Authors: Russell Barlow
Categories: Review Articles, numerals, base, anchor, quinary, decimal, linguistic numeral systems
Source: Philosophical Transactions of the Royal Society B: Biological Sciences
Authors: Russell Barlow
This article details how the notion of the numeral base has been used by linguists, both in the description of individual languages and as a broader concept for comparing different linguistic numeral systems. Although common linguistic understandings of ‘base’ are generally not the same as mathematical definitions of the term, they may help us understand the different ways in which languages encode numbers. In the hopes of supporting efforts in comparative linguistics as well as facilitating interdisciplinary studies of numeral systems, this article presents a typology of the various phenomena that may all be considered to be a ‘base’.
This article is part of the theme issue ‘A solid base for scaling the structure of numeration systems’.
Linguists are interested in systematic patterns in the structure of languages, including the ways in which different elements may be combined to create meaning. Thus, among other things, linguists examine the various ways in which words are arranged into sentences in different languages and the different ways in which words are themselves composed of smaller linguistic elements. Languages often employ conventionalized means of referring to exact quantities, and these linguistic expressions—called numerals—may exhibit systematic patterns whereby elements are combined to create numerical meaning. As such, numeral systems are also of interest to linguists, who have long used the term base (along with related terms like decimal, quinary and vigesimal) to describe certain structural patterns found in these linguistic systems. However, the use of these terms in the linguistic literature does not always match their use in other fields. Moreover, these terms are often used differently in the typological linguistic literature, which is concerned with comparing features across multiple languages, from how they are used in the descriptive linguistic literature, which is concerned with describing individual languages.
This article aims to clarify the different yet related features that linguists have in mind when employing terms like base, both when describing individual languages and when conducting typological studies comparing numeral systems across multiple languages. I organize the various features that may be considered a numeral base in linguistic systems into a proposed typology, hopefully improving conceptual clarity both within linguistics and across disciplines.
Descriptions of individual languages are written by various kinds of professional linguists, PhD students, pedagogues, missionaries, government employees and other individuals who are not necessarily adhering to a terminological standard. Thus, while terms like base, quinary and decimal are commonly found in the descriptions of linguistic numeral systems, they are not necessarily understood in the same way by all who use these terms. Unfortunately, such terms are rarely defined in grammatical descriptions, allowing us only to infer what was meant by a given author. In the best cases, a grammatical description contains a long list of numerals in the perhaps the words for ‘1’ through ‘100’, as well as ‘1000’ and even some higher numbers. Other times, however, we are only given a few numerals and a statement like ‘Language X has a quinary numeral system’. What does this mean?
Although rarely defined in the descriptive linguistic literature, terms like quinary (or base-5) and decimal (or base-10) often reflect a shared understanding by linguists of what is important in the study of linguistic strategies of referring to exact quantities. In short, linguists have frequently used the term base in a broader sense than is used in mathematics and other fields (for the mathematical understanding of base, see [1]). Perhaps the broadest general linguistic understanding of base can be summarized as
The term numeral is to be understood here in a linguistic a numeral is a word that refers to an exact quantity [2]. This description specifies the systematic use of the base for two reasons. First, language users may have various means of referring to quantities, including ad hoc or idiosyncratic formulations. Therefore, to be considered to be part of ‘the language’, these numerical expressions must be conventionalized.^1^ Second, in order to be considered a base, the numerical value in question should be used with some degree of systematicity (or regularity) within the system. For example, a linguist would probably not describe a system as ‘base-5’ if the number 5 is used in the formation of only one numeral other than 5.^2^ At the same time, however, complete and utter regularity is rare within the grammatical domains of natural language, so most linguists would probably accept a system as ‘base-5’ even if there is one exception to the ‘base’-defining rules.^3^ Thus, terms like systematic and regular should be understood as scalar numeral systems may be more or less systematic or regular. The degree of systematicity or regularity required for identifying a base may vary according to author or description.
Several linguists have acknowledged the incongruence between the mathematical notion of the ‘base’ and many linguists’ tacit understanding of the term in a linguistic context. For example, Lynch [7]
Although terms like base-5 and quinary are commonly used in describing linguistic numeral systems, it seems to be the case—by the mathematical definition of base—that no natural human language makes regular and exclusive use of a conventionalized base-5 system. Putative cases of base-5 systems generally instead reflect what might be considered sub-bases of 5 (e.g. a base-10 system with a sub-base of 5 or a base-20 system with a sub-base of 5).^4^
Similarly, linguists sometimes use the term binary to refer to systems that use the number 2 as an element for constructing higher numerals despite not using the higher powers of 2 (e.g. 2² = 4) in a similar way. In New Guinea, for example, where such systems are relatively common, it is unusual for people to use these ‘binary’ means for counting higher than 4 or 5. Often, there are no conventionalized terms for such higher numbers. Sometimes reference is made to a word meaning ‘hand’ for numbers greater than or equal to 5. For example, Holzknecht [10] refers to the Markham languages (Austronesian family, Papua New Guinea) as employing ‘binary number systems’, to which Ross [11]
Although linguists describing individual languages do not always define their terms, typological linguists, whose job is to compare linguistic features across multiple languages, must work with clearly defined terms that can be applied crosslinguistically.
In the typological literature, a commonly cited definition for base is that of Comrie [13]: ‘the value n such that numeral expressions are constructed according to the pattern . . . xn + y, i.e. some numeral x multiplied by the base plus some other numeral’.^5^ Hanke [15], citing Comrie [13], refers to this sort of numerical expression as an additive-multiplicative base. Other mentions of this definition are commonly found [16–19].
Comrie’s definition thus requires the presence of both addition and multiplication. In an earlier definition, Greenberg [20] appeals only to multiplication, defining the base as a ‘serialized multiplicand’, which, in turn, he defines as ‘a number whose successive multiplication by at least two other numbers results in serialized products which are either expressed as simple lexemes or as a product of the multiplicand and multiplier, and such that each serialized product is also a serialized augend or minuend’.
Another definition of base comes from Hammarström [5,12]:^6^
These definitions are more restrictive than some commonly used interpretations of base in the descriptive linguistic literature. Whereas some linguistic descriptions apply the term base to cases where the number in question is only used in additive operations, these definitions require a base either to be used (also) in multiplicative operations or to have special designations occurring at one or more multiples of itself. This is reminiscent of mathematical definitions of base, in which it is required that one or more powers of the base be specially defined [22]. In linguistic terms, we would thus expect a base-5 system to have a word for 5² (i.e. 25) that is not derived from any other numerals, similarly to how in base-10 systems—such as the one used in English—there is an underived word for 10² (i.e. 100; e.g. hundred). However, such mathematically defined base-5 systems are essentially unattested among natural languages. Thus, commonly described ‘quinary’ systems—provided they have conventionalized expressions for sufficiently large numbers—tend to switch to using 10 or 20 (or both) as elements for building numerals with values higher than 10 or 20.^7^
On the other hand, the descriptive uses of terms like binary, quinary, decimal and so on (as discussed in §2) tend to be more in keeping with the notion of cycles, as put forward by Salzmann [23]:
There have been several references to Salzmann’s [23] use of cycle in the linguistic literature [11,19,20,24,25]. A very early definition of base that seems kindred to this notion of cycle is that of Conant [26]: ‘… a number from which [one] makes a fresh start, and to which [one] refers the next steps in [one’s] count …’. Similarly, Stampe [27] ‘The base number of a number system, then, must be defined as that number from which counting starts over. In the vast majority of languages, at least originally, it is the highest of the simple numbers.’
Plank [16] makes a helpful distinction between what he calls a construction-base and what he calls a cycle-base:
Finally, we may mention Hurford’s [28,29] work, since it has been influential in crosslinguistic accounts of numeral systems. Although he uses the term base, Hurford [29] is generally concerned with the formal composition of individual complex numerals; thus, he provides a pragmatic suggestion for identifying a base in a given ‘Let us label the highest-valued word in a numeral construction the “base-word”.’
Plank’s [16] terminological distinction does an excellent job of distinguishing the two broad uses of the term base in the linguistic literature. However, each term is capable of being more or less strictly defined. In this section, I further define and typologize these two broad categories, showing how they can be related to Pelland’s [30] term anchor, which is a broader category that includes not only bases but also other numerical values that are used in structuring numeral systems. The definitions I provide below represent a narrower interpretation of ‘anchor’, meant to apply to linguistic numeral systems in particular. First, I would define construction base as
A construction base (or anchor) is a numerical value to which an arithmetic operation is applied so as to form a numeral.^8^ Table 1 provides abstract examples of systems in which different numbers serve as anchors.
It may be helpful to identify and define several subcategories of construction base (or anchor), such as the following.
A regular anchor is a numerical value n that functions as the anchor of at least half the numerals other than n within a cycle.^9^ For an additive anchor, a cycle could be defined as the set of numerals greater than (n)·(a) and less than or equal to (n)·(a+1), where a is a positive integer. For example, the number 10 (= n) meets the requirements of a regular additive anchor if it is employed as the additive anchor in at least five (= n÷2) numerals greater than 10 (= (n)·(1)) and less than or equal to 20 (= (n)·(1+1)).^10^ The number 10 (= n) would also be considered a regular additive anchor if it occurred, for example, in five (= n÷2) numerals greater than 70 (= (n)·(7)) and less than or equal to 80 (= (n)·(7+1)), provided that the operation of addition in those numerals is applied to a numeral derived from 10 (e.g. 72 = (7·10)+2). In practice, we may be mostly interested in the range from n+1 to 2n.
For a subtractive anchor, a cycle could be defined as the set of numerals greater than 0 and less than n or the set of numerals greater than or equal to (n)·(a) and less than (n)·(a+1), where a is a positive integer. For example, the number 5 (= n) meets the requirements of a regular subtractive anchor if it is employed as the subtractive anchor in at least three (≥ n÷2) numerals between 0 and 5 (= n).^11^ As another example, 10 (= n) meets the requirements of a regular subtractive anchor if it is employed as the subtractive anchor in at least five (= n÷2) numerals greater than 10 (= (n)·(1)) and less than or equal to 20 (= (n)·(1+1)), provided that the operation of subtraction in those numerals is applied to a numeral derived from 10 (e.g. 17 = (2·10)−3). Regular subtractive anchors are probably very uncommon in numeral systems.
For a multiplicative anchor, a cycle could be defined as the set of multiples of n between nᵃ and n⁽^a^^+1^⁾, inclusive, where a is a positive integer. For example, the number 10 (= n) meets the requirements of a regular multiplicative anchor if it is employed as the multiplicative anchor in at least five (= n÷2) multiples of 10 (= n) between 10 (= n¹) and 100 (= n⁽¹^+^¹⁾).^12^ For this definition, we may wish to accept systems in which multiplication by 1 is not overtly expressed (e.g. systems in which n is expressed as ‘10’, as well as systems in which n is expressed as ‘1·10’).
Although theoretically possible, languages are not known to employ divisive anchors (there are, however, a few attestations of derivation by means of multiplication by a fraction, as in ‘100·½’ for ‘50’). It is unclear how the cycle for a regular divisive anchor should be defined. Likewise, it is unclear what, if anything, should be considered a ‘cycle’ for an exponential anchor (or base). Table 2 provides some additional abstract examples of regular anchors using different operations.
A canonical anchor is a numerical value n that functions as the anchor of all numerals in a cycle aside from the start and end values of the cycle. For example, 5 can be considered a canonical additive anchor if it functions as the additive anchor in all of {6, 7, 8, 9}. As another example, 5 would be considered the canonical additive anchor if it occurs as the additive anchor in all of {11, 12, 13, 14}, provided that the operation of addition in those numerals is applied to a numeral derived from 5 (e.g. 12 = (2·5)+2). It is not required for the ‘end values’ (e.g. n, 2n) to be formed with n as an additive anchor. Thus, 5 would be considered a ‘canonical’ additive anchor in all three of the following sets of
As another example, 10 can be considered a canonical multiplicative anchor if it functions as the multiplicative anchor in all of {20, 30, 40, 50, 60, 70, 80, 90}. It may, but need not, also occur as a multiplicative anchor in the ‘end values’ of the cycle, as, for example, ‘10·1’ for ‘10’ and ‘10·10’ for ‘100’.
Perhaps the most ‘canonical’ type of numeral system is one in which the same value functions as a canonical additive anchor in the system’s additive cycles and as a canonical multiplicative anchor in the system’s multiplicative cycles.^13^ A canonical anchor is a particular type of regular anchor. Table 3 provides additional abstract examples of canonical anchors using either addition or multiplication.
A sporadic anchor is a numerical value n that functions as the anchor of less than half the numerals other than n within a cycle. All anchors that are not regular are sporadic. Table 4 provides abstract examples of sporadic additive anchors.
A hapax anchor is a numerical value n that functions as the anchor in exactly one numeral other than n within a cycle. A hapax anchor is (usually) a particular type of sporadic anchor.^14^ Table 5 provides abstract examples of hapax additive anchors.
Thus, all anchors (or construction bases) are either regular or sporadic. Canonical anchors are a subtype of regular anchors. Hapax anchors are a subtype of sporadic anchors. (However, exceptionally, the anchor value 2 may be both hapax and regular.)
Terms like augend, minuend, multiplicand and dividend can be expressed in terms of various types of construction bases or anchors, as shown in table 6.
Continuing with Plank’s [16] terminological distinction, a definition for cycle base may be formulated as
A cycle base is a regular construction base to which an arithmetic operation is applied so as to form another regular construction base. The cycle base is thus closer to the mathematical notion of a base. In practice, in linguistic numeral systems, the arithmetic operation used in regular construction bases is almost always—if not always—addition; and the (secondary) arithmetic operation used in forming one or more other regular construction bases is almost always—if not always—multiplication. Thus, the notion of cycle base is practically synonymous with the notion of additive-multiplicative base [15]. However, a cycle base could, in theory, represent an additive-additive base (in other words, an additive cycle base). Subcategories of cycle bases could include those shown in table 7.
I question the degree to which additive cycle bases are used in conventionalized systems. Although recorded (to some degree) for some languages with a construction base of 5 (e.g. Southwest Tanna, Austronesian family, Vanuatu), these formulations might be ad hoc.^15^
The multiplicative cycle base is the ‘canonical’ cycle base. Linguistic numeral systems that employ a multiplicative cycle base are the systems most widely accepted—both by linguists and by researchers in other disciplines—as representing ‘true’ base systems.
Unlike the operations of addition and multiplication, exponentiation is perhaps never overtly expressed in linguistic numeral systems. Thus, although a word meaning ‘17’ may be composed of morphemes meaning ‘10’ and ‘7’ (addition), and a word meaning ‘30’ may be composed of morphemes meaning ‘3’ and ‘10’ (multiplication), I know of no language in which a word meaning ‘100’ is composed of morphemes meaning ‘10’ and ‘2’. When languages do employ special designations for powers of a base, this is generally done by means of an atomic term, underived from any other numeral, as in English hundred and thousand, rather than something like ten-to-the-two and ten-to-the-three.^16^
Consider, for example, Mandarin, in which the arguments of the operations of addition and multiplication (although not the operators themselves) are transparently stated in complex numerals. In the Mandarin numeral shí qī ‘17’ (literally ‘10[+]7’), the additive construction base is the element shí ‘10’. In the Mandarin numeral sān shí ‘30’ (literally ‘3[·]10’), the multiplicative cycle base is (also) the element shí ‘10’. And in the Mandarin numeral sān shí qī ‘37’ (literally ‘(3[·]10)[+]7’), the element shí ‘10’ serves as a base simultaneously for addition and for multiplication. However, 10² is expressed as bǎi ‘hundred’, which is an opaque term—that is, it contains neither the exponential cycle base shí ‘10’ nor the exponent èr ‘2’.^17^
Why do linguists call things bases that are not mathematical bases? The terminology may be unfortunate, but the reason suits the goals of the field. Numerical elements are combined in languages in interesting ways, often irrespective of any mathematical base. Indeed, many linguistic numeral systems lack bases entirely (in the sense used in other fields) and yet contain structured patterns. Many of these structures recur across the world’s languages, such as using 2 or 5 as elements for constructing higher numerals, whereas other logically possible structures are all but unattested—for example, using 3 or 7 in similar ways. To investigate only mathematical bases would overlook many of the recurring patterns found in numerical expressions in the world’s languages.