4
Mathematical Symbols and the Architecture of the Grammar of Mathematics

Mathematics is organised through vast sets of relations between symbols. These relations arise through the complexing of symbols into expressions and of expressions into statements. This strong ability for complexing is highly significant for the power of mathematics in physics and in science more broadly. As will be discussed in Chapters 5 and 6, it allows large sets of technical knowledge to be related coherently and offers a means through which new knowledge can be built. However, it also reflects a large distinction in the overall grammatical organisation of mathematics in comparison to language. In particular, it suggests that the intrinsic functionality of mathematics—conceptualised through metafunctions—is somewhat different to that of language. It is the purpose of this chapter to explore this difference by building a model of mathematics’ overall grammatical organisation. First, it will focus on the lower levels of mathematics concerned with the internal organisation of symbols to show that these maintain their own structures derived from their own systems. Second, it will combine this understanding with the description of statements and expressions from the previous chapter to present the hierarchy of units in mathematics. And, finally, it will bring together all of these to characterise the overall architecture of mathematics in terms of its metafunctional organisation. From this broader model, we can begin to understand the specific functionality of mathematics in relation to language and to appreciate why it is used.

4.1 Mathematical Symbols

We will start by focusing on two texts, to see that there is still some variation not accounted for in the previous chapter. An example of this is the use of subscripts to distinguish between different types of E (glossed as energy) at beginning of Text 4.1, a mathematical text written by a teacher on a white board in a high school physics class:

$$\begin{array}{llllll}\Delta E_{emitted}\hfill & =\hfill & E_i-E_f\hfill & E_i\hfill & =\hfill & \mathrm{initial}\hfill \\ \hfill & \hfill & \hfill & E_f\hfill & =\hfill & \mathrm{final}\hfill \end{array}$$

$$\begin{array}{lll}\Delta E_{3\to 2}\hfill & =\hfill & -1.5--3.4\hfill \\ \hfill & =\hfill & 1.9\mathrm{eV}\hfill \\ \hfill & =\hfill & 1.9\times 1.6\times {10}^{-19}\hfill \\ \hfill & =\hfill & 3.04\times {10}^{-19}\mathrm{Joules}\hfill \end{array}$$

Text 4.1 High school classroom whiteboard

This text is concerned with calculating the energy E when an electron moves between two levels in a hydrogen atom. To do this, it distinguishes between four instances of energy: E_i and E_f, glossed as the initial and final energy during a transition, E_emitted, the energy emitted in a transition between two levels, and E_3®2, the energy emitted specifically in the transition between level 3 and level 2. The subscripts indicate different instances of the same technical symbol. As well as the subscripts, in the first and the third line, the use of the Greek character Δ modifies E. Δ is usually glossed as change, so that ΔE_3®2 would be read as the change in energy from 3 to 2. This character is related to other modifications such as the trigonometric functions sin and cos shown in bold in Text 4.2, from a senior high school textbook:

A car of weight 20,000 N rests on a hill inclined at 30 degrees to the horizontal. Find the component of the car’s weight:

• (a) perpendicular; and
• (b) parallel to the plane of the hill.

Solution

From Figure 3.49 [not shown] we see that:

• (a) The component of weight perpendicular to the plane is given by:

  $$\begin{array}{lll}{F}_{prep}\hfill & =\hfill & Wcos\theta \hfill \\ \hfill & =\hfill & 20,000cos30=17,300\mathrm{N}\hfill \end{array}$$

• (b) Similarly, the component of weight parallel to the plane is given by:

  $$\begin{array}{lll}{F}_{prep}\hfill & =\hfill & Wsin\theta \hfill \\ \hfill & =\hfill & 20,000sin30=10,000\mathrm{N}\hfill \end{array}$$

Text 4.2 Warren (2000: 117)

Each of the characters noted earlier will be classed provisionally as different types of modifiers. There are numerous other modifiers that occur throughout the texts under study. Indeed, they form a valuable component of the discourse, performing a host of different functions within the texts. The justification for grouping the various modifications together is twofold. First, these characters cannot sit on their own in an expression. That is, each of the following equations are ungrammatical:

• (4.1) *Δ = y
• (4.2) *sin = y
• (4.3) * ₂ = y

Second, binary operations cannot occur between a modifier and its head. Thus each of the following expressions is also ungrammatical:

• (4.4) *Δ × y
• (4.5) *sin × y
• (4.6) * y_×2

These two characteristics distinguish modifiers from symbols such as x, y, 2, π, etc. This distinction sets up two distinct functional elements: those that can sit on their own in an expression and can be related to other symbols through binary operations, and those that cannot. Beginning with the subscript relation, e.g. E_i, we will call E a Quantity, reserved for elements that can sit on their own and enter into a binary operation, and the subscript a Specifier, that cannot. Thus we would analyse E_i as Quantity^Specifier. As the Quantity^Specifier structure contains two distinct functions that are not recursive, this relation is a multivariate structure. The multivariate nature of these functions is in contrast to most of the grammar described in the previous chapter and has an impact on the overall architecture of mathematics, to be discussed toward the end of the chapter.

There are a range of modifiers that perform their own functions. We began with the Specifier noted earlier, as it is the odd one out within this system. Specifiers can only modify a single symbol, such as E_i, meaning they cannot modify symbols complexes, e.g. *(5 E)_i, or symbols with other modifications, e.g. *(Δ x)_i. This is in contrast to other modifiers, such as sin and Δ, which can modify whole complexes, e.g. $\sin \left(\frac{n\pi x}{L}\right)$ and $\Delta \left(m\vec{v}\right)$. In addition, Specifiers can only modify pronumerical symbols, such as E, not numbers, such as 5. For these reasons, Specifiers will be split from the rest of the modifiers as an optional feature.

The other modifiers will be grouped as unary operations. Unary operations are similar to binary operations (+, −, ×, etc.) in that their insertion usually changes the value of the expression they are in. However, unlike binary operations, unary operations only necessitate one symbol.

The symbol that is being operated on we will call the Argument. The unary operator will take the function Operation. Thus sin 3 will be analysed as Operation^Argument, where sin is the Operation and 3 is the argument. As mentioned, a key feature of unary operations is that they can modify multiple symbols at once. That is, the Argument can be realised by a symbol complex. This means that would also be analysed as Operation^Argument.

These elements give us the basic structure of symbols. They may occur on their own, such as E, in which case they are analysed purely as a Quantity. Or they may include a subscript such as in E_i, giving the analysis Quantity^Specifier. And, finally, they may include a unary operation such as sin E_i which is represented through the structure shown by 4.7.

• (4.7)
  

At this point, it is worth pausing for a moment to show a full analysis of a statement and its symbols (excepting covariate structures). To see each element, we will use the constructed example: $y=5\cos \frac{E_i}{mvr}\approx 0.5$. This analysis will show the statements’ univariate structure through its 1^2^3 … configuration, its thematic structure through its Theme^Articulation configuration, the expression structure through α^β, and structure of its symbols through Operation, Argument, Quantity, Specifer.

(4.8)

$$y=5\cos \frac{E_i}{mvr}\approx 0.5$$

At the levels of both statement and symbol, there are multiple structures. At the level of statement, a univariate structure is shown by 1 ⁼ 2 ^≈ 3 and a periodic structure by Theme^Articulation₁ ^Articulation_2. At the level of symbol, a univariate structure is shown within the second expression by the α^´ β between 5 and $\cos \frac{E_i}{mvr}$, and by the $\frac{\alpha }{\beta \left({\alpha }^{\times }{\beta }^{\times }\gamma \right)}$ structure of $\frac{E_i}{mvr}$, while a multivariate is shown by the Quantities, Specifier and the Operation^Argument sequence.

Continuing the description, we have so far distinguished between Specifiers that modify Quantities and are shown by subscripts, and Operations that modify Arguments. To capture the fact that the structures Quantity^Specifier and Operation^Argument can enter into binary operations of multiplication, addition, division, etc. we will now use the term symbol as the class label that includes the affixation of the modifiers (either Specifier or Operation) onto the elements that realise the Quantity. That is, for example, symbol is used to capture the entirety of sin x1. This is similar to the label word in English being used to capture the grouping of root morphemes with prefixes or suffixes. This means that symbols are realised by Quantities plus optional modifiers.

As mentioned previously, there are a number of different types of unary operations. In fact, within the entire field of mathematics the range of unary operations is vast, far too large to account for in this book. However, within the restricted register of physics teaching before calculus is introduced, the number of modifiers is relatively modest. The operators accounted for in this description are as follows:

change	Δ	as in:	Δx
factorial	!	as in:	x!
absolute value	|…|	as in:	|x|
summation	Σ	as in:	Σ x
sine	sin	as in:	sin x
cosine	cos	as in:	cos x
tangent	tan	as in:	tan x
positive	+	as in:	+x
negative	-	as in:	-x
generic	e.g. f(…)	as in:	f(x)

The first distinction is between unary operators that come after the Argument (suffixual), those that come before the Argument (prefixual) and those that occur on both sides of the Argument (circumfixual). There is only one type of each of suffixual and circumfixual unaries and so these can be generated first. The suffixual operator is the factorial, shown by!, as in 5!. Factorials indicate that each positive whole number (integer) between 0 and the number in the Argument are multiplied together. For example, 5! = 1 × 2 × 3 × 4 × 5 = 120. The circumfixual operator is known as absolute value and is denoted by |…|, as in | x |. Whatever the sign of the Argument, whether positive (above zero) or negative (below zero), the absolute value shifts the sign to positive. For example, both |5| and |−5| are equal to 5: |5| = 5; |−5| = 5. All other unary operators are prefixual, coming before the Argument.

The prefixual operators include three distinct types: the trigonometric operations of sine, cosine and tangent (e.g. sin x, cos x, tan x) and the change operator shown by Δ (e.g. Δ x) and the summation operator Σ (e.g. ∑ x). The trigonometric operators can be grouped under a single feature through their agnation patterns. Given a right-angled triangle with three sides termed opposite, adjacent and hypotenuse, each of the trigonometric operators are equal to a relation between two of them:

• (4.9) $\sin \theta =\frac{opposite}{hypotenuse}$
• (4.10) $\cos \theta =\frac{adjacent}{hypotenuse}$
• (4.11) $\tan \theta =\frac{opposite}{adjacent}$

More succinctly, the three operators can be related in a single equation through:

• (4.12) $\tan \theta =\frac{\sin \theta }{\cos \theta }$

Second, the change operator Δ gives the numerical difference between two instances of an Argument. This is usually formalised as Δ x = x2 − x₁. For exam ple, if x2 = 5 and x₁ = 3 the change in x would be shown through: Δ x = x2 − x₁ = 5 − 3 = 2. As Δ necessitates a symbol that can change, it cannot modify numerals (e.g. 5) nor can it modify pronumerals known as constants (symbols that don’t change such as π). The elements it can modify are known as variables.

Third, the summation operator Σ indicates the sum of all different instances of the Argument. For example, if there are three instances of x: x1 = 3, x₂ = 5 and x3 = 7, the sum of all x is shown by ∑ x = x₁ + x₂ + x₃ = 3 + 5 + 7 = 15.

Each of the unary operators so far can be equated with a specific set of binary relations. For example, a change in x indicated by Δ x is equal to the difference between x at two specific points, or Δ x = x₂ − x₁. Indeed most unary operators across the field of mathematics appear to encode sets of relations such as this to greater or lesser specificity. This fact in part explains their existence. They are used to encode sets of relations in a relatively economical way, not dissimilar to the function of technicality in language (Halliday and Martin 1993, similar to Lemke’s thematic condensation 1990: 96). In LCT terms (Maton 2014), the distillation of these relations indicates relatively strong semantic density (Maton and Doran 2017a, b), allowing physics to efficiently describe quite complex relations.1 The distillation of univariate relations into multivariate operators will be taken up again in Chapter 5 in relation to their role in knowledge building.

There is one set of unary operators, however, that do not encode a specific set of relations. These have been termed generic operations and are most commonly indicated by f (…), such as f (x) in, though others can be used, such as g (x). These operators do not encode any specific set of relations within the broader grammar of mathematics, but rather work as general operators whose meaning shifts instantially with each text. More field-specific operators can be found, such as Ψ (x, t) in Ψ (x, t) = Ae^{i(kx − ωt)} used in quantum physics (from Young and Freedman 2012: 1333). However, the relations these operators encode are constrained by field, not the grammar, and generally allow a larger set of relations for different situations in comparison to the entirely grammaticalised operators such as Δ.

The final pair of unary operators concern signs distinguishing positive (+) and negative (−). These operators most obviously occur in statements that appear at first to have two binary operations in sequence, or a binary operation that links only a single symbol. An example of this is shown twice in an equation from Text 4.1: Δ E_3®2 = −1.5 − −3.4. Here there is a negative sign before 1.5 not linking it to anything else, and two negative signs between 1.5 and 3.4. Although having the same form, these negative signs are not the binary operators of subtraction. Rather, they are unary operations that distinguish between positive and negative. This is justified by the fact that they can come before a single symbol without following another, as shown earlier. The positive sign (+) can do the same as the negative; however, it is much less common, owing to the fact that positive is default for numbers, and so does not need a sign in the unmarked case. Unary operators such as these can only occur for positive and negative; there are no correlates for multiplication × or division ÷.

From the discussion earlier, we can now set out the options for unary operations as in Figure 4.1.

In addition to specifying each type of modification, this network shows that modifications are optional. Expressions can simply include a single Quantity as the symbol, as in both expressions in x = 0.3, or alternatively, they can include a unary operation, a Specifier or both, as in x₁ = 0.3, sin x = 0.3 or sin x₁ = 0.3. This network also specifies that when there is a Specifier, the Quantity necessarily must be a pronumeral (such as E, as opposed to a numeral such as 4) and when there is a unary operation, the Quantity must necessarily be a value (distinct from a unit). These will be introduced and discussed in more detail below.

It was mentioned previously that unaries can operate on Quantities with a Specifier. That is, for example, it is acceptable for an expression to include sin E_i. It is also possible for unaries to operate on other unaries—e.g. to produce something like sin Δ x. This possibility for unaries to be repeated indicates a recursive system. However, recursive unaries such as this are unusual. In particular, it is unusual for recursive operators that repeat the same choice, e.g. sin (sin x) at least in the data under study. Nonetheless, they are grammatical. This creates an issue, as specifying a recursive system without probabilities or stop rules suggests that unary operations are better described as a univariate structure rather than as the multivariate structure suggested earlier. However, the relatively rare instantiation of recursive unaries means that in the vast majority of situations there are only two distinct functions appearing and that these have considerably different agnation patterns. For this reason, it seems preferable to stand by the multivariate analysis for unary operations, while accepting the possibility for recursion, albeit unusual.

This completes the description of modifiers in symbols. We can now place these systems in relation to those developed in the previous chapter. In that chapter, we saw that symbols can complex with any number of other symbols to form large symbol complexes. This complexing involves

Figure 4.1 Symbol modifications

linking symbols through binary operations. Any type of symbol, whether they include modifications or not, can be included in these symbol complexes. This means that when developing a network of symbols, the system describing modifications are simultaneous to those describing symbol complexing. Importantly, the entry condition for choosing symbol complexing and symbol modification is the same. Both symbol complexing and symbol modification occur at the level of symbol. The system of EXPRESSION TYPE, determining whether there is a symbol complex or not, looks outwards to the external relations between symbols; the systems of SPECIFICATION and UNARY OPERATION that organise the modification of symbols, look inwards to the internal structure of symbols. As each of these systems are simultaneous at the level of symbol, they can be placed in the same system network. In this network, we can also include the choice of symbols that realise Theme and Articulation, as well as Theme ellipsis and final or medial Articulation. Thus, Figure 4.2 presents the entire network for the level of symbol.

This symbol network complements the statement network given in Figure 3.9 in the previous chapter. These two networks account for all the variation from the level of symbol up. They describe the internal structure of symbols, their complexing into expressions and the complexing of expressions into statements. These networks represent two levels in the architecture of mathematics. As these make up the bulk of the variation in mathematics, it is important that we now consider the relation between them.

4.2 Layering in Mathematics

The discussion in this and the previous chapter has so far focused in detail on variation within statements, expressions and symbols, and has developed structural and systemic models for each. Although there is still one more area of the grammar to cover, it is worth taking a step back and viewing the grammar as it stands. In particular, we can focus on the interaction between statements, expressions and symbols and characterise the hierarchy of levels they sit in. Two levels of networks have been used to describe mathematics so far: statement and symbol. Systemically, expressions have arisen within the symbol network as complexes of symbols, and within the statement network as parts of the statement. In one sense then, the relationship between statements, expressions and symbols is straightforward: statements contain expressions and expressions contain symbols. This rather simple characterisation, however, clouds the organising principles of these levels and their interaction.

In English (and to this point every language described in the Systemic Functional tradition, see Caffarel et al. 2004a, Martin and Doran 2015), the scale of units—e.g. morpheme, word, group/phrase and clause—are organised hierarchically in a rank scale. Structurally, this rank scale is organised in terms of constituency, i.e. a relation of parts to wholes. A clause contains one or more groups, groups contain one or more words and words contain one or more

Figure 4.2 Network of symbol

morphemes (Halliday and Matthiessen 2014). Paradigmatically, there is a tendency for preselection from higher units to lower units. For example, options in the clause network tend to preselect options in the group/phrase network.

The rank scale proposed by Halliday for English involves multivariate structures associated with the experiential component of the ideational meta-function (Halliday 1965). However, for mathematics, this chapter has argued that the overarching structural organisation relating statements, expressions and symbols is not multivariate, but univariate. Although there is internal variation within symbols that is best described multivariately, this internal variation has no bearing on a symbols’ relation with its higher levels (it does, however, impact on lower levels, to be discussed next). In the description, statements are complexes of expressions, and expressions are complexes of symbols. This hierarchy of levels in mathematics is not one of multivariate constituency, but of univariate complexing (or interdependency). In this sense, the hierarchical scale in mathematics is more like the layering that occurs in larger clause complexes in English than it is like constituency within a clause.

On the other hand, like the constituency-based rank scale of English and unlike clause complexing, mathematics has obligatory levels with distinct sets of choices. Any mathematical statement makes choices at both the level of statement and the level of symbol. Mathematics thus has a scale with obligatory levels. Following from the previous paragraph, however, these obligatory levels are not a multivariately based rank scale, but one based on univariate layering. We thus have an obligatory set of levels based on univariate layering. This hierarchy we will call a nesting scale. As shown earlier, only two networks are needed to account for the variation from the level of symbol up. As there are only two networks, only two nesting levels are needed: symbol and statement; the level of expression is not needed.

The nesting scale comes about through choices made on two levels. At the level of symbol, symbols can complex with other symbols through binary operations such as ×, ÷ and −. These symbol complexes can in turn complex to form statements. The relations of binary operations, however, are not available for the complexes that form statements. At this level, Relators such as = and > are used. Thus, there are two mutually exclusive sets of relations that form different sized units. From this, two distinct levels are justified. The term nesting is used purely to distinguish levels based on a univariate structure from ranks based on a multivariate structure.

The choice of relations at two univariate levels is comparable to those within and between verbal groups in English verbal group complexes. Within the verbal group, there is a serial tense system, built on a hypotactic univariate structure (Halliday and Matthiessen 2014). For example, (Notation follows Martin et al. (2010). Superscript 0 indicates present tense, − indicates past tense, + indicates future tense, perf. indicates perfective aspect, imp. indicates imperfective aspect.)

In addition, whole verbal groups can complex with other verbal groups to form larger verbal group complexes. The choice of relations between verbal groups is not the same as those within the verbal group. Within the verbal group, the univariate relations build serial tense: had been going to run. The relations between verbal groups, on the other hand, build phase, began to run, conation, try to run, or modulation, tend to run. The choices of different types of verbal group complexing are relatively independent of tense choices within the verbal group. This means that serial tense choices can occur within verbal group complexes, such as in will have tried to have been running. With two sets of choices linking two units, two layered levels occur based on the univariate structure:

With two sets of choices relating two different units—tense between auxiliary and finite verbs in the verbal group, and phase/conation/modulation between verbal groups in the verbal group complex—two levels occur associated with the verbal group.

In English, the organisation of these verbal group levels is distinct from the layering that occurs in clause complexing. Clauses can be complexed through either expansion, (such as coordination: I ordered stew and she ordered scrambled eggs), or through projection (such as reported speech in: I asked that we go to the restaurant). Importantly, the choice of both projection and expansion is available at any layer of a complex. For example, both (4.13) and (4.14) are grammatical. In (4.13) projection, shown with square brackets [], is in the outer layer, with expansion, shown with braces { }, in the inner layer:

• (4.13) [Halliday said that {grammatical categories are not theoretical and that they are ineffable}]

In contrast, in (4.14) expansion is in the outer layer with projection in the inner layer:

• (4.14) { [Halliday said that grammatical categories are not theoretical] and [he said that they are ineffable] }

Thus the nature of layering in mathematics and English verbal group complexing is distinct from English clause complexing. In mathematics and verbal group complexing, there are distinct sets of choices at different levels, whereas this is not the case for clause complexing. However, in contrast to the English verbal group, in mathematics the choice of the highest layer (statement) is obligatory. That is, a statement necessarily must include two expressions, but an English verbal group may occur on its own in a clause without being in a verbal group complex. Thus, in mathematics, there is obligatory univariate layering (a nesting scale), while in the English verbal group there is not. Rather, the English verbal group offers an optional set of levels. The three different forms of layering can be distinguished as in Table 4.1.

Table 4.1 Types of layering

	Univariate layering	Distinct choices at each level	Choices at highest layer obligatory
English clause complex	Yes	No	No
English verbal group/verbal group complex	Yes	Yes	No
Mathematics symbol/ statement	Yes	Yes	Yes

As Table 4.1 shows, it is only because mathematics involves univariate layering, distinct choices at each level and an obligatory choice at the highest layer, that a nesting scale is needed. Such a scale would not be needed if any of the earlier noted criteria were not met.

As mentioned earlier, only two nestings are needed: statement and symbol. The network of symbol accounts for the internal structure of symbols, their complexing into expressions and whether or not these symbols/expressions are thematised. The network of statement includes the complexing relations between expressions and covariate relations. It is possible this higher level network could be called expression rather than statement; however, since two expressions must necessarily complex into a statement and this complex has its own Theme-Articulation structure, the label statement is preferred. Regardless of what they are called, only two nesting levels are needed. If we name them statement and symbol, the term expression no longer has any formal meaning in terms of the paradigmatic networks. Informally, however, it will continue to be used to refer to symbol complexes and to either side of a statement. The nesting hierarchy of symbols and statements can be represented diagrammatically as in Figure 4.3.

Figure 4.3 Nesting scale of mathematics

The nestings of statement and symbol broadly correspond to the ranks of statement and component in O’Halloran’s (2005) grammar. As described earlier, there is no specific level corresponding to O’Halloran’s expression, nor is there any equivalent level to O’Halloran’s clause rank. Under the description developed in this chapter, O’Halloran’s clause (e.g. F = ma) is a minimal statement with only two expressions.

O’Halloran also shows the high degree to which optional layering can take place. Working with a rank scale, O’Halloran describes this optional layering as rankshift. As the description being built in this chapter uses univariate nesting, we will deploy the term layering for O’Halloran’s rankshift. Thus, nestings indicate the obligatory levels, while layering indicates the optional levels. Nesting is to layering as rank is to embedding:

Types of Levels

multivariate-based obligatory:	rank
multivariate-based optional:	embedding
univariate-based obligatory:	nesting
univariate-based optional:	layering

It must be noted that optional layering is not available at all levels. Indeed, it is only within the nesting of symbol that layering can occur. Statements have no possibility of optional layering. That is, statements cannot occur within statements. Any insertion of a Relator such as = necessarily happens at the same level as every other Relator in the statement. Symbol complexes, on the other hand, can have quite deep and complicated optional layering, as shown in (4.15). Square brackets have been added to (4.16) to show the different optional layers within the expression:

• (4.15) $K+U=\frac{1}{2}m{v}^2-\frac{GMm}{r}$
• (4.16) $[K+U]=\left[\left[\frac{1}{2}\right]m\right[v^2\left]\right]-\left[\frac{\left[GMm\right]}{r}\right]$

On the left side of the equation, there is only a single optional layer. On the right-hand side, there are three optional layers. These optional layers occur at the obligatory nesting of symbol. Both the optional layering and the obligatory nesting are shown with their different relations in the Table 4.2.

When looked at from the point of view of language, the univariate nesting scale that occurs in mathematics is ‘exotic’. This is because in Systemic Functional descriptions of language, the obligatory hierarchies are organised around multivariate rank scales. It remains to be seen whether nesting scales are a broader feature of certain types of semiotic system. It may be the case, for example, that the broader family of symbolism including chemical symbolism, linguistic symbolism and formal logic symbolism are organised

Table 4.2 Obligatory nesting and optional layering in an equation

around nesting scales such as this (discussed in Chapter 7). We will see in Chapter 6 that when viewed through field, the univariate organisation of mathematics has a significant impact on the structuring of knowledge in physics. If it turns out that other symbolic systems are indeed organised univariately like mathematics, this field-based perspective could provide an explanation for their uptake and evolution alongside language.

The nesting scale is not the only hierarchy needed to account for mathematical symbolism. The following section will describe a network for types of element, which work at a level below symbol. This network derives from the preselection of the type of Quantity needed by various unary types. From this, it will become clear that a small rank scale based on a multivariate structure is needed to account for the variation in types of symbol. This will mean the architecture of mathematics involves two interacting hierarchies based on different structures, with the level of symbol facing both ways.

4.3 Types of Elements

Section 4.1 showed that only some symbols can take a unary operation. For example, Specifiers cannot occur on numerals, *2₁, but they can occur on pronumerals that use the Roman or Greek alphabet, E1. To account for this, we need to distinguish between these different types at a unit we will call element. Elements realise Quantities, and thus occur at a lower level than symbol. Elements themselves do not have any internal structure, but rather are justified through preselection from higher levels. In this way, they are similar to morphemes in relation to words in English; they form the lowest level of the description and make up the constituents of symbols. We can use sin x to distinguish between the levels of symbol and element, through the analysis shown in (4.17).

• (4.17)

  	sin	X
  symbol class	unary
  symbol structure	Operation	Argument
  		Quantity
  element class		pronumeral

Under this analysis, the entire symbol sin x is of the class [unary] at the level of symbol, realised by the Operation^Argument. The Argument is conflated with the Quantity, which is realised by the element class [pronumeral]. The ‘sin’ does not need an analysis at a lower level as it has been lexicalised at the level of symbol. There is no possible variation within sin that necessitates its own system. This section will be concerned with developing the system of different types of elements, before moving to a consideration of the relationship between the level of symbol and element in the following section.

The first distinction is between elements known as [units] and all others, termed [values]. The distinction occurs in the equation F_perp = 20,000 sin 30 = 10,000 N from Text 4.2 given at the beginning of the chapter. The final N (glossed as Newtons) is unlike the other numbers and pronumerals. It is a unit of measurement (in this case, the unit of Force, F in the equation). Each physical quantity will have its own unit or set of units. For example, in the standard units of physics (know as SI units) mass is measured in kilograms (kg), length in metres (m), time in seconds (s). These units can complex through binary operations of multiplication, division, etc., just as other symbols can. For example, acceleration is measured in m/s² (metres per second squared), and the universal gravitation constant (G) is measured in Nm²/kg² (Newton metres squared per kilogram squared). They do not, however, occur within a unary operation—e.g. neither *sin N nor *Δm occur. Units are usually not written in italics (whereas pronumerals are) and they predominantly occur after the final numerical solution of a quantification.2 An example of this is shown in an excerpt from Text 4.2, with the unit N occurring at the end of the second line:

$$\begin{array}{lll}{F}_{prep}\hfill & =\hfill & Wsin\theta \hfill \\ \hfill & =\hfill & 20,000sin30=10,000\mathrm{N}\hfill \end{array}$$

All other elements are classed as [values]. The basic distinction within [values] is between [numerals]: 1, 2, 7.3, 1,049,274, etc., and [pronumerals]: x, y, π, κ, etc. Grammatically, they can be distinguished through preselection at the higher level of symbol: pronumerals can take a Specifier, E₁, whereas numerals cannot, *2₁. This distinction is important for the discussion of genre in the following chapter. Two genres known as quantifications and derivations are differentiated by whether the final Articulation within the final statement involves numerals (quantification) or pronumerals (derivation).

Within [pronumerals], we can distinguish [variables] from [constants]. [variables] are pronumerals that could potentially be replaced by a number of different numerical values, whereas constants are those that can only be replaced by a single number. An example of a [constant] is π, which has an unwavering value of 3.1415 … In contrast, F and W in the equations mentioned earlier are considered variables, as in principle, any number could replace them. This distinction is bolstered by the fact that the unary operator change, Δ, necessarily requires a variable: Δ F is acceptable, whereas *Δp is odd at best.

Finally, within variables we can distinguish [scalars] and [vectors]. Notionally, [vectors] are numbers with a direction, whereas scalars do not include a direction. For example, if a force was to occur, it necessarily occurs in a particular direction (up, down, left, right or somewhere in between), thus it is a vector. Mass, on the other hand, is a directionless scalar; it does not occur in any direction, but is simply a physical quantity. The direction for vectors are often specified after the units in a quantification, such as downwards in W_Mars = 180 N downwards. Vectors are often indicated through being bold: e.g. F or through an arrow placed above them, e.g. $\vec{F}$.3

Various distinctions could be made within [numerals]; however, this will be left for future description as needed. The system network for element is shown in Figure 4.4.

Figure 4.4 System of ELEMENT TYPE

4.4 Levels in the Grammar of Mathematics

It was suggested earlier that the relationship between symbols and statements involves two levels on a nesting scale. This was based on the fact that symbols complex into statements through a univariate structure and that two networks were needed for these levels. The statement network showed complexing involving Relators such as = and >, whereas the symbol network indicated complexing that utilised binary operators such as +, − and ×. With each potential complexing relation only available at specific levels (Relators between expressions and binary operators between symbols), two nesting levels were justified. In addition, this chapter has shown that as well as the univariate complexing, there is also internal structure within symbols that is best described multivariately. This multivariate structure preselects distinct types of element to be placed within each symbol, with the network of elements occurring at a level below symbols. Importantly, this element network does not have the same relation to symbols as symbols have to statements. There is no univariate complexing of elements that make up symbols—we cannot say sin(y 2) where y 2 does not indicate a binary relation such as multiplication or power. Rather, a more fruitful avenue is to view elements as constituents of symbols. This is similar to viewing morphemes as constituents of words in English. This comes about through the internal multivariate structure of symbols. Different components of symbols perform different functions, with some of these components preselecting certain types of element at the level below. Thus, due to the multivariate nature of symbols and their constituency relation with elements, the development of an element network sets up a rank scale of the type more commonly associated with language. This rank scale has the symbol as the highest level and the element as the lowest.

Under this formulation, an entity such as sin x sits at the rank of symbol, while the x within it sitting at the rank of element. As discussed previously the sin is lexicalised at the rank of symbol and so does not need to be accounted for at the rank of element. Combining this rank scale with the nesting scale, we see that the symbol plays two roles. It is the lowest level of the nesting scale below statements, but is also the highest level of the rank scale above elements. It is both a rank and a nesting. With the symbol facing both above and below, the hierarchy of levels in mathematics can be viewed as an interaction between a nesting and rank scale, represented in Figure 4.5.

Figure 4.5 Levels in mathematics

We can now combine the rank scale with the obligatory nesting scale and its optional layering to represent a single statement. Table 4.3 shows each level within the equation v_y = v sin θ − at.

Although only containing three levels, the janus-faced nature of mathematical symbols means the hierarchy of mathematics is somewhat more complicated than for any language described systemically to date. There is an interaction between obligatory nestings and obligatory ranks, with the potential for further optional layering involving symbol complexes. As discussed previously, an architecture such as this might be viewed as exotic from the perspective of language as, to this point, descriptions of languages have not necessitated an obligatory nesting scale, let alone one that coexists with a rank scale. It remains to be seen whether interacting scales such as this is a feature of many families of semiotic resources, or whether it is a feature of only mathematics. What is clear, however, is that the two scales are possible and can occur simultaneously.

This discussion of the levels within the grammar of mathematics concludes the grammatical description proper. Each relevant network and their structural organisation has been introduced, with the possible variation at each level comprehensively described. It is now time to take a step back and view the grammar as a whole. From this, we can approach the questions

Table 4.3 Nesting, layering and rank in an equation

of how to interpret mathematics in terms of theoretical concepts such as metafunction. Indeed it is from this perspective that we will see the most striking feature of mathematics in relation to the Systemic Functional model of language.

4.5 Metafunction in the Grammar of Mathematics

The description put forward in this and the previous chapter has attempted to treat the grammar of mathematics on its own terms. To do this, it has taken the axial relations of system and structure as the primary basis upon which semiotic description holds. This has meant that broader phenomena such as metafunction, strata and rank have not been assumed at the outset. The challenge set forth was to independently justify these phenomena. Accordingly, this approach has the potential to produce different architectures for mathematics than for English. If we wish to understand the specific functionality of distinct semiotic systems, however, such an approach is necessary. The discussion earlier on the levels in mathematics exemplifies this. In building a scale of levels on distinct patterns of systems and structures rather than assuming a rank scale, a unique set of levels have been developed. A two-level rank scale complements a two-level nesting scale, with the level of symbol situated in both. This is in contrast to O’Halloran’s (2005) hierarchy that posits a four level rank scale. The difference has come about through distinct methodologies and motivations. O’Halloran proposes a language-based view of mathematics which carries over the rank scale from language; this description takes an axial view of mathematics and builds a distinct set of levels.

As it is for rank, so it is for metafunction. Following the same principles that determined the distinct levels of mathematics, this section will be concerned with building a model of metafunction in mathematics. As stressed throughout, metafunctions are not assumed, but must be justified. Evidence for metafunctionality is drawn from two sources: i) relative paradigmatic independence or interdependence and ii) structural similarity or dissimilarity (presented in Chapter 2). If an area of the grammar has both the potential for relatively independent variation with other areas and a distinct type of structural realisation, evidence of a metafunctional component exists.

From this basis, we will see that the architecture of mathematics is dominated by the ideational metafunction. In particular, the logical component permeates the grammar and builds the nesting scale on which most sets of choices exist. The other component within the ideational metafunction we will call the operational component. In comparison to the logical component, this component is relatively small and is in some ways subservient to the logical. It is nonetheless responsible for the development of the rank scale. Textual variation comes about at both levels of the nesting scale, organising the information flow. Most strikingly, there appears to be no evidence for an independently motivated interpersonal component. Each of these observations will be considered in turn, before turning to a discussion of the overall functionality of mathematics that this analysis suggests.

The Logical Component

The predominant structural organisation of the grammar is a univariate structure. Statements are built from an indefinitely iterative complex of expressions and expressions are built from an indefinitely iterative complex of symbols. This is reflected in the paradigmatic organisation of the statement where all choices are potentially recursive (see Figure 3.9 in the previous chapter). Each new expression necessitates a new choice in the type of Relator and vice versa. Below, at the level of symbol, recursion is also dominant. Symbols regularly complex into highly intricate expressions. Indeed, it is through the obligatory complexing of symbols into expressions and expressions into statements (that is, the complexing of symbol complexes) that the nesting scale arises. The recursive and univariate organisation of the mathematics is pervasive.

The structural similarity of each of these systems, being univariate, suggests each could be part of a similar functional component. This is augmented by the fact that in mathematics the prototypically recursive systems are in general independent of those that produce the multivariate and periodic structures. The choice of the number of expressions in a statement is independent of the choice of their organisation in terms of Theme and Articulation. That is, the sequence of expressions does not determine the choice of Relator.⁴ At the level of symbol, the system of EXPRESSION TYPE that determines the complexing relationships between symbols is simultaneous and thus independent of the choices within symbols that are multivariate. That is, any unary operation such as sin, cos, Δ etc. can occur with any binary operation, such as ×, ÷ and +. The system of EXPRESSION TYPE is also closely intertwined with the system of COVARIATION. The covariate relations, in conjunction with the system of STATEMENT TYPE, determine the possible types of expression. This leads to the potentially indefinitely iterative nature of the COVARIATION system, whereby any number of covariate relations may occur. Due to this close interaction of COVARIATION and EXPRESSION TYPE and the fact that the COVARIATION system is indefinitely recursive, leading to a structure more closely related to univariate structures than multivariate structures, COVARIATION can also be considered part of this component.

All areas of the grammar that are organised through an interdependency structure (univariate plus covariate) are almost entirely independent of those organised multivariately. That is, there is a large group of systems that have structural similarity and paradigmatic independence from other systems. Thus these systems fulfil the criteria for being grouped into a distinct functional component.

Given the fact that this component has recursive systems as one of its hallmarks and is primarily organised through a univariate structure, this component appears most similar to the logical component within English (Halliday 1979). We can thus responsibly classify these systems as being part of the logical metafunction. The systems that constitute the logical component are as follows:

statement

• STATEMENT TYPE
• REARTICULATION
• COVARIATION
• COVARIATE MULTIPLICITY

symbol

• EXPRESSION TYPE

The logical metafunction dominates the grammar, colouring most other systems. As discussed previously, this is seen through the potentially recursive unary operations (classed as operational) and the iterative Articulations (classed as textual) brought about by tension with the potentially iterative expressions. The logical metafunction is also responsible for the nesting scale in mathematics. However, as we will see, the rank scale comes about through the other component of the ideational metafunction, the operational component.

The Operational Component

Simultaneous to the logical system of EXPRESSION TYPE at the level of symbol are the systems of UNARY OPERATION and SPECIFICATION. Both of these systems are realised through a multivariate structure. The system of SPECIFICATION distinguishes between the functions of Quantity and Specifier, while the system of UNARY OPERATION gives both the Operation and Argument. As these systems are paradigmatically independent of the systems in the logical component and are realised through a distinct structure, a case holds for these systems to form their own functional component. This component is entirely responsible for the rank scale in mathematics. It is through preselections within both SPECIFICATION and UNARY OPERATION that the development of a system at this lower level is justified. Moreover, the multivariate structure resulting from these systems provides the constituency relation between symbols and elements that distinguish this scale from the nesting scale derived from the logical component. As the distinctions within the rank of element are preselected from systems within a single component, and there are no simultaneous systems within this rank, the entire set of choices within the rank of elements can be classed as part of the same component.

The multivariate structures in this component are similar to those within the experiential metafunction in English. Notionally the experiential meta-function is concerned with construing our experience of the outside world (Halliday 1979). For example, in English, the experiential system of TRANSITIVITY divides the clause into material, mental and relational clauses (Matthiessen 1995), broadly construing the realms of doing, thinking and being. Although in an axial description the notional ‘meanings’ of categories are not privileged, they are helpful when labeling. At the level of symbol, it is difficult to reconcile the choices of UNARY OPERATION or SPECIFICATION as in some way construing our outside world. In this sense, the label ‘experiential’ is somewhat awkward. This component is more concerned with operations on elements in symbols than with construing the experiential world. Thus, to more easily capture this nature, this component will be called the operational component. The systems included in the operational component are as follows:

symbol

• UNARY OPERATION
• SPECIFICATION

element

• ELEMENT TYPE

At this point it is pertinent to note the ineffability of semiotic categories (Halliday 1984). The component labelled earlier as operational is justified as a distinct component through its paradigmatic and syntagmatic organisation, not through its notional meaning. By labelling this component ‘operational’ we emphasise the differences between this component and the experiential component in English. These differences include the considerably distinct sets of choices that each component includes: the operational component of mathematics does not include choices for TRANSITIVITY, CIRCUMSTANTIATION, CLASSIFICATION, EPITHESIS, QUALIFICATION, EVENT TYPE or ASPECT as it does in the grammar of English (Halliday and Matthiessen 2014: 87). Conversely, the experiential component of English does not include choices of UNARY TYPE or SPECIFICATION as occurs in the operational component of mathematics. Indeed, aside from their multivariate structure, there is little in common between the two components. Thus, distinct labelling is appropriate. It remains to be seen whether the systems captured under the experiential component in English and those in the operational component in mathematics are in some sense part of the same component in the broader scheme of semiosis (or, indeed, whether metafunction is a useful category at all in the broader description of semiosis). To determine this, detailed axially motivated descriptions of inter- and multi-semiosis would need to be developed. What this will uncover, or, indeed, what this would look like, is at this stage unclear.

This aside, the operational component will be grouped with logical component as parts of the ideational metafunction. This allows the mathematical system to be characterised as one built primarily by the ideational metafunction. They are both organised syntagmatically through a particulate structure and notionally allow mathematics to represent the world.

In Section 4.1, it was shown that each unary operation can be equated with a set of logical relations. For example, the change operator Δ, when operating on a variable, is defined as Δx = x₂ − x₁. The unary operation Δ thus distils the logical relation x₂ − x₁.⁵ Indeed all unary operators distil a set of logical relations that can be applied to a range of symbols.

This suggests that the multivariate component of mathematics has developed as a grammaticalisation of large sets of logical relations. As mathematics progresses through higher levels of schooling, more unary operations are introduced, distilling ever-increasing sets of logical relations. Thus ontogenetically speaking, it appears that the operational component develops out of the logical. Indeed, as students move into calculus, increasingly multivariate structures are built upon growing sets of logical relations. The distillation of logical relations into the multivariate structure is similar to the development of technicality in English. The operational component both condenses and changes the nature of the logical relations (Halliday and Martin 1993: 33). Thus although the operational component of the grammar grows through the years, it is based on and in some sense in service of, the logical metafunction. This provides another justification for both components to be aspects of the same ideational metafunction, fulfilling complementary roles. The final set of variation relates mathematics to the textual metafunction.

The Textual Component

In addition to the variation involved in the logical and operational components, there is a small set of choices that organise the information flow of the text. These systems include the choice of Theme, its possible ellipsis and the ordering of Articulations. Each of these choices are independent to choices in the logical and operational components. The Theme or any Articulation may be any expression involving any complex of symbols or unary operations. Similarly, any set of symbols can be elided if they are Theme. Further, the Theme-Articulation structure was described as a periodic structure, with the potential for indefinitely iterative Articulations under pressure from the logical component.

It was said earlier that the nesting scale develops through the logical component, while the rank scale derives from the operational component. The paradigmatic choice of which symbols are thematised and which are placed in the Articulation occurs at the level of symbol. However, these choices realise the Theme and Articulation structures deriving from the level of statement. As well as this, all variation at the rank of element, and all choices at the rank of symbol that preselect types of elements are accounted for by the operational component. Thus the bundle systems organising the Theme-Articulation choices and those of ellipsis sit firmly within the nesting scale.6

As mentioned earlier, these systems are independent of all choices in the logical and operational components. As well as this, they sit in the nesting scale but have a distinct periodic configuration to the logical univariate and the operational multivariate structures that produces the nesting and rank scales. For these reasons, it seems appropriate to consider these systems to be part of a separate component altogether. As this component is concerned with the information flow of mathematics, it can be termed the textual metafunction for mathematics. The systems comprising this metafunction are as follows:

symbol

• THEME
• THEME ELLIPSIS
• ARTICULATION

Against an Interpersonal Component

From an axial perspective, there is no evidence to propose an interpersonal component in mathematics. The three components outlined so far, the logical, operational and textual, account for all of the systems within this grammar. There are no apparent systems realised through a prosodic structure, nor are there any other systems paradigmatically independent of those already accounted for. Looking notionally, there are no systems that appear to give similar meanings to those of NEGOTIATION or SPEECH FUNCTION, nor those that give the evaluative meanings of APPRAISAL or the power and solidarity dimensions of VOCATION. Nor are there any systems that give choices comparable to those of MOOD. Without paradigmatic independence or distinct syntagmatic structures, there is no reason to suggest a distinct interpersonal component in mathematics.

In light of this description, however, it is pertinent to consider a number of important insights made by O’Halloran (2005) regarding the possibility of an interpersonal component. First, O’Halloran points out that there are some similarities in the meanings between certain Relators in mathematics and interpersonal constructions in English. In particular, she suggests a system of POLARITY to distinguish between the positive polarity of = and the negative polarity of ≠ (often glossed as not equal to) (2005: 100, 115). POLARITY in English is considered an interpersonal system (Martin 1983, though not without some contention, see Halliday 1978: 132 where it is treated as experiential and Fawcett 2008 where it is its own functional component), thus O’Halloran considers the distinction between = and ≠ to also be interpersonal in mathematics. Along similar lines, we could consider Relators ≈ and ∼, both glossed as approximately equal to, to notionally give some sort of meaning of GRADUATION, another interpersonal system (Martin and White 2005). These could also therefore form part of an interpersonal component. However, developing an interpersonal component based on similarities in meanings such as these goes against the principled axial description built in this and the previous chapter. Arguments along these lines rely on notional reasoning that analogises from English. In contrast, if looking axially, we see that each of these Relators, ≠, =, ≈ and ∼ are entirely dependent on choices within the STATEMENT TYPE network, which was classified as part of the logical component. Thus, the choices giving rise to these Relators are firmly within the logical component. They do not form a distinct paradigmatic system simultaneous with those of other components, nor are they realised by a distinct type of structure. Therefore, they do not constitute a distinct functional component. They do, however, suggest that the meanings of polarity and graduation have in some sense been ‘ideationalised’ when translated into mathematics. Meanings that would be made through the interpersonal metafunction in English are made through the ideational metafunction in mathematics.

An interpretation along these lines allows an understanding of the quantification of certainty through probability, statistics and measurement errors, in relation to linguistic modality. As O’Halloran (2005: 115) states,

In mathematics, choices for MODALITY in the form of probability may be realised through symbolic statements or measures of probability; for example, levels of significance: p < 0.5 (where the notion of uncertainty is quantified) and different forms of approximations.

Viewed from the description developed in this chapter, the meanings of modality have been ideationalised in a similar way to the polarity and graduation meanings discussed earlier. What would be expressed interpersonally in language through graded modalisation of probability (e.g. The hypothesis is probably true), is expressed through an ideationally organised mathematical statement (such as through p-values: e.g. p < 0.05 used to determine the likelihood of a hypothesis being true or false). What is interpersonal in language can be seen as quantified and ideationalised in mathematics. Although statistical mathematics is not studied in detail in this description, it is possible that this system has developed largely to ideationalise what would otherwise in language be fuzzy interpersonal measures of modalisation.

A second important observation by O’Halloran regards interactions between mathematics and language in instances such as Let x = 2. The use of Let before the mathematical statement indicates the construction is a discourse semantic command (demanding goods and services). On the other hand, without the Let, the mathematical statement is arguably more similar to a discourse semantic statement (giving information). Thus, the Let affects the speech function, giving it variability in interpersonal meaning.

In regards to whether this constitutes evidence for an interpersonal component in mathematics, the grammar developed in this description only considers mathematics in isolation; it does not look at the interaction of mathematics and language. From this perspective, the introduction of language into the statement is immaterial to a discussion of the functionality internal to the system of mathematics. However, it does raise an important challenge that has yet to be fully solved. The introduction of language appears to contextualise the mathematics, transposing the speech-functional meanings from language across to the mathematics. This raises the question of how then we are to model metafunctionality across intersemiotic systems, or indeed across semiosis in general. Arguing that mathematics does not have an interpersonal component internal to the system does not preclude the possibility that mathematics occurs in texts with resources that do engender interpersonal meaning. More broadly speaking, with multimodal texts, the various functionalities of each semiotic resource are likely to contextualise one another. In the case of Let x = 2, as mathematics does not have the ability to distinguish between speech-functions, it appears that language is being ‘imported’ as necessary to make these meanings. Studying the internal functionality of different semiotic resources could provide insights into why some are used in conjunction with others.

One final point regarding the lack of an interpersonal component in mathematics concerns its relation with the register variable of tenor. In Systemic Functional studies, it is generally accepted that there is a meta-functional ‘hook-up’ with different register variables: shifts in field tend to impact ideational meanings, shifts in tenor tend to impact interpersonal meanings and shifts in mode tend to impact textual meanings. But without an interpersonal component, this register-metafunction correlation could potentially be put at risk. This, however, brings us back to the point made earlier that mathematics is rarely used in isolation. Other semiotic resources such as language, images and gesture, are regularly used alongside mathematics in various contexts. Multimodal texts that include mathematics are likely to shift interpersonal meanings across multiple resources. Thus tenor would be realised multimodally.

In short, the lack of an internally motivated interpersonal component in mathematics does not preclude mathematics from being involved in interpersonal meaning in a multimodal text. What it does suggest is that mathematics cannot produce variations in interpersonal meaning of its own accord; it must do so in interaction with other semiotic resources. Interpersonal meanings from one resource are likely to be ideationalised when translated into mathematics. It is possible this ideationalising feature of mathematics is a large reason for its powerful role in academic disciplines. The role of mathematics in building academic knowledge will be discussed in relation to genre and field in Chapters 5 and 6.

The Function-Level Matrix for the Grammar of Mathematics

With the discussion of metafunctionality in mathematics, the grammatical description has been completed. We can now bring together the metafunctions, level hierarchy and systems of mathematics into a single function-level matrix shown in Table 4.4.

Table 4.4 Function-level matrix for the grammar of mathematics

Nesting	Rank	Logical	Textual	Operational
statement		STATEMENT TYPE REARTICULATION COVARIATION COVARIATE MULTIPLICITY
symbol		EXPRESSION TYPE	THEME THEME ELLIPSIS ARTICULATION	UNARY OPERATION SPECIFICATION
	element			ELEMENT TYPE

This table shows the most salient features of the architecture of the grammar of mathematics:

• Three functional components:
  • ⚬ textual
  • ⚬ logical
  • ⚬ operational (but no interpersonal);

• Two level hierarchies:
  • ⚬ a nesting hierarchy involving statements and symbols
  • ⚬ a rank scale involving symbols and elements. The level of symbol operates on both hierarchies;

• Logical and textual systems are confined to levels on the nesting scale;
• Operational systems are confined to levels on the rank scale.

4.6 An Axial Description of Mathematics

This and the previous chapters set the goal of developing a model of mathematics based on axial principles. They took the paradigmatic and syntagmatic axes as primitive and sought to justify larger features of the descriptive architecture from these. To this end, they did not assume macrotheoretical categories such as metafunction or rank, but rather looked to derive these from the systems and structures apparent in the grammar. In doing this, they have produced a description that is in a number of ways ‘exotic’ when compared to language. Rather than a single rank-hierarchy based on constituency, the grammar has shown there are two hierarchies in play, catering to different types of variation. The nesting hierarchy is derived from the logical component and affords the possibility for informational organisation through the textual component. On the other hand, the rank scale organises the systems of the operational component. These three components, the logical, operational and textual, were shown to be the only metafunctional variables needed to account for the entirety of the grammar. No interpersonal systems were found. Indeed, it was suggested that part of the reason for using language in interaction with mathematics is to make use of language’s interpersonal meanings. Mathematics, for its part, was argued to ‘digitally’ ideationalise meanings that may otherwise have been expressed interpersonally in graded systems in language. The specific meanings made by mathematics, and their role in the knowledge building in physics will be explored in relation to the register variable field in Chapter 6.

The approach taken in this chapter has attempted to utilise a descriptive methodology that can show the functionality of mathematics on its own terms. Axis was chosen as the primitive in relation to i) its potential to be used as the basis for deriving other characteristics of the grammar and ii) its generalisability across semiotic systems. If we wish descriptive and theoretical categories such as metafunction and rank to continue to have utility in the future of Systemic Functional Semiotics, they must be justifiable. The description offered here has shown that metafunction and rank are indeed productive notions, allowing large segments of mathematics and other semiotic resources to be characterised and generalised. But without a principled axial foundation for determining and distinguishing ranks, metafunctions and other categories, descriptive semiotics runs the risk of emptying these terms of meaning, and making everything look like English.

This and the previous chapter has looked at mathematics in isolation on a small, grammatical scale. The following chapter comes at mathematics from a different angle, genre. As part of this, language will be brought into the picture, providing an avenue for understanding intersemiotic relations. With views from both grammar and genre, Chapter 6 will consider mathematics’ role in knowledge building in relation to images and language from the perspective of the register variable field.

Notes

1. More specifically, they indicate stronger discursive semantic density (Maton 2014 chapter 9). That is, the relations being encoded have nothing to do with the object of study, but rather are relations entirely internal to the system of mathematics. It is only once the unary operations have been placed within the field of physics (or another field) and operate on technicality within that field that the operators add meanings relating to the object of study (strengthening the ontic semantic density).

2. Often units will be accompanied with a direction, such as downwards in W_Mars = 180 N downwards, from Text 3.1 given in the previous chapter. This depends on whether the variable being measured is a vector or a scalar (discussed next). Directions are not taken as part of the grammar of mathematics described here as they cannot complex with binary operations, nor can they vary their position like all other symbols, and they are almost always realised by language, not a specific mathematical symbol. They appear best described as a linguistic element emergent from the intermodality between mathematics and language.

3. Particular types of vector known as unit vectors, such as those used in certain coordinate systems (e.g. for spherical coordinates) are not included in this grammar. They are more commonly used in vector calculus, which is beyond the scope of the physics used as the corpus for this description. However, if they were to be included, the system distinguishing vectors and scalars would be simultaneous with the system distinguishing constants and variables.

4. The exception is swapping [greater-than] (e.g. >) with [smaller-than] (e.g. <) in certain cases, discussed in Chapter 3.

5. More strictly, the unary operation Δ distils a dummy relation of subtraction between two instances of the same symbol (shown through different specification subscripts). This relation can be applied to any symbol, so Δ x = x2 − x₁, Δ y = y₂ − y₁, Δ z = z 2 − z 1, etc.

6. There is another set of choices not described here that could also form part of this component at the level of symbol. This involves the distinction between symbol complexes such as $\frac{m{v}^2}{r}$ and $m\frac{v^2}{r}$. Ideationally, these symbol complexes are the same. However, they are organised marginally differently in a way that appears to give some sort of textual meaning. Further description is needed to incorporate this variation into the grammar.

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