Intuitive Periodicity in Numerical and Temporal Sequence

If I may be excused in what follows for relying upon a heavily intuitive approach, 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 solely, 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, as a measure of Time, but also as possessing the characteristics of a form of spatial distribution (at least terms of the intuition of the idea of space, rather than its objective perception as such), and the ways in which such qualitative relations are manifested as intuitive periodicity. Employing this concept, I raise the issue that there are there fundamental aspects of our intuitive apprehension of numerical scales which go beyond the constraints of formal mathematical or analytical enquiry, and which are not fully accounted for within the terms of exclusively quantitative explanations.

In conventional rational and scientific terms an 'integer' (whole number) is taken to represent a stable index of (pure) quantity – a logical object which is the bearer of an intrinsically-held ('integral') value – and which determines it to behave proportionally in its relations with all other integers. However, in the page Radical Affinity etc., I have presented empirical evidence to show that integers, when compared with their corresponding values across a series of diverse number-radices (or 'bases'), have tendencies to behave at variance with the principle of rational proportionality that coventionally informs our approach to the analyses of quantitative systems. These tendencies are revealed through a comparison of the differences between the logarithmic ratios of successive values in exponential series when expressed across a range of alternative radices from binary to nonary, in comparison with their 'identical' values in decimal. Those excercises reveal that the presumed intrinsically-held value of an integer is in fact proportionally inconsistent with that of its corresponding ('identical') values expressed through alternative numerical radices. This implies that the proportional characteristics of integers arise, not according to conventional understanding as a factor of their integral ('transcendental') values, but are rather determined extrinsically according to the context of the specific set of rules governing the expression of values in the particular numerical radix we happen to be working in (e.g., binary, octal, decimal, hexadecimal, etc.).1

This is in effect to say that the proportional (logical) characteristics pertaining to integers (whether expressed as decimal or as alternative radical values) must be determined locally according to the expression of values within a restrictive group of available permitted digits – the membership of each respective group varying according to the particular radix employed (0-9 in decimal, 0-7 in octal, for instance). As a consequence of this restrictive, 'intimate', grouping of available digits governing the expression of values in any particular radix, the proportional determinants affecting values in any single radix are incompatible with and non-transferable to their corresponding values expressed in any alternative radix. In the light of the evidence revealed (for the first time) at Radical Affinity etc., these extrinsically determined qualitative (or 'behavioural') properties of the natural numbers cannot be dismissed as the ephemeral products of superstition or esoteric thinking, but are now revealed as empirical items which present problems for the exercise of conventionally held rational principles, in so far as those principles are understood to operate 'universally'.

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 proportionately, that is, 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?


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, as 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 '10000', 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 so difficult 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 – 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 conceptual rather than concrete and substantial? If there are options of progression from the small to the large by quantum leaps rather than 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 life-spans 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?2 For instance, if I first imagine the value '10' and it 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.3 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. 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 the suggestion made earlier 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 understanding, 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 life-span (of course the latter is 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 life-span.

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 My intuition tells me that the year 2012 is rather 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?) comes 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

  1. For further technical elaboration on this point, see the Analysis section of the page: Radical Affinity etc. [back]
  2. 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. [back]
  3. 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 for the intuitive conception of the temporality of our life-span, 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]
  4. 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 counter-intuitive. 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]
  5. 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]

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