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 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 '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 which 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 numbers are never more than conceptual indices of value (or the abstract marks which represent those concepts); and therefore it would be more accommodating to experience to consider numbers as relational groups in series (rather than as discrete entities) having notional rather than substantial value, and with particular dispositional properties determined extrinsically (between concepts), according to the relative sizes of the groups.

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, inhering in 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 concept derived, not physically, but metaphysically.

To address this fundamental aspect of integers as relational concepts, 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 concepts 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 are not indices of absolute meanings, but their meanings are derived mainly extrinsically. In a comparative sense, numbers, as ideographic sign-forms (and not actual objects), can neither be adequately considered as indices of absolute, self-contained values. In overly simple terms, and in terms of common apprehension, a '3' followed by a '3' exhibits a different syntactical relationship, or qualitative disposition, from that of a '3' followed by an '8'.

The page Radical Affinity etc. compares the logarithmic differences between successive exponentials of the decimal x=10, across the range of corresponding values for the number radices binary to nonary (base9). The resulting eight graphs display series of variegated peaks and troughs, rather than the rational straight-line distribution which would be exhibited if one plotted the graph of the decimal distribution. Each series displays a quite unique syntactical distribution, and which (importantly) cannot be explained on any rational principles. If one refers to the graph in the octal section of that page, both the exponential values '4' and '10' are revealed as significant peaks of inconsistency in the relations of octal and decimal. One can see that the graph begins with an expected (rational) straight-line segment corresponding to the exponential values: z=1, and z=2; but then deviates with the distinctly 'bullish' orientation of the graph at the point z=4 (repeated also at z=10, and to a lesser degree at z=7). One can also clearly see the 'bearish' return to normality beginning at z=5 and ending at z=6 (followed by a less pronounced dip at z=8). There is no rational explanation for these deviations – the graph is quite startling in its display of non-rationality in the series (I should stress that these are empirical findings, beyond dispute3), and is more evocative of a stellar constellation than what we are accustomed to expect from a mathematical relationship. I have abstracted the shape of the graph in the image below in order to illustrate this impression:

graph-shape of the logarithmic differences between successive octal exponentials

Figure 1

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 grown-ups, 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

  1. 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]
  2. Aristotle, The Politics, Sinclair, T. A. (tr.), Penguin, 1981, Bk. II/ii, pp.103-6.[back]
  3. The graphs are distributions of the logarithmic differences between successive exponentials in each radical series. Therefore, their failure to produce logically consistent distributions for any numerical radix except that of decimal, points to a failure in the principle of logarithmic proportionality, when applied across diverse number-radices. See the Analysis section of Radical Affinity etc.; and Integers & Proportion for further elaboration. [back]

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