Mathematics, Substance and Surmise: Views on the Meaning and Ontology of Mathematics (2015)
The linguistic status of mathematics
Micah T. Ross^{1 }^{ }
(1)
National Tsing Hua University, Hsin Chu, Taiwan
Micah T. Ross
Email: micah@micahross.com
Abstract
Mathematicians have often discussed mathematics as a language. Common linguistic categories have analogous mathematical objects. A comparison of linguistic categories and mathematical objects is developed with reference to the early history of mathematics.
1 Introduction
A modern aphorism declares mathematics a universal language. Because this aphorism presents a complicated philosophical proposition which assumes definitions of mathematics, language and universality, the appearance of the aphorism and its ultimate source merit identification. A well-known instance of this aphorism occurs in the 1997 movie Contact, when the character Ellie Arroway states “Mathematics is the only true universal language.” Carl Sagan did not include the phrase in the 1985 novel [37], but the dialog echoes [34, p. 18], which describes the composition of the Pioneer plaque in “the only language we share with the recipients: science.” Before he drafted the Pioneer plaque, Sagan had addressed the question of extraterrestrial communication and explicitly cited Hans Freudenthal [35, p. 150] who had designed the artificial language Lincos to contact alien life.^{1} In his work, Freudenthal developed, but did not cite, a brief description of an earlier artificial language, Astraglossa, by Lancelot Hogben.^{2} Both Freudenthal and Hogben identified mathematics as the key to extra-terrestrial communication. In contrast, Sagan initially identified science as a language. In so doing, Sagan took up a line of metaphor which leads back to Galileo. In a famous metaphorical passage of Il Sagiattore, Galileo described natural philosophy as written in a book, which he identified as the universe, “in mathematical language, and the characters are triangles, circles and other geometrical figures (in lingua matematica, è i caratteri son triangoli, cerchi, ed altre figure geometriche)” [19, p. 25]. Semioticians may note, and mathematicians may decry, the exchange of the signifier (mathematics) for the signified (natural philosophy), but Sagan frequently used language as a metaphor. Sagan claimed that “DNA is the language of life,” “neurons are the language of the brain,” and “science and mathematics” are the language of all technical civilizations [36, p. 34, 277, and 276]. Regardless of whether Sagan recanted the materialist heresy in the screenplay of Contact, the construction of artificial languages for the expression of mathematical statements differs from the proposition that mathematics is a language, universal or otherwise.
While Sagan may have been its most powerful publicist, the aphorism does not lack supporters. In 1981, Davis and Hersh confirmed the currency of the aphorism among mathematicians [11, pp. 41–43] and even tentatively proposed an analogy between grammar and mathematics [11, p. 140, 156]. Davis and Hersh proposed that “mathematical adjectives are restrictors or qualifiers” but declined to press the grammatical analogy. Others had taken up that standard before. In 1972, Roger Schofield appended an extended grammatical analogy to a chapter titled “Sampling in Historical Research” [38, pp. 185–188]. Schofield used the analogy as a pedagogical tool to acquaint social scientists with mathematical expressions and identifies the nouns, adjectives, verbs, and adverbs of mathematics. The analogy of subscript indices with adjectives has been adopted in at least one other introduction [33, p. 11]. In these instances, the aphorism appears as a metaphor to encourage mathematical novices.
For some, though, the aphorism ascended to dogma. In 1972, Rolph Schwartzenberger wrote that
My own attitude, which I share with many of my colleagues, is simply that mathematics is a language. Like English, or Latin, or Chinese, there are certain concepts for which mathematics is particularly well suited: it would be as foolish to attempt to write a love poem in the language of mathematics as to prove the Fundamental Theorem of Algebra using the English language” [39, p. 63].
Despite the clarity of the sentiment, the aphorism appeared alongside appeals to unnamed authorities and a tacit acceptance of the so-called Sapir-Whorf Hypothesis. Perhaps Schwartzenberger numbered Alfred Adler among his colleagues because in the same year, Adler declared that
Mathematics is pure language—the language of science. It is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms. (In a spoken language, there exist words, like “happiness,” that defy definition.) It is also an art—the most intellectual and classical of the arts” [1, p. 39].
Adler afforded some compromise, but a published conversation between Jean-Paul Changeux and Alain Connes drove the point further. Changeux opened with a bold assertion and followed with a related question: “Mathematical language is plainly an authentic language. But is it therefore the only authentic language?” Connes answered “It is unquestionably the only universal language” [8, p. 10]. The strength of the sentiment outweighs the clarity of the qualifications.
The aphorism may have entered academic parlance because it neatly juxtaposes unconscious assumptions about mathematics and language. These unconscious assumptions ultimately derive from romantic notions about both mathematics and language. By comparing these unstated assumptions, the aphorism demonstrates the verbal artistry of a poet, a fact confirmed by an earlier popular version of the dictum. In 1939, Rukeyser published the poem “Gibbs,” which posthumously inserted the phrase into the mouth of Josiah Gibbs [31]. Poets are not wont to cite sources, but in 1942, Rukeyser developed her interests into a biography and popularized the aphorism by titling a chapter “Mathematics is a language.” In this chapter, Rukeyser reports the origin of the phrase:
At home, in New Haven, Gibbs was preparing for a journey to Buffalo. A story is told of him, the one story that anyone remembers of Willard Gibbs at a faculty meeting. He would come to meetings—these faculty gatherings so full of campus politics, scarcely veiled manoeuvres, and academic obstacle races—and leave without a word, staying politely enough, but never speaking.
Just this once, he spoke. It was during a long and tiring debate on elective courses, on whether there should be more or less English, more or less classics, more or less mathematics. And suddenly everything he had been doing stood up—and the past behind him, his father’s life, and behind that, the long effort and voyage that had been made in many lifetimes—and he stood up, looking down at the upturned faces, astonished to see the silent man talk at last. And he said, with emphasis, once and for all:
“Mathematics is a language.” [32, pp. 279–280].
Rukeyser may have held this sentiment, but Lynde Wheeler, Gibbs’ student, revoked both poetic license and semiotic insight:
[H]e was nevertheless a very regular attendant at faculty meetings. It was in connection with his infrequent participation in the discussions at these meetings that those of his colleagues not on the mathematical or physical faculties gained the greater part of their knowledge of him. There is ample evidence that his comments on such occasions were pithy and to the point; but because he always approached every problem, whether of science or of the ordinary affairs of life, from an original and frequently an unconventional point of view, some of his contributions to these discussions were not always appreciated in a sense he intended. Thus his injection of the thought, “mathematics is a language,” into a discussion of the revision of the language requirements for the bachelor’s degree was in some quarters regarded as an irrelevance; whereas for Gibbs it meant that if language is a means of expressing thought then mathematics is obviously in the same category in the educational scheme as Greek or Latin or German or French, and logically the claims of no one of them can be ignored in planning a balanced course of study [44, p. 173].
Rukeyser may have drawn inspiration from Nathanael West’s phrasing of the aphorism in Miss Lonelyhearts, “Prayers for the condemned man will be offered on the adding machine. Numbers constitute the only universal language” [43, p. 30]. Because West addressed themes of religion, despair and disillusionment, he may have intended a satire of the Adamic language.
Before West, Joseph Fourier described analytical equations in terms of language. “There may not be a more universal and simple language, more free of errors and obscurities, that is to say, more worthy to express the invariable relations of natural objects” (Il ne peut y avoir de langage plus universel et plus simple, plus exempt d’erreurs et d’obscurités, c’est-à-dire plus digne d’exprimer les rapports invariables des êtres naturels) [15, p. xiv]. However, Fourier did not describe mathematics as a language. Fourier praised analytical equations at the expense of other mathematical systems. In the same passage that praised analytical equations for their universality, Fourier limited their domain to the representation of natural objects. Aside from Galilean references to the universal language of nature, earlier authors rarely described mathematics as a language, universal or otherwise. Language was the domain of the arts. The motif of a universal language predates its identification with mathematics. In 1662, John Evelyn claimed that “For Picture is a kind of Universal Language” [14, p. 140], and in 1736 James Thomson sang in “Liberty” that “Music, again, her universal language of the heart, Renew’d” [41, pt. 4, ll. 246–7]. Perhaps the aphorism reflects a peculiarly modern anxiety. West developed a pessimistic view of numeric language, and Rukeyser inverted the cold connotations of mathematics in support of an unconventional hero. A universal language of mathematics appeared as an element of a post-religious age when alien contact seemed a real possibility. For their predecessors, language remained a religiously charged topic and most references to a universal language recall the linguistic state of man before the confusio linguarum at the Tower of Babel described in Genesis 11:2, which may be traced to the Sumerian epic Enmerkar and the Lord of Aratta.
An analysis of the validity of the proposition that “mathematics is a language” depends on the definitions of both mathematics and language. Because neither field has a universally accepted definition, the semantic status of either is open to the fallacy of special pleading. Linguists often limit the definition of languages to vocal expression or to potentially descriptive communication or to other constraints; in contrast, semioticians generally consider signs in any medium or mode capable of forming a language. Thus, by some semiotic definitions, mathematics is a language because it forms a system of symbols which communicates to a community. Francophone philosophers distinguish these uses with the division between langue and langage but the distinction does not survive in English. The validity of the aphorism has ontological ramifications. The status of mathematics as a language could instantly clarify other discussions. For example, languages are not discovered by their speakers; they are developed. If the evaluation of the aphorism is momentarily abandoned, an expansion of the aphorism according to linguistic methods elucidates differences between mathematics and language. These differences, in turn, may be clarified by recourse to the historical development of mathematics.
2 Mathematics as a Language
Without an axiomatic definition of languages or mathematics, a direct proof of the equivalence between these two abstractions cannot attain Euclidean certainty, but an empirical relationship between the two ideas may yet be evaluated. A structuralist method may be adapted from the proofs of congruent triangles. Given the proposition that mathematics is a language, if the two ideas are congruent, then the elements of the two ideas ought to be congruent. In other words, if mathematics is a language, it ought to have categories analogous to linguistic categories. If mathematics lacks an analog to any of these categories, mathematics may yet possibly be a language by some definition, but it would differ from the languages under the domain of linguistics.
Linguists generally hold that every language has open classes of words called syntactic categories which can be identified as verbs, nouns, and adjectives. Because this tenet has empirical support but no rationalistic necessity or justification, challenges to its universality occasionally arise. Thus, claims arise that a given language lacks verbs, nouns, or (most frequently) adjectives. These claims fall into two categories: constructed languages created to challenge linguistic assumptions and descriptions of natural languages. Constructed languages do not relate to the aphorism which depends on a naturalistic understanding of language and the descriptions of verbless, nounless, or adjectiveless languages often permit other descriptions and are not universally accepted. Those intent on establishing a linguistic outlier to justify the aphorism might more soundly insist on a semiotic definition of language.
A language may have categories other than verb, noun, and adjective, but these three categories seem minimally necessary [12, vol. 2, p. 37–8, 62]. Languages assign words to these categories independently of their meaning. Thus, English may express the idea “he is good,” but another language may express goodness with a noun. (Even English permits the construction “he is a saint.”) Yet another language may express this meaning with a verb. (Here, English struggles. Perhaps, in some contexts, “he behaves” expresses the same sentiment.) Nor must a particular word remain in a single category. One may butter bread, or bread a chicken, or be too chicken to complete a dare. Each change of category creates a new lexeme. The two words may be homophonous, but they are categorically separate. If mathematics is a language, it ought to have categories analogous to verbs, nouns, and adjectives.
3 The Verbs of Mathematics
In the case of verbs, mathematics conforms easily with natural languages. Some grammatical systems divide verbs into stative verbs (which describe states of being) and dynamic verbs (which describe actions). Modern mathematical expressions present a ready analogy to both. Mathematical relations such as “is greater than,” “is less than or equal to,” or “is a subset of” resemble stative verbs, whereas operations such as addition or subtraction indicate the actions of mathematics.
Languages apply different morphological processes to verbs in order to mark tense, aspect, modality, or evidentiality. In modern mathematical expressions, these morphological processes are largely absent. The rules governing orders of operation encode a local, relativistic tense, but two unrelated operations cannot be compared.^{3} Not all languages communicate the same information about verbs, nor did any Adamic language once contain all possible information about verbs. Languages develop within cultures, and collective choices determine which features of verbs are developed. Mathematics, too, develops within cultures, and the aspectless nature of operations may reflect the value which mathematical culture places on a rhetorical preference for abstraction and generalization.
Just as every language has a category of verbs, every axiomatic mathematical system necessitates a small set of fundamental operations. The number of fundamental operations has changed throughout the history of mathematics. Johannes de Sacrobosco reported nine fundamental operations of numeration, addition, subtraction, duplation, mediation, multiplication, division, progression, and the extraction of roots in De Arte Numerandi (c. 1225). For the past millennium, the trend has been to reduce the number of fundamental operations. Most modern texts identify the four fundamental operations as addition, subtraction, multiplication and division but limit their range to arithmetic. However, modular expression has recently appeared as a fundamental operation of arithmetic, and current mathematical hearsay attributes the insight to Martin Eichler. Different sub-fields of mathematics specify different fundamental operations, and the number of these operations can vary within a field. For example, as developed by Gödel [20, p. 35], set theory needs eight operations, but later set theorists [24, pp. 178–180] have increased that number (even while retaining the name “Gödel operations”). Other branches of mathematics have clarified that fundamental operations need not resemble algorithmic executions. Numerical analysis includes “table lookup” as a fundamental operation [45, p. 1]. Just as modern linguists risk imposing the grammar and categories of their native language on the languages they study, when modern mathematicians attempt to identify the fundamental operations in ancient mathematics, they hazard projecting modern mathematical structures onto ancient mathematics.
3.1 Mesopotamia
Although Mesopotamian scribes relied heavily on lists, no published cuneiform tablet lists the fundamental operations of mathematics. Two approaches have been used to identify the fundamental operations, with different results. In his 1935 landmark study of cuneiform mathematics, Otto Neugebauer undertook a mathematical approach toward the identification of the fundamental operations of Babylonian mathematics [27, pp. 4–8]. Neugebauer recognized that lists and tables formed a large portion of the education of scribes and derived a set of fundamental Babylonian operations from collections of tablets used for pedagogy. Neugebauer noted that the structure of these texts resembled non-mathematical economic texts of supplies, lists, and inventories. Thus, he discounted nearly 100 tablets of metrical conversions as non-mathematical. Then, he sorted the mathematical tablets according to their contents. The resulting categories included tables of multiplicative reciprocals, multiplication, squares, and cubes. A small sample of texts also tabulated geometric progressions, but the orthodoxy of geometric progression as a fundamental operation was not clear. Thus, Babylonian mathematicians had composed reference works for four or five fundamental operations. However, in the course of their computations, Babylonian mathematicians also employed addition and subtraction. Perhaps, these operations were too intuitively obvious to merit the composition of standard reference texts, but their inclusion would increase the number of fundamental operations to six or seven.
Recently, Christine Proust proposed that although the metrical tables do not conform to modern lists of fundamental operations, these tables shared a genre with Babylonian mathematics [29]. Babylonian scribes associated the shape and size of a tablet with the use for the writing on that tablet. Within the Babylonian system of cultural axioms, then, the decision to separate the tables of metrological conversion was unwarranted. Objections to the inclusion of metrological conversions as fundamental operations illustrate the seductive power of cultural perspective. A modification of the textual corpus allows modern mathematicians to state the fundamental operations of Babylonian mathematics succinctly, but numerical analysts are not held to the same standard. The exclusion of “clerical” elements by Neugebauer also reveals the rhetorical presumption that mathematics is proscriptive and executive but not descriptive. That is, by removing troubling elements from the history of mathematics, the illusion may be maintained that mathematics embodies certain (potentially universal) rules and the instructions for using these rules but denies evidence of the connection of mathematics to the culture that created it.
Jens Høyrup adopted a philological strategy for identifying the fundamental operations of Babylonian mathematics [23, pp. 18–32]. This approach focused on the distinct vocabulary for eight operations: two types of addition, two corresponding types of subtraction, three types of multiplication, and one geometric construction, of which a special case yields squares, cubes, and their roots. Babylonian mathematicians separated addition into two operations and used different verbs for the two processes. One type of addition applied to operations which were not considered commutative. The verb (wasābum) for this operation took the augend as an object and expressed the addend as the object of a preposition (ana). The summation retained the identity of the augend, albeit with a greater magnitude. The other type of addition was considered commutative and merely united two quantities with a conjunction (u). The second operation applied to addends which were mathematical abstractions, as was the result. These two types of addition spawned analogous types of subtraction, also marked by word choices. The verb (nasāḫum)for the inverse of the first type of addition took the subtrahend as the object, marked the minuend with a noun (libbi) used adverbially, “from the heart of,” and reported the result as the remainder (šapiltum). The inverse of the second type of addition compared two magnitudes but seems to have been open to some variation in the Babylonian phrasing: A can “go beyond” (itter) B, or B can “be smaller than” (matûm) A.
Three words denoted separate operations of multiplication. The development of Babylonian mathematics left traces on the words chosen for calculations. In the first type of multiplication, Babylonian mathematicians used a word from their cultural predecessors, the Sumerians. This Sumerian verb (a.rá) described an operation analogous to modern multiplication. This verb used the multiplicand as the subject and the multiplier as the object. The terminology of this construction seems to have arisen from the measurement of areas. A purely Babylonian category of multiplication seems to have resulted from the determination of volumes. The verb for this operation (našûm) took the multiplicand as the subject and denoted the multiplier by a preposition (ana). Although it may have begun as a volumetric construction, this type of multiplication extended to constants, metrological conversions, volumes found from a base and height, and areas of pre-existing figures. A third verb (esēpum) denoted duplation. Unique among the multiplicative operations, duplation has an inverse, which could be indicated by either the verb (ḫepûm) or the noun (bāmtum). However the operation of bisection was denoted, it constituted another fundamental operation even though no generalized division formed a fundamental operation for Babylonians. Instead, the identification of a multiplicative inverse constituted a fundamental operation. Because Babylonians used a Sumerian phrase for reciprocals (igi # gal.bi), reciprocality, like the first type of multiplication, had probably been long recognized as a fundamental operation.
In modern mathematics, geometric constructions form a category independent of arithmetical functions. Since Euclid, geometers have limited themselves to fundamental constructions, often denoted by their tools: the compass and the straight-edge. Euclidean geometers connect two points with a line, draw circles with a given center and a point on the circumference, identify the point at the intersection of two lines, locate the point or points at the intersection of a line with a circle, and specify the point or points at the intersection of two circles. Mathematicians recognize these constructions, but they are not arithmetic operations, fundamental or otherwise. Babylonian mathematicians did not subdivide their mathematics in the same way. They “constructed” rectangles to solve arithmetic problems. To be sure, these geometric constructions were often mental constructs, but given two magnitudes, Babylonian reckoners could construct a rectangle with sides of those magnitudes. The terminology of this operation is complex, but the two magnitudes serve as objects of a causative verb, “to make hold” (šutakullum). The expressions for square numbers either use this verb but repeat the same magnitude or insist on the equality (mitḫartum) of the magnitudes. Through this construction, the operations for squares and square roots share the same vocabulary. The only question is whether the side or the area of the relation is known. Another magnitude can even be added to the expression to invoke the determination of cubes or cube root, but the fundamental operation remains the same geometric construction.
Neugebauer adopted a mathematical approach which ignored textual genres, and his conclusions conformed to the expectations of modern mathematicians. Whatever criticisms might be raised against this approach, this effort managed to establish that a pre-Greek civilization conceived of mathematics beyond simple arithmetic. Høyrup opted for an etymological method and clarified the mathematical meaning of abstract expressions through reference to natural language. His conclusions depend on the use of Babylonian words in literary contexts which employ natural language. Nonetheless, his careful study of word choice reveals conceptual differences between Babylonian mathematics and modern mathematics. The ability to isolate mathematical differences by reference to Babylonian language calls into question whether Babylonian mathematics existed as an entity ontologically separate from natural Babylonian language. Proust extended the connection beyond language to reunite Babylonian mathematics with Mesopotamian culture. Perhaps the fact that Babylonian mathematicians and modern mathematicians differ in their assessment of the number of fundamental operations challenges the premise that mathematics is universal.
3.2 Egypt
If mathematics constitutes a universal language, similar constructions should appear throughout mathematical texts. In other words, the mathematics of one culture should resemble the mathematics of another. Whereas Babylonian mathematics is known from hundreds of brief texts, Egyptian mathematics is known from a small handful of texts, particularly the Rhind Mathematical Papyrus (pBM 10057 and 10058; hereafter, RMP). The philological and mathematical approaches of Høyrup and Neugebauer are echoed in the publishing history of the RMP. When Peet edited the RMP, he explicitly acknowledged that “[t]here is only one truly fundamental process in arithmetic, that of counting” [28, p. 11], but he focused on an elucidation of the vocabulary of mathematics. When Chace reedited the same papyrus, he declared that Egyptian mathematicians had a “thorough understanding of the four arithmetical operations,—addition, subtraction, multiplication, and division” and proceeded to demonstrate the existence of these operations [7, p. 3]. Like the cuneiform corpus, no Egyptian mathematical text explicitly addresses the question of fundamental operations, but an empirical understanding may be gathered from the RMP. Sometimes, Egyptians identified operations similar to those used by Babylonians. In some ways, they differed. In every case, though, the conception of Egyptian mathematics is directed by the expectations of the reader.
The RMP contains examples of addition and subtraction. Usually, a preposition (ḥr) indicates the operation of addition. The augend stands as the noun to which the preposition phrase is a complement; the addend is the object of the propositional. This expression may be expanded to include a verb (either wʒḥ, di, or dmḏ), in which case the augend serves as the object of the verb, and the addend again serves as the object of the preposition. The two instances (“Problem 41” and “Problem 42”) which use the verb di to denote addition also relate to the area of circles. However, not every detail of preservation relates to mathematical conceptions. As Jöran Friberg has hypothesized, the RMPmay be a “recombination text” which excerpts selections from other texts and recombines them into a new text. The variations in verb choice may reflect geographical or temporal variations in dialect among the source material [17, p. 26]. Whereas the methods of Høyrup work well for many texts which share the same tradition, they do not apply equally well to a single text which draws from varying traditions: the Egyptian choice of verbs for addition has eluded mathematical clarification. Whereas a preposition denotes addition, Egyptians considered subtraction to be a verbal activity (either ḥbi or skm). The subtrahend is the object of the verb, and the minuend is indicated as the object of a preposition (either n or ).
Modern analyses of Egyptian mathematics often focus on the algorithm behind multiplication because it differs from modern techniques. Nevertheless, it can be introduced easily. Babylonian multiplication and modern multiplication work in roughly the same way: a memorized value is assigned to each pair of digits, and those values are combined in the case of multiple digits. Throughout the RMP, though, multiplication is accomplished by successively doubling up to half of the multiplier and subsequently summing those doublings which correspond to a binary expression of the multiplier. As in Babylon, doubling formed a fundamental operation of Egyptian mathematics. Unlike Babylon, direct multiplication seems to be a secondary operation. Peet noted the primacy of doubling among Egyptian calculation, but he omitted to mention that the Egyptians themselves seem not to have had a word for doubling [28, p. 15]. A verbal phrase (wʒḥ tp) indicates the generalized, secondary operation of multiplication, but this phrase uses a preposition (m) to indicate the multiplicand and prepositional phrase (r spw) to indicate the multiplier. Literally translated, the phrase implies counting by a number other than one: “incline the head with the multiplicand for the multiplier’s number of times.” The mental image behind the phrase seems to be that the reckoner would nod his head with each increment of the multiplicand, as if counting aloud by sixes or elevens. (Against this interpretation, though, wʒḥ had long been connected to addition and tp has a wide range of metaphorical uses.)
Despite this descriptive metaphor, the RMP never executes multiplications in this way. Scribes are not expected to count by arithmetic progressions; they are expected to double efficiently. The same verb also denotes division, again with a preposition (m) to indicate the divisor and the dividend as the object of an infinitive in a prepositional phrase (r gmi). (Literally translated, the phrase reads: “incline the head with the divisor to find the dividend.”) As it did in the case of multiplication, the RMP preserves variations in the vocabulary of division. Another verb (nis) takes the dividend as its object and indicates the divisor by a preposition ( ). Another fundamental operation of Egyptian multiplication appears in “Problem 69” of the RMP: the increase of a multiplier by tenfold. In Egyptian hieroglyphics, each power of 10 necessitated an independent grapheme. By exchanging the graphical expression of a number, Egyptian mathematicians had recourse to multiplication by 10 as a fundamental operation. Although such multiplication is tabulated and occasionally employed, no specific term defines the operation. Because both doubling and multiplication by 10 are repeated throughout the admittedly limited corpus of Egyptian mathematics, they may be accepted as fundamental operations. The acceptance of these operations as fundamental suggests an explanation for “Section ” and “Section ” of RMP. Rather than merely simplistic tables of division, these “problems” may be explained as tables of inverse operations for fundamental operations of Egyptian mathematics. By interpreting these passages as inverse tables of fundamental operations, Egyptian mathematics shares similarities with Babylonian mathematics.
This interpretation derives from a consideration of tables as the fundamental element of composition in the RMP, and it holds a certain well-ordered allure. The RMP uses two fundamental operations and begins with two tables of inverses. However, the interpretation depends on an assumption that the composition of the RMP exhibits a (pre)-Aristotelian unity within the papyrus. The author of the papyrus does not guarantee a cohesive structure to its composition. In fact, the same internal evidence (variations in verb choice, repeated problems, thematic grouping of contents) which suggests that the papyrus was composed as a recombination text could be used to dismiss this reading. As stated before, the conclusions about the structure of Egyptian mathematics depend on the expectations of the analyst. Difficulty in the reconstruction of the mathematical ideas underlying the composition of the papyrus reveals limitations in the understanding of Egyptian mathematical culture.
Other tabular sections of the RMP have passed largely unconsidered. Like the Babylonian tables of metrological conversion, they have been assumed to be pedestrian in nature. However, given that “table lookup” is accepted as a fundamental operation of numerical analysis and similar metrological tables formed a large part of Babylonian mathematics, these portions of the RMP may preserve operations fundamental to Egyptian mathematics. “Problem 47” reports subdivisions of 100 quadruple grain measures (4-ḥqʒt) into simple measures (ḥqʒt). “Problem 80” and “Problem 81” present tables for the conversion of fractions of grain measures (ḥqʒt) into volumetric measurements (hnw). Oddly, the RMP uses idiosyncratic Egyptian units of area but includes no tables of conversion. Whether this absence represents an oversight of the scribe, his limited source material, or an exclusion of area conversion from the fundamental operations again depends on assumptions about the state of Egyptian mathematics.
Although the RMP opens with a somewhat grandiose title which promises “the model for inquiring into affairs, for knowing all which is unclear [and deciphering] every mystery,” it does not explicitly identify any fundamental operations. The assumption that the fundamental operations remained constant among all Egyptian mathematicians reflects another set of assumptions about mathematics, language, and culture. The Moscow Mathematical Papyrus (pGolnischev, henceforth MMP) contains fundamental operations which do not appear in the RMP. Specifically, the scribe of the MMP assumes that squares and square roots can be found directly. The RMP and MMP are both documents of the Middle Kingdom but do not share the same fundamental operations.
The eleven operations (addition, subtraction, duplation, mediation, ten-fold multiplication, decimation, identification of multiplicative inverses, two types of metrological conversion, squares, and square roots) identified here as fundamental to Egyptian mathematics derive from a comparison of the Egyptian mathematical texts with Babylonian mathematical texts. To be clear, the assumption of a parallel development between Egypt and Babylon invites as many potential risks as the assumption that mathematics developed in a continuous unbroken chain from the RMP to the modern rules of arithmetic. However, at times, the perspective suggested by the Babylonian mathematical texts clarifies the compositional style of the RMP. In the case of Babylon, historians of mathematics have access to a wide corpus of texts in several genres. Recent examinations have revealed how modern editors have fit this corpus to support mathematical assumptions drawn from modern mathematics.
3.3 Universality
In order to establish an analogy between language and mathematics, the number of the minimal set of operations is not particularly important, but the variations between Babylonian, Egyptian, and modern mathematics suggest that if mathematics is a language, different mathematical communities communicate in divergent dialects. Moreover, within Babylonian and Egyptian mathematics, mathematical texts must be understood through recourse to the metaphors of natural language. Little surprise, then, that the mathematics of different cultures reflect variations between those cultures. Just as they spoke different languages and wrote in different scripts, they held different fundamental mathematical concepts. Even the fields of modern Western mathematics, in which mathematicians may be assumed to have had similar educations and access to the same publications, develop different mathematical objects. In the case of modern mathematics, at least, the objection may be raised that the mathematicians address different problems. The objection applies less easily to ancient mathematics. As Friberg noted [17, pp. 25–26], the types of problems solved by Egyptians and Babylonians are roughly similar. Nonetheless, the mathematics of these ancient cultures differ no less than their languages and cultures.
Although variations among the fundamental operations erode the notion of universality, the fundamental operations of each mathematical culture fill a category analogous to the syntactic category of verbs. Indeed, the fundamental operations may be repeated, combined, or repeated and combined into more complicated expressions. Beginning in the nineteenth century, mathematicians developed special functions so extensively that operations resemble an open category.^{4} Essentially, special functions enabled a morphology of verbal phrases. In this way, every function from the fundamental operations to trigonometric functions may be considered analogous with a lexeme in the syntactic category of verbs. To make an analogy with the field of genetic linguistics, which seeks to compare languages grouped together by similarities, much work remains to be done in “genetic mathematics,” particularly among those systems which did not originate from a classical root. Some similarities between Babylon and Egypt might be explained by contact mathematics, but a monogenesis of mathematics remains as elusive as a universal grammar or the monogenesis of language.
4 The Nouns of Mathematics
The category of nouns seems readily congruent with several mathematical categories. Three categories of mathematical objects present an analogy to nouns. Perhaps sets have the strongest claim to being universal nouns but the primacy of sets in mathematics derives from the labors of Bourbaki to establish all of mathematics as a logical development of set theory. A concerted intellectual effort has been applied to describing all mathematics in terms of sets. Notwithstanding the potential universality of set theory, this undertaking is a modern endeavor. Of course, mathematics could have been conducted with no awareness of set theory. After all, Vedic hymns were composed before Pāṇini elucidated Sanskrit grammar. However, an analysis of pre-modern mathematics in terms of set theory invites an attempt to establish continuity retroactively. The descriptions of ancient mathematics from this perspective would have been foreign to people who worked in those paradigms.
The mathematical category of geometrical objects also conforms to the linguistic category of nouns. The geometric objects of Babylon are not so different from the geometric objects of Euclid’s Elements but the techniques of proof and problem solving differ greatly. Cuneiform mathematics recognizes right angles, can identify similar angles, but does not compare dissimilar angles. Mesopotamian mathematicians could identify the orientation of axes and calculate the areas of triangles, squares, and trapezoids or solve for the lengths of line segments of their constituent parts [23, pp. 227–31]. The geometric systems of Babylonian scribal schools differ from the axiomatic system of Euclid in that Babylonians preferred to solve geometric problem through arithmetic or algebraic techniques. Mesopotamian mathematicians were capable of stating geometric rules but they often invoked these rules as arithmetic statements. For example, ancient Babylonians proposed and solved right triangle problems but they relied more on the transformation of Pythagorean triples than geometric constructions. The Nile Valley shared this algebraic Mesopotamian approach. Egyptian mathematicians added a new geometric object, the sqd, or the reciprocal of a slope of an inclined surface but they retained the preference for algebraic methods. Moreover, from a linguistic perspective, the terminology of Babylonian and Egyptian geometry and the statement of problems depends heavily on natural language and the terms for ordinary objects.
As mathematical objects which resemble nouns, only numbers would please ancient mathematicians as much as their modern counterparts. After all, mathematics is often described as the study of numbers. Just as the verbs of a language govern nouns, so operations (and functions) govern numbers. Numbers even operate as nouns in natural languages. For example, in English, “four” is a countable noun which refers to the digit four or any grouping of four units (among other specialized uses), but the classification “countable” belies a significant difference between languages and mathematics. In mathematics, a countable set is a set with the same cardinality as the natural numbers. That is, every element in the set may be paired with a natural number. In linguistics, a countable noun can be modified by a number; an uncountable noun cannot. An uncountable noun is sometimes called a mass noun. For example, a farmer may have two cows which produce milk. In order to discuss the two gallons of milk each cow produces, a measure word quantifies the uncountable noun “milk.” Some languages (such as Chinese and Japanese) have developed systems of measure words, but English preserves only a few measure words. The previous example may be expanded to introduce such a measure word and illustrate how countability, in the linguistic sense, can modify meaning. Thus, the two head of cattle come from different breeds and produce milks of different quality. In this way, countable and uncountable nouns are grammatical categories which do not depend on the nature of the signified object.
4.1 Countability and Infinity
One obvious difference between nouns and integers is that integers are countably infinite, but no language actually has an infinite number of nouns. Although the number of nouns is theoretically infinite, the phonemes of a language are a closed category, and most languages have only a few thousand nouns. Thus, linguists describe the category of nouns as open. That is, limitless new nouns can be generated. The morphology of language is more complicated than the morphology of mathematics. To generate a new noun, a language must add an affix (a morpheme attached to a word stem to form a new word); mathematics must merely increase the value by 1. The complexity of linguistic morphology, though, can be productive. A simple set of rules can be used (sometimes recursively) to generate as many lexemes as are needed from a limited set of phonemes. Linguistic communities differ somewhat in the upper bound for the length of a lexeme, but eventually noun phrases replace unwieldy lexemes. In the case of mathematics, the patience and memory of the reckoner forms a practical limitation.
The difference between the open set of nouns and the infinite set of integers forms one of the earliest ontological problems of mathematics. Because different cultures answered the challenge differently, the expression of large numbers has become a standard opening among historians of mathematics. In Egypt, each power of ten necessitated a new hieroglyphic symbol. A coiled rope denoted a hundred; a lotus flower stood for a thousand. Far from a primitive notation of tallies, scribes displayed an insight into the intersection of cognition and mathematics by subitizing these graphemes. In other words, they arranged the signs into a pattern which allowed the reader to perceive the number of units quickly rather than tallying up the individual marks [13, pp. 521–522]. As noted in the comparison of verbs and operations, replacing one set of subitized units with another set of similarly subitized units rendered multiplication by 10 a primary operation. Each power of ten took its name from a relatively common noun. Whereas the names for the numbers from 1 through 10 must be reconstructed from later alphabetic writings, rendering the vocalizations of large numbers presents little challenge, and the system seems to have had an upper bound in the millions.^{5} Amazingly, a document from the dawn of Egyptian history preserves an instance of a tally in the millions. King Narmer, who may have unified Egypt, counted 1,422,000 goats captured in the course of his wars and inscribed this number on a ceremonial macehead. Egyptians rarely had to count into the millions. The number was rarely used, and as the Egyptian language developed, meiosis weakened the meaning to “many.” The millions took their name from the god of multiplicity, and the signs for expressing numbers beyond nine million are unclear (Tables 1 and 2).
Table 1
Hieroglyphics for powers of 10.
Table 2
Egyptian subitization patterns.
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
⋯ |
⋯ |
|||||||
⋯ |
⋯ |
⋯⋅ |
||||||
⋅ |
⋅ ⋅ |
⋯ |
⋯⋅ |
⋯ |
⋯ |
|||
⋅ ⋅ |
⋯ |
⋯ |
||||||
⋅ ⋅ |
⋯ |
This problem of rendering mathematical infinity in a natural language is not a feature of hieroglyphic writing or some “primitive” state of Egyptian mathematics. Every culture faces this problem. Although modern English speakers might offer a “googol” or “googolplex” as an astoundingly large number, few can name the number which precedes it. Most reckoners, ancient or modern, assume that numbers beyond frequently encountered measurements exist but blissfully ignore them. The Greek number naming system did not originally extend beyond 10,000, but Classical Sanskrit introduced a new word for each power of ten up to 10^{12}. Like the modern scientific notation, the name for the unit of trillions refers to twelve—the power of ten which the unit denotes. Although no number larger than one trillion has been found in Classical Sanskrit, greater units could have been generated by the same rule. An early Buddhist composition, the Lalitavistara Sūtra, describes a number generation contest between the mathematician Arjuna and Buddha. Arjuna named the units up to 10^{53} but Buddha expanded the method geometrically to 10^{421}before sagely abandoning the contest in pursuit of wrestling and archery. Like linguistic communities, mathematical communities are irrepressible at communicating necessary information. Archimedes expanded the traditional Greek system beyond 10,000 in much the same way as Arjuna, but Archimedes bested both Arjuna and Buddha by naming numbers up to 10. Perhaps his wrestling partner and archery coach were busy. Archimedes trumped Buddha, but he recognized the limitations of human numeracy: he never employed numbers larger than 10^{63}.
Whereas Egypt, India, Greece, and Rome generated new words to describe the ever increasing units, Babylon employed a different strategy. Although the Babylonian system of numerals is often described as “base 60” or “sexagesimal,” this characterization is not perfectly accurate. Cuneiform numerals did not contain a set of 59 unique digits. Rather, the Babylonian notation used vertical wedges for units and horizontal wedges for tens. Just as the Egyptian scribes organized their hieroglyphic graphemes for readability, the Babylonian scribes subitized both the vertical wedges of the units and the horizontal wedges of the tens. The positional notation, however, was sexagesimal. For numbers greater than 60, the vertical wedge was reemployed as 60. This system of notation lacks a radix point, so only context distinguishes 141 from 8,460 or 507,600, or some other summation of the corresponding powers of 60 (Tables 3, 4 and 5).
Table 3
Cuneiform powers of 10.
Table 4
Mesopotamian subitization patterns (rotated ≈ 30^{∘} counterclockwise for tens).
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
⋯ |
⋯ |
⋯ |
||||||
⋯ |
⋯ |
⋯ |
||||||
⋅ |
⋅ ⋅ |
⋯ |
⋯ |
⋯ |
⋯ |
|||
⋅ |
⋅ ⋅ |
⋯ |
||||||
⋅ |
⋅ ⋅ |
⋯ |
Table 5
Babylonian positional notation, where n depends on context.
60^{ n} |
60^{ n+1} |
60^{ n+2} |
60^{ n+3} |
… |
[1-59] |
[0-59] |
[0-59] |
[0-59] |
… |
Introductions to the history of mathematics often dwell on this ambiguity, but polysemy is not the interesting point of floating-radix, sexagesimal notation. Languages permit ambiguities and clarify them by context and convention. Two other points deserve greater notice. First, the cuneiform notation is entirely removed from the verbal expression of either numeric grapheme, namely the two vertical wedges and the horizontal wedge. In fact, the trend toward a sexagesimal base is not reflected in the spoken languages of Mesopotamia. The words for seventy, eighty, and ninety derive from the words for seven, eight, and nine. The lexemes for numbers continued to invoke new units with the change of the powers of 10 (meatum for 100, lı̄mum for 1000). Even though the grapheme for one hundred consists of one vertical wedge and four horizontal wedges, the lexeme marked a change based in the powers of ten and did not relate to the appearance of the grapheme. Nor did the graphemes represent a bijection with the lexemes. A single vertical wedge could be read (and perhaps vocalized) as išten (one) or šūšum (sixty, from šeššum, meaning six). Secondly, the inclusion of this numbering system into cuneiform compositions embedded a secondary semiotic system into the language. This embedding relates directly to mathematical ontology. These cuneiform signs were not shorthand for a verbalization. Their interpretation and manipulation relied on an independent set of rules. While these two changes do not constitute a definition of a mathematical entity, they may serve as a bellwether that mathematical expressions have separated from natural language.
4.2 Number and Zero
Like countability, linguists acknowledge number as a grammatical category which can potentially apply to any category of word, but nothing requires the application of this grammatical category number to any word. Languages vary in their identification of grammatical number. For the most part, English marks only singular or plural. Egyptian and Greek had fossilized “dual” constructions for paired objects, such as hands or eyes. Sanskrit and most Semitic languages never abandoned the category of dual. Linguists have even described languages which use trial and quadral as grammatical categories. Some languages include more than one type of plural category, such as the grammatical category “paucal” which refers to a small number of items. In contrast to most natural languages, the nouns of mathematics—that is, numbers—cannot be marked according to whether they are singular, plural, or some other variation. From a linguistic perspective, most mathematical expressions are “numberless.” This numberless aspect of integers reflects an assumption about numbers as mathematical entities: because they are distinguished only by quantity, when that quantity is the same, the entities are interchangeable. Although identifying a language with no grammatical category of number invites teleological errors similar to searching for a language without verbs, nouns, or adjectives, a numberless language seems plausible because some languages, such as Chinese, essentially limit the category of number to pronouns. For example, the pronoun (tā) has a plural form (tāmen), but (men) cannot be appended to common nouns. This limitation does not preclude such languages from discussing numbers. Linguistic communities are irrepressible at communicating necessary information. However, the extreme truncation of grammatical number suggests that such a language could easily exist. Somewhat paradoxically, then, if mathematics is assumed to be a language, from a grammatical perspective, it constitutes a numberless language.
The grammatical state of numberlessness differs greatly from a mathematical state of numberlessness. Mathematical numberlessness has become another standard opening for surveys of the history of mathematics: the story of zero. The standard story of zero acknowledges the Babylonian need to introduce a place holder to separate the various powers of sixty in the floating-radix, sexagesimal notation but then retracts this “achievement” on the grounds that the Babylonians never fully realized the glory of their accomplishment. The standard story gives short shrift to the Egyptians, who, because they wrote with clumsy pictographs, were incapable of writing nothing. Greece is chastised pro forma for its shortsightedness, and India and the Maya receive snippets of Orientalizing adulation to confirm a “global perspective” which conforms to modern, Western political biases. All of which is important, of course, because zero defines “true” mathematics. Without recognition of the importance of zero, the fundamental operations of arithmetic cannot be stated because zero cannot be excluded from divisors without explicit acknowledgement of its existence. As a result, natural numbers could not be separated from the integers, set theory could never develop, and civilization would never advance. The story varies in the choice of historical anecdotes which it articulates, but the perspective rarely changes.
The frequent emphasis on the story of zero reveals the extent to which the history of mathematics has been pressed into a Whiggish view of mathematical development. By limiting the analogy between language and mathematics to modern mathematics, the comparison of nouns and numbers reveals several similarities. First, the integers correspond to nouns. The identification of zero and its exclusion from divisors forms a key grammatical rule which creates categories among the mathematical nouns. Natural numbers can be divisors; the integers are limited to dividends. The rational numbers, then, present an analogy to clauses. The divisor stands as the head of a noun phrase; the operation of division functions as the verb; and the dividend fills the role of an object. The analogy defines the natural numbers, integers, and rational numbers with a small set of grammatical rules. Mathematics may be rebuilt by considering mathematical objects as noun phrases of increasing complexity; set theory survives; and civilization may flourish.
However, as noted in the consideration of the mathematical verbs, neither Babylon nor Egypt recognized division as a fundamental operation. Babylonians considered multiplicative inverses to be a fundamental operation. The Babylonian tables of multiplicative inverses do not report the lack of a multiplicative inverse for zero, but they also usually omit numbers relatively prime to 60, such as 7 or 13. In this way, Babylonians classified commensurate and incommensurate inverses much like modern arithmeticians classify rational and irrational numbers.
On the other hand, because Egyptian mathematicians did not use sexagesimal notation, they were not equally concerned with commensurability. Egyptian mathematicians had two types of fractions. The first of these fractions derived from the inverse of a fundamental operation, namely doubling. This set of fractions are the erroneously named “Horus-eye” fractions and consist of the first six mediations, or, in more modern mathematical parlance, the first six powers of [30]. These fractions, and their graphemes, developed from a system of volumetric fractionalization. Perhaps, at one time, Egyptian mathematicians would have approximated all fractions by adding successive mediations. Just as a binary representation could approximate any multiplier, a summation of dyadic fractions can approximate any rational number with an arbitrary degree of precision. By the time of the earliest Egyptian texts on mathematics, though, this fractional system had been limited to six units and restricted to measures of grain. The second set of fractions invoked multiplicative inverses directly by setting the grapheme for a mouth over the digits of the number for which the inverse was sought. Most historical surveys of mathematics do not rigorously separate these two classes of fractions in their explanations of “Egyptian fraction.” These modern surveys introduce “Egyptian fractions” as unitary fractions, or aliquot fractions, or fractions with a numerator of one, often for the purpose of introducing mathematical exercises. No such entity existed for the Egyptian reckoner.
For Egyptian mathematicians, multiplicative inverses did not constitute an artificially limited set of rational numbers. Instead, the multiplicative inverses constituted another set of mathematical entities, over which the fundamental operations also reigned. Instead of mixed numbers and improper fractions, Egyptian numbers had an integer portion and an inverse portion.^{6} When these inverses were doubled by multiplication, the number of the multiplicative inverse was halved. When those numbers were odd, equivalent expressions for the multiplicative inverses of odd numbers had been tabulated. One example of such a table forms the beginning of the RMP. This table omits zero for two reasons. First, zero is even. Secondly, zero has no multiplicative inverse. The expectation that Egyptians might distinguish between natural numbers and integers imposes modern categories as much as the insistence that they used “fractions.” In fact, an Egyptian word nfr has been found to indicate the starting point in technical drawings and as the final statement of closed accounts (pBoulacq 18) [25, pp. 113–115]. Through names such as Nefertiti, nfr (good) has become one of the few relatively widely known Egyptian words. However, in its uses related to zero, nfr relates to another usage for negation [3].
4.3 Negation
Languages employ a wide variety of techniques for negation, with the negation of the main clause seemingly universally permitted. Depending on the language, though, both nouns and verbs can be negated. Likewise, both nouns and verbs can have positive and negative polarities. The method of negation may vary with tense, mood, or aspect. Negations may rely on grammatical constructions, intonation, or contextual cues. In English, ambiguities relating to these shades of meaning often enable humor or trickery. (“Have you stopped beating your wife?”) The grammar of negation depends on an axiomatic system, but this system reflects culture as much as mathematics. Some languages (such as Greek or French) permit doubly negated constructions as intensifiers. Other languages reject such constructions as contradictory. Still other systems permit such constructions for different purposes. Western constructivist mathematicians distinguished double negations from positive statements, but the earlier Buddhist Madhaymaka school developed the interpretation of a negation of a negation in the details of paraconsistent logic. For them, a doubly negated statement might undermine the categorizations and logic on which the statement and its negation were built. In logical operators, A ≠ A. In a concrete example, A may be the statement, “Harry is a dog;” A would be “Harry is not a dog;” A would be the double negative “Harry is not not a dog.” Perhaps Harry is a wolf. Grammarians and schoolmarms to the contrary, contemporary English often employs exactly such constructions when modifying the scope of negation. Thus, if logicians insist that A ≠ A is nonsensical, their complaint echoes those who would dismiss the example as ungrammatical because of its double negative. Such constructions occur in English and are meaningful. In linguistic terms, even negations are potentially subject to contrastive focus reduplication. If it is allowed that Harry is not not a dog, it must also be permitted that Harry is not a dogdog. The denial of the potential value of such constructions divides the function of negation in language and mathematics. This functional division results from a deliberate limitation of the rhetorical range of mathematical expressions.
Whereas languages often develop negation to reflect the negation of contingencies and possibilities, modern mathematicians depend on negation according to formal logic (which has diverged from natural language). If mathematical negation is infrequent relative to linguistic negation (or at least, the potential for linguistic negation), this fact relates to a tacit division made between natural languages and mathematics. Linguists assume that language is communicative, but communication can be descriptive or executive. Mathematical texts often compose the definitions of terms in descriptive language but operate in executive language. By nature, definitions are positive. Likewise, useful executive expressions also tend to be positive because prohibitive imperatives produce few changes in mathematical expressions. Often, mathematical negation is categorical and depends on formal logic. For example, many proofs arrange ideas to generate a contradiction. At other times, though, mathematicians consider conditions which have a value of zero to express a special property. Like Egyptian scribes, modern mathematicians still find value in equating negation with zero. Taken in the other sense, though, as the identification of an additive inverse, negation parallels the third syntactic category, adjectives.
5 The Adjectives of Mathematics
In the case of adjectives, mathematics differs greatly from natural languages, but the adjectives of natural languages vary more than other syntactic categories. In some languages (like Chinese), adjectives resemble verbs in their negation or their inflections for tense. In other languages (like Latin), adjectives resemble nouns because they are inflected for gender, number, and case. In a small number of languages, adjectives have the properties of nouns and verbs. In another small group of languages (like English), adjectives are distinct from both nouns and verbs. Adjectives perform two tasks [12, vol. 1, p. 113]. First, they state a property. Secondly, they specify the referent of a noun. That is, in the phrase “the green leaves,” the adjective specifies those leaves and not the brown ones. In some languages, adjectives perform other tasks, such as serving as a parameter in comparative constructions or functioning adverbially, but stating properties and specifying referents are fundamental for adjectives. Whereas a rough parallel could be forced into place for both verbs and nouns, the mathematical analog to an adjective eludes simple identification.
5.1 Properties of real numbers
Certainly, adjectives exist with mathematical definitions. Prime, negative, and square all have precise definitions which reflect the mathematical culture which became interested in these properties. These adjectives usually derive from the natural language in which mathematical texts are composed. Generally, mathematicians rely on mathematical definitions of adjectives from their natural language to perform the first task of adjectives, stating a property about numbers. The mathematical expression of these ideas may be standardized, but it develops with reference to the natural languages of mathematical communities. Despite the familiarity of the opening phrase of many proofs “Let p be a prime,” this phrase constitutes an English statement. A German mathematician could compose “Es sei p eine Primzahl.” Lest the term “prime” be considered a linguistic universal among mathematics, Japanese renders the same idea as with no reference to Western terminology.^{7} And, as in natural languages, these definitions evolve contemporaneously with their use. Pre-Greek mathematicians indicated no special interest in prime numbers, but Mesopotamian mathematicians excluded numbers non-commensurate with 60 from tables of reciprocals. The underlying ideas are similar, but the goals of the mathematicians are not. Even in modern mathematics, the definition of prime has evolved [6]. No notation analogous to an adjective class indicates that a number is prime, but the analogy of mathematics to language invokes more than semiotic similarity of notation to script. Some portion of modern mathematics is still rendered in natural language.
If mathematics constitutes a language in more than a semiotic sense, the linguistic elements need not be limited to notation, but some notations do communicate mathematical properties. For example, a preceding “minus sign” denotes negative numbers. Some users of this sign elevate it to distinguish it from the operator for subtraction. Like many elements of mathematical notation, this sign is overdetermined. A simple horizontal line indicates negativity, subtraction, or the complement of two sets. These uses are clearly related, but the composers of mathematical texts are free to introduce new notations with ad hoc definitions. For example, a bar over a variable may indicate the mean value of that variable, a finite sequence, algebraic closure, or a complex conjugate. Egyptian mathematical texts occasionally indicated subtractive quantities with red ink, but cuneiform texts did not have recourse to this method. In fact, cuneiform mathematics did not acknowledge a class of negative numbers. The lack of a given mathematical adjective does not indicate the lack of mathematics. Different languages do not exclusively contain adjectives which map bijectively to the adjectives of other languages. These differences in language reflect differences in culture.
Linguistic descriptions of adjectives often acknowledge differences in the semantic content of adjective classes. The core semantic types form a set of four adjective classes: dimension, age, value, and color. Not all languages have all four adjective classes, but in languages with few adjectives, the adjectives usually embrace these types [12, vol. 2, pp. 73–76]. Just as the mathematical analog to verbs lacks most aspects of natural language and the analog to nouns lacks the grammatical category of number, mathematical adjectives present an unusually narrow range of adjective classes. Ultimately, all mathematical adjectives relate to dimension.
The small set of dimensional adjectives of mathematics may be limited to positive and negative. The other frequently used dimensional adjectives—greater, lesser, and equal—have been described as “relations.” The atemporality of mathematical verbs obviates the second class of adjectives and key difference between numbers and nouns nearly precludes mathematical adjectives of the third class. Because every number has a magnitude and an implicit ordering, any given number always has a single fixed relationship to every other number. Even when nothing can be stated about the relationship between two numbers because nothing is known about them, they exist in a common frame of reference. The same is not true of nouns. Broad categories of nouns may be classified together and marked by grammatical gender, but no relationship exists between the categories, let alone a global ordering. No implicit relationship exists between the nouns “crocodile” and “heresy,” nor can any linguistic relationship be stated without reference to an external category such as gender. Mathematical comparisons, on the other hand, relate to dimension but allow no accommodation for the value judgments of the third adjective class. If a metaphorical interpretation of the fourth adjective is permitted, mathematics does preserve a limited set of adjectives related to abstract principles. These adjectives, such as figurate, prime, or perfect, rarely merit mathematical notation. Rather, these conditions are described abstractly in the portion of mathematics composed in natural languages. Mathematical objects may be identified with or excluded from the set of elements which share those descriptions. Although rarely abbreviated to mathematical notation, these adjectives also represent a restriction of natural language. Because a number is objectively prime or square or negative, comparatives and superlatives do not apply to these categorizations.
5.2 Referents of mathematical objects
Mathematicians have made use of the implicit ordering of numbers. As noted in the discussion of the aphorism, Schofield identified the subscript index of variables as a set of mathematical adjectives. Schofield was correct that this use of indices fulfilled the second task common to adjectives—they specify the referent of a noun. Insofar as each of these indexes invokes another instance of the same set of values, this use resembles the noun-like adjectives of natural languages. (Another explanation might equate such variables with an open category of pronouns or determiners, but these categories are not as universally used in natural languages as adjectives.) The meaning of subscript indices, however, can vary. In one case, the indices may refer to the values returned for an ordered progression of a function through the integers; in another case, they may merely represent the individual values associated with the elements of a group in no particular order. Either case specifies the referent of a mathematical object, namely the variable.
Despite their similarity to noun-like adjectives, these purely mathematical indexes do not state a property, the first task common to adjectives. Rather, these indices introduce new mathematical nouns which fulfill predetermined characteristics. The mathematical use of indices does not reveal the property which each element satisfies. Consider the variable b. Various values which satisfy b may be identified and a set B emerges. The elements b _{1}, b _{2},⋯ b _{n}_{ }, and so on may be identified as b _{1} = 4 and b _{2} = 64, but whether b _{n}_{ } = any even number, any square number, or any power of 2 is not communicated by the mathematical adjective. The subscript adjective functions more like a determiner (such as an article, demonstrative, or quantifier) than an adjective because it serves only to separate the instances of the noun which fill the category.
A similar notation, superscript, often relates to exponentiation. In early European mathematical texts, repeated multiplications were laboriously repeated, but modern notation indicates number of multiplications by superscript numbers. This practice has liberated mathematical expressions from a connection to the dimensions of the physical world and ended the special privilege once extended to the category of square and cubic numbers. The notation considers all powers analogous, even though specialized adjectives may be drawn from natural language to discuss squares and cubes. Because this notation refers to a set of operations performed on the numbers, this practice resembles the verb-like adjectives of natural languages. Although exponentiation is acknowledged as an operation, the notation n ^{2} is commonly rendered into English as “n squared.” The analogy of superscript notation of exponentiation with mathematical adjectives is limited, though, because exponentiation reveals a mathematical property of the range, not the domain. In effect, the verb-like mathematical adjective of exponentiation functions like a variable because mathematical nouns are distinguished only by magnitude.
5.3 Variables and Generalization
Thus, mathematical adjectives, whether noun-like or verb-like, do not state the property of a mathematical noun directly but, rather, function as referents which indicate membership in a set of mathematical objects which share a property. Mathematicians still must distinguish between the nouns which satisfy this property. Although mathematicians cannot discuss the category of squares or primes directly, they can generalize the properties of these categories through variables. Thus, a square number is s ^{2}, with s being an element of the set of integers; a prime p is any integer for which the statement m × n = p implies that m or n (but not both) must be p. A strong tradition of shared notation for variables has arisen. Thus, n is often an integer and p is often a prime, despite the fact that variables are a relative late-comer to mathematical expressions. Ancient mathematical statements consider problems phrased in terms of known and unknown sides, areas, and quantities on which operations are performed. Francois Viète (1540–1603) formulated the idea of variables which occur in an analytical argument but to which no specific numerical value has been assigned. Despite the fact that he displayed a linguistic awareness by assigning variables to vowels and parameters to consonants, Viète considered mathematics an art (ars) [42, p. 7]. When mathematicians use variables to represent values which satisfy a category, these variables often reflect the mathematical adjectives derived from natural language. Thus, p is often prime; n is often an integer. Adjacent letters often introduce other mathematical objects of the same category. A proof which requires two primes usually invokes primes p and q; two integers are often m and n (with o omitted due to its typographical similarity to 0.) In this way, variables also resemble determiners because they enable mathematicians to refer to “this prime” and not “that prime.” Historically, determiners have been classified among adjectives, but the linguistic consensus has changed. Mathematicians could achieve the same communicative goals with subscripts (such as p _{1} and p _{2} for p and q), but they rarely employ this rhetorical strategy.
Mathematicians have avoided developing adjectives analogous to natural languages for good reason. Mathematics is assumed to express a generalization [11, pp. 134–136]. Generalization may be a rhetorical goal of mathematicians, but it is not a grammatical feature of a language. This feature, along with formalization, underpins the universality towards which mathematicians aspire [11, pp. 137–140]. The development of mathematical adjectives to distinguish between instances of the same number risks undermining the goal of generalization. The only property relevant in numeric statements is magnitude, and this property is inherent in the lexeme, or “name,” of the mathematical entity itself. The sign of the number, whether it is positive or negative, forms an extremely truncated class of adjective which relates directly to magnitude and may be equally well expressed by operators. Set theory enabled the formalization of many adjectives which had once been expressed in portions of mathematical texts which closely resembled natural language. The fact that this technique of description was not invoked by early mathematicians but was developed deliberately may be evidence of the linguistic status of mathematics.
6 Mathematical Language
The analogy of operations with verbs demonstrates that the verbs of mathematical systems have developed with reference to natural languages and vary by culture. The analogy of numbers with nouns shows that mathematical expressions are grammatically numberless. A search for mathematical adjectives reveals that while mathematicians use adjectives to describe relationships among numbers, they create these portions of mathematical compositions in something very near natural language because distinguishing between two instances of the same magnitude runs contrary to the presumed goals of mathematics. In each case, mathematics differs from natural languages not so much by what can be stated in a mathematical way but by which simplifications and truncations have been applied to a natural language. If a natural language produces an axiomatic system, then correspondences with the language in which the axioms originated ought to be found in that system. This proposition seems to be confirmed by the fact that throughout the development of this history of mathematics, mathematical texts are clarified by consideration of similarities between mathematical texts and non-mathematical texts in the same language.
A related proposition states that if a language corresponds to an axiomatic system, then that system, in some way, must have produced that language. Taken as a hypothesis, this proposition has buoyed the hopes of a universal grammar. Presumably, according to this perspective, at some far off point of abstraction, linguistic, mathematical, and cognitive systems converge. Concurrent with the pursuit of a universal grammar, though, linguists endeavored to mathematize their descriptions and analysis of languages. As early as 1914, Leonard Bloomfield proposed that each sentence contained two constituents and each of these constituents in turn contained two constituents [2, p. 110]. Roman Jakobson reworked phonological oppositions as another set of binary expressions [9]. In retrospect, these models better reflect the successes of binary expressions in the computing of Claude Shannon and the expansion of mathematical modeling in the sciences than an empirically derived structure of language. For example, the assumption of binary models does little to explain the Japanese trend toward a tripartite division, as expressed by the determiners (kono, indicating something close to the speaker), (sono, indicating something close to the listener), and (ano, indicating something removed from both the speaker and the listener) and the related pronouns (kore), (sore), and (are). For some time, the mathematicization of language was a useful analogy, but analogies (and mathematical models) are descriptive creations, not prescriptive discoveries. These tools are not reifications of reality and may be abandoned when they are no longer useful.
The analogy between language and mathematics reveals some connections but an analysis of the metaphor reveals unstated presumptions. The preservation of an analogy between nouns and numbers necessitates the exclusion of the grammatical category of number in a way not found among natural languages. Despite the faults of the analogy, an analysis of the aphorism that mathematics is a language reveals that mathematics resembles a recently identified linguistic practice—controlled natural language [18, p. 4]. On the whole, mathematics permits fewer categories and developments than natural languages.^{8} Operators lack tense, mood, and other aspects. Numbers lack grammatical number. Adjectives may be limited to sign. This systematic reduction of linguistic complexity fits the description of a controlled natural language. Some computer systems use controlled natural languages to weld the precision of machine languages with the intelligibility of natural languages. The resulting languages stand between constructed languages like Astraglossa and Lincos and natural languages. The role of controlled natural languages is not limited to numeric topics. Commentaries, citations, and scholastic compositions also conform to the strategies of controlled natural language. Just as the traditions of commentary and citation vary between cultures, so do differences mark mathematical systems. As a comparison of Egyptian and Babylonian mathematics shows, different cultures adopted different mathematical conventions, just as different branches of mathematics adopt different conventions to this day. In the case of antiquity, these differences reflected differences not only in mathematical culture but also in language.
In the case of mathematics, several cultures have developed a similar set of linguistic strategies for the study of quantities. Whether the decision to control natural language for the discussion of quantities stemmed from a common origin or whether multiple cultures adopted the same tool for the problem is not obvious. Mesopotamia and Egypt seem to share similar perspectives on the study of quantities. However, the mathematics of one culture may not be immediately understood by members of another culture despite modern claims of universality. The greatest difficulty in understanding mathematical expressions comes not from mathematical cultures geographically distant from modern mathematicians, but from cultures temporally removed from modern mathematics. The first reason for this trend is that modern mathematics has been an international affair for more than a century. Most mathematics have been done in a handful of languages relative closely related to each other. Even those recent works of modern mathematics undertaken in non-Western languages have been inspired by “Western” mathematical culture. The second reason for this trend is that mathematics is often learned through sharp value judgments of “correct” and “incorrect.” Coupled with a belief in technological progress, the foreign mathematics of antiquity too easily looks “wrong,” or “primitive.” Among the variants of the aphorism, a limited version proposed by Yuri Manin is the most defensible: “[T]he basis of all human culture is language, and mathematics is a special kind of linguistic activity [26, p. 17].” The modern understanding of these conventions is a reflection of the understanding of mathematical culture and the process of controlling different base languages.
Acknowledgements
Academics frequently praise books, but scholarly reflections seldom suit the tastes or developmental needs of children. Michel de Montaigne seems apt to inspire future Nietzsches when he declares that “obsession is the wellspring of genius and madness.” In contrast, [10] by Philip Davis allows those who doubt their genius but prize their sanity to pafnuty, that is, “to pursue tangential matters with hobby-like zeal.” Children as young as eleven have been known to incorporate pafnutying into their pedagogical formation. Through this slim volume, Davis tames madness with whimsy and humanizes genius by levity. Should an early exposure to the admixture of language and mathematics distract the student, that is, if the child becomes a recidivist pafnutier, youthful exuberance may be regulated by [11]. If the present argument resembles too much Montaigne’s madness or Davis’ skeptical classicist, it must be realized that despite several introductions and a shared institutional affiliation, this offering did not result from tutelage under Davis. Rather, it is offered as an homage to a past and present inspiration.
References
1.
A. Adler. Mathematics and creativity. The New Yorker, 47(53):39–45, feb 1972.
2.
L. Bloomfield. An introduction to the study of language. H. Holt, 1914.
3.
M. Brose. Die mittelägyptische nfr-pw-negation. Zeitschrift für Ägyptische Sprache und Altertumskunde, 136(1):1–7, aug 2009.
4.
E. A. T. W. Budge. The Gods of the Egyptians. Methuen, London, 1904.
5.
E. A. T. W. Budge. An egyptian hieroglyphic dictionary, volume 1. John Murray, London, 1920.
6.
C. K. Caldwell and Y. Xiong. What is the smallest prime? Journal of Integer Sequences, 15(9), 2012. Article 12.9.7 and arXiv:1209.2007.
7.
A. B. Chace. The Rhind mathematical papyrus: British Museum 10057 and 10058. Mathematical Association of America, Oberlin, 1929.
8.
J.-P. Changeux and A. Connes. Conversations on mind, matter, and mathematics. Princeton University Press, 1995.MATH
9.
E. C. Cherry, M. Halle, and R. Jakobson. Toward the logical description of languages in their phonemic aspect. Language, 29(1):34–46, 1953.CrossRef
10.
P. J. Davis. The thread: a mathematical yarn. Birkhäuser, Boston, first edition, 1983.
11.
P. J. Davis and R. Hersh. The mathematical experience. Birkhäuser, Boston, 1981.
12.
R. M. W. Dixon. Basic Linguistic Theory. Oxford University Press, Oxford, 2010–2012.
13.
T. W. R. E. L. Kaufman, M. W. Lord and J. Volkmann. The discrimination of visual number. The American Journal of Psychology, 62(4):498–525, oct 1949.
14.
J. Evelyn. Sculptura, or, The history, and art of chalcography and engraving in copper. G. Beedle and T. Collins, Oxford, 1662.
15.
J.-B.-J. Fourier. Théorie analytique de la chaleur. F. Didot, 1822.
16.
H. Freudenthal. Lincos; design of a language for cosmic intercourse. Studies in logic and the foundations of mathematics. North-Holland Publishing Company, Amsterdam, 1960.MATH
17.
J. Friberg. Unexpected links between Egyptian and Babylonian mathematics. World Scientific, 2005.MATHCrossRef
18.
N. E. Fuchs, H. F. Hofmann, and R. Schwitter. Specifying logic programs in controlled natural language. In Specifying logic programs in controlled natural language, volume 94.17 of Berichte des Instituts für Informatik der Universität Zürich; Institut für Informatik der Universität Zürich, 1994.
19.
G. Galilei. Il saggiatore. Giacomo Mascardi, Rome, 1623.
20.
K. Gödel. The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory. Annals of mathematics studies. Princeton University Press, Oxford, 1940.
21.
B. Gunn. Notices of recent publications. The Journal of Egyptian Archaeology, 3(4):279–286, oct 1916.
22.
L. Hogben. Astraglossa, or first steps in celestial syntax. Journal of the British Interplanetary Society, 11(6):258–274, nov 1952.
23.
J. Høyrup. Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin. Sources and studies in the history of mathematics and physical sciences. Springer, New York, 2002.CrossRef
24.
T. Jech. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer, Berlin Heidelberg New York, 2003.
25.
B. Lumpkin. Africa in the mainstream of mathematics. In A. B. Powell and M. Frankenstein, editors, Ethnomathematics: challenging eurocentrism in mathematics education, SUNY series, reform in mathematics education., pages 101–118. State University of New York Press, Albany, 1997.
26.
Y. I. Manin. Good proofs are proofs that make us wiser. In M. Aigne, editor, Berlin intelligencer: International Congress of Mathematicians, Mitteilungen der Deutschen Mathematiker-Vereinigung, page 16–19, Berlin, aug 1999. Springer.
27.
O. Neugebauer. Mathematische Keilschrift-Texte, volume 1 of Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik. J. Springer, Berlin, 1935.
28.
T. E. Peet. The Rhind mathematical papyrus, British museum 10057 and 10058. The University Press of Liverpool and Hodder & Stoughton, London, 1923.
29.
C. Proust. Mesopotamian metrological lists and tables:forgotten sources. In F. Bretelle-Establet, editor, Looking at It from Asia: The Processes that Shaped the Sources of History of Science, volume 265 of Boston Studies in the Philosophy of Science, pages 245–276. Springer, 2010.
30.
J. Ritter. Closing the eye of horus: The rise and fall of horus-eye fractions. In J. M. Steele and A. Imhausen, editors, Under One Sky: Astronomy and Mathematics in the Ancient Near East, volume 297 of Alter Orient und Altes Testament, pages 297–323. Ugarit-Verlag, Münster, 2002.
31.
M. Rukeyser. A turning wind, chapter Gibbs. The Viking Press, 1939.
32.
M. Rukeyser. Willard Gibbs. Doubleday, Doran & Company, 1942.
33.
R. P. Runyon, A. Haber, and P. Reese. Student workbook to accompany Fundamentals of behavioral statistics. Addison-Wesley Publishing Company, fourth edition, 1980.
34.
C. Sagan. The cosmic connection: an extraterrestrial perspective. Anchor Press, Garden City, NY, 1973.
35.
C. Sagan. The recognition of extraterrestrial intelligence. Proceedings of the Royal Society of London. Series B, Biological Sciences, 189(1095):143–153, may 1975.
36.
C. Sagan. Cosmos. Random House, New York, 1980.
37.
C. Sagan. Contact: a novel. Simon and Schuster, New York, 1985.
38.
R. Schofield. Sampling in historical research. In E. Wrigley, editor, Nineteenth-century society: essays in the use of quantitative methods for the study of social data, Cambridge Group for the History of Population and Social Structure, pages 146–190. Cambridge University Press, 1972.
39.
R. L. E. Schwarzenberger. The language of geometry. Mathematical Spectrum, 4(2):63–68, 1972.
40.
K. Sethe. Von Zahlen und Zahlworten bei den alten Ägyptern und was für andere Völker und Sprachen daraus zu lernern ist. K. J. Trübner, Strassburg, 1916.
41.
J. Thomson. Britain: being the fourth part of Liberty, a poem. A. Millar, Oxford, 1736.
42.
F. Viète. In artem analyticem isagoge. Mettayer, Tours, 1591.
43.
N. West. Miss Lonelyhearts, a novel. New Directions Books, 1933.
44.
L. Wheeler. Josiah Willard Gibbs, the history of a great mind. Yale University Press, New Haven, 1951.MATH
45.
E. T. Whittaker and G. Robinson. The calculus of observations; a treatise on numerical mathematics,. Blackie and Son Limited, London, first edition, 1924.
Footnotes
1
The initial proposal of Lincos most closely approximated the aphorism in its claim that “mathematical expressions and formulae belong to a language different from that of the surrounding context…The syntactical structure of ‘mathematical language’ differs enormously from that of all natural languages” [16, p. 6].
2
For the initial proposal of Astraglossa, see [22]. Hogben presumed the recipients of the messages to be Martians and Astraglossa depends on two-way communication. Hogben most nearly approximated the aphorism by writing that “[n]umber will initially be our common idiom of reciprocal recognition; and astronomy will be the topic of our first factual conversations” [22, p. 260].
3
For example, the order of operations suggests that in 7 × (3 + 2) the addition happens before the multiplication. Independent mathematical statements exist outside of a temporal framework. Without a context established by natural language, the mathematical statements and 7 × 5 = 35 have no relationship to each other. In contrast, natural languages can often easily suggest a temporal relationship, such as “Six was increased by two to make eight; Seven will be multiplied by five to find thirty-five.”
4
By contrast, the category of Babylonian and Egyptian operations is clopen. Both categories are closed because new Babylonian or Egyptian mathematical texts cannot be created but they are also open because new texts might be discovered or the reconstructions might be reassessed. For example, Plimpton 322 has yielded recent surprises.
5
Popular histories of mathematics often reproduce weakly cited counter-evidence. According to one edition, Chapter 64 of the Book of the Dead tallies 4,601,200 gods [4, p. 164]. Other editions confirm only one million. A grapheme introduced in the Ptolemaic era may express a sense of totality, or cyclical completion [40, p. 12, n. 6], or it may stand for millions, thereby increasing the old grapheme of the god Ḥeḥ to tens of millions [21, p. 280]. Occasionally, decorative hieroglyphs combine the graphemes of large numbers into images which may be read as “a hundred thousand million years” or even “ten million hundred thousand million years” [5, p. 507, s.v. ḥeḥ], but these graphemes are more likely artistic hyperbole than mathematical quantity.
6
The fraction appears idiosyncratically as the only portion of an Egyptian number which is not easily classified as either an integer or an inverse. (The fraction is sometimes cited as another example but this fraction is actually a ligature of and .) This idiosyncrasy may be resolved by morphology. The fraction was vocalized as rwy, a grammatical dual which literally means “the two parts.” The grapheme is better interpreted as rather than a ratio of 2 to 3.
7
The naked inclusion of non-Western languages too often provokes consternation. The phrase may be transliterated as “kō ha sosū da to ı̄mashō” and translated as “Let’s say that is a prime.” The grapheme derives from a pictogram of hands braiding raw silk and the semantic range of the sign includes plain, poor, foundation, and root. The use of kō as a variable is admittedly somewhat contrived for modern Japanese. Probably, a variable from the Roman alphabet would be used, most likely a in slight contrast to the Western preference for p. Mathematics has become an international endeavor.
8
The obvious exception is Indian mathematics which has developed an overabundance of mathematical synonyms. Because Indian mathematical texts are almost exclusively poetic, these variations may be explained by causa metris. However, in many cases, this explanation is facile. To date, a systematic study of whether the variation in vocabulary of Indian mathematics remains a desideratum. Only careful comparison will reveal if these variations indicate regional variations, the temporal development of Sanskrit, or nuances among mathematical objects.