The Limits of Rationality

The pages linked under this section are intended as a complement to the main content of this site – they refer to some recent (and ongoing) research projects, which in some sense were provoked in tandem with the main content, though they are independent from it, and still very much work-in-progress. At any rate, I hope it might offset some of the heavy seriousness of my principle exposition, which is rather inescapable. These notes remain fairly discursive, but also, considering the depths of the subject matter, reasonably concise and, I hope, accessible.

Recreational Mathematics

I should say first of all that I am not a professional mathematician, and so the documents linked in this section which relate to maths functions have no claim to mathematical authority. They are observations which perhaps could only have arisen outside of conventional mathematical circles, and they do not entail highly sophisticated algebraic propositions or arguments – it's not 'rocket-science'. However, I believe they elicit some rather different behavioural characteristics of numbers and number sequences, than those which have been typical of formal mathematical enquiry.

In the following I use both the terms 'number' and 'integer' interchangeably, although I should point out that my empirical investigations are concerned only with the set of natural numbers, i.e., those positive whole numbers (including zero) we conventionally employ to count. Natural numbers are a subset of the category integers, as the latter also includes negative whole numbers, with which I am unconcerned. I am concerned however with the technical definition of an integer (i.e., as an entity in itself, whose properties are generally understood to be self-contained), a definition hence inherited by the sub-category natural numbers.

We are familiar with the term 'irrational numbers' in maths – referring to examples such as √2, or π, and which implies that the figure cannot be expressed exactly as the ratio of any two whole numbers (e.g., 22/7 – a close rational approximation to π), and therefore does not resolve to a finite number of decimal places, or to a settled pattern of recurring digits. They are relatively few in number, and the restricted application of the term 'irrational' conveys upon other numbers – natural (whole) numbers and finite fractions – certain rationality. That is to say that numbers are taken for indices of pure quantity; which means that any integer can be equivalently expressed in terms of its constituent units (to say '5' is for all purposes equivalent to saying '1+1+1+1+1' – the former is simply a more manageable expression). So there is an assumed rational proportionality (or proportional invariance) governing the system of the natural numbers; and by virtue of this we can depend upon them as signifiers of pure quantity, untroubled by issues of quality.

The research which is reproduced in these pages shows however that the rules of proportional invariance pertaining to quantitative systems apply only in a restricted sense; that is, only in so far as we maintain our calculations within the terms of a particular numerical radix (e.g., in decimal, or in binary, or in octal, for instance). One cannot apply the conditions of proportionality between integers obtaining in a decimal system to their corresponding values in an alternative radix and achieve a consistency of ratios across the two systems. This problem is one that has not been previously reported, as far as I am aware, and so it cannot easily be stated verbally without showing empirical proof.

The page Radical Affinity etc. (and associated pdf file) presents a series of numerical datasets of the decimal exponential series x0, [...], x10 , beginning with the decimal value x=10 (extended for x=(2, [...], 9) in the pdf), in comparison with corresponding series from all number bases from binary to nonary (base9). It then displays tables and graphs of the values of the logarithmic differences between successive exponential values in each series; i.e., employing the derived radical logarithms (logb) for each respective radix (baseb). In each case, with a few exceptions, the graphs reveal a failure of logical consistency. The ratios between successive exponentials of, for instance, 128 (=1010) when treated as octal logarithms, display a series which cannot be determined on any rational principles. The problem arises due to the fact that octal logarithms (log8) are derived from 'common' or decimal logarithms (log10), according to the formula: log8x=log10x/log108. If one performs the same exercise for successive exponential values in the decimal series, and produces a series of graphs showing the distributions of values for constant values of x, with the exponential index z occupying the horizontal axis, the results are a series of horizontal straight lines at y=Logx. In the examples for the radical series described above, however, horizontal straight lines occur only in a limited number of cases1. The distributions revealed are mostly irregular series of variegated peaks and troughs displaying proportional inconsistency. The page Radical Affinity etc. (and associated pdf) therefore fully explicates the problem stated verbally in the preceding paragraph.

This poses problems for uses of the logarithmic function as logarithms express common ratios of proportion, and logarithms for diverse number bases (logb) are conventionally assumed to be perfectly derivable from 'common' logarithms (log10). If the logarithmic differences between successive exponentials in, for instance, the octal series: 12z8 (derived where the decimal value of x=10 – see graph below, and the octal section of Radical Affinity etc.) do not produce a horizontal straight line, then these values are not proportionally consistent with their corresponding values in the decimal series: 10z10 , whose logarithmic differences do produce a horizontal straight line (in the graph below, the logarithmic differences, expressed as r, occupy the vertical axis, and z the horizontal). This explication of a failure inherent in the logarithmic function undermines the accepted principles of rational proportionality pertaining between diverse number radices and indicates that rationality operates effectively only under formally circumscribed limits, where previously no such limits had been perceived.

Graph to show logarithmic differences between octal correspondents of sequential exponentials of x=10 (decimal).

r=(log8xz) – (log8xz-1) , for x=128

In view of these findings it is reasonable to conclude that there are empirically verifiable qualitative (or 'behavioural') properties arising out of the relational (group) characteristics of particular integers; otherwise, the restrictive proportional rules which appear to be 'native' to individual numerical radices would be empirically impossible, or absurd, and that therefore this undermines the standard assumption of absolute proportional invariance between numbers. However, I feel that it would be a mistake to consider such behavioural properties of numbers inhering mysteriously as intrinsic properties of integers themselves. Contrary to the standard definition of an integer (i.e., as an 'integral whole', or entity in itself), numbers are primarily conceptual items, and as such do not really have the status of phenomenal objects capable of holding any intrinsic properties, aside from their notional quantities. Therefore, if they also exhibit empirical behavioural properties, it is likely that these arise out of the sequential relationships between numerical characters (digits), and with respect to their relative frequency as members of a limited group of available characters. The fact that in binary, for instance, the available characters are limited to '0' and '1', means that an instance of '1' in binary is quite differently potentiated from the same instance in decimal, even though the values 12 and 110 are quantitatively identical.

This analysis leads us to conclude that the exercise of rational proportionality (proportional invariance) in terms of quantitative understanding, as a governing principle, with universal applicability (therefore across diverse numerical radices), entails a basic technical misapprehension: it fails to perceive that the ratios of proportion obtaining in any quantitative system will depend implicitly on the terms of a signifying regime (i.e., the restrictive array of select digits at our disposal – 0-9 in decimal, 0-7 in octal, for instance), the proportional rules of which will vary according to the range of available signifying elements, and the relative frequency ('potentiality') of individual elements therein.

An Inconvenient Truth Revealed

It is perhaps unfortunate that this recognition of the principle of variant proportionality between numerically equal integer values when expressed across diverse number radices (which has gone entirely unnoticed by mathematicians since Napier's invention of logarithms 400 years ago) was not made prior to the emergence in the late 20th Century of digital computing and digital information systems, for, as I will attempt to show in what follows, the issue has serious consequences for the logical consistency of data produced within those systems.

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. The assessment of quantitative and qualitative differences at the level of the observable world retains its accuracy despite at some stage involving 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 implicit assumption here is that there is 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.

However, as should now be clear from the analysis indicated above, 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 (in the case of binary, limited to two members). It does not derive from a set of transcendent logical principles arising elsewhere and having universal applicability (which will come as a surprise to many Mathematicians and Information Scientists alike). The research now revealed at Radical Affinity etc. (and associated pdf) shows that, without any doubt, the proportional (logical) ratios between particular values expressed in binary do not correspond seamlessly to the ratios between the same values when expressed within decimal or octal for instance, as these must be determined uniquely according to the member-ranges of their respective permitted digit groups; one consequence of which of course is the variable relative frequency of specific individual digits when compared across radices.

It follows that the proportional relationships affecting quantitative expressions within binary, being uniquely and restrictively determined, 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 (and hence digital) system of codes, being subject to the same restrictive determinations, cannot therefore be applied, with logical consistency that is, to conventional or analogue representations of the observable world, as this would be to invest binary code with a transcendent logical potential which 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 which 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); which 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.

Logical Inconsistency is Inherent in Digital Information Systems

The extent of the problem of logical inconsistency is not limited however to that of the effects upon data arising from transformations of existing analogue information into digital format. Unfortunately, it is not a saving feature of digital information systems that, although not quite fully consistent with traditional analogue means of speaking about and depicting the world, they nevertheless result in a novel digitally-enhanced view of the world through which they are able to maintain their own form of technologically-informed consistency. Rather, logical inconsistency is a recurrent and irremediable condition of data derived out of digital information processes, once the data is treated in isolation from the specific processes under which it is derived.

The principle that it is possible to encode information, from a variety of non-digital sources, into digital format, and to reproduce that information with transparency, depends implicitly on the idea that logic (i.e., proportionality) transcends the particular method of encoding logical values, implying that the rules of logic operate universally and are derived from somewhere external to the code. This principle is now shown to be insupportable, in view of the fact that, in the context of natural numbers, the ratios between sets of numerical values, when compared with the ratios of corresponding values expressed across diverse number-radices, are as a general rule found to be proportionally inconsistent (see: Radical Affinity etc.), implying that the rules of proportionality (i.e., logic) are instead derived uniquely and restrictively according to the internal characterological requirements of the specific code-base employed. This tells us that the principle widely employed in digital information systems – that of the seamless correspondence of logical values whether they be expressed as decimal, octal, hexadecimal, or as binary values2 – is now revealed to be hopelessly flawed in mathematical terms.

As I have indicated in the previous section above, this makes problematic the assumption of consistency and transparency in the conversion of analogue information into digital format. However, the problem of logical inconsistency as a consequence of the non-universality of the rules of the various code-bases employed is not limited to the (mostly unseen) machine-level translation of strictly numerical values from decimal or hexadecimal values back and forth into binary ones. The issue also has a bearing at the programming level: the level at which data objects are consciously selected and manipulated, and at which computational algorithms are constructed. Even at this level – at which most of the design and engineering component of digital information processing takes place – there is an overriding assumption that the logic of digital processes derives from a given repository of functional objects which possess universal logical potential, and that the resulting algorithmic procedures are merely instantiations of (rather than themselves constituting unique constructions of) elements of a system of logic which is preordained in the design of the various programming languages and programming interfaces.

But there is no universal programming language and, in addition to that, there are no universal rules for the formulation of computational procedures, and hence of algorithms; so that each complete and functional algorithm must establish its own individual and unique set of rules for the manipulation of its requisite data objects. Therefore, the data which is returned as the result of any algorithmic procedure (program) owes its existence and character to the unique set of rules established by the algorithm, from which it exclusively derives; which is to say that the returned data is qualitatively determined by those rules (rather than by some non-existent set of universal logical principles arising elsewhere) and has no absolute value or significance considered independently of that qualification.

To clarify these statements, we should consider what exactly is implied in the term 'algorithm', in order to understand why any particular algorithmic procedure must be considered as comprising a set of rules which are unique, and why its resultant data should therefore be understood as non-transferable. That is to say, when considered independently from the rules under which it derives, the resultant data possesses no universally accessible logical consistency.

Not all logical or mathematical functions are computable3, but the ones which are computable are referred to as 'algorithms', and are exactly those functions defined as recursive functions. A recursive function is that in which the definition of the function includes an instance of the function 'nested' within itself. For instance, the set of natural numbers is subject to a recursive definition: 0 is a natural number defines the base case as the nested instance of the function – its functional properties being given a priori as a): wholeness; b): serving as an index of quantity; and c): having a successor. The remainder of the natural numbers are then defined as the (potentially infinite) succession of each member by another (sharing identical functional properties) in an incremental series.4 It is the recursive character of the function which makes it computable (that is, executable by a hypothetical machine, or Turing machine). In an important (simplified) sense then, computable functions (algorithms), as examples of recursive functions, are directly analogous in principle to the recursive function which defines the set of the natural numbers.

The nested function has the property of being discrete and isolable, these characteristics being transferable, by definition, to each other instance of the function. In these terms, recursive functions have the (perhaps paradoxical) characteristic of 'countable infinity' – as each instance of the function is discrete, there is the possibility of identifying each individual instance by giving it a unique name. In spite however of its potential in theory to proceed, as in the case of the natural numbers, to infinity, a computable function must at some stage know when to stop and return a result (as there is no appreciable function served by an endlessly continuous computation). At that point then the algorithm must know how to name its product, i.e., to give it a value; and therefore must have a system of rules for the naming of its products, and one which is uniquely tailored according to the actions the algorithm is designed to perform on its available inputs.

What is missing from the definition given above of the algorithm defining the natural numbers? We could not continue to count the natural numbers (potentially to infinity) without the ability to give each successive integer its unique identifier. However, we could neither continue to count them on the basis of absolutely unique identifiers, as it would be impossible to remember them all, and we would not be able to tell at a glance the scalar location of any particular integer in relation to the series as a whole. Therefore, we must have a system of rules which 'recycles' the names in a cascading series of registers (for example, in the series: 5, 25, 105, 1005, etc.); and that set of rules is exactly those pertaining to the radix (or 'base') of the number system, which defines the set of available digits in which the series may be written, including the maximum writable digit for a single register, before that register must 'roll-over' to zero, and either spawn a new register to the left with the value '1', or increment the existing register to the left by 1. We can consider each distinct number-radix (e.g., binary, ternary, octal, hexadecimal etc.) as a distinct computable function, each requiring its own uniquely tailored set of rules, analogously with our general definition of computable functions given above.5

For most everyday counting purposes, and particularly in terms of economics and finance, we 'naturally' employ the decimal (or 'denary') system of notation in the counting of natural numbers. The algorithmic rules which define the decimal system are therefore normally taken for granted – we do not need to state them explicitly. However, the rules are always employed implicitly – they may not be abandoned or considered as irrelevant, or our system of notation would then become meaningless. If, for instance, we were performing a series of translations of numerical values between different radices, we would of course need to make explicit the relevant radix in the case of each written value, including those in base10, to avoid confusion. The essential point is that, when considering expressions of value (numerical or otherwise) as the returned results of algorithmic functions (such as that of the series of natural numbers, or indeed any other computable function), the particular and unique set of rules which constitute each distinct algorithmic procedure, and through which data values are always exclusively derived, are indispensible to and must always be borne in mind in any proportionate evaluation of the data – they may not be left behind and considered as irrelevant, or the data itself will become meaningless and a source only of confusion.

In the broader context of data derived through digital information processes, it is essential therefore to the proportionate evaluation of all resultant data, that the data be qualified with respect to the particular algorithmic procedures through which it has been derived. There is no magical property of external logical consistency which accrues to the data simply because it has been derived through a dispassionate mechanical procedure (algorithm) – the data is consistent only with respect to the rules through which it has been processed, and which therefore must be made explicit in all quotations or comparisons of the data, to avoid confusion and disarray.

Such qualifications however are rarely made these days in the context of the general melee of data-sharing which accompanies our collective online activity. The alacrity with which data tends to be 'mined', exchanged, and reprocessed, reflects a special kind of feverish momentum which belongs to a particular category of emerging commodity – much like that attached to oil and gold at certain stages in the history of the United States. Our contemporary 'data-rush' is really concerned with but a limited aspect of most data – its brute exchangeability – and has little opportunity to reflect upon the actual relevance of any data to its purported real-world criteria.

Conclusion

It was stated above that computable functions (algorithms) performed upon data values are defined as recursive functions, and are analogous, as a matter of principle, to the recursive function which defines the set of natural numbers. Logical consistency in digital information processes is therefore directly analogous to proportional consistency in the set of natural numbers, which the research reproduced at Radical Affinity etc. now reveals as a principle which depends locally upon the rules (i.e., the restrictive array of available writable digits) governing the particular numerical radix we happen to be working in, and cannot be applied with consistency across alternative numerical radices. We should then make the precautionary observation that the logical consistency of data in a digital information system must likewise arise as a unique product of the particular algorithmic rules governing the processing of that data. It should not be taken for granted that two independent sets of data produced under different algorithmic rules, but relating to the same real-world criteria, will be logically consistent with each other by virtue of their shared ontological content. That is to say that the sharing of referential criteria between independent sets of data is always a notional one, which requires each set of data to be qualified in terms of the rules under which the data is derived.

Nevertheless, since the development of digital computing, and most significantly for the last three decades, computer science has relied upon the assumption of logical consistency as an integral, that is to say, as a given, transcendent property of data produced by digital means, and as one ideally transferable across multiple systems. It has failed to appreciate logical consistency as a property conditional upon the specific non-universal rules under which data is respectively processed. This technical misapprehension derives ultimately from a mathematical oversight, under which it has been assumed (at least since the invention of logarithms 400 years ago) that the proportional consistency of a decimal system might be interpreted as a governing universal principle, applicable across diverse number radices. The evidence presented in Radical Affinity etc. (and associated pdf) suggests however that these are rather tacit assumptions that are no longer empirically substantiated.

May 2016

  1. In the resulting distributions horizontal straight lines are found to occur only where the decimal value of x (prior to its conversion to baseb) is equal to the value b, or to b2 or b3 (also, it is assumed, by extension to bn). [back]
  2. In terms of the largely unseen hardware-instruction (machine-code) level, digital information systems have made extensive use of octal and hexadecimal (base16), in place of decimal, as the radices for conversions of strings of binary code into more manageable quantitative units. Historically, in older 12- or 24-bit computer architectures, octal was employed because the relationship of octal to binary is more hardware-efficient than that of decimal, as each octal digit is easily converted into a maximum of three binary digits, while decimal requires four. More recently, it has become standard practice to express a string of eight binary digits (a byte) by dividing it into two groups of four, and representing each group by a single hexadecimal digit (e.g., the binary 10011111 is split into 1001 and 1111, and represented as 9F in hexadecimal – corresponding to 9 and 15 in decimal). [back]
  3. One explanation given for this is that, while the natural numbers themselves are 'countably infinite', the number of possible functions upon the natural numbers is uncountable. Any computable function may be represented in the form of a hypothetical Turing machine, and as individual Turing machines may be represented as unique sequences of coded instructions in binary notation, those binary sequences may be converted into their decimal correspondents, so that every possible computable function is definable as a unique decimal serial number. The number of possible Turing machines is therefore clearly countable, and as the number of possible functions on the natural numbers is uncountable, there must be more functions than there are computable functions. See: Section 5 of: Barker-Plummer, D., Turing Machines, The Stanford Encyclopedia of Philosophy, Summer 2013 Edition, Edward N. Zalta (ed.): http://plato.stanford.edu/archives/sum2013/entries/turing-machine/ (accessed 09/12/2014).

    Turing's formulation of the Turing machine hypothesis, in his 1936 paper: On Computable Numbers..., was largely an attempt to answer the question of whether there was in principle some general mechanical procedure which could be employed as a method of resolving all mathematical problems. The question became framed in terms of whether there exists a general algorithm (i.e., Turing machine) which would be able determine if (another) Turing machine Tn ever stops (i.e., computes a result) for a given input m. This became known as the "Entscheidungsproblem" or "Halting problem" – Turing's conclusion was that there was no such algorithm. From that conclusion it follows that there are mathematical problems for which there exists no computational (i.e., mechanical) solution. See: On Computable Numbers, with an Application to the Entscheidungsproblem; Proceedings of the London Mathematical Society, 2 (1937) 42: 230-65: http://somr.info/lib/Turing_paper_1936.pdf. See also: Ch. 2, pp.45-83 of: Penrose, R., The Emporer's New Mind, OUP, 1989; as well as pp.168-177 of the same, with reference to Diophantine equations and other examples of non-recursive mathematics. [back]

  4. The principle of recursion is nicely illustrated by the characteristics of a series of Russian Dolls. It is important to recognise that not all of the properties of the base case are transferable – for instance, zero is unique amongst the natural numbers in not having a predecessor. [back]
  5. For a discussion of these criteria in relation to Turing machines, see the section: Turing Machines & Logical Inconsistency, in the page: Mind: Before & Beyond Computation. [back]

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