# Somatic Inscription (growing-up with numbers)

I once had a maths teacher who had six fingers on one hand, and this led me to wonder that perhaps this physical anomaly was the source of his extra ability in mathematics, as surely he would have had to work harder at it. Would this bonus of a vestigial digit have encouraged in him a deeper understanding of the centrality of *decimal* notation to conventional numerical description?

In certain native Mesoamerican cultures – e.g., the *Pamean* in Mexico – they count by using the *spaces between the fingers*, rather than the fingers themselves. Hence they are limited to eight. Theirs is an *octal* number system.

Counting is made less abstract by employing one's fingers and thumbs. As we possess ten digits, this suggests an implicit rationale for our adoption of decimal notation. From 'one' to 'ten' is straightforward, or apparently so. We do not begin at zero. Simple counting begins at the idea of *something*, rather than *nothing*. Formal (abstract) arithmetic requires that we acknowledge the zero position. However, we do not devote a finger digit to zero because: a) in simple terms fingers indicate positive values; and b) if we did we would only be able to count up to 9. Would it help if we had an extra finger?

This may seem frivolous, but I am trying to emphasise the kinds of difficulties that *zero* presents to the intuition, and in particular when it comes to bridging between the enumeration of real physical objects on the one hand, and abstract numerical notation on the other.

## Something or Nothing, or The Double

For the infant the world consists of *doubles* – its own and its mother's body; the two breasts; its two parents. Not only that, most of its significant bodily features are also duplicated. Eventually it will confront its self-image duplicated in the mirror. All meaning for the child is therefore constructed on the basis of pairings, which also imply *division*.^{1} The idea of the 'singular' is the threat of the loss of meaning, and also the loss of being. The singular must also be paired, if not with its double, then with zero.

It takes *two* – you cannot have something *or* nothing; you must have at least something *and* nothing, for the concept of 'something' to acquire any meaning at all.

My own body is not a coherent unit (in the sense of being a stable self-sufficient entity); it subsists in a series of shifting pairs, in a sort of continuous symbiotic flux. Its identity is at best a convenient fiction. If I were a one-legged cyclops it might be different.

Aristotle had remarked that the idea of 'unity' typically ascribed to individuals (in the somewhat remote Classical sense of the unity of mind and body – the seat of virtue) is not a characteristic that guarantees for an individual any degree of self-sufficiency. In his heirarchy of socio-political entities – individuals, households, and the state (the *polis*) – individuals are the least self-sufficient of the three, while at the same time exhibiting the greatest degree of unity. For Aristotle, self-sufficiency is a factor which increases only in proportion to the *plurality*, rather than the unity, of the social body, through the co-dependency of diverse characters and roles. The supposed unity of individuals is therefore, in Aristotle's terms, a mark of their *dependency* (as well as their dependability), and is inversely related to their capacity for self-sufficiency.^{2}

## 'One' – On The Edge of Being

In the page entitled Intuitive Periodicity in Numerical & Temporal Sequence, it was noted that as a condition of our intuitive apprehension of numerical scales beginning at zero, there is an implicit ambiguity between the integers '1' and '0', which I referred to there as 'binary instability'. The digits '0', '1', and '2' are in a unique relationship, and one that is not shared, by vertical correspondence to successive quantum exponentials (it is suggested that '100' corresponds vertically to both '1' and '0'). The unit '1' lies 'at the edge of being', so to speak, and this dynamic insubstantiality has a significant bearing upon '2', exposing it to division.

This relationship presents us with a precarious dynamic, and one that seems to me rather untenable. As soon as we settle for 'one', as an imaginary locus of meaning, i.e., as the guarantor of referential unity, we are threatened with its loss, with the draining of its substance. The unity of the singular is a mythical one which, for as long as we pursue it, will expose us to the scenario of *diminishing returns*. We should remember also that the idea of unity implied in 'one', is offset by *the other One* – the locus of our point-of-view in apprehending the number scale, at some radial distance from the curve, and suspended in nothingness (see: Intuitive Periodicity etc.).

If we are seeking an index of numerical value which encapsulates a sense of meaning (otherwise a sense of being, or positivity), we might avoid the dichotomy implicit in the relation of '1' and '0', and refer less precariously to √2. The latter represents a more intuitive division of '2' to its root, rather than to its discrete constituents, albeit geometrically rather than arithmetically.

## Digitalia

The concept of unity is central to our system of numerical notation – we count positive digits (fingers) as units, i.e., as discrete entities (rationally proportional, and possessing absolute, or intrinsic, value). The Oxford English Dictionary's definition of the word *integer* is *"whole number; thing complete in itself"*, and its etymology suggests the idea of something untouched, having intrinsic value – its properties are understood to be entirely self-contained. An integer's value, that is, is expressed independently of its relations to other integers. This definition obscures the fact that the system of the natural numbers is never more than an index of quantity serving the intellect; and therefore that it would be more accommodating to experience to consider numbers as *relational groups in series* (rather than as discrete entities in their own right) having notional rather than substantial value, and with particular dispositional properties determined extrinsically; i.e., *between* the individual elements of a group, according to the relative size of that group.

In conventional approaches to quantitative understanding, the graphical representations of numbers (e.g., '6', or '3', or '7') are treated as arbitrary marks – their qualitative differences (as ideographic sign-forms, or *glyphs*) are considered as merely coincidental to the values they represent, and these differences (essential, for instance, in any child's induction to the world of numbers) are resolved under the principle of *rational proportionality* that governs any mature, or scientific, understanding of quantitative value. My argument in these pages is that the principle of rational proportionality, while certainly convenient to various instrumental approaches to the observation and measurement of the processes of Nature, is a principle based upon a *teleological* assertion of unity, stability, or substance, that is inherent to the integer '1'. In other words, this principle asserts the integer '1' as a concrete index of quantity, having absolute, intrinsic value, when in fact it is a relational construct derived, not physically, but metaphysically.

To address this fundamental aspect of integers as relational constructs, it might be helpful to consider instances of numerical sign-forms by quasi-linguistic methods. Linguistics is concerned with *correspondences*, that is, the *syntactical* relationships between phonemes and between graphemes. Viewed in linguistic terms, the interpretation of integers as indices of absolute value is a purely *semantic* interpretation – numeric values are understood to reside as intrinsic properties of numbers as objective entities, and the relational syntax of numbers as ideographic constructs is forgotten. One of the benefits of a structural linguistic approach to language is its emphasis on meaning as derived through context and syntax – words rarely function as indices of self-contained meaning, but their meanings are derived mainly extrinsically. In a comparative sense, numbers, as ideographic sign-forms (with syntactic dependencies), can neither adequately be considered as indices of absolute, self-contained values.

## Goldilocks and The Three Bears

Of undoubted significance are the roles played by numbers, relations of scale, and repetition in children's fiction, particularly that for the very young (under fives). It is as if one's formative consciousness progressed through series of comparisons of number and scale, and surely, in relation to grownups, a child's dominant experience is one of diminution – all drive and ambition is focused on the number of my years and my size, i.e., numerically and subjectively. *"When I am BIG"*, everything I might wish for becomes a theoretical possibility, including, that is, *whom I might become* (as Alice discovered, her actual and original proportions were the only guarantee of her original identity).

Perhaps the most commonly encountered narrative numerical phenomenon is the *three* – the triplet – suggesting that the transition from dyadic to triadic relationships – invoking such impulses as competition and choice, rivalry and favouritism, etc. – is one that demands frequent cognitive reprocessing for the child. It is certainly far more complex a matter than the simple enumeration of objects for the purposes of counting, possessing, exchanging, etc. A child's experience is affected or 'stitched together' in terms of the qualitative relations of numbers. The number of a thing is of primary significance – it is never coincidental.

In *Goldilocks..* the archetypal family triangle is distanciated (by species) and through Goldilocks the reader identifies with the *fourth* position – as an outsider/intruder, who disturbs the natural order, but who sleeps through the consequences. This intrusion is made possible by the device of a temporal delay – the cooling of the porridge. Hence the cognitive shift from *three* to *four* adds a further layer of complexity to experience – a complication which involves the temporal dimension.

March 2012

### Footnotes:

- Of course, it is more complicated than this. There are implicit
*triadic*relationships here too, that is, as soon as the child acquires consciousness of its own body as a separate entity. My chief concern is to point out the initial importance of the*pair*as the minimal requirement in the construction of meaning, and the sense of being. [back] - Aristotle,
*The Politics*, Sinclair, T. A. (tr.), Penguin, 1981, Bk. II/ii, pp.103-6.[back]