number in a published computation. The endeavor was short-lived, however, and Cardano soon retreated into orthodoxy, feeling ‘‘obliged to offer geometrical demonstrations to support his algebraic arguments’’ and, in the case of imaginaries, ultimately failing (Green, p. 101). This decision would complicate his entire approach to mathematics, precluding him from accounting for not only imaginary and negative numbers but the concept of zero as well, at least to his and his traditional colleagues’ satisfaction. A contemporary of Cardano, Descartes (cited in Pycior, 1997) likewise rejected imaginary numbers on geometric grounds, but Descartes went a step further: Even algebraic symbols that admitted geometric interpretation were to be seen as provisional, never an end in themselves, for algebra was not a field of study in its own right but a tool to be developed and used in the service of geometry. As Pycior explained, ‘‘[Descartes] translated geometric problems into equations, but—because his aims were geometric and perhaps also because he did not ‘regard ...an equation as an adequate definition of the curve’ [Boyer, History of Analytic Geometry, 88]—he refused to accept roots coming from algebraic formulas as final solutions’’ (Pycior, p. 85). Owing to the enduring influence of Descartes, this framing of algebra as ever the handmaiden of geometry figured centrally in the intellectual milieu of Hamilton’s era. At the same time, a growing number of mathematicians set aside the perennial debate over algebra’s muddled foundation in favor of its notational economy and practical utility. Some of the more applied thinkers, taking their cue from such notable figures as William Oughtred, inven- tor of the slide rule, tended to simply ignore imaginary roots altogether. Others were content to allow them to crop up in equations as long as they easily canceled in a subsequent step, as in

pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi x ¼ 5 þ 3 þ 5 3 ¼ 10 280 LITTLE & BRANKER

Others returned to the question of foundations, but argued that algebra was capable of being developed into a science of its own, independent of and on par with geometry. By the 18th century, a fundamental reformulation of what it meant to do mathematics was underway (Heeffer, 2008), and by the turn of the 19th, those in favor of an independent, symbolic algebra had accrued many powerful adherents, past and present. Although the work of Francois Vie`te, Thomas Harriot, John Pell, John Kersey, and to some degree Oughtred significantly contributed to the enterprise, its leading exponent was found in Oughtred’s student, John Wallis, a contemporary of Newton and a member of the early Royal Society. Pycior (1997) brilliantly captured the gist of Wallis’s ambition, which serves as an emblem of the rise of symbolic algebra; we can therefore do no better than to quote her at length:

This new view entailed a redefinition of the relationship between algebra and geometry. Following Oughtred and Descartes, Wallis had used algebra extensively in the study of geometry. But, more important, in the Treatise of Algebra, he enunciated the general theme of algebra’s superiority over geometry based on algebra’s economy, intellectual transparency, and abstractness. On a more specific level, he recast the traditional concept of proportion as an algebraic concept. Algebra was exceptionally economical and lucid because it was symbolical. As Bacon and his scientific followers juxtaposed scientific prose to ornate prose, Wallis juxtaposed symbolical algebra to traditional geometry. In the Treatise he discussed his earlier Conic Sections: New Methods Exposed, in which he had defined the parabola, ellipse, and hyperbola as algebraic equations rather than slices of a cone. In a rather complete turning of the tables of mathematical tradition ...he displayed algebra as the preferred approach to conics. (p. 121)

It is in this light, following the likes of Wallis and company, that we find Hamilton in 1837, for he too advocated a science of algebra independent of geometry, one that would proceed by symbolic reasoning, the logical manipulation of symbols without any recourse to geometric proof. But unlike his predecessors, save perhaps Abbe Buee, Hamilton insisted that it was not only algebra’s dependence on geometry but also its reliance on the concept of quantity, the cornerstone of traditional algebra, that was impeding the field’s advance, especially with regard to imaginary numbers. He therefore revised the very basis of modern algebra, shifting its foundation from the concept of quantity, whose theoretical misgivings will be explained in the following section, to that of time, a murky construct that Hamilton made palpable by way of an analogy to movement along a path. As Hamilton developed his treatise, he reasoned analogically to invest this time-based algebra, known as his theory of moments, with the same rules that govern the real number system; so his new theory, though peculiarly founded on the concept of time, would nonetheless behave like the old algebra. In the end, it has the look and feel of traditional algebra, but its symbols refer to fundamentally new mathematical objects. The task of the second part of his treatise was to couple these objects, these moments, in a way that allowed him to accomplish his goal: the rigorous expression of imaginary numbers as pairs of real numbers. Here again, he carefully extended by way of analogy the rules governing single moments—themselves fashioned in the image of real numbers— to his theory of moment couples, achieving a fundamentally new algebraic universe capable of handling a wide array of new mathematical problems while remaining dexterous in the old algebraic worldview. It is a feat worthy of considerable study. In the following sections, we focus on but one of its many aspects: how analogy participated in the whole ANALOGY IN NEW ALGEBRA 281 endeavor, serving both to spatialize Hamilton’s key concepts and extend common logical ground across otherwise disparate conceptual domains.

SPATIALIZING KNOWLEDGE AND TIME

Hamilton (1837) opened his preliminary essay by dividing algebra into three schools of thought: the practical, the philological, and the theoretical. Whereas the practical school gives priority to the instrumental value of algebra in solving scientific problems, the philological school values algebra’s formal expression, a language capable of symmetry and beauty. But of traditional algebra, he wrote, ‘‘So useful are those rules, so symmetrical those expressions, and yet so unsatisfactory those principles from which they are supposed to be derived ...’’ (p. 3). As Hamilton saw it, the problems plaguing algebra were logical in nature, spurred by the perplexing concept of magnitudes ‘‘less than nothing,’’ which, when introduced into even the most basic mathematical expressions, had a tendency to spiral out of conceptual control. In his introductory remarks, Hamilton offered a sense of the mounting incredulity felt at the time:

It requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine of Negatives and Ima- ginaries, when set forth (as it has commonly been) with principles like these: that a greater magnitude may be subtracted from a less, and that the remainder is less than nothing; that two negative numbers, or numbers denoting magnitudes each less than nothing, may be multiplied ... and that the product will be a positive number ...; and that although the square of a number ...is therefore always positive, whether the number be positive or negative, yet that numbers, called imaginary, can be found or conceived or determined, and operated on by all the rules of positive and negative numbers, as if they were subject to those rules, although they have negative squares, and must therefore be supposed to be themselves neither positive nor negative, nor yet null numbers, so that the magnitudes which they are supposed to denote can neither be greater than nothing, nor less than nothing, nor even equal to nothing. (pp. 2–3)

He therefore concluded that only by way of the third school, the theoretical, which prizes neither ease of application nor fine expression but clear reasoning above all else, could algebra have any hope of overcoming its conceptual faults and developing into a science of its own. And yet, Hamilton was quick to point out that clear reasoning is not enough. He drew an analogy between creating knowledge and constructing a building to emphasize algebra’s shaky foundation despite its countless practical and aesthetic successes. As Thagard and Beam (2004) explained, this comparison sponsors a variety of correspondences between the architect and the scholar—chief among them, sturdy-base=indubitable-knowledge and build=justify—which have been used to conceptualize the advance of knowledge and orient oneself to an object of study since at least the time of Descartes, who wrote:

Throughout my writings I have made it clear that my method imitates that of the architect. When an architect wants to build a house which is stable on ground where there is a sandy topsoil over underlying rock, or clay, or some other firm base, he begins by digging out a set of trenches from which he removes the sand, and anything resting on or mixed in with the sand, so that he can lay his foundations on firm soil. In the same way, I began by taking everything that was doubtful and throwing it out, like sand; and then, when I noticed that it is impossible to doubt that a doubting or thinking 282 LITTLE & BRANKER

substance exists, I took this as the bedrock on which I could lay the foundations of my philosophy. (cited in Thagard & Beam, p. 505)

Following Descartes, Hamilton insisted that, for algebra to be regarded as a science on par with geometry, algebraists must reason from ‘‘grounds’’ (p. 3), later ‘‘a basis’’ (p. 6), as incontrovert- ible as Euclid’s parallel lines, of which ‘‘no candid and intelligent person can doubt the truth’’ for it ‘‘involves no obscurity nor confusion of thought, and leaves in the mind no reasonable ground [emphasis added] for doubt, although ingenuity may usefully be exercised in improving the plan of the argument’’ (p. 2). The algebraist must always be on guard for ‘‘Reasonings of his Science [that] seem anywhere to oppose each other, or become in any part too complex or too little valid for his belief to rest firmly upon them [emphasis added] ....’’ (p. 1), for in contrast to geometry, ‘‘confusions of thought, and errors of reasoning, still darken the beginnings of Algebra ...’’ (p. 2). Nowhere in Hamilton’s (1837) discourse is there room for the philologist’s coherentist epistemology or its attendant framing of knowledge as a web, which, in broadening the epistemic gaze, would threaten Hamilton’s agenda by holistically evaluating the merits of an algebraic system rather than cleaving the theoretical from the rest and examining it from its foundation up. There is a flexibility to coherentism, a valuing of internal consistency as some measure of epistemic validity, which Hamilton explicitly abjured. He warned his readers against the ‘‘growing tendency’’ among mathematicians to adopt

one or other of those two different views, which regard Algebra as an Art, or as a Language: as a System of Rules, or else as a System of Expressions, but not as a System of Truths, or Results having any other validity than what they may derive from their practical usefulness, or their logical or philological coherence [emphasis added]. (p. 3)

Where would algebra be if in fixating on the coherence of its signs, it found it ‘‘cannot look beyond the signs to the things signified?’’ (p. 1). So confident was Hamilton in his foundation- alism that he readily conceded the practical and philological value of algebra, insisting nonetheless that it must be rebuilt from the ground up:

It must be hard to found [emphasis added] a Science on such grounds as these [negative and imaginary magnitudes; emphasis added], though the forms of logic may build up from them [emphasis added] a symmetrical system of expressions, and a practical art may be learned of rightly applying useful rules which seem to depend upon them. (p. 3)

As long as reasoning, clear and practical and coherent though it may be, rests on the concept of magnitude, there can be no science of algebra. In what we regard as the most innovative move of his 130-page treatise, he responded to this emerging exigence by shifting the conceptual foundation of algebra from magnitude to time, a relatively nebulous concept, but one, as we will see, Hamilton clarified through a second analogy that spatializes time in culturally familiar ways. Whereas geometers have space, argued Hamilton (1837), algebraists have time. In his new algebra, B>A therefore signifies not that B is larger than A but that B is ‘‘later’’ than A (p. 9); likewise, the expressions B ¼ A and Btimes. Although natural language expressions in some cultures evince a conceptual system that places the past in front of the body because it alone can be ‘‘seen,’’ the analogy usually produces the opposite orientation, exemplified in English, by which ‘‘[e]very day conversations provide many examples that seem to imply that people locate the past behind them and the future in front of them, and that they think of time as a movement from one (past) location to another (future) location’’ (Santiago et al., p. 512). In the same way, to proceed in Hamil- ton’s algebra from moment A to B in the case of B>A, we must move ‘‘forward’’ in time; to ‘‘arrive’’ at A from B, we step ‘‘backward’’ (pp. 6, 15, 67–68). Hamilton in fact referred to these movements as ‘‘steps’’ or ‘‘mental acts of transition’’ by which we ‘‘pass’’ between moments and sometimes ‘‘return’’ from them (p. 20). Moments, then, are not only ‘‘earlier’’ or ‘‘later’’ than other moments; they also ‘‘precede’’ and ‘‘follow’’ them, and can be ‘‘nearer’’ to some moments and ‘‘farther’’ from others in what Hamilton called ‘‘the con- tinuous progression of time’’ (pp. 15, 58). Just as waypoints, from the perspective of a moving observer, appear to pass by continuously, so too do moments in Hamilton’s (1837) algebra: We pass from moment to moment, and in doing so, time passes us by. In this way, the passage of time ‘‘flows’’ for Hamilton, never fixed but always forming, always in the ‘‘process of generation’’ (pp. 3–4). Set in the larger intellectual context, time so construed offers theoretical algebra precisely the foundation it needs to develop into a science of its own, for the flow of time—the idea that moments necessarily ‘‘coincide,’’ ‘‘precede,’’ or ‘‘follow’’ all other moments in linear progression—was, for Hamil- ton, which he claimed for all, an ‘‘intuitive truth, as certain, as clear, and as unempirical as this, that no two straight lines can comprehend an area’’ (p. 5). What is more, Hamilton speculated that this notion of time is ‘‘more deep-seated in the human mind’’ (p. 5) than the notion of space, suggesting that if anyone needs a more surefooted basis upon which to build a science, it is the geometer, not the new algebraist. The mathematical implications of this new conceptual scheme are striking, for they directly affect how mathematicians are to read the new algebra. Zero no longer means ‘‘nothing’’ but ‘‘coincident’’ (p. 17): A – B ¼ 0 means not that A and B are the same size but that they occur at the same time, A being neither subsequent nor precedent to B. What is more, there are no negative numbers in Hamilton’s theory, only ‘‘contra-positives,’’ which signify particular ordinal relationships between pairs: B – A < 0 means B precedes A in the flow of time (p. 18). Because the inequality simply expresses a positional relationship between two moments, it does not entail the perplexing idea of quantities ‘‘less than nothing,’’ which Hamilton saw as the bane of traditional algebra. Likewise, a contra-positive such as a ¼7 signifies a mental step seven units earlier in time (from a position unknown), not seven units less than nothing, and the 284 LITTLE & BRANKER mysterious operation of multiplying by a negative number, now framed as a contra-positive, becomes the sensible procedure of simply reversing the direction of progression.

EXTENDING FROM SINGLES TO COUPLES

As we saw in the previous section, analogy functioned substantively in Hamilton’s (1837) preliminary essay by spatializing his conception of mathematical knowledge. As the architect approaches the construction of a building, so too does the mathematician approach his work: building a firm foundation, first and foremost, upon which one can then erect a sound structure, material in the architect’s case, logical in the mathematician’s. Although the analogy was the dominant epistemological scheme of the time, it nonetheless placed Hamilton at paradigmatic odds with mathematicians who subscribed to a coherentist epistemology and, less so, with mathematicians who privileged what Hamilton called the practical and philological schools of mathematical thought over fundamentally theoretical concerns. Analogy also spatialized the cardinal concept of Hamilton’s new algebra: Moving beyond a purely temporal description, Hamilton came to understand time by way of an analogy to movement along a straight path, which afforded him two concrete qualities—direction and distance—that would prove essential to the development of moments, steps, and progression. In this section, we examine the function of analogy in the second part of his treatise, his development of algebraic couples, also known as moment couples. After boldly departing from the real number system and its engrained concept of quantity to develop a new theory of algebra based on time, or moments—a theory that nonetheless preserves many of the essential properties of the real number system for strategic use later—Hamilton inched his way from the theory of moments to his theory of moment couples, rigorously extending by analogy the essential properties of conventional algebra to his new mathematical universe. Here we can see analogy functioning not by ‘‘daring leaps,’’ as in Gross’s (1990) account of Euler or in the more general imagery we use to describe flashes of insight, but as a series of gradual extensions. Just as Hamilton (1837) developed moments in his preliminary essay, so too did he develop their analogue, moment couples, in the second part of his treatise. In keeping with his science of pure time, he began his second essay by inviting readers to imagine two moments in time, A1 and A2, referring to them as the primary and secondary moments, respectively, but he insisted that readers not think of either moment as following, preceding, or coinciding with the other in ‘‘the common progression of time’’ (p. 87). They are, in other words, independent of each other. Primary moments, he explained, should be compared only with other primary moments, and secondary moments likewise. ‘‘We may also speak of this primary and this secondary moment,’’ he continued, ‘‘as forming a couple of moments, or a moment-couple, which may be denoted thus, (A1,A2)’’ (p. 87). We can, in fact, think of an infinite number of moment couples denoted by (B1,B2), (C1,C2), and so on, and we can compare them primary to primary, secondary to secondary, such that

ðB1; B2ÞðA1; A2Þ¼ðB1A1; B2A2Þ Moments and moment couples are different mathematical objects, but Hamilton provided a one-to-one correspondence between moments and a subclass of moment couples, allowing his reader to view moment couples as an enlargement of the theory of moments. In other words, ANALOGY IN NEW ALGEBRA 285 moment couples encompass moments, which are in turn fashioned in the image of real numbers: What is traditionally understood as 1 or 2 or 7 is, in Hamilton’s algebra, represented by (1, 0), (2, 0), or (7, 0), respectively. Also, although Hamilton’s algebra signals a radical departure from tradition in the sense that his objects signify elements of time, not quantity, he nonetheless strives to make his new algebra logically consistent with the old so it can be effectively used by mathematicians to solve a wide range of problems. In particular, Hamilton made moment couples amenable to the same mathematical operations—the same rules of algebra—as those that govern moments. They can be added, multiplied, powered, and the like in recognizable ways: Just as 1 þ 3 ¼ 4, so too does (1, 0) þ (3, 0) ¼ (4, 0), for one example. In the language of analogy studies, moments and moment couples share higher-order relational correspondences: Predicates, such as adding and powering, and not object attributes, such as bigger or smaller, describe their commonalities and thus the commonalities that span and structurally align the two parts of Hamilton’s treatise. In more general terms, moments and moment couples are ana- logues, different but functionally similar mathematical objects in their respective theories of algebra; and because they are also structurally aligned with the real number system, they retain a commensurate relationship with traditional algebra, giving them wide versatility. Throughout his essay, Hamilton proceeded in this way, extending by way of analogy from the theory of singles to that of couples, occasionally making explicit—that is, in natural language rather than implicitly in the mathematics—the analogical relationships that span both theories. Another example of this can be found in the case of subtraction that precedes zero. In his preliminary essay, Hamilton carefully developed the concept of opposition for moments, which is procedurally identical to the traditional concept of negation though not bound by the concept of quantity:

Again, to denote a relation which shall be exactly the inverse or opposite of any proposed ordinal relation a or b, we may agree to employ a complex symbol such as oaorob, formed by prefixing the mark, o (namely, the initial letter O of the Latin word Oppositio, distinguished by a bar across it, from the same letter used for other purposes,) to the mark a or b of the proposed ordinal relation; that is, we may agree to use oa to denote the ordinal relation of the moment A to B, or ob to denote the ordinal relation of C to D, when the symbol a has been already chosen to denote the relation of B to A, or b to denote that of D to C .... (p. 19)

In other words, if the transition from moment A to moment B is termed a, then the return transition, ‘‘against’’ the flow of time if you will, from moment B to moment A, should be termed oa. Hence, the expression 0 3 ¼ o3 signifies not the perplexing idea of three units less than nothing, but a relational idea: three units in the opposite direction of the flow of time indi- cated by positive numbers. And likewise for moment couples: ‘‘Employing a notation analogous to that explained for single steps’’ (p. 89), Hamilton concluded:

ð0; 0Þða1; a2Þ¼ða1; a2Þ

In which, because the concept of quantity has been replaced by that of time, (–a1,–a2) is understood as the opposite, not the negative, of (a1,a2). In other words, as 0 3 ¼ o3 in his theory of moments, so too does (0, 0) – (3, 0) ¼ –(3, 0) in his theory of moment couples—yet another analogical relation found deep within the technical recesses of Hamilton’s mathematical advance. 286 LITTLE & BRANKER

The final example we wish to present is that of multiplication, which shows analogy working more deeply in Hamilton’s (1837) text than it has in our previous cases. Because addition, subtraction, and the like have all followed a similar pattern—that of combining primaries with primaries, secondaries with secondaries—we might expect multiplication to follow suit, which would result in the following definition for couples:

ða1; a2Þðb1; b2Þ¼ða1b1; a2b2Þ

This definition is procedurally straightforward, primaries still combining with primaries (a1 with b1) and secondaries with secondaries (a2 with b2), and it is, in fact, consonant with the rest of Hamilton’s theory of moment couples. In other words, if Hamilton were to adopt it, his new algebra would remain internally consistent. However, the definition would breach fundamental properties of traditional algebra, namely the associative and distributive laws, and thus sever Hamilton’s work from the conventions of mainstream mathematics. A property of all real numbers, the associative law states that it does not matter how numbers are grouped when they are added or multiplied. For example, ð4 þ 5Þþ8 ¼ 4 þð5 þ 8Þ or more generally, ða þ bÞþc ¼ a þðb þ cÞ Also governing the behavior of real numbers, the distributive law states that one can multiply a number by the sum of two numbers (called addends) by multiplying each addend separately and then adding the products. This is more readily understood in its traditional symbolic form: aðb þ cÞ¼ab þ ac Along with a handful of other properties, the associate and distributive laws authorize what can and cannot be done with real numbers. They prescribe the propositional backbone to what every student, and a fortiori every professional algebraist, comes to understand as the general behavior of numbers. Had Hamilton therefore adopted (a1, a2)(b1,b2) ¼ (a1b1, a2b2) as his definition of multiplication, his decision would have had the deleterious effect of making his new algebra incommensurate with not only his theory of moments but also the real number system in general. The result would have been a new mathematical universe, internally consistent but essentially isolated from the rest of mathematical thought and rendered very difficult to use by his colleagues. Hamilton accordingly looked elsewhere for an understanding of couple-couple multiplication, settling on the following definition:

ða1; a2Þðb1; b2Þ¼ða1b1 a2b2; a2b1 þ a1b2Þ Though procedurally more complicated than the previous option—primaries no longer interact- ing only with primaries, nor secondaries with secondaries—this ingenious definition preserves the associative and distributive laws of algebra, among others, and therefore enables mathematicians to see moment couples as logically similar to moments and real numbers. Though novel objects, they retained to a large degree the look and feel of real numbers. This allowed mathematicians to enlist the new algebra to solve equations much more easily than if Hamilton had breached the associative and distributive laws and created a new mathematical universe with altogether esoteric sensibilities. ANALOGY IN NEW ALGEBRA 287

In our earlier cases, we saw analogy functioning as a higher-order process made visible at the syntactic level of formal expression: Just as 1 þ 3 ¼ 4, so too does (1, 0) þ (3, 0) ¼ (4, 0); and as 0 3 ¼ o3 in his theory of moments, so too does (0, 0) – (3, 0) ¼ –(3, 0) in his theory of moment couples. In the present case, however, the equation for multiplication may not appear analogous to multiplication of two real numbers—it does not seem to resemble the real number case of a being multiplied by b—but it is: Preserved are those deeper relational properties, the associative and distributive laws of traditional algebra principally among them, which enabled Hamilton’s new objects to behave customarily—that is, as real numbers do—while nonetheless offering mathematical options that far exceed the conceptual limitations of real numbers. These are precisely the sort of properties that Baake (2003) alluded to in his Colvan reference to the ‘‘deep structural similarities’’ that unify systems of imaginary mathematics (p. 93). In broad strokes then, and pace Colvan, we can see in the details of multiplication, analogy functioning as a higher-order process, one that preserves not the kind of simple relationship suggested by pro- portional analogies or syntactic similarities but a deeper system of nested relationships, which is akin to what many scholars see in the sciences as the mark of modern reasoning (Gentner, Holyoak, & Kokinov, 2001).

CONCLUSION

The conspicuous omission of research-level mathematics from technical studies of analogy lends a kind of prima facie credence to the debilitating conceptual scheme so prevalent today that places the rigorous mathematician at one pole of an intellectual continuum, the creative artist at the other. It is a scheme with a long and robust history, as Snow (1959) addressed in his Two Cultures.As we have shown in this initial foray, however, Hamilton’s (1837) work reveals a fundamental intermingling of analogy and mathematics: Analogy functioned substantively in Hamilton’s preliminary essay by spatializing his conception of mathematical knowledge. As the architect approaches the construction of a building, so too does the mathematician approach his work: building a firm foundation upon which one can then erect a sound structure, material in the architect’s case, logical in the mathematician’s. Even though the analogy was the dominant epistemological scheme of the time, it nonetheless placed Hamilton at paradigmatic odds with mathematicians who subscribed to a coherentist epistemology and, less so, with mathematicians who privileged what Hamilton called the practical and philological schools of mathematical thought over fundamentally theoretical concerns. Analogy also spatialized the cardinal concept of Hamilton’s new algebra: Moving beyond a purely temporal description, Hamilton came to understand time by way of an analogy to movement along a straight path, which afforded him two concrete qualities—direction and distance—that would prove essential to the development of moments, steps, and progression. Even in research-level mathematics, it seems, our symbol use is sometimes derived from everyday, embodied experiences (Lakoff & Nunez, 2000). Analogy also served substantively in Hamilton’s subsequent essay on moment couples. After boldly departing from the real number system and its engrained concept of quantity to develop a new theory of algebra based on time, which nonetheless preserves many of the essential properties of the real number system for strategic use later, Hamilton inched his way from the theory of moments to his theory of couples, rigorously extending by analogy the essential properties of conventional algebra into his newly created world of moment couples. Here we can see analogy 288 LITTLE & BRANKER functioning not by ‘‘daring leaps’’ as in Gross’s (1990) account of Euler or in the more general imagery used to describe flashes of insight, but as a series of gradual extensions. And gradual though they may have been, they served an important ontological function in this case by under- writing the very existence of the objects that would inhabit Hamilton’s new mathematical universe, a universe that would profess a new algebra independent of geometry. In other words, analogy did not function alongside Hamilton’s mathematics, nor was it exegetical to it (Boyd, 1993). It was not enlisted to solve a particular problem within an already established mathematical framework, as with Euler (Gross). In Hamilton’s case, analogy served as a kind of mathematical reasoning itself, underlying and structuring the development of Hamilton’s technical argument and the new mathematical reality that his argument brought into ideational existence. Hamilton’s work, and imaginary numbers more generally, certainly merits further investi- gation. For the most part, Hamilton’s couples behaved like conventional numbers, which allowed mathematicians to bring their existing sensibilities to bear in his new theoretical space, simultaneously offering those same mathematicians a rigorous definition of square roots of negative integers at a time when the very idea of negative integers, interpreted conventionally as quantities, was the subject of much debate. Out of this balancing of sameness and difference—a sameness in how the symbols are manipulated, a difference in what they mean—arises the question of polysemy: Was the reception of Hamilton’s work facilitated by the potential mul- tivalency available to his symbols, expressions such as A < Bora ¼7, which would have looked familiar, even functioned in familiar ways, but which in fact signified for Hamilton, and for those who gleaned the full significance of his preliminary essay, a new conception of number predicated on time? Such a question is beyond the scope of this paper, but it serves as a possible entry point, one among many, for future studies as we begin to acknowledge that in the technical recesses of professional mathematics, no less than in the sciences, rhetoric makes a significant contribution of both epistemic and ontological effect.

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Joseph Little is an associate professor of English at Niagara University, where he directs the first-year writing program. His research focuses on the function of analogy in technical discourse.

Maritza M. Branker is an assistant professor of mathematics at Niagara University. Her research interests include complex analysis and the history of complex analysis.