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 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 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 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!

S O M R

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Integers and Proportion

In Measure & Rule it was stated that Science, as part of its general empirical approach, presents quantitative information as if it were neutrally derived, on the basis of a hypothetical 1:1 relationship between the units of measurement and the object measured. This is the basis of scientific objectivity – the units of measurement are understood to be qualitatively neutral. The mechanical understanding of natural phenomena inevitably involves some attention to qualitative criteria, and to assessments and descriptions frequently involving the use of figurative language. This unavoidably entails a degree of subjective interpretation; however, we can resolve the uncertainty implicit in qualitative description so long as the units by which we measure natural phenomena remain proportionally invariant. Proportional invariance in quantitative assessment is guaranteed if numbers can be relied upon to function as indices of absolute value. Hence the importance of the concept of numbers as integers, where value appears as an intrinsic, self-contained (‘integral’) property – having no bearing or influence upon, and likewise being uninfluenced by, other adjacent or proximate values.

In the title page of this section, I have outlined a technical critique of this received definition of integers, based upon the relative frequency of individual digits with respect to the limited range of digits available within any particular numerical radix. Relative frequency is a property that must be determined extrinsically, i.e., between integers, based upon the group characteristics of the governing radix, where the principle of variance is that of the restrictive array of available digits through which values may be represented (0-9 in decimal, 0-7 in octal, for instance). For instance, a ‘1’ (or a ‘0’) in binary notation exhibits a higher relative frequency than its corresponding instance in decimal, or in any other radix, even though the corresponding values are numerically equal. Similar relative comparisons can of course be made between other available digits in any combination of radices. The case of binary is exemplary as it opens up the possibility of interpreting the either/or relationship of ‘1/0’ in terms of the ‘true/false’, or ‘positive/negative’ distinction; hence, a distinction based upon qualitative criteria (in their simplest forms), rather than strictly quantitative ones. If this tendency towards qualitative potential is extrapolated in principle from the binary example, with respect to the variable relative frequency of digits when transposed across a diversity of radices, then we have an analytical basis for understanding the qualitative potential of integers (if indeed we are content to continue with that designation, with its overtones of ‘intrinsicness’), and a justification for rethinking their conventional interpretation as bearers of integral, or absolute, values.

Does the issue of relative frequency exhaust this critique? There may in addition be a meaningful epistemological critique of proportional invariance (rational proportionality), which considers the nature of integers primarily as symbolic constructs (rather than as phenomenal objects in their own right), and their rootedness in the requirement for the cognitive organisation of concrete categories of objects – suggesting thereby their association with customary expectations of use and scale with respect to those individual object categories. Hence, in terms of our everyday requirement for numbers to assist in the counting, possessing, and exchanging of objects in various categories, those numbers will always in practice bear some qualitative relationship to the categories of objects they organise and measure. The principle of rational proportionality eschews such qualitative dependencies, based as it is upon the idea of integers as abstract formal entities with self-contained values, whose objectivity transcends their application to any specific instance of quantity. Therefore, while an insistence upon absolute proportional invariance may satisfy the discourses of mathematics and the empirical sciences, and may also constitute a mercantile advantage to the extent that it gives vent to the smooth liquidity of exchange mechanisms, when applied to the measurement and quantification of tangible real-world objects, the abstract formality of the units of measurement is antithetical to the common purpose of maintaining adequate categorial distinctions based upon customary expectations of use and scale.

There is a further aspect to the technical critique referred to above, but which concerns the issue of an integer’s relative position, rather than its frequency. The conventional definition asserts integers as stable entities of fixed intrinsic value, so that the movement between each integer and the next is a series of rational ‘jumps’ in value, each equal to the unit ‘1’. This permits individual expressions of value to be considered as discrete wholes, whose values are unaffected by adjacent or proximate values. If however we consider the simple reciting of numbers in sequence (or the sequential counting of a series of objects), this implies a cognitive process, rather than a purely mechanical one. Each progressive counting instance entails an awareness of the preceding value (by memory), and of the succeeding value (by projection); so that a counting instance of, say, ‘5’, involves a mental oscillation between each of the adjacent values ‘4’ and ‘6’. In this sense our instance of ‘5’ loses its rational stability as a fixed index of value, and emerges as a dispositional effect of each of its adjacent values. In these terms it should not be discounted that it perhaps also makes an important difference, in terms of its dispositional properties, or its relative potential, whether a ‘5’ is the maximal available integer (as in the case of senary – base-6), or whether it occupies a near median position, as in the case of decimal. Furthermore, and independently of any excercise of consecutive counting, we might also wish to consider the dispositional effects of non-consecutive adjacent values within any sequence of digits.

It ought to be clear that these features of numerical behaviour cannot be explained in terms of unbounded rational principles – only by acknowledging that rationality operates within formally circumscribed limits – the ratios that hold between the integers in a decimal notation, while internally consistent, cannot be assumed to be proportionally consistent with those between numerically equal values expressed in any alternative radical notation, as the ratios between decimal values must be strictly determined by the rules governing their expression within decimal notation. We are accustomed, for most quantitative purposes, to working within the decimal rational schema; but we misconstrue the conditions of proportionality operating restrictively within the decimal schema if we assume that those conditions operate universally across diverse numerical radices; or, as a correlative to that assumption, that rational proportionality is somehow given in the very act of measuring.

This should not perhaps be too surprising to us when we consider that numbers are primarily tools of the human understanding, deriving originally from the need to quantify and manage objects having at least notional concrete existence, and which therefore entail qualitative dependencies. Numbers do not possess any phenomenal properties in their own right – they are derivations under the general concept of number, and are therefore bound by intuitive as well as rational processes. What makes the idea seem at first strange and esoteric is the legacy of nearly 400 years of modern scientific attempts to reduce Nature to a set of objective mechanical principles, and the requirement of an absolute system of measurement in order to achieve this. The twin imperatives of measurement and calculation predisposed Enlightenment Science and its descendents to caricature the qualitative determinants of numbers as superstitious, fanciful, and archaic.

Precision Points, or Intervals?

We are accustomed to the practice of representing a sequence of consecutive integers as fixed points on a linear scale (such as in the numerical divisions that populate the axes in a graph showing a mathematical equation). Positions on the axes lying between fixed points would therefore indicate fractional values of integers – suggesting the integers here are serving merely as ‘gatekeepers’ for the (more unruly) rational and real numbers. This describes a typical linear distribution. However, the precise alignment of integers with discrete points on a linear scale emphasises the role of integers as abstract entities, and is incompatible with some of the ways in which we commonly represent the natural numbers (a major sub-set of the integers). For instance, to accommodate certain roles of the natural numbers in describing the (not strictly linear) apportionment of numeric value corresponding to familiar divisions in space and time, we would need to identify those numbers, not with precise fixed points on the scale but with the whole interval lying between two fixed points.

In identifying certain periods of time, or regions in space, we are accustomed to using whole numbers to represent entire periods or regions (the word ‘zone’ covers both uses). We may speak of ‘Week 1’ to cover any point in time in a particular 7-day duration, for instance; or use a numeric description for concentric zones in space, such as in the London Underground map, where ‘Zone 2’ describes any point between the lines of concentric division separating it from Zones 1 & 3 (stations occurring precisely on the line are understood to occupy both adjacent zones simultaneously). In these examples the number applies to a whole interval in time or region in space, and there is no meaningful place for the idea of a fractional part of any integer. This suggests that integers may be variously employed in ways that are either inclusive or exclusive with respect to the rational and real numbers.

The observation that points lying exactly on the line dividing two adjacent zones occupy both zones simultaneously suggests a partial analogy with the description of sequential counting given above, where ‘5’ was described as involving a mental oscillation between ‘4’ and ‘6’. To explore this ‘wave’ analogy further – in physics, for instance, we might wish to consider a continuous sine wave in terms of its discrete single iterations, and thereby to identify them consecutively as iterations 1, 2, 3, etc., and so to dissociate the numeric notation from any particular point of amplitude of the sine wave. Such dissociation would be impossible for any precise positional point on the horizontal axis, so that in this exercise integers would also need to be associated with the entire parabolic interval of the sine wave. By exploring these kinds of ‘periodic’ or ‘zonal’ features in certain ways of using integers it may help to illuminate for us why it is that the received definition of the integers as discrete entities with transcendental properties is so clearly inadequate at explaining the varied roles and functions that integers perform for us.

Logarithms – The Imperative of Calculation

Logarithms were first developed as late as the 17th century by John Napier (1614), and later by Leonard Euler.¹ The inspiration for this development was the desire to make complex numerical calculations simpler and more manageable, i.e., by reducing geometrical math progressions (number series based upon multiplication or division) to arithmetic progressions (those based upon addition or subtraction). In the absence of electronic aids to calculation, this made it possible to systemise handwritten complex calculations (and especially, for instance, financial calculations of compound interest) by reference to logarithmic tables. So, where one needs to perform complex calculations involving large integers, large differences between values of x may be expressed as much smaller differences in terms of Log(x).

The function relies on the principle that numerical differences can be reduced or ‘crunched’ according to a set of essential ratios. The most useful of these ratios are common logarithms (log₁₀), binary logarithms (log₂), and natural logarithms (log_e) – e being the irrational mathematical constant developed by Euler, defined as the “asymptotic quadrature of the hyperbola”, and which is approximately equal to 2.71828 (log_ex is the inverse of the exponential function: e^x). The logarithmic roots of diverse number bases (log_b) are understood to be perfectly derivable from ‘common’ (decimal) logarithms, according to the formula: log₈x=log₁₀x/log₁₀8.

The function of logarithms is perhaps best illustrated through the idea of scaling invariance, and the application of this principle to the observation of self-similarity in the repetition of natural forms in ascending or descending scales, i.e., as fractals. As a fractal replicates, or cascades, its entire global structure throughout the distribution of its parts, logarithms may be used to express the proportional identities relating successive magnifications of the fractal – for example, in the ratios of scale between the segmented chambers of the Nautilus shell. Nature may indeed exhibit such regularities of proportion across scale in the physical properties of natural forms organically and inorganically. However, such features are characteristics of the structural and developmental properties of natural physical forms, which may be observed empirically. On the other hand, the derivation of radical logarithmic values from common (decimal) logarithmic ones according to the above formula implies that the resulting radical ratios are perfectly consistent with those in decimal – an assumption of proportional invariance that does not result from any empirical observation, but rather from the exercise of a metaphysical tendency. That is to say, it is not so much a reflection of proportional invariance, but an enforcement of it. After all, integers, which represent the form and the method of that enforcement, are but the reified characters of modern abstract arithmetic notation, having as it were ‘forgotten their roots’ as symbolic constructs, formed out of the requirement to organise qualitative categories of objects with respect to intuitive criteria of use and scale. Hence, proportional invariance asserts qualitative indifference a priori, essentially in a manner that promotes all objects ruthlessly towards the system of their exchange.

The Historical Context

Napier’s development of the logarithmic principle was the result of his having spent most of the last twenty years of his life calculating:

N = 10⁷(1−10^–7)^L

for values of N between 5 and 10 million, in order to reveal L as a ratio, which he called the ‘artificial number’, and later ‘logarithm’ (literally meaning ‘proportional number’). So, the logarithmic principle, and the later development of Euler’s constant that it engendered, entailed from the beginning the (exclusive) employment of the seventh power of ten, and its inverse: 10^–7 (Napier later went on to employ 10^–5 also). Of course, within the decimal rational schema, each successive power of 10 is proportionally consistent with each other exponential of 10, and the logarithmic difference of each exponential step is equal to 1. However, according to the analysis that the ratios of proportion found within the decimal schema are unique to decimal, there is no guarantee that the ratios between the 7th power of ten and other powers of ten will be consistent with the ratios between their corresponding values when expressed in any alternative numerical radix² – after all those ratios must be determined uniquely according to the restrictive character-set of the governing radix, and not according to some external arbitrary principle.

It should be noted that the historical development of Napier’s method occurred at a time of revolutionary changes in terms of mathematical and of philosophical understanding. In seeking to enhance the mechanical understanding of Nature and the Universe, Enlightenment scientists sought an antidote to centuries of Classical (Aristotelian) philosophical thinking, in which certain knowledge of natural forms and processes depended essentially upon the exercise of the intuition. In Classical understanding, knowledge derived from sense data was incomplete without the addition of deductive reasoning involving intuitive, or universal, principles. Empirical science, from the 17th century onwards, sought to disgorge this dependency by establishing new paths to certainty based upon an atomistic approach to knowledge – the progressive measurement and tabulation of inexhaustible raw empirical data. This required a complete revision in methodology, involving a severing of the connection between knowledge and intuition. Reliable, mechanical knowledge of cause and effect in Nature should only now be acquired through the painstaking enumeration of every conceivable experimental instance of a known phenomenon, to arrive at certain knowledge of its principle causes by inductive elimination. Hence, the inductive methodology established by Francis Bacon sought to establish a tabula rasa for scientific knowledge through the method of observational parsimony, eschewing teleology and the acquired habits of syllogistic reasoning – in fact, an attempt to eschew any input from the mind save for its clinical ability to observe and record the data provided from direct sense impressions.³

In this context, a theory which acknowledged an intuitive dimension to the understanding of numerical quantity would have been regarded as archaic or superstitious – as counter-revolutionary, and an unnecessary hangover from the dark ages. Such a theory would have detracted from the primary instrumental function of numbers in facilitating progressive measurement and quantification, and the establishment of empirical certainty over the mechanics of true causes and effects.

In view of this enthusiastic revisionism in the development of early modern science, we can understand the urgency, and the temptation, of the reductions to scalar identities through which logarithms seem to facilitate the rational understanding of Nature, and of numbers. However, we should understand this not only as a revelation of new knowledge previously inhibited by the habits of thought instilled in philosophical doctrine (Bacon’s “idols of the mind”)⁴, but also as the forging, by force of will, of a new kind of indifference, through the subordination of qualitative relations and the dominance of quantitative ones. A consequence of this indifference is that it precipitates a radical shift – a fault line – between the ‘human’ and the ‘scientific’; between the mechanical-physical and the philosophical-ethical-spiritual; between the rational and the intuitive; and which will henceforth render these as incompatible discourses.

January 2013
(revised: 5 April 2021)

Footnotes:

Napier, John, Mirifici Logarithmorum Canonis Descriptio (“Description of the Wonderful Rule of Logarithms”), in: Ernest William Hobson, John Napier and the invention of logarithms, 1614, Cambridge UP, 1914. [back]
The fact that these ratios are in fact inconsistent with those in decimal is shown to be objectively the case by the emprical excercises represented in the page: Radical Affinity and Variant Proportion in Natural Numbers. Refer to the statistical assessment of the variation factors in the results on that page, as well as the Analyis section, for a discussion of the fact that ratios between the seventh power of 10 and other powers in the exponential series appears frequently as a point of high elevation when those values are transposed into the alternative numerical radices from binary to nonary (base-9). [back]
Bacon, F., Novum Organum, Or True Directions Concerning the Interpretation of Nature (1620), Constitution Society: http://www.constitution.org/bacon/nov_org.htm (accessed 18/01/2015). [back]
Ibid., Aphorisms [Book One], CXV. [back]