An Important Mathematical Oversight

The original intention for this website was to encourage public awareness of an historical medical crime, one that has remained a tightly-kept British state secret now for more than five decades. The matter is of enormous public interest, not least because the motivation behind the crime itself was that of advancing scientific research into areas that would come to provide the seminal knowledge behind much of the technological progress of the last half-century. My investigation into the matter inspired a parallel enquiry into some of the fundamental principles that underpin that scientific and technological impulse.

There are therefore two principle concerns of this website, and if there is acknowledged to be a substantive connection between them, that has inevitably to do with late 20th Century developments in science and information technologies, and more broadly with the idea of an burgeoning technocracy – the suggestion of a growing alliance between corporate technology and state power – one which might be judged to have atrophied the powers conventionally assigned to liberal-democratic institutions. This link therefore serves as a segue to emphasise the equal importance, to my mind, of what is going on in the X.cetera section of the site, so that that section should not appear, from the point of view of the other, as some kind of 'afterthought'.

X.cetera is concerned with a problem in mathematics and science to do with the way we think about numbers. As a subset of the category defined as integers, elements in the series of the natural numbers are generally held to represent quantities as their absolute, or 'integral', properties. It is argued that this conventional understanding of integers, which is the one widely held amongst mathematicians and scientists adopting mathematical principles, is the cause of a significant oversight with regard to changes in the relations of proportion between numbers when numerical values are transposed out of the decimal rational schema into alternative numerical radices, such as those of binary, octal, and hexadecimal, etc.

On the page: The Limits of Rationality it is argued that the relations of proportion between integers are dictated principally by their membership of the restricted group of characters (0-9) as defined by the decimal rational schema; and that corresponding ratios of proportion cannot be assumed to apply between numerically equal values when transposed into alternative numerical radices having either reduced (as in binary or octal, for instance) or extended (as in hexadecimal) member-ranges.

This is shown to be objectively the case by the results published at: Radical Affinity and Variant Proportion in Natural Numbers, which show that for a series of exponential values in decimal, where the logarithmic ratios between those values are consistently equal to 1, the corresponding series of values when transposed into any radix from binary to nonary (base-9) results in logarithmic ratios having no consistent value at all, in each case producing a graph showing a series of variegated peaks and troughs displaying proportional inconsistency.

These findings are previously unacknowledged by mathematicians and information scientists alike, but the import of the findings is that, while the discrete values of individual integers transposed into alternative radices will be ostensibly equal across those radices, the ratios of proportion between those values will not be preserved, as these ratios must be determined uniquely according to the range of available digits within any respective radix (0-9 in decimal, 0-7 in octal, for instance); one consequence of which of course is the variable relative frequency (or 'potentiality') of specific individual digits when compared across radices. This observation has serious consequences in terms of its implications for the logical consistency of data produced within digital information systems, as the logic of those systems generally relies upon the seamless correspondence, not only of 'integral' values when transcribed between decimal and the aforementioned radices, but ultimately upon the relations of proportion between those values.

Information Science tends to treat the translation and recording of conventional analogue information into digital format unproblematically. The digital encoding of written, spoken, or visual information is seen to have little effect on the representational content of the message. The process is taken to be neutral, faithful, transparent. While the assessment of quantitative and qualitative differences at the level of the observable world necessarily entails assessments of proportion, the digital encoding of those assessments ultimately involves a reduction, at the level of machine code, to the form of a series of simple binary (or 'logical') distinctions between '1' and '0' – positive and negative. The process relies upon a tacit assumption that there exists such a level of fine-grained logical simplicity as the basis of a hierarchy of logical relationships, and which transcends all systems of conventional analogue (or indeed sensory) representation (be they linguistic, visual, sonic, or whatever); and that therefore we may break down these systems of representation to this level – the digital level – and then re-assemble them, as it were, without corruption. Logic is assumed to operate consistently without limits, as a sort of 'ambient' condition of information systems.

In the X.cetera section I am concerned to point out however that the logical relationship between '1' and '0' in a binary system (which equates in quantitative terms with what we understand as their proportional relationship) is derived specifically from their membership of a uniquely defined group of digits limited to two members. It does not derive from a set of transcendent logical principles arising elsewhere and having universal applicability (a proposition that, despite its apparent simplicity, may well come as a surprise to many mathematicians and information scientists alike).

As the proportional relationships affecting quantitative expressions within binary are uniquely and restrictively determined, they cannot be assumed to apply (with proportional consistency) to translations of the same expressions into decimal (or into any other number radix, such as octal, or hexadecimal). By extension therefore, the logical relationships within a binary system of codes, being subject to the same restrictive determinations, cannot therefore be applied with logical consistency to conventional analogue representations of the observable world, as this would be to invest binary code with a transcendent logical potential that it simply cannot possess – they may be applied to such representations, and the results may appear to be internally consistent, but they will certainly not be logically consistent with the world of objects.

The issue of a failure of logical consistency is one that concerns the relationships between data objects – it does not concern the specific accuracy or internal content of data objects themselves (just as the variation in proportion across radices concerns the dynamic relations between integers, rather than their specific 'integral' numerical values). This means that, from a conventional scientific-positivist perspective, which generally relies for its raw data upon information derived from discrete acts of measurement, the problem will be difficult to recognise or detect (as the data might well appear to possess internal consistency). One will however experience the effects of the failure (while being rather mystified as to its causes) in the lack of a reliable correspondence between expectations derived from data analyses, and real-world events.

So that's some of what X.cetera is all about.. If you think you're 'ard enough!

PDF DOWNLOADS

Download my 165-page report
[pdf – 1.9MB]:

Download my Open Letter to the British Prime Minister & Health Secretary
[pdf – 612KB]:

The Limits of Rationality
(an important mathematical oversight)

[479KB]:

Radical Affinity and Variant Proportion in Natural Numbers
[2.0MB]:

Mind: Before & Beyond Computation
[470KB]:

Dawkins' Theory of Memetics – A Biological Assault on the Cultural
[339KB]:

Randomness, Non-
Randomness, & Structural
Selectivity

[273KB]:

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

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Footnotes:

  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]

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