Radical Affinity & Variant Proportion in Natural Numbers

[N.b. the tabular data on this page is not well-formatted for printing purposes – please use the pdf version.]

The term 'Radical Affinity' stems from an investigation into the properties of natural numbers – their tendencies to behave, according to characteristics of their radices (or 'bases'), in ways previously unacknowledged in the analyses of quantitative systems. In the title page of this section I raised some concerns over conventional approaches to quantitative understanding, with respect to the definition of an 'integer', and to the principle of rational proportionality governing integers in the denotation of numeric value. This inquiry begins from an empirical comparison of values in exponential series across a limited range of diverse number radices (base2 to base9), in terms of the logarithmic ratios of sequential values in each exponential series, relative to the ratios of corresponding values in the decimal series. While the logarithmic ratios of sequential values in the decimal series are naturally consistent, and would produce graphs consisting of horizontal straight-lines, in the case of each of the radical series reproduced here the distributions revealed are mostly irregular series of variegated peaks and troughs, displaying proportional inconsistency. In other pages in this section (and in the Analysis section below), this inquiry is treated more discursively; the following exercises attempt to explicate these concerns in basic empirical terms.

Problemata

The following datasets are intended to explore comparisons between the decimal exponential series (100, [...], 1010) with its corresponding series in a range of radices from binary to nonary (base9). In what follows I have used the term 'z' to refer to the exponential index, and the term 'b' to refer to the radical index, or 'base'. The decimal series is represented by sequential values of s=10z10. Values in each respective corresponding radix are represented by sequential values of s=xzb ; i.e., for x=(1010)b ; z=(0, [...], 10) ; b=(2, [...], 9). Generally, s is equal to xz, and is employed here to represent the exponential series of any radix, whereas x retains association with the initial decimal value (e.g., 101).

In the decimal series, for z=(0, [...], 10), s=(1, [...], 10000000000).

The following tables show distributions of values corresponding to s=10z10 , in terms of s=xzb , for each of the respective radices:

z s=10z10 s=1010z2 s=101z3
0 1 1 1
1 10 1010 101
2 100 1100100 10201
3 1000 1111101000 1101001
4 10000 10011100010000 111201101
5 100000 11000011010100000 12002011201
6 1000000 11110100001001000000 1212210202001
7 10000000 100110001001011010000000 200211001102101
8 100000000 101111101011110000100000000 20222011112012201
9 1000000000 111011100110101100101000000000 2120200200021010001
10 10000000000 1001010100000010111110010000000000 221210220202122010101

Table 1(a)

s=22z4 s=20z5 s=14z6 s=13z7 s=12z8 s=11z9
1 1 1 1 1 1
22 20 14 13 12 11
1210 400 244 202 144 121
33220 13000 4344 2626 1750 1331
2130100 310000 114144 41104 23420 14641
120122200 11200000 2050544 564355 303240 162151
3310021000 224000000 33233344 11333311 3641100 1783661
212021122000 10030000000 554200144 150666343 46113200 20731371
11331132010000 201100000000 13531202544 2322662122 575360400 228145181
323212230220000 4022000000000 243121245344 33531600616 7346545000 2520607101
21110002332100000 130440000000000 4332142412144 502544411644 112402762000 27726678111

Table 1(b)

Clearly, within the terms of each respective series, the ratio: sn/sn-1 is constant for each value of z:

sn/sn-1 = 1010 = x1b

and in a graphical representation with z as the horizontal axis, would produce horizontal straight-lines at y=1010 , and y=x1b .

However, for the non-decimal series, if we calculate sn/sn-1 dividing the figures according to base10 rules (i.e., treating them as if they were decimal values) instead of baseb rules, in each case the resulting series becomes inconsistent above a certain (variable) value of z.

Tables 2&3 below display the resulting distributions of values of (sn/sn-1)10 for each series s=xzb (the expression '(sn/sn-1)10' is used here simply to imply that the sequential radical values of xzb are divided as if they were decimal values):

(sn/sn-1)10
z [ s=1010z2 ] [ s=101z3 ] [ s=22z4 ] [ s=20z5 ]
0 - - - -
1 1010 101 22 20
2 1089.207920792 101 55 20
3 1010 107.930693069 27.454545455 32.5
4 9010.072000655 101 64.121011439 23.846153846
5 1098.781452499 107.930686774 56.392751514 36.129032258
6 1010.008080065 101.000589126 27.555447702 20
7 9010.720064805 165.161950272 64.054313251 44.776785714
8 1010.000000001 101.003496315 53.443411218 20.049850449
9 1097.912088781 104.846159379 28.524266590 20
10 9017.207279337 104.334590762 65.313129759 32.431626057

Table 2

(sn/sn-1)10
z [ s=14z6 ] [ s=13z7 ] [ s=12z8 ] [ s=11z9 ]
0 - - - -
1 14 13 12 11
2 17.428571429 15.538461538 12 11
3 17.803278689 13 12.152777778 11
4 26.276243094 15.652703732 13.382857143 11
5 17.964536025 13.729928961 12.947907771 11.075131480
6 16.207086510 20.081882857 12.007320934 11
7 16.676027065 13.294115285 12.664634314 11.622932272
8 24.415732638 15.415932157 12.477130193 11.004828431
9 17.967452971 14.436710488 12.768596866 11.048259227
10 17.818855798 14.987188276 15.300084870 11

Table 3

If we examine the ratio sn/sn-1 logarithmically, we can more simply employ subtraction rather than division in determining the series.

Generally, logbx is given by:   log10x/log10b.

As it is conventional to derive radical logarithms from decimal logarithms, we may do so for the values of s=xzb given in Table 1 above, which allows us to express the ratio sn/sn-1 in terms of:

r = (logbxz) – (logbxz-1)

The following 8 subsections display tables showing the values for logbxz [logbs] and r for each radical data series (binary to nonary) given in Table 1, i.e., for the initial decimal value of x=10 (the linked pdf document: Radical Affinity etc. repeats the same exercises for the starting values of x from 2 to 9 – see pp.24-101). Graphical representations of the tabular data in terms of r against z are displayed as vertical and horizontal axes respectively (n.b. the vertical axes in these graphs are not at a constant scale).*

* Some may find it surprising, or erroneous, that the values of z in the horizontal scales in the following graphs are not aligned with the divisional markers, but between these points. I must confess to being a novice at the use of the 'chart' function in Microsoft Excel, and so did not override this default configuration when initially entering the data, which is the configuration most suitable when creating bar-charts, for instance, rather than linear distributions of precise values. The subsequent decision not to override the default configuration for these graphs, resulting in an unconventional display, was made with regard to the fact that the precise alignment of integers with divisional markers is characteristic only of a certain limited definition of integers, as discrete points of value on a linear scale. With consideration to the scope of this investigation, and its limitation to the sphere of natural numbers – as 'wholes' (excluding fractions) – the resulting unaligned scale of values helps to accommodate certain roles of natural numbers in describing the (not strictly linear) apportionment of numeric value corresponding to familiar divisions in space and time, for instance. 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 'week1' 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 'zone2' 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 physics, for instance, we might wish to consider a continuous sine wave in terms of its discrete single iterations, and then 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 a dissociation being impossible for any precise positional point on the horizontal axis). By allowing attention to such 'periodic' or 'zonal' features in certain ways of using integers we may perhaps help illuminate and further explain their apparent proportional inconsistencies.

x=10

Binary

z s=1010z2 logbs r
0 1 0 -
1 1010 9.980139578 9.980139578
2 1100100 20.069203241 10.089063663
3 1111101000 30.049342819 9.980139578
4 10011100010000 43.186665738 13.137322919
5 11000011010100000 53.288354486 10.101688748
6 11110100001001000000 63.268505605 9.980151119
7 100110001001011010000000 76.405932289 13.137426684
8 101111101011110000100000000 86.386071867 9.980139578
9 111011100110101100101000000000 96.486618692 10.100546825
10 1001010100000010111110010000000000 109.625083660 13.138464968

log102 = 0.301029995664

Slideshow

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

r=(log2xz) – (log2xz-1) , for x=10102

Ternary

z s=101z3 logbs r
0 1 0 -
1 101 4.200863730 4.200863730
2 10201 8.401727460 4.200863730
3 1101001 12.663002651 4.261275191
4 111201101 16.863866381 4.200863730
5 12002011201 21.125141519 4.261275138
6 1212210202001 25.326010559 4.200869040
7 200211001102101 29.974535395 4.648524836
8 20222011112012201 34.175430634 4.200895239
9 2120200200021010001 38.410313290 4.234882656
10 221210220202122010101 42.640743808 4.230430518

log103 = 0.4771212547197

Slideshow

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

r=(log3xz) – (log3xz-1) , for x=1013

Quaternary

z s=22z4 logbs r
0 1 0 -
1 22 2.229715809 2.229715809
2 1210 5.120395666 2.890679857
3 33220 7.509882226 2.389486560
4 2130100 10.511244865 3.001362639
5 120122200 13.419963781 2.908718916
6 3310021000 15.812096612 2.392132831
7 212021122000 18.812708520 3.000611908
8 11331132010000 21.682678615 2.869970095
9 323212230220000 24.099737559 2.417058944
10 21110002332100000 27.114388128 3.014650569

log104 = 0.602059991328

Slideshow

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

r=(log4xz) – (log4xz-1) , for x=224

Quinary

z s=20z5 logbs r
0 1 0 -
1 20 1.861353116 1.861353116
2 400 3.722706232 1.861353116
3 13000 5.885722315 2.163016083
4 310000 7.856362447 1.970640132
5 11200000 10.085150978 2.228788531
6 224000000 11.946504094 1.861353116
7 10030000000 14.308626795 2.362122701
8 201100000000 16.171526676 1.862899881
9 4022000000000 18.032879792 1.861353116
10 130440000000000 20.194587325 2.161707533

log105 = 0.698970004336

Slideshow

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

r=(log5xz) – (log5xz-1) , for x=205

Senary

z s=14z6 logbs r
0 1 0 -
1 14 1.472885940 1.472885940
2 244 3.068028002 1.595142062
3 4344 4.675042050 1.607014048
4 114144 6.499318847 1.824276797
5 2050544 8.111365354 1.612046507
6 33233344 9.665953809 1.554588455
7 554200144 11.236461588 1.570507779
8 13531202544 13.019752124 1.783290536
9 243121245344 14.631889245 1.612137121
10 4332142412144 16.239391402 1.607502157

log106 = 0.7781512503836

Slideshow

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

r=(log6xz) – (log6xz-1) , for x=146

Septenary

z s=13z7 logbs r
0 1 0 -
1 13 1.318123223 1.318123223
2 202 2.727909971 1.409786748
3 2626 4.046033194 1.318123223
4 41104 5.459584413 1.413551219
5 564355 6.805781229 1.346196816
6 11333311 8.347382756 1.541601527
7 150666343 9.677002975 1.329620219
8 2322662122 11.082721287 1.405718312
9 33531600616 12.454713875 1.371992588
10 502544411644 13.845937269 1.391223394

log107 = 0.8450980400143

Slideshow

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

r=(log7xz) – (log7xz-1) , for x=137

Octal

z s=12z8 logbs r
0 1 0 -
1 12 1.194987500 1.194987500
2 144 2.389975000 1.194987500
3 1750 3.591046402 1.201071402
4 23420 4.838484485 1.247438083
5 303240 6.070033515 1.231549030
6 3641100 7.265314311 1.195280796
7 46113200 8.486225483 1.220911172
8 575360400 9.699963563 1.213738080
9 7346545000 10.924806260 1.224842697
10 112402762000 12.236628843 1.311822583

log108 = 0.9030899869919

Slideshow

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

r=(log8xz) – (log8xz-1) , for x=128

Nonary

z s=11z9 logbs r
0 1 0 -
1 11 1.091329169 1.091329169
2 121 2.182658339 1.091329170
3 1331 3.273987508 1.091329169
4 14641 4.365316677 1.091329169
5 162151 5.459743807 1.094427130
6 1783661 6.551072976 1.091329169
7 20731371 7.667472316 1.116399340
8 228145181 8.759001215 1.091528899
9 2520607101 9.852322719 1.093321504
10 27726678111 10.943651889 1.091329170

log109 = 0.9542425094393

Slideshow

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

r=(log9xz) – (log9xz-1) , for x=119

Proportional Graphs

The first graph below shows the distributions represented individually above, in a single graph with a proportional vertical axis for the full range b=(2, [...], 9). The three graphs in the subsequent section show the relationships between the distributions for the range b=(3, [...], 9), with expanded vertical scales (r):

Slideshow

Graph to show comparison of the complete radical series, with proportional vertical axis.

r=(logbxz) – (logbxz-1) , for x=(1010)b

Graphs to show relations of close sequential groups

Slideshow

Graph showing proportional comparison of ternary, quaternary, and quinary series.

Comparison of ternary, quaternary, & quinary series.

Slideshow

Graph showing proportional comparison of senary and septenary series.

Comparison of senary & septenary series.

Slideshow

Graph showing proportional comparison of octal and nonary series.

Comparison of octal & nonary series.

Variation Factors

To measure the degrees of variation in proportion which are exhibited in the values of r in each radical series, I have taken as a baseline the value of ru given for z=1, then calculated the increase factor for rv at z=7 (as this appears as a frequent point of high elevation): (rv/ru); and, if rmax > rv , the increase factor for rmax at z=n: (rmax/ru):

b ru rv rmax rv/ru (z=7) rmax/ru (z=n)
2 9.980139578 13.137426684 13.138464968 1.316   1.316 (z=10)
3 4.200863730 4.648524836 1.107 -
4 2.229715809 3.000611908 3.014650569 1.346   1.352 (z=10)
5 1.861353116 2.362122701 1.269
6 1.472885940 1.570507779 1.824276797 1.066 1.239 (z=4)
7 1.318123223 1.329620219 1.541601527 1.009 1.170 (z=6)
8 1.194987500 1.220911172 1.311822583 1.022   1.098 (z=10)
9 1.091329169 1.116399340 1.023

r=(logbxz) – (logbxz-1) , for x=(1010)b

These variation factors are represented as percentage-increase in the graphs below. The horizontal axis represents the value b. It is clear that the values for rv/ru and rmax/ru are virtually identical for the radices represented by b=(2, 3, 4, 5, and 9); i.e., excepting b=7, and the two adjacent radices b=6 and b=8.

rv/ru +% (z=7)

Graph of log-variation percentages in radical-exponential distribution series(1)

rmax/ru +% (z=n)

Graph of log-variation percentages in radical-exponential distribution series(2)

Analysis

Much of what has been written across the other pages in this section was influenced by the findings of an earlier version of the pdf document Radical Affinity etc., which focused initially on a comparison of values in the decimal exponential sequence: 10z10, for z=(0, [...], 10), with their octal correspondents: 12z8 (see graph of the octal series). That document was then extended to similar comparisons for the series of number radices from binary to nonary, for values in each radix corresponding to 10 in decimal. Those exercises (presented above) reveal that, while the ratios between successive values in a progressive exponential sequence (e.g. 10z10 or 12z8) calculated by division, according to the rules of the specific radix, will be constant: (xn)b / (xn-1)b = (x)b , and would be represented graphically by a horizontal straight line at y=(x)b (i.e., with z as the horizontal axis); the treatment of the same ratios rendered through the logarithmic function (which derives all values from common logarithms - log10) results in series of ratios which are no longer constant – excepting of course those of base10 itself. This is true for each of the number radices from binary to nonary, which are shown to be inconsistent with decimal, as well as being inconsistent with each other, and is evidenced by the graphs represented above, which in each case display a series of variegated peaks and troughs. Those exercises were then further extended to consider the same series of radical correspondents for the decimal values of x between 2 and 9 (see pages 24-101 of Radical Affinity etc.).

The same calculations performed for any value of x in decimal (without involving a derived radical logarithm), result in horizontal straight lines at y=Logx. In the exercises above there are no instances of horizontal straight lines; however, in later exercises, for values of x other than 1010, horizontal straight lines in general result only where the decimal value of x is equal to the value b (see for example p.24 of Radical Affinity etc., where x=2, b=2). There are three exceptions to this in the result sets: occurring where x is equal to b2 or b3 (see Comments on p.33 & p.53).

The logarithmic function was developed in the 17th century by John Napier (and later by Leonard Euler) on the premise of invariant proportion between integers, and only on this basis can the derivation of radical logarithms from 'common' logarithms be assumed to be unproblematic. For example, in the example of the octal series, the deviation from the horizontal affecting the values z=3 onwards, is the result of applying logarithms to the octal series, i.e., the essential proportionality between integers on which the logarithmic function is based is lost. If we represent these ratios without the use of logarithms, i.e., by dividing successive exponentials in the series 12z8 along base8 rules, we of course end up with a horizontal straight-line at y=128. It is the derivation of the octal logarithmic values from 'common' decimal logarithms that introduces these deviations. This undermines the role of the logarithmic function in expressing common ratios of proportion across diverse number radices, and suggests that the rules of proportionality between integers apply only in a restricted sense, according to the particular range, or according to the 'group characteristics', of the select digits at our disposal. This failure inherent in the logarithmic function undermines the accepted principles of rational proportionality pertaining between diverse number radices and indicates that rationality operates effectively under formally circumscribed limits, where previously no such limits had been perceived. To account for this it is necessary to consider exactly what it is that defines an 'integer', as the behaviour of the values in the octal series, as well as in other series, suggests that integers are unable to fulfil their customary role as stable indices of intrinsic value.

A comparison of the set of all eight series may help eliminate some erroneous or misleading explanations. The most striking comparison to make in the shapes of the various distributions is that between the ternary and nonary series, as both series feature significant peaks at z=7, as well as featuring comparable smaller peaks to either side of z=7:

Graph of the logarithmic differences between successive ternary exponentials

r=(log3xz) – (log3xz-1) , for x=1013

Graph of the logarithmic differences between successive nonary exponentials

r=(log9xz) – (log9xz-1) , for x=119

A comparison of the two elevations at z=7 in absolute terms shows the scale of the increase in the base9 series is only ≈5.6% of that in the base3. However, in relative terms (i.e., taking into account the difference in the baseline value at z=1) the difference is between a ≈10.7% increase in the ternary series, and a ≈2.3% in the nonary, so that the latter increase is proportionally ≈21.6% of the size of elevation in the ternary series.

This high individual peak is not a characteristic which is repeated in the other series, however, with the exception of the septenary and octal series – in the former the peak occurring at z=6, with larger adjacent peaks than those in the previous examples. The binary, quaternary, quinary, and senary series all feature more regular variations, with some signs of patterning, particularly in those of the binary and quaternary series. The octal series shows quite a unique distribution, especially in view of its elevated peak at z=10.

The statistical assessment of the variation factors with respect to the maximum (rmax) and minimum (ru) variations exhibited in each distribution, and their comparison with the variation factor at z=7 (rv), shows that in four of the eight series (b=2, 3, 5, and 9) rmax is found to occur at rv (for b=2, an equivalent maximum value recurs at z=4, 7, and 10 – the value of r at z=10 is only ≈0.008% higher than that at z=7, which I have assumed to be negligible); while in the quaternary series (b=4) the increase factor at rv is only ≈0.5% lower than its maximum, which occurs at z=10. The effect of this is that the values for rv/ru and those for rmax/ru are identical across 4/8 of the series, and almost identical across 5/8 – the exceptions being b=7 and its two adjacent radices b=6 and b=8. This similarity is displayed in the two graphs above showing percentage increase factors for rv/ru and rmax/ru (the horizontal axis represents the value for b in these two graphs). For instance, where b=4 the value of rv is a ≈34.6% increase on the value of ru.

The identity of shape across the majority of these two distributions (see graphs above) gives empirical proof of the qualitative uniqueness of the integer '7' (i.e., within the context of this particular series starting from the decimal x=10)1, and which is irreducible to rational or quantitative principles. A further observation is that in 3/8 of the series (b=2, b=4, and b=8) rmax occurs at z=10 (n.b., in b=2 and b=4, rmax and rv are virtually identical), so that both z=7 and z=10 appear to be 'potentiated' in comparison to other values of z. It is noticeable also that the distribution of peaks where b=2 (i.e., at z=4, z=7, and z=10) is also a characteristic found where b=4, and where b=8, although in the latter case the peaks do not exhibit the regularity of elevation found in the former two cases.

Viewing the set of distributions as a whole, it is noticeable that there is an absence of close correspondence or consistency between any two distributions (where they exhibit similarities of shape for instance, they disagree in scale). However, the statistical analysis reveals the frequency of z=7 as the point at which a half of the distributions reach maximum variation, and at which an additional one is very nearly at its maximum. The recurrent significance of z=7 and z=10 therefore seems to exclude any explanation of variation in terms of random or chaotic factors – an explanation which is also resisted by evidence of patterning in some of the distributions.

In the notations we have employed in the foregoing there are two values – z and b – which represent series of progressive whole numbers. In the conventional understanding of the meaning of an 'integer' (i.e., an entity which is self-contained, qualitatively neutral, and whose value is determined intrinsically) we should expect such series to behave proportionally. If we limit our calculations to decimal notation, there might never be an indication that integers behave in any other way (this possibly explains how these phenomena have escaped the attention of mathematicians during the 400 years since the development of Napier's method). However, in the comparisons outlined above, there are exposed proportional inconsistencies both in the sequence z=(0, [...], 10), and in the sequence b=(2, [...], 9) – the former by comparing the logarithmic differences between sequential values in any individual radical sequence, and the latter in the absence of formal consistency between the shapes of the graphs of successive radical distributions (compare, for instance, the distributions for b=6 & b=7).

Numbers: Objects or Concepts?

I have discussed elsewhere some of the theoretical problems with the concept of integers (see: Somatic Inscription & Integers & Proportion). Integers are conventionally understood to represent marks of intrinsic (or 'absolute') value, on the basis that any integer can be broken down, proportionally, into its constituent single units. The integrity of the unit '1' is fundamental to this understanding – it is taken as a fundamental objective unit of value, capable of expressing any value which is its multiple unambiguously. The problem with this understanding is that the unit '1' is not an object with measurable properties of physical extension, but an idea, or concept, which stands in, by a sort of tacit mental agreement, as an index for value. As such it is a character which lacks a stable substantial basis, unless, that is, we assert that certain mental concepts possess transcendental objectivity. If we consider the unit '1' in the context of binary notation, for instance, we perceive that, in addition to its quantitative value, it is also invested with the quality of 'positivity', it being the only alternative character to the 'negative' '0'2. Can such qualitative properties be contained within the transcendental objectivity of the unit '1', considering that they do not similarly apply to '1' in decimal? In this case clearly not, as the property arises only as a condition of the restrictive binary relationship between the two digits. Hence it appears as a necessary conclusion that there are dynamic, or context-specific, attributes associated with particular integers which are not absolute or fixed (intrinsic), but variable, and which are determined extrinsically, according to the relative frequency of individual elements within the restricted range of available characters circumscribed by the terms of the current working numerical radix.

The results which have been elaborated above in the comparisons of diverse number radices add empirical weight to this critique. Numbers, it appears, are subject to behaviours, according to their relative position within a limited series of available values; that is, according to their relative values. It makes a difference to an instance of '1', in terms of its relative frequency, or its dispositional value, whether it is 1 in base3 or 1 in base10, for instance, even though 13 and 110 are quantitatively identical. I have tried to show also, in the statistical assessment of sequential radical distributions above, how this instability does not arise from random or chaotic factors, but rather determines certain patterns of affinity, within a given context of variation (as exemplified by the heightened potential of z=7 within the context of x=10)3. This is the case, as I understand it, because numbers, rather than behaving as stable transcendental objects, are affected by context-specific dynamic and dispositional properties, as well as by rational principles. In fact, rational principles – the ability of numerical values to express conformable ratios – do not operate as universal principles consistently governing all numerical notations, but depend locally on the terms of the particular numerical radix we are employing. The rational principles governing a decimal system are incompatible with those governing an octal or a binary one, as is clearly shown in the distributions exhibited in the foregoing exercises.

Clearly, issues of relative frequency must be determined extrinsically – from the relations between integers as notional quantities, or between the individual members of relational groups of digits (as quantities) in series. From the conventional rational (and, it has to be said, deeply metaphysical) viewpoint of integers as self-contained transcendental objects whose values are determined intrinsically, the constitutive effect of numerical behaviour upon value, of quality upon quantity, will forever be shrouded in mystery.

I mentioned above that, as numbers do not in themselves occupy any physical dimensions, they should be treated firstly as conceptual items. While concrete objects may exist in variable quantities, the representation of their quantities in terms of number is always an abstraction, deriving historically from the need for the cognitive organisation of objects, for the purposes of counting, possessing, exchanging, remembering, etc. The elevation of numbers to the status of transcendental objects is a phenomenon of cultural sophistication, predicated on the rational distinction between quantitative and qualitative forms of knowledge, but which attempts to resolve the abstraction implicit in numerical description, by hypostatising that which should correctly be understood as a fundamentally conceptual category (number), into something idealised as a concrete entity (integer), by asserting integers as proportionally invariant, or qualitatively indifferent.

But to return, momentarily, to our 'crude' requirement of the cognitive organisation of objects, we employ a series of representational characters (signs) as numbers, bearing notional quantities, which acquire 'concreteness' only in relation to real or theoretical objects. We do not require numbers to be concrete 'in themselves'. Such a requirement only arises out of mathematical discourse. In fact we tend to characterise certain categories of objects within certain preferred scales of value, so that the number of a thing bears an intuitive connection to the type of object, and the use we intend to make of it. For instance, I might possess four apples, and ten-thousand pins. In the case of the apples, my intention is most likely to eat them. However, if instead I possessed ten-thousand apples, then I would probably have acquired them with the intention of selling them for profit. The point I am trying to make is that, in terms of our cognitive understanding, the number of a thing is not quantitatively absolute, but relative to the use-value of the thing, and to an accustomed scale of quantity. In other words, numbers, as applied to the organisation of particular categories of concrete objects, acquire qualitative dependencies in relation to common expectations of use and scale.

Two Criticisms of Rational Proportionality

In the foregoing we have acknowledged certain difficulties in the exercise of rational proportionality, firstly with respect to the vagaries exhibited in the results of the empirical exercises which form the earlier sections of this web-page, resulting in a critique of the role of integers as bearers of fixed intrinsic properties; and secondly, in a discussion of the rootedness of concepts of numerical value and scale in the need for the cognitive organisation of categories of concrete objects. The first criticism is technical in nature, the second epistemological.

Technical Critique:

It was noted that the proportional relationships governing sequential exponential values in the decimal system are inconsistent with those operating within alternative numerical radices, as evidenced by the failure of the logarithmic function in expressing common ratios of proportion between corresponding sets of values. It was concluded that the basic unit integer '1' exhibits a variable potentiality depending on its relative frequency within a restrictive range of possible values – this range varying naturally within the terms of each respective radix. The technical misapprehension implicit in the idea of rational proportionality is that it assumes the value of the entity '1' to be invariable across all systems, as it were, unaffected by issues of relative frequency; in other words, its value is understood to be determined intrinsically. Relative frequency, however, is a property determined extrinsically. While each numerical radix is proportional only within its own terms, the misapprehension implied in rational proportionality is that it assumes logical consistency as a transcendental principle, with universal applicability – it fails to perceive that the ratios of proportion obtaining in any quantitative system will depend implicitly on the limited terms of a signifying regime; that is, on the restrictive array of select digits at our disposal.

Epistemological Critique:

The system of rational proportionality presupposes a linear scale of numerical value which transcends its application to any concrete circumstance, i.e., as possessing a priori validity in an ideal, or Platonic sense. The exercise of rational proportionality with respect to quantity is then only a matter of applying pre-existing abstract principles of pure proportion to the real world of objects. The system is infallible in its own terms. However, it ignores the fact that quantitative systems arise in the interaction of cognitive understanding with real-world objects – the mind responds to concrete requirements of counting, possessing, exchanging, remembering, etc., with concepts of value and scale, and in accordance with a variety of categorical objectives. With respect to any individual category, the 'units', or individual items, are irreducible to the units of another category, unless we employ a neutral system of translating, and thereby exchanging, one category for another. In any exchange relationship, the ratios between the various categories of objects and the means of exchange will fluctuate on their own terms, and independently of one another, and will therefore be incommensurable with each other in any absolute or permanent way. Rational proportionality can only operate effectively to the extent that all categories of objects are rendered as equivalent to one another, for instance, in the measure of their economic value, as exchangeable items. It emerges, therefore, as a form of understanding which is tied to the processes of exchange, and in the service of enhancing the velocity of exchange. It bears no relationship to the differential forms of interaction with object categories, and subverts use-value in the name of exchange-value. More importantly, neither can it further quantitative understanding with respect to any object-category in its particulars.

These two criticisms are not intended to be exhaustive, but are an initial response to the problems arising from the failure of the principle of rational proportionality, as evinced by the exercises in the foregoing sections, as well as in the extended exercises in the pdf document: Radical Affinity etc. I do not suggest here that these criticisms provide a full or adequate explanation of the instances of variable proportionality revealed in these exercises. Attempts at analytical explanation tend to be predisposed towards rational solutions; and after all it was necessary to employ the distinctly rational method of logarithmic comparison in order to expose the failures in rational proportionality as the principle underpinning that method.

Having got thus far, and in the light of these results, it may be that we must accept that rationality operates only within well-defined limits; that it is ultimately undermined as an overarching principle in the analysis of quantitative systems. It begins to appear as a contingent, relational property, which fluctuates according the terms of a signifying regime (i.e., according to its numeric or indexical syntax), rather than as a necessary precondition with universal, or absolute, applicability. This suggests that a fully rational explanation of these results may be unjustifiable, or unnecessary.

Further historical and epistemological enquiry should help towards understanding of how the selective inheritance of certain notions from Classical mathematics lent support to the overvaluation of the principle of rational proportionality, and the hypostatisation of integers as Platonic entities with intrinsic (absolute) properties. A major impulse, from the 17th Century onwards, towards the mechanical understanding of the laws of Nature, required the establishment of systems of absolute measurement and quantification; which in turn enforced the rational dichotomy between quantitative and qualitative forms of knowledge. While this may have been historically and scientifically necessary to the development of modern empirical science in its infancy, from our own developed information-based technological perspective, which requires a seamless correspondence between diverse numerical radices (binary, octal, decimal, hexadecimal, etc.), the continued disregard for their rational non-conformability becomes an issue of urgent critical importance.

April 2015

  1. The same qualitative characteristics do not apply to instances of z=7 in the series which take the decimal values of x from 2 to 9 as their starting point, as shown in the analysis of variations across these subsequent series in pp.104-114 of the linked pdf document: Radical Affinity etc. (see also Note 3 below). [back]
  2. This characteristic is of course what enables digital computer systems to employ binary code principally to convey a series of processing instructions, rather than merely as an index of quantity. [back]
  3. As a further example, see the distributions of variation factors given for the series where x=9, on p.112 of Radical Affinity etc.. In this context there is displayed a heightened frequency of elevated values of r where z=6, which occurs in 4/8 of the series (i.e., for b= 2, 4, 5, & 8). [back]

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