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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 universal adoption of decimal notation (our characteristic approach from positivism leads us to privilege the phenomenal digits, rather than their intervals). 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.

There is an implicit dialectic here—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, (e.g., as a guarantor of referential unity in thought and language), 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.).

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 (analytic rather than synthetic). 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 members of relational groups in series (rather than as discrete Platonic entities in their own right), having notional rather than substantial value, and with particular dispositional properties that arise extrinsically—i.e., according to the restrictive range of nominal digits available within the terms of the current working radix (0-9 in decimal, 0-7 in octal, for instance).

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 accidental 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 commodious to various instrumental (exploitative) approaches to the observation and measurement of processes in Nature, is a principle based upon a transcendental assertion of unity, stability, or substance that inheres in the integer ‘1’. In other words, this principle asserts the integer ‘1’ as a concrete index of quantity, having absolute, intrinsic value, while failing to judge that its supposed value in those terms can only be derived metaphysically. For mathematical objects to achieve reality for the subject, they cannot be be exhibited to the mind in the idea of transcendental substance, but must take the form of an image or illustration, which is to say, as an empirical intuition.³

It does not require an exceptional philosophical insight to notice that the expression of integers according to the decimal rational schema requires: firstly (as noted above), an empirical intuition of available finger-digits based upon somatic characteristics common to most of the species; and secondly, a formal synthesis with a set of rules for the naming of discrete units, which determines that ‘9’ should be the maximum writable digit (rules that are formally incompatible with those that apply in the case of octal notation, for instance). The formal definition of the integers as self-contained entities—hence with values that transcend their particular system of notation—which has supported abstract mathematical theory and discourse for four centuries, is one that by necessity neglects and remains insensitive to these fundamental requirements involving both intuition and synthesis in the creation of a formal system that enables us to count beyond the ten digits we have in front of us.

Are mathematical judgements analytic or synthetic?

Rationalist and scientific understanding holds that the certainty of mathematical judgments is formally established prior to their application to real world criteria. In this sense, arithmetic statements are required to have validity on their own terms, without reference to external criteria such as the evidence from sense data, or knowledge not already implied in the formal rules of arithmetic. For this reason, the mathematical logicists Gottlob Frege and Bertrand Russell held that statements of arithmetic were analytic in nature, and as such they offer grounds for certainty in scientific judgments independently of any appeal to metaphysics (notwithstanding the received wisdom with regard to the natural numbers shared by these analytic philosophers was itself indebted to metaphysics).

Both Frege and Russell, in their unsatisfactory attempts to prove that arithmetic may be deduced from pure logic, had objected to Kant’s assertion in his Critique of Pure Reason in which he wrote that the expression: “7 + 5 = 12” is an example of a synthetic proposition (i.e., that the concept of the sum ‘12’ involves an element of knowledge or intuition which is not intrinsically implied within the additives on the left side of the equation).⁴

“[T]he self-evident propositions as to the relation of numbers, are certainly synthetical but not universal, like those of geometry [...] That 7 + 5 = 12 is not an analytical proposition. For neither in the representation of seven, nor of five, nor of the composition of the two numbers, do I cogitate the number twelve.”⁵

Against this we might argue that it is elementary that the sum of the additives is 12, and that Kant is being overly pedantic. But that would be to miss Kant’s point: which is that it requires something additional, by way of a synthesis in intuition, to arrive at the concept of 12—whether that intuition consists of the image of the digits of one’s hands, or of our customary formalisation of that image in the rules the image has long provided for the expression of values according to the decimal rational schema.

In the epoch of Kant’s Critique it was uncommon for mathematicians to engage in converting between decimal and other arbitrary number bases—the practice did not become widespread until the 19th Century. For this reason Kant illustrates the process of arriving at the number 12 by reference to the image of his hands and fingers, without referencing the specificity of the decimal rules as a synthetic element in the equation. Nevertheless, it is clear that it is the fact that the rules of decimal are employed heuristically when computing the sum: 7 + 5, but that the rules are in no sense endemic to those values per se (as they might just as well be octal values) that leads Kant to insist that it requires a synthesis with some intuition not strictly contained in (7 + 5) in order to arrive at the concept of 12 as their sum.

A key concern of the Critique is the question of how mathematical knowledge is possible. Kant states that an a priori condition for the subject’s conception of number in general is its experience of internal temporal succession, as the pure intuition of a series of countable moments in time.⁶ In other words, there cannot be a purely cerebral form of mathematical experience for the subject, one that answers only to the rules of pure logic. The idea therefore that statements involving functions upon numbers can proceed analytically, purely on their own terms, is to defy the subject’s comprehension of those terms, which after all is dependent upon a synthesis with a pure intuition of time as its foundation (as well as with any empirical intuitions involved in the cognition of space-occupying units as the constituents of assembled quantities, such as in the image of one’s fingers). This is the basis for Kant’s claim that statements of arithmetic, and all other forms of mathematical judgment, are unavoidably synthetic a priori in nature.

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
(revised: 23 July 2026)

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. 103-106. [back]
Cf. Alfredo Ferrarin’s Pure Intuition in Mathematics: Historical Origins of a Misunderstanding, Studi Kantiani, XXV, 2012, pp. 32-33. [back]
Cf. Russell, B., The Philosophy of Logical Atomism (1918), Routledge, 2010, p. 128; Frege, G., The Foundations of Arithmetic (1884), Austin, J. L. (tr.), 2nd edition, Harper, New York, 1953, pp. 17-24. [back]
Kant, I., Critique of Pure Reason, Guyer, P. & Wood, A.W. (trans.), Cambridge University Press, 1998, p. 288 (A164/B205). (Available at: http://archive.org – accessed 22/07/2026.) [back]
Kant, I., Ibid., pp. 274-275 (A142-143/B182). [back]