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[ PDF version ]

[ PDF version ]

[ PDF version ]

Radical Affinity and 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 sequential exponentials of x=10, transposing the series of those values across a limited range of diverse number radices (base-2 to base-9), in order to examine the logarithmic ratios of sequential values in each radical series, relative to the ratios of corresponding values in the decimal series. While the logarithmic ratios of sequential values in the decimal exponential series are naturally consistent, and result in a graph consisting of a horizontal straight line at y=1, in the case of each of the radical series reproduced here the distributions revealed are 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.

Problematic

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 (base-9). 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=10^z₁₀ . Values in each respective corresponding radix are represented by sequential values of s=(x^z)_b; i.e., for x=10₁₀ ; z=(0, [...], 10); b=(2, [...], 9). Generally, s is equal to (x^z)_b, and is employed here to represent the exponential series of any radix, whereas x retains association with the initial decimal value (10¹).

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

The following tables show distributions of values corresponding to s=10^z₁₀ , in terms of s=(x^z)_b 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: s_n/s_n–1 is constant for each value of z:

s_n/s_n–1 = 10₁₀ = (x)_b

and in a graphical representation with z as the horizontal axis, would produce horizontal straight lines at y=10₁₀ , and y=(x)_b .

However, for the non-decimal series, if we calculate s_n/s_n–1 dividing the figures according to base-10 rules (i.e., treating them as if they were decimal values) instead of base-b 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 (s_n/s_n–1)₁₀ for each series s=(x^z)_b (the expression ‘(s_n/s_n–1)₁₀’ is used here simply to imply that the sequential radical values of (x^z)_b 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 s_n/s_n–1 logarithmically, we can more simply employ subtraction rather than division in determining the series.

Generally, log_bx 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=(x^z)_b given in Table 1 above, which allows us to express the ratio s_n/s_n–1 in terms of:

r = (log_bs_n) – (log_bs_n–1)

The following 8 subsections display tables showing the values for log_bs [i.e., log_b(x^z)_b] 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.23-103). Graphical representations of the tabular data in terms of r against z are displayed as vertical and horizontal axes respectively in the following graphs (n.b. the vertical axes in these graphs are not at a constant scale).¹

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

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

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

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

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

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

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

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

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):

Graphs to show relations of close sequential groups

Variation Factors

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

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=(log_bx^z) – (log_bx^z–1), for x=(10₁₀)_b

These variation factors are represented as percentage-increase as the vertical axis in the graphs below. The horizontal axis represents the value b. It is clear that the values for r_v/r_u and r_max/r_u 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.

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: 10^z₁₀, for z=(0, [...], 10), with their octal correspondents: 12^z₈ (see graph of the octal series). That document was then extended to similar comparisons for the series of numerical 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. 10^z₁₀ or 12^z₈) calculated by division, according to the rules of the specific radix, will be constant:
(xⁿ)_b/(x^n–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 that are no longer constant – excepting of course those of base-10 itself. This is true for each of the numerical 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 pp.23-103 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.32 & p.52).

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 12^z₈ along base-8 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, i.e., according to the particular limited range and group characteristics of the select digits at our disposal within each respective radix. 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:

A comparison of the two elevations at z=7 in absolute terms shows the scale of the increase in the base-9 series is only ≅5.6% of that in the base-3. 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 that 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 (r_max) and minimum (r_u) variations exhibited in each distribution, and their comparison with the variation factor at z=7 (r_v), shows that in four of the eight series (b= 2, 3, 5, and 9) r_max is found to occur at r_v (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 r_v is only ≅0.5% lower than its maximum, which occurs at z=10. The effect of this is that the values for r_v/r_u and those for r_max/r_u 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 r_v/r_u and r_max/r_u (the horizontal axis represents the value for b in these two graphs). For instance, where b=4 the value of r_v is a ≅34.6% increase on the value of r_u.

The identity of shape across the greater part 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),² 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) r_max occurs at z=10 (n.b., in b=2 and b=4, r_max and r_v 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 notable that there is an absence of close correspondence or consistency between any two distributions, despite some similarities in shape over portions of certain distributions. 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. Hence, the recurrent significance of z=7 and z=10 seems to exclude any explanation of variation in terms of random factors such as rounding errors – an explanation which is also resisted by the sheer scale of some of the elevations, and by the 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 the 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).

Are natural numbers the bearers of integral, self-contained values?

I have discussed elsewhere some of the theoretical problems with the concept of integers (see: Somatic Inscription & Integers & Proportion). The natural numbers are a subset of the integers (which also includes negative whole numbers), and, as implied in the name, they have acquired a conventional definition in terms of their ‘integrity’, as the bearers of intrinsic, self-contained values. It is noted that the first use of the term ‘integer’ dates from the late 16th Century.³

At the time of this emergence, the presumption that the integers were the bearers of stable intrinsic values was predicated on the singular integrity of the primary unit ‘1’. There had at that point been no explicit theorisation of the real numbers, and so the unit ‘1’ was perceived as an indivisible whole, capable of expressing any larger quantity unambiguously by simple addition or multiplication. It may be noted therefore that the remaining members of the series of the natural numbers do not share in the property of indivisibility associated (somewhat archaically) with the primary monad, as they are at least divisible into their component units. Would it nevertheless remain reasonable to assert that the natural numbers are the bearers of intrinsic values on the basis that they are composed only of whole ‘integral’ units?

We may infer that the counting of units originates as an ancient practice (e.g., from the requirements of quantification and exchange of produce in agrarian societies), prior to the development of any of the systems (Greek, Roman, Arabic, etc.) involving the naming of individual numbers as recognisable characters. Such practices hence proceeded by way of a unary notation, with a tally of simple vertical marks made upon some impressionable material, involving a direct one-to-one correspondence between each inscribed mark with each discrete item in a set of objects to be counted. As the system of unary notation is physically equinumerous in its members with the sets of objects it numerically represents, reliable judgments of comparative magnitudes, and hence their ratios and proportions, could be derived by comparing the relative space occupied by their respective tallies.

The integrity that we customarily attribute to the unit ‘1’, and by extension to the set of integers in general, arises from this ancient ability to stand in one-to-one correspondence with a single item in a set of objects to be counted – the inscribed vertical mark ‘Ι’ possesses a single analytic (intrinsic) property – inasmuch as it serves to index a single unit within a set of multiples, without aiming to express their total as a unique cardinal character. The single tally mark has however no meaning or intrinsic value whatever unless it relates to the physical act of counting some object actually present to sensory experience.

What if, for instance, we introduce into the spatial distribution of unary tally marks a minimal, rudimentary form of codification, such as by the familiar ‘five-bar gate’ convention of taking every fifth mark and crossing the preceding four marks diagonally with it? The practice facilitates a more compact and efficient comparison between large numbers of objects segmented into groups of five than with the original unbarred series of marks. Assessments of proportion between sets of objects might still be made by comparing the relative space occupied by their respective tallies; but as the five-bar gate involves an arbitrary compression of space in the distribution of units that is not reflected in the original unbarred series (placing emphasis on the complete five-bar gate as the effective quantitative unit), the two systems of notation are now proportionally incommensurable with each other.

Thus, by considering quantities in this primitive sense as space-occupying distributions of unnamed units we can see that by introducing the simplest form of arbitrary rule for the codifying of those quantities we have embarked upon a process of abstraction from the original act of counting which implies that relations of proportion between values expressed within that system come to be determined uniquely by the rules that define it, and which will render those relations of proportion inconsistent with those between corresponding, numerically ‘equal’ values expressed by way of an alternative system according to different rules of codification.⁴

Developments in arithmetic and sophisticated methods of accounting involve varying degrees of abstraction from the original act of counting, and flexible means for the calculation and expression of total values, and hence of the naming of the natural numbers according to a cascading series of registers (i.e., in decimal notation: the ones, the tens, the hundreds, etc.). It is by virtue of the fact that the available digits 0-9 in decimal are ‘recycled’ across this series of registers that we are able to tell at a glance the scale (and hence the proportion) of one value in comparison with another.⁵ But it is important to recognise that our ability to apprehend scale in this sense is entirely attributable to a synthesis between our original nameless units and a set of rules for the naming of successive numbers that is both arbitrary and external to the idea of intrinsic value originally attached to those units solely by virtue of their direct correspondence with the objects of sensory experience. That is to say, our apprehension of scale and proportion between values abstractly expressed within the decimal rational schema is dedicated to the rules which define that schema and to the limited array of writable digits within it – there is no magical or metaphysical transfer of intrinsic value which survives the codification (naming) of values by means of this arbitrary set of rules.

There is a degree of false confidence therefore in the belief that reliable assessments of proportion can be made between members of the natural numbers if it is taken as implicit (based upon our received definition of integers) that the value of each natural number is inherent to itself and as such transcends its juxtaposition or comparison with other values. I made the point earlier that this confidence in assessments of proportion has survived during the four centuries since the emergence of our current definition of the natural numbers only due to the fact that we generally maintain our calculations within the decimal rational schema; and that this convention having been assumed to be ‘natural’ and thus taken for granted has resulted in the blindness of mathematicians towards inconsistencies in relations of proportion when integers are transposed out of the decimal schema into their numerically ‘equal’ values across a range of alternative radices.

If the relations of proportion between specific numerical values are seen to change when those values are expressed as their ‘equal’ values in an alternative radix, then it is inescapable to conclude that those values themselves cannot be considered as transcendent and wholly intrinsic to their identity as integers or as natural numbers – that those values are in some degree relative to their expression within the limited range of available characters as circumscribed by the terms of the current working numerical radix.

The results that 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 positions and values within a limited series of available digits. It makes a difference to an instance of ‘1’, in terms of its relative frequency, or its dispositional value, whether it is 1 in base-3 or 1 in base-10, 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).⁶ 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 happen to be 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.

These 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 this analysis provides 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 ultimately be unjustifiable, or inappropriate.

Further historical and epistemological enquiry should help towards an understanding of how the selective inheritance of certain notions from Classical mathematics lent support to the overvaluation of the principle of rational proportionality, by hypostatising that which should correctly be understood as a fundamentally conceptual category (number), into something reified as a concrete entity (integer), asserting the latter as proportionally invariant, or qualitatively indifferent. A major impulse, from the 16th 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 theoretically necessary to the development of modern empirical science in its infancy, from our own developed information-based technological perspective, which mistakenly assumes a seamless correspondence between diverse numerical radices (binary, octal, decimal, hexadecimal, base-64, etc.), the continued disregard for their rational non-conformability becomes an issue of urgent critical importance.

November 2011
(revised: 4 April 2026)

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Footnotes:

1. Some may find it surprising, or erroneous, that the values of z in the horizontal axes 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 ‘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 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. [back]
2. The same qualitative characteristics do not apply to instances of z=7 in the series taking the decimal values of x from 2 to 9 as their starting point, as shown in the analysis of variations across these further series in pp.104-114 of the linked pdf document: Radical Affinity etc. (see also Note 4 below). [back]
3. The first use of ‘integer’ as a noun is attributed to Thomas Digges in 1571 in his A geometrical practise named Pantometria (ref: http://jeff560.tripod.com/i.html) although his use was limited to positive whole numbers, making it synonymous with the natural numbers (excluding zero). The inclusion of the negative integers as well as zero in the number line is most likely attributed to Leonard Euler in his Elements of Algebra, 1771. [back]
4. The Greeks saw that there could be no consistent theory of proportion commensurable with the distinction between discrete (arithmetic) and continuous (geometric) form of magnitude. Discrete magnitudes involve the arithmos, or individually named numbers, in a staggered scale of incremented whole units. Geometric magnitudes are continuous (homogenous), infinitely divisible, and as such could only be represented diagrammatically. My description of an ancient (pre-Greek) system of unary notation is one that employs nameless units that are in that sense homogenous (i.e., as distinct from the arithmos), and whose magnitude is most readily apprehended through their space-occupying distribution, rather than by any nominal face-value they exhibit. The Greeks’ distinction is useful as it reminds us that the rules of proportion do not apply axiomatically; i.e., independently of the specific formal rules through which quantities are expressed. [back]
5. The recursive nature of the rules that uniquely define the expression of values within decimal notation is discussed in more detail in the title page of this section: The Limits of Rationality. A functional analogy is drawn in principle with the design of recursive digital algorithms (Turing machines), and this issue of the non-transferability of decimal values owing to their formation according to a unique and arbitrary set of rules is understood to have corresponding implications in terms of the logical inconsistency between shared sets of digital data; i.e., where each dataset is derived under a different domain and under different algorithmic rules. [back]
6. 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, occurring in 4/8 of the series (i.e., for b= 2, 4, 5, & 8). [back]

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