"Invention" and "Discovery" as Modes of Conceptual Integration: The Case of Thomas Harriot

"Inventing" something, nowadays, means causing it to be; "discovering" something means perceiving that it already is. The logical opposition that currently pairs these terms, however, seems not to have done so consistently in early-modern usage. There, "discover" primarily meant uncover or reveal—as often a matter of showing to a second party as perceiving for oneself. "Invent," meanwhile, retained its Latin sense of find. If we go back far enough in early-modern English, we arrive at a point where people could speak colloquially of the pleasure of inventing (coming upon) what God or nature had discovered (shown) to them, whereas we might now idiomatically say almost the reverse. Yet we cannot necessarily infer that they simply or straightforwardly meant the reverse. Perhaps all we can really be sure of, with regard to early-modern "invention" and "discovery," is that both denote aspects of the experience of novelty; but with mutual and shifting imbrications of agency and subjectivity.

"Conceptual integration" or "blending," as explored by recent work in cognitive science, offers a useful way of understanding novelty in terms both of individual experience and of cultural or categorical change. If we sense that "making" and "finding" are interestingly overdetermined in early modernity, we may want to know about some recent efforts to describe these phenomena in cognitive terms.1

The focus of the current collection is on our "understanding of understanding" as well as our understanding of the early-modern period, with the presumption that these two matters are intertwined and mutually illuminating. Recent contributions of cognitive linguists to the former issue, I believe, offer new opportunities for insight into the latter.

One early-modern person who was known for his involvement in both invention and discovery—however exactly we are to understand these terms—was the Elizabethan scientist and explorer Thomas Harriot.2 "Imployed in discovering" is how Harriot characterizes his role as naturalist, ethnographer, and linguist for the 1585 Virginia expedition; "invention" is the word used of his mathematical and scientific work by his friend William Lower.3 Harriot’s name has been familiar to literary studies ever since New Historicism first emboldened scholars to read historical and literary texts into each other; he has gained a belated notoriety as an agent of colonialism, and has figured often in works of literary criticism and cultural history devoted to the subject.4 What has found little place in these discussions, however, is serious consideration of Harriot’s Algonquian-language experience—let alone its potential relationship to his mathematical innovations.5

In what follows, I hope to use the theory of conceptual blending to make some sense of Harriot’s multiple and seemingly disparate experiences of novelty. That is to say, I hope to show that they are really one experience, in a way that is theoretically, as well as historically, informative. I do not claim that Harriot’s Algonquian translation work must have informed his mathematical thought—but I think I can offer a reasonable account of how it could have: how his invention of algebraic techniques might have flowed from his discovery of Algonquian syntactic relationships, and how his "discovery" of mathematical principles may reflect an inventive mind at work, knitting together the conceptual relationships of early-modern mathematics with those suggested by his encounter with an unfamiliar language and culture. In sum, I will argue here for Harriot’s work as an exemplary, if exotic, blend of early-modern invention and discovery; and for the theory of conceptual blending as a useful, though relatively unfamiliar, way of talking about the latter.

Letters and Numbers

One of the most consequential of all historical developments in mathematics, it has been argued, was "the adoption, in the decades around 1600, of literal symbolism—that is, the use of letter symbols to represent arbitrary or unknown numbers."6 Harriot was an advanced figure in this development; his work had greater symbolic abstraction than that of the early algebraic pioneer Viete, and it preceded that of the more celebrated Descartes.7 Jacqueline Stedall, the leading authority on Harriot’s mathematical work, has written that "as early as 1600 he was already writing purely symbolic mathematics. With hindsight this can be seen as an early manifestation of a more general trend, but at the same time it can be recognized that it sprang from Harriot’s innate interest and ability in such matters, the evidence for which begins with his Algonquian alphabet of 1585."8 The latter, she says, "was not just a phonetic alphabet but something very like a phonetic algebra"9—a system in which observed relationships among phonemes are systematically correlated with observable relationships among the letter shapes that "encode information about the position and formation of each sound in the mouth."10 The question for us, here, is what relationship may obtain between Harriot’s mathematical innovations and his empirical encounter with Algonquian languages.

Regrettably, the record of Harriot’s Algonquian experience is scant. Nonetheless, we do have some testimony about how the peculiarities of Algonquian grammar struck some other early-modern Englishmen. John Eliot (1604-90), while noting the frequently substantial length of individual Algonquian words on account of "the many syllables which the grammar rule requires," also says the language "doth greatly delight in the compounding of words … to speak much in few words."11 Roger Williams (1603-83) notes a grammatical form that he likens to the ablative absolute in Latin, and that enables the Narragansetts to "comprise much in little."12 Peter Stephen Du Ponceau (1760-1844) writes that "the mind is lost in the contemplation of the multitude of ideas thus expressed at once by means of a single [Algonquian] word, varied through moods, tenses, persons, affirmation, negation, transitions, etc., by regular forms and cadences."13 John Pickering (1777-1846) notes "a new manner of compounding words from various roots, so as to strike the mind at once with a whole mass of ideas."14

Some categories in Algonquian grammar seem particularly pertinent to Harriot’s mathematical work. There is, for example, what Eliot calls the "suffix animate mutual"—relevant because of the intrinsic reciprocity of the relations among elements in algebra—and "the suffix form advocate, or ‘instead’ form, when one acteth in the room or stead of another."15 There is also a "distributive" inflection: distributives "express the number of things taken at a time, as ‘each one, two at a time, every third one, four apiece’," which seems similar to what algebraic variables and coefficients do. William Jones cites its use in multiplication: "ne ‘sw’, three; … ne’swi na’nese’nw’, three is taken thrice at a time."16 Distributives are indicated syntactically "by means of reduplication."17 As John Eliot says, "when the action is doubled, or frequented, &c., this notion … is expressed by doubling the first syllable of the word: as Sa-sabbathdayeu, every Sabbath."18 This inflectional category, or categorizing inflection, is interesting in light of Harriot’s algebra, where the multiplication of a quantity by itself any number of times is not indicated in the Cartesian way, with superscript numerical exponents (e.g. "a4"}, but by reduplication (e.g. "aaaa"). The pattern of quantitative distribution can extend to any number of "dimensions" whatsoever, and is not limited to the three that can be illustrated by length, area, and volume. "Harriot was the first to devise a notation in which any number of unknown quantities could appear to any power."19

The imaginative and conceptual extension necessary to distribute specified shares or reflections of hypothetical quantity a from the second- and third- to a "fourth dimension"20 may to some extent correlate with what would be needed to extend one’s grammar, in conversation, to accommodate the unaccustomed consideration of the Algonquian "fourth person." This is the "obviative" inflection, which obliges a speaker to observe and indicate the "relative contributions of two third persons to action within a sentence or utterance."21 Leonard Bloomfield describes it in terms of "identity and non-identity of objects,"22 which of course is a primary concern of algebra. Algonquian grammar indicates whether a given verb implicates the listener or the speaker—in that order, which in itself is noteworthy—or a "proximate" third party, or an "obviative" other party, or any combination of these. The constant reassigning of grammatical markers for the latter two categories, necessary as one’s discourse plays over a wide field of subjects and entities, is called by linguists "proximate shifting."

Laura Ann Buszard, who studies the phenomenon in Potawatomi Algonquian discourse, explains that proximate shifts occur "when there is a focus on a particular character, or the narrative is presented from a particular character’s viewpoint," and that the effect is "to shift our attention or focus to a secondary character, or to represent that character’s point of view."23 Harriot records at least one Algonquian-language conversation with this kind and degree of complexity. "It was told me for strange news," he writes, that one being dead, buried and taken up again … showed that although his body had lain dead in the grave, yet his soule was alive, and had travelled far in a long broad way, on both sides whereof grew most delicate and pleasant trees, bearing more rare and excellent fruits then ever he had seen before or was able to express, and at length came to most brave and fair houses, near which he met his father, that had been dead before, who gave him great charge to go back again and show his friends what good they were to do to enioy the pleasures of that place, which when he had done he should after come again.24

In her account of this sort of phenomenon as it occurs, with varying effects, in different Algonquian languages, Buszard relies on the theory of mental spaces: "the use of these grammatical constructions reflects a difference in the mental space configurations for foreground and background."25 Indexing the participants in a discourse situation, providing "a conceptualization of viewpoint structure,"26 is in fact the original purpose of mental space theory in linguistic analysis; as Ilana Mushin says, it is "designed to be able to track prepositional information through different belief spaces and so represent how information is distributed through different viewpoints."27 It is notable that Adrian Robert also employs the mental-space model in his discussion of how we track "the specific elements of an imagined configurational situation" in mathematical proofs.28

Harriot would have had to reckon with the way that Algonquian grammar forces a speaker to mark at every moment the array of viewpoints involved in a speech-situation—and that is just taking into account the human ones; Algonquian languages inflectionally distinguish animate from inanimate entities, but in a given context almost anything may become grammatically animate.29 The cultural historian Howard S. Russell observes that, as "every man, bird, beast, flower, fruit, or even rock had its role and special value," a traditional Algonquian man "felt himself in the presence of living entities who were as conscious of his existence as he of theirs."30 The discursive world into which Harriot entered, in venturing to converse with the native people of the Carolina Outer Banks, would have presented him repeatedly with the challenge of organizing semantic relationships among the entities participating or implicated in the discourse, variably foregrounding their presence or absence, and reshuffling the mental space configurations necessary to do so. Such juggling of spaces, I think, suggestively prefigures the algebra that Harriot pioneered.


Mental spaces, on the view of cognitive linguists, behave in two important ways: they link together to form networks which are salient or active for a period of time (as when one mentally co-ordinates variables in a mathematical problem, or persons in a narrative) and they also at times integrate or blend, as when a relationship of identity—whether exact and literal, or approximate and figurative—is found between them. Like poststructuralist paradigms, blend-theory is a constructivist model of meaning. In place of the deconstruction of terms, though, as a conception of what happens when perceptive analytical thinkers scrutinize or escape reductive dichotomies, blend-theory offers the unpacking or decompression of mental scenarios, or "frames," that have been combined and compressed to form any instance of meaning under examination.

The process of conceptual blending tends toward the condensing or compression of what is diffuse into an integrated, more tractable form. Concision in writing is an example of this compression: when we say that a student has boiled down her first draft, or has synthesized the readings she was assigned, we are talking about these phenomena. Compression is bound by some constraints: for example, a preference that the things compressed should not be irreparably distorted, and that the new conceptual synthesis should lend itself readily to unpacking or expansion.31 Harriot’s mathematical innovations included one manifestation of such compression in his devising of "canonical equations"—a repertoire of equation forms assembled so that complex higher-order equations could readily be solved if manipulated into fitting one of these patterns.32 These equations, along with Harriot’s shift from verbal to symbolic algebra, allowed far greater reaches of mathematical quantity to be choreographed into a smaller space than had previously been possible.33 Harriot’s canonical equations might be seen as accomplishing within the constraints of mathematical exactness something akin to the linguistic phenomenon of "chunking" complex predications into pronouns or into concise new nouns.34

Another principle of blending is that notions of similarity often reflect a perception of shared spatial topology between the things in question; we tend to think in terms of scenarios derived from our embodied experience, so things seem significantly similar to us when they share a quality that impinges on that experience. What’s true of thought in general is also true of mathematical thought in particular, as it attends to the growing, shrinking, disappearance, displacement etc. of hypothetical quantities. With respect to literary and cultural studies, the attention that blend-theory pays to scalar or gradient relationships as a factor in meaning is one reason why it better accommodates the articulation of some insights than a critical discourse concerned with dichotomies would.35 With respect to Harriot, I have wondered if a symbolic notation he devised for indicating linear, planar, or cubic quantity might not have been suggested by the Algonquian dimensional classifiers: grammatical inflections that indicate whether an item is elongated, flat, or voluminous.36 The salience of spatial relations in blend-theory provides new theoretical grounds for exploring this kind of hypothesis.

In the domain of human meaning, as opposed to that of mathematics, blending involves conceptual frames, and frames clash. Sometimes a conceptual network brings together two things that have the same frame.37 Sometimes, though, it brings together mental spaces that have strongly clashing frames, and invites us to draw select relevancies from each frame, ignoring some others, in order to construct a new, blended conception. A blended conception that uses the frame from one input to organize elements from another is called a "single-scope" blend.38 An example would be Sir Walter Raleigh’s comment that minds retain the permanent trace of their early education just as leather drinking vessels retain the "savour" of their first liquor.39 This metaphor uses a culturally familiar fact about leather drinking vessels to give force to an argument about education; but it does not use the idea of education to make an argument about drinking vessels. If it did—if Raleigh drank from a new mug and declared it to have been thereby "educated"—he would be running the blend backwards, and offering something like poetry. A blend that communicates inferences back and forth in this way is called "double-scope."

Harriot shows a remarkable capacity for double-scope blending. In his mathematical work it appears, for instance, as a farsighted flexibility about how the multiplication of binomials should determine the positive or negative valence of their terms—whence his mnemonic rhyme: "Yet lesse of lesse makes lesse or more; / Use which is best; keep both in store."40 Gilles Fauconnier and Mark Turner, the exponents of blend-theory, write: "We stress that failing to satisfy a governing principle [such as that of preserving each frame of reference intact] does not necessarily mean that the resulting blend fails; on the contrary, constructing a useful double-scope blend often depends upon finding a suitable way to relax governing principles."41 In this respect, there is a suggestive resonance between the Algonquian "absentative" inflection—which indicates that the thing referred to is absent or does not exist—and Harriot’s innovation of the predicative or algebraic zero.42 About the latter, Alfred North Whitehead has written that "it is not going too far to say that no part of modern mathematics can properly be understood without constant recurrence to it."43 Like an Algonquian absentative, Harriot’s zero paradoxically indicates a "this" that isn’t here. I am not implying that Algonquian native speakers would have experienced the absentative inflection as paradoxical. But Harriot very likely would have. By that very token, it is possible that he learned from his Algonquian hosts how to be analytically comfortable with the enabling paradox of a zero that could anchor algebraic expressions and arbitrate among them.

Visible Bullets

One of the very few other seventeenth-century Europeans who left a record characterized by "insight and understanding of the Indians of the Southeast"44 was John Lederer, a multilingual physician originally from Hamburg, whose three journeys of American exploration began and ended in the Algonquian-speaking zone known to Harriot, each of them beginning from a different one of the rivers flowing into the Chesapeake Bay.Lederer travelled as far as the Appalachian mountains, encountering mainly Iroquian- and Siouan-speaking groups along the way, but there was contact between these groups and the coastal Algonquians, and there was at least some cultural continuity across the groups. Even "the remoter Indians," he records, customarily carried "their currant Coyn of small shells, which they call Roanoack."45 That, of course, is also the name of the coastal island where Harriot lived in 1585. Impressed by their "reason and understanding,"46 Lederer observed that the people among whom he travelled were able to "supply their want of letters" in three ways: with "Tales," with "Emblemes or Hieroglyphicks," and with "Counters."47

The "counters" mentioned here may be pertinent to Harriot’s mathematical thought. Lederer says: them orderly in a Circle when they prepare for Devotion or Sacrifice, and that performed, the Circle remains still; for it is Sacreledge to disturb or to touch it: the disposition and sorting of the straws and reeds, shew what kind of Rites have there been celebrated, as Invocation, Sacrifice, Burial, etc.48

We have here the picture of a landscape decked with geometric forms carefully constructed to encode information in the spatial relationship among their elements, and in particular we have the picture of one recurring shape: a large circle sketched in small straight lines. This exemplifies Harriot’s insight that "a continuum is an aggregate of tangencies."49

The "pebbles" mentioned by Lederer are also important, as giving rise to another geometric form in the landscape. "Where a Battel has been fought," he writes, "or a colony seated, they raise a small Pyramid of these stones, consisting of the number slain or transplanted."50 It is striking that, on this account of what sounds like memorial cairn-building, the number of stones is underscored as centrally important. It is also significant that the construction is referred to as a "pyramid," rather than, say, a pile. Since a pyramid must be built from a base of given dimensions, to make this shape with a given number of stones—the number of people commemorated—one would need to know the numerical ratio of side to base to full three-dimensional figure; that is, how many stones per side would give the base for a pyramid containing x stones. Lederer’s testimony, if we take it seriously, suggests at least a practical tradition among the indigenous people for reckoning these ratios, a cultural feature that would have been highly convergent with known interests of Harriot’s.

Scholars have observed, among his papers, some notes on the stacking of what he called "bullets," which in Harriot’s time meant "a small ball," not necessarily considered as a missile.51 "Concerning piling," the notes run, "there are two questions: one, the number of bulletes to be piled being geven with the forme of the ground plat, to know how many must be placed in every ranke, with how many rankes in the said ground plat. The second—a pile being made—to know the number of bulletes therein conteyned."52 These bullets are generally presumed to have been the cannonballs on Raleigh’s ships, but even assuming that they were, this need not have been the first or only context in which Harriot considered the question, either practically or theoretically. Harriot’s report that the Algonquians understood infectious diseases to be the result of "invisible bullets" cast by the English has been widely noted. Here, by contrast, we have a suggestion that Harriot could have learned from the Algonquians how to make numbers visible through the stacking of bullets.

This extremely simple point connects very directly with some of Harriot’s most advanced and abstract mathematical work. He was interested, not just in triangles and pyramids, but in triangular and pyramidal numbers, the ratios or "constant differences" between the number of units in edges, faces, and solid figures. This is what Harriot called his "doctrine of triangular numbers" or "Magisteria magna."53 It is an approach to constant difference interpolation: the relationship of series and continua. According to Stedall,

The key to Harriot’s method, as his title suggests, is an understanding of the properties of what he called ‘triangular numbers’, and so we begin by looking at what Harriot himself knew or discovered about such numbers … The simplest triangular numbers are those that arise from stacking pebbles or dots in equilateral triangles … If such triangles are imagined stacked vertically in decreasing order of size, they form triangular pyramids (or tetrahedra) and the number of pebbles or dots in successive pyramids are clearly sums of consecutive triangular numbers … Harriot called these numbers ‘pyramidal’.54

For example, the number of pebbles in triangles (tiers, counting from the top) of increasing size are 1, 3, 6, 10, 15, 21; if one counts the number of pebbles contained in the new pyramid formed each time a new base-tier is added, one has 1, 4, 10, 20, 35. As Stedall says, "the number of pebbles or dots in successive pyramids are clearly sums of consecutive triangular numbers." (That is, 1 + 3 = 4, 1 + 3 + 6 = 10, 1 + 3 + 6 + 10 = 20, 1 + 3 + 6 + 10 + 15 = 35, etc.) "Each triangle (or pyramid) is defined by the number of objects in a single side, 1, 2, 3, 4, 5 … for which reason these numbers were sometimes called ‘laterals’ or ‘sides’ or ‘roots’. These in turn are made up from simple units 1, 1, 1, 1." When arranged in rows, as they are in some of Harriot’s manuscripts, this array of numbers "is now usually known as Pascal’s triangle," though Blaise Pascal was born 63 years after Harriot. The numbers in each row give the binomial coefficients, as Harriot found in his algebraic work. Moreover, "the lateral numbers exhibit a constant difference (namely, 1), the triangular numbers a constant second difference, pyramidals a constant third difference, and so on. Harriot’s theory was concerned with any sequence of numbers that has constant first, second, third, or higher differences." The calculation of "constant differences" is involved in the interpolation of data points between other data points, as Harriot undertook to do in his determination of meridional parts ("the adjustments needed at each degree of latitude in order to calculate an accurate position on a constant compass bearing")—a practical instance of the relationship between a geometric continuum and a mathematical series.55

Of course, Harriot did not need to learn about pyramidal piles of simple units from the Algonquians. But what is an invention, if not the recognition of the possibility of novelty in the familiar? Christopher Miller and George Hamell have written of an "identity of berries and beads" that is attested in many Algonquian languages56—a conceptual equivalence that greatly enhanced the value of European beads as trade goods, as they powerfully combined a supernatural value associated with berries and another value associated with materials like crystal and copper, which shared qualities of brightness and hardness with the beads.57 As Miller and Hamell observe, "what were ‘toys’ to Verrazzano and ‘trash’ to John Smith were to the Woodland Indians powerful cultural metaphors that helped them to incorporate novel items and their bearers into their cognitive world"; it is generally by such a process, the same authors affirm, that "the unknown is made known."58 Thus a hard round thing, for the Algonquians, was not just a pebble or a bullet, but a representation of the very unit of significance and complexity. And so it is for Harriot, as he imagines stacking up triangular numbers.


One of the few scholars to have considered Harriot’s mathematics in relation to any aspect of his New World experience is Amir Alexander. Emerging from Alexander’s account as particular concerns for Harriot are the elusive notions of void in physics and nullity in mathematics, and their relations respectively to physical substance and geometric continuum. And emerging as a distinct and consequential innovation of Harriot’s is his solution of the Mercator problem— how to chart a straight course on a curving Earth—by dividing the curve into small sections and positing a straight line in each, the result being an approximated curve "composed of an infinite number of straight segments, each bearing a fixed ratio to its predecessor."59 All of these matters are cited by Alexander in support of a claim that Harriot’s mathematics express or reflect a culturally specific but at the same time ubiquitous "narrative of Elizabethan expansionism," wherein "supposedly smooth surfaces were found to be lined with deep cleavages, leading to great wonders in the interior."60

I am, at one level, in agreement, and at another dissatisfied. I think that Alexander has pinpointed, with what he calls a "narrative," just the kind of basic scenario, grounded in embodied physicality, that cognitive theorists call a "conceptual frame" and treat—persuasively, in my opinion—as an unconscious organizing principle and basic building-block of thought. Alexander has noted the seeming recurrence of a particular conceptual frame, and has put the case, plausible in principle, that this frame exerted a powerful influence in Elizabethan culture and thought. It seems to me, though, that he has greatly overstated the cultural specificity of the conceptual frame that one could call "finding a passage" or "discerning something hidden"; and also that his use of Harriot’s mathematical ideas to confirm what he posits to be true of Harriot’s whole society does little justice to the specificity of Alexander’s own consideration of those mathematical ideas, or even, finally, to their importance as ideas.

In this topic, I have tried to treat Harriot’s work more flexibly and pragmatically. Blend-theory, it seems to me, paradigmatically allows such an approach. It is a nuanced account of culture and its creations that takes seriously both individual subjectivity and individual agency. Harriot lived and worked at the intersection of two historical beginnings: that of English America, and that of modernity in science and mathematics. He was a key innovator, an individual historically significant for the power of his thought; not just "power" in the sense now most common in the humanities, but in the sense of an extraordinary ability to hold many things in the mind, to integrate them where possible, and to represent those dynamics abstractly. There was much, in the cultural landscape of "Virginia" and its challenge of language-decipherment, that Harriot may or might have learned about meaning-construction and thought processes: about possibilities of calculation latent in shells, straws, and stones; about solving the riddle of an unknown expression; about the linking, sorting, substitution, and distribution of mental spaces; about the logic of another language, the logic of his own, and the possibility of thinking outside language altogether. As he blended old world and new, I have argued, so Harriot may have blended mathematics and linguistics, invention and discovery.

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