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 that 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 Xcetera section of the site, so that that section should not appear, from the point of view of the other, as some kind of afterthought.

Xcetera 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 numerical values, i.e., when those 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 otherwise 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 Xcetera 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 Xcetera is all about.. If you think you’re ‘ard enough!

S O M R

[ PDF version ]

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Download my 173-page
report: Special Operations
in Medical Research
[pdf – 1.9MB]:

Download my Open Letter to the British Prime Minister & Health Secretary
[pdf – 363KB]:

The Limits of Rationality
(An important mathematical oversight)
[863KB]:

Radical Affinity and
Variant Proportion in
Natural Numbers
[2.53MB]:

Mind: Before & Beyond Computation
[643KB]:

Dawkins' Theory of Memetics – A Biological Assault on the Cultural
[508KB]:

Randomness, Non-
Randomness, & Structural Selectivity
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Intuitive Periodicity in Numerical and Temporal Sequence

If I may be excused in what follows for relying upon a heavily intuitive approach, but I am chiefly concerned with advocating an approach to the understanding of quantitative systems which acknowledges that there are aspects, generally speaking, of our approach to numbers and numerical scales which are not primarily, or not entirely, rational, logical, and proportional. I am interested in the ways in which we habitually conceive of numbers, both large and small, not strictly as measures of pure quantity, that is, for instance, with respect to measurements of Time, but also as possessing the characteristics of a form of spatial distribution – that we might conceive of numbers as occupying an ordered ‘space’ of their own, on a distributed line or curve. In this I am adopting a properly Kantian notion of ‘space’ as a form of primary intuition (rather than as a perceptible object in its own right) formed necessarily prior to the sensory perception of material objects within space. While the notion of space as an intuition in Kant serves as a precondition for the later cogniton of objects in space through the modes of sensory perception, the idea I am alluding to is that of a notion of an otherwise empty space reserved purely for the natural numbers – existing ‘in the mind’s eye’ so to speak, uncorrupted by issues of sense-perception. The idea that the set of the natural numbers might be presented to the mind with the features of a spacial distribution suggests that numbers might thus display certain dispositional properties, as pertaining to the particular spatial location of individual numbers relative to other numbers. As members of a predefined and ordered distribution (albeit one based upon their nominal quantitative values), in this sense numbers cannot be considered as free-floating or freely-selectable quantities, and therefore for as long as they remain – in the mind’s eye – the occupiers of predetermined positions in a spatial distribution, they will always retain certain qualitative dependencies in their relations to other members in the same distribution.

Such qualitative dependencies can only be understood as being derived out of the relationships between numerical values – they cannot be understood as the properties of numbers held integrally, according to the formal definition of an integer. To proceed with this enquiry therefore requires that we suspend or reserve our attachment to the formal definition of an integer inherited from classical mathematics. That definition presupposes that integers be considered as entities in themselves, whose properties are entirely analytic (self-contained).

Imaginary Numbers

If one tries to conceive of a very large number, say ‘one million’, not as it might commonly be conceived as a mass of individuals occupying an area (say, the population of a city), but rather in terms of the position of this integer on a linear scale, which also includes, for the purposes of the exercise, the numbers ‘0’, ‘10’, ‘1000’, etc.; how does this line of integers appear to the intuition? Is it possible for the intuition to conceive of such a line in its entirety? If it is possible to do so, what sort of shape does it have? If such a conception is readily available to us, either completely or in parts or segments, is this conception something universal, which is shared by all, or is it a matter of the individual imagination? I ask these questions because it occurs to me that it is a technical impossibility, I think, to render such a line objectively and proportionally without the lower value integers collapsing into a single point. Therefore, if we want to ascend the scale (one might wish to think how to convert the £10 we have in the bank into our first million, for instance), from the lower numbers all the way up, what sort of mental exercise is involved?

Let’s try something a little less ambitious. Let’s try to conceive of ‘ten thousand’, while of course not sacrificing the idea of linear continuity, that is, the possibility of progressing from our meagre £10, to this perhaps more realistic inflation. If we want to make the ‘ten thousand’, but would still like to keep in mind the option of further progression, does this make the mental acrobatics any easier?

Quanta

It does not seem, to my intuition at least, that it is possible to make these mental adjustments in terms of a continuous straight line, because as we approach the higher values the lower values will become compressed and we will lose our proportional foothold on the scale. Some kind of curve is necessary. In fact, as we add consecutive zeros onto the end of our target figure, it appears to me that I must ascend up the scale by a sort of quantum helix. And at any particular point on this helix I have the option of either following the helical gradient gradually upwards (that is, mostly horizontally), or instead of taking a quantum leap vertically onto the next elevation. If all one has to do, in imaginary terms, is add imaginary zeros to the value as desire dictates, in order to jump from say ‘100’ to ‘100,000’, in qualitative terms doesn’t this make the whole exercise rather facile? If making your first million is such an easy conception, but in fact much harder to materialise, what are the implications of this arrangement for our psyches?

This problem, if it is a problem, seems to me to be a characteristic (or a symptom) of an undue reliance on numerical description as a concrete index of quantity. Of course, we need numbers to represent quantities reliably – we would have difficulty making any kind of exchange without that. But are there aspects of numerical description which the above hypothesis of a quantum helix suggests may be primarily intellectual rather than concrete and substantial? If there are options of progression from the small to the large by quantum leaps rather than by steady linear progressions, to what extent do imaginary drives influence quantitative expectations? How do numbers relate to our mundane experience, for instance, to the proportionality of our lifespans in the grand scales of historical or geological time, rather than simply to other abstract values in the numerical scale? Do we establish any patterns in our treatment of small-scale numbers which can be applied vertically by way of imaginary quantum up-scaling? In trying to ascertain what sort of treatment or ‘shape’ we intuitively give to numbers, it may help if we first concentrate on what is going on in the early stages of the curve.

From Zero to One-Hundred

If we accept that this hypothesis of a quantum helix is such an inescapable feature of the imagination, or rather the intuition – i.e. as a psychological necessity in order for us to conceive of certain large numbers in scalable terms, how are the low numbers, say from ‘0’ to ‘100’, represented on the curve? What sort of position do we take in relation to the curve when considering these values?¹

For instance, if I first imagine the value ‘10’ and its relationship to ‘0’, and then proceed in leaps of ten through ‘20’, ‘30’, ‘40’, etc., up to ‘100’, I imagine my point of view at some radial distance from the curve, and as I progress my focus shifts to different segments of the curve, and in doing so my viewpoint rotates, in a clockwise fashion.² However, the curve is not circular as such, but it veers more sharply at certain points. For instance, for the periods in between the values ‘0’ and ‘10’, ‘10’ and ‘20’, ‘20’ and ‘30’ respectively, the curve is relatively smooth; but at the precise points of the values ‘10’, ‘20’, and ‘30’, it bends significantly, more significantly at ‘20’, and most at ‘30’. By the time we have passed ‘30’ the curve has more than doubled back on itself from the direction it took between ‘0’ and ‘10’. Between ‘30’ and ‘100’ there are no further significant bends in the curve, its progress being relatively smooth, so that at point ‘100’ we are now situated somewhere in a position vertically above the origin at ‘0’, though without necessarily being able to focus simultaneously on both ‘0’ and ‘100’ (so ‘100’ is in fact only theoretically above ‘0’ – it may also appear to have replaced it).

A further observation is that as we progress above ‘30’ in the direction of ‘100’, the ten-unit periods appear to become more compressed, so that ‘50’ is proportionally significantly closer to ‘100’ than it is to ‘0’ (considering the distance along the curve we have already travelled). The distance between, for instance, ‘80’ and ‘100’ is a fraction of that which we travelled between ‘0’ and ‘20’.

One-Hundred and Beyond

As the curve continues beyond the value ‘100’ it mimics the curve which is theoretically below it and which represented the distribution of integers between ‘0’ and ‘10’; but, due to the quantum elevation, there is a necessary adjustment in scale by a factor of x100, so that what occupied the positions ‘2’, ‘3’, ‘4’, etc., is now occupied by ‘200’, ‘300’, ‘400’, respectively.³ Eventually we reach the value ‘1000’, which is in the position theoretically above the original value of ‘10’. So it appears that in ascending by a single elevation of the helix, we have added two (not one) zeros to our original integers. If we continue by the same process upward we may project that each successive elevation of the helix is of the same factor (x100) of the values in the preceding elevation.⁴ The value ‘100’ then has a particular significance and importance in this intuitive schema, and this reaffirms analyses made frequently elsewhere in these pages that integers have irreducible qualitative properties in addition to, and in association with, their quantitative values.

If we return to our earlier observation that we perceived the figure ‘100’ to be theoretically above the origin of ‘0’, we notice that there now appears an ambiguity, as in fact if we remove the two zeros from the value ‘100’ we return not to ‘0’, but to ‘1’. There is no such ambiguity between the successive quantum values ‘10’ and ‘1000’. The only means of removing this ambiguity is effectively to remove ‘0’ from the scale altogether, which seems unreasonable as zero is indispensible in terms of abstract numerical notation. In mathematical terms ‘0’ is as valid a natural number as any other, and although there does not seem to be a role for negative integers in this intuitive schema, removing zero does not seem to be a valid option. Intuitively then, as revealed by the schema, there is an implicit ambiguity between ‘0’ and ‘1’, which means that henceforth the value ‘1’ will be affected by binary instability – ‘1’ will always, in terms of our intuitive apprehension, represent simultaneously both ‘1’ and ‘0’. Only an overarching and imposing rationalism can save it from this existential and ontological uncertainty.

We have to assume either that the curve begins abruptly at ‘0’, or, more intuitively, that at that point the line ‘plunges’ into nothingness. In view of the ambiguity between ‘1’ and ‘0’, the force of this is felt strongly at ‘1’ which has the effect of draining the number of its substance. As we descend the curve in the direction of zero, the last whole digit which feels secure is ‘2’, though the proximity of ‘2’ to both ‘1’ and ‘0’ suggests that it may be under constant threat of division, and this places some special significance on √2 (approximately, 1.414).

For a tabular representation of the ascending values in the quantum helix, and their vertical correspondences, see the document linked above as Quantum Elevations.

The Periodicity of Historical Time

The method I have tried to describe above seems to be applicable both to abstract numbers and also to the conception I have of the duration of my lifespan (of course the latter tends to be limited to the initial elevation of the helix). Can we apply a similar intuitive method to the passing of historical time? I say ‘historical’ time because decades and centuries are measured similarly in decimal notation, whereas small-scale time periods make use of duodecimal (base-12), and also sexagesimal (base-60). In addition the weekly cycle employs octal (base-8). The relevant question is whether historical time obeys the same intuitive logic as the time of our lifespan.

In the latter case we have a clear conception of a point of origin – our birth – and there is an equivalent conceptual zero at the beginning of our abstract number scale. The problem with historical time is that we are less certain of an absolute beginning. Even though western societies conventionally accept the birth of Christ as the zero registration for date notation, we are aware that this is by no means the actual beginning of time. And certainly, I cannot place this date at the base of our curve, and thereby deduce our current position to be in an equivalent position to 2012 on our number scale (i.e., approximately two tenths of the way round the second elevation, theoretically above ‘20’).⁵ Intuition suggests the year 2012 should rather be in an equivalent position to 12 on the number scale. Similarly, I project that the year 2100 will mark the end point of a complete cycle, just as ‘100’ does abstractly (however, I do not feel confident projecting further than this). Somehow then, the year 2000 (or was it 2001?) has come to represent a ‘new beginning’ in Time.

The same cannot be said, at least when viewed retrospectively, for the beginning of the second millennium (i.e., AD1000/1), or even for the birth of Christ. I do not perceive any similar helix extending backwards in time to cover these dates, but instead a rather steady incline stretching back as far as I care to take it. The same helical rules do not apply. I do perceive some change in inclination after the end of the first millennium, but it is not severe. What is striking however, is that this incline begins to level out at some point around the 17th Century, and finally, turns its first major corner at the years 1900-1920 – the subsequent trajectory of the 20th Century being virtually at right angles to that which preceded it – and approaching the beginning of the 21st Century with another significant bend, so that at least the first decade of the 21st Century proceeds in a reverse direction to that of the 19th..

May 2012
(revised: 29 April 2024)

Footnotes:

I am not suggesting here that this schema is one that is deliberately, or even consciously, chosen. In referring to the schema as a ‘psychological necessity’, I am interpreting it as a necessary precondition for the mental apprehension of numerical scale, prior to any involvement of the will, or the senses. It is not a feature therefore that we could choose to think independently of, or even one that we should be routinely aware of, in the analytical way in which I am trying to present it here. Although I have described the feature here as an “intuitive schema”, it would be more accurate, in terms of Kant’s Critique of Pure Reason, to refer to it as a ‘schema for the intuition’. [back]
The fact that this point of organisation exists at some radial distance from the curve, and at some elevation to it, both in terms of abstract numerical sequence, and also in terms of the relation the curve may have to the temporality of our lifespan, is significant, as the implication must be that in the case of the latter there is a position of notional immortality, suspended in nothingness, and which exerts a variable centrifugal/centripetal force upon the curve itself. [back]
In this respect, the frequent references made in the Old Testament to the ages attained by certain biblical characters being in the order of starkly unrealistic factors of 100 certainly become more comprehensible. [back]
At least this was my initial projection, assuming that the single exponential increases (10ⁿ⁺¹) at each position ‘10’ and ‘100’ would be reflected at each successive elevation of the curve. However, after further consideration, it seems that this projection results in various vertical mappings of integers which are counterintuitive. The document linked above as Quantum Elevations describes this problem in more detail and presents two sets of tables, the first defining the initial ‘problematic’ projection, and the second its preferred solution. [back]
In terms of abstract numbers, the initial elevation of the curve adds two successive exponentials of 10 (the double zero), the first at ‘10’, the second at ‘100’. Therefore ‘2000’ as an abstract number, rather than that of historical time, occurs in the second elevation, in the position which corresponds to ‘20’ on the first elevation. The conflict between abstract intuitive notation and historical intuitive notation is that ‘2000’ can occur at all in the position corresponding to ‘0/1’, as this position is normally only occupied by integers beginning with a ‘1’, not a ‘2’. [back]