## Recreational Mathematics

*rationality*. That is to say that numbers are taken for indices of pure quantity; which means that any integer

^{1}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 vitue of this we can depend upon them as signifiers of pure quantity, untroubled by issues of quality.

*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.

*x*

^{0}, [...],

*x*

^{10}, beginning with the decimal value

*x*=10 (extended for

*x*=(2, [...], 9) in subsequent sections), 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 (log

*) for each respective radix (base*

_{b}*b*). In each case, with a few exceptions, the graphs reveal a failure of logical consistency. The ratios between successive exponentials of, for instance, 12

_{8}(=10

_{10}) 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 (log

_{8}) are derived from 'common' or decimal logarithms (log

_{10}), according to the formula:

**log**. 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

_{8}*x*= log_{10}*x*/log_{10}8*x*, with the exponential index

*z*occupying the horizontal axis, the results are a series of horizontal straight lines at

*y*=Log

*x*. In the examples for the radical series described above, however, horizontal straight lines occur only in a limited number of cases.

^{2}The distributions revealed are mostly irregular series of variegated peaks and troughs displaying proportional inconsistency. The linked document therefore fully explicates the problem stated verbally in the preceding paragraph.

*common ratios of proportion*, and logarithms for diverse number bases (log

*) are conventionally assumed to be perfectly derivable from 'common' logarithms (log*

_{b}_{10}). If the logarithmic differences between successive exponentials in, for instance, the octal series: 12

^{z}

_{8}(derived where the decimal value of

*x*=10 – see graph below, and the octal section on p.11 of the aforementioned document), do not produce a horizontal straight line, then these values are not proportionally consistent with their corresponding values in the decimal series: 10

^{z}

_{10}, 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.

*r*= (log

_{8}

*X*

^{z})-(log

_{8}

*X*

^{z-1}), for

*X*=12

_{8}

*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 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 1

_{2}and 1

_{10}are quantitatively identical.

## An Inconvenient Truth Revealed

*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.

*transcends*all systems of conventional analogue (or indeed

*sensory*) representation (be they linguistic, pictorial, 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.

*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) relationships within binary do not correspond seamlessly to those found within, for instance, decimal or octal, 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.

*logical*determinations 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.

*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

*in*consistency 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.

*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 values

^{3}– is now revealed to be hopelessly flawed in mathematical terms.

*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.

*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.

^{4}, 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.

^{5}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.

*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.

*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, 2

__5__, 10

__5__, 100

__5__, 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.

^{6}

*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

*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.

*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.

- 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 entirely self-contained), a definition hence inherited by the sub-category natural numbers. [back] - In the resulting distributions horizontal straight lines are found to occur only where the decimal value of
*x*(prior to its conversion to base*b*) is equal to the value*b*, or to*b*^{2}or*b*^{3}(also, it is assumed, by extension to*b*). [back]^{n} - 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] - 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*T*ever stops (i.e., computes a result) for a given input_{n}*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] - 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] - For a discussion of these criteria in relation to Turing machines, see the section: Turing Machines & Logical Inconsistency, in my other post: Mind: Before & Beyond Computation. [back]