Yuri Tarnopolsky                                                                                                          ESSAYS
13. On Numbers

 order. Confucius. US tax code. combinatorial culture. poset. complexity. bureaucracy.

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                                             Essay 13. On Numbers

If not reason then vision is definitely something we share with animals. It reduces the need of thinking because, unlike the mental space, the Euclidean space around us is ordered. We can take the largest peach and go to the closest seat almost automatically. The order of the space means that for every two spheres and two distances we can tell with decent accuracy which one of two is larger. If vision does not help, we can measure the differences and convert them into the numerical food for thought.

In search for a landmark on the flat vastness of the combinatorial culture (see Essay 12), some simple measure, like distance, height, width, time, and quantity—anything numerical—could greatly help. As a matter of fact, such measure exists, and of course it is money. We can buy the cheapest air ticket with our eyes closed.

With a numerical measure on hand, we can compare values of different things as if we actually saw the landscape of values. The search for the highest or the lowest point of the combinatorial landscape (or landfill) could become quite mechanical. Money performs its function because it is number, and rational (i.e., integers and fractions) numbers used in commerce are perfectly ordered: for any two different money values we can tell which one is larger than the other. Money, like any number, brings order and sense of direction into our otherwise chaotic life, so that we can navigate it under clear star-studded skies and not in blind fog and can find a good deal on air ticket, hotel, and computer memory. By reducing everything to the simple one-dimensional space of price, money softens the unbearable complexity of the world we have created. In other words, money introduces a kind of geometry in our life. With money we are relieved to be closer to animals and need intelligence more for earning than for spending.

There are things, however, that have no price tag for. Despite all its totalitarian might, money does not measure political power (at least, not completely), beauty, truth,  knowledge, and virtue, although all can be occasionally bought and sold. The parameters of human nature that meant so much for Montaigne, do not do too well on the market of modern democracy, except for power and beauty.

As far as beauty is concerned, there is a simple procedure of ordering: beauty pageant. The contestants are compared with each other and lined up as ordered set. The place in the competition is a number but it has no absolute meaning, because somebody with a lower place can still win in another competition. All contests are relative. All money is absolute, and no collective judging at the pageant of money is necessary.
We are moving toward the market price for health and life, but general ideas are still difficult to evaluate in terms of money, and moral qualities are even more so. But if we do not have any quantitative measure, how can we choose between values, behaviors, and ideas that are not listed in religious commandments as do and don't?

Consumer ratings and polls play the same role in evaluating quality of goods, performers, politicians, sports personalities, and authors as beauty pageants, and with the same limitations. They work by placing the objects of rating in ordered sets. This can be done if the relation, for example, "more" or "better," can be established for any two objects.

 The knowledge of what is good and what is bad, whether true or false, reminds of  force in physics: it directs the movement.

 The problem with a diverse pluralistic democracy is that there are many different ethical standards. Another  problem is that corporate standards can override the personal ones. With ethics there is so much confusion that the modern society, drowning in combinatorial flood, seems to abandon the risky ethical standards at all. Money offers simplicity.

It turns out that the non-monetary numerical currency has always been used to maintain social order. In authoritarian societies, however, the price list was short, written by a single hand, and designed to stand for a long time.

Teachings of Confucius, who lived around 500 B.C., seem to be directly aimed at controlling   kongfuzicombinations, search, novelty, excess, and chaos. His main idea was that preservation of order was the best way to happiness. If we start to implement this concept, we have to maintain  the same order today as a year ago, and so, going back step by step, we come to the oldest known order, which for Confucius was embodied in the writings of ancient sages.  The ideas of Confucius are just combinations of terms and anybody could express them or rediscover. They can be found in Plato, ancient Indian philosophy and, probably, in any modern chicken soup for the soul. All we need is a set of terms, and moral terms are more or less universal, I think, because, with the exception of  worshiping a superhuman being, they find roots in animal behavior. Whether dogs worship their uber-canine masters I don't know, but I doubt it.

Confucius was as contradictory as any major religious teacher and this is why his mostly non-religious system, actually, became a religion. The same happened with Marx and Lenin. If a book is free of ambiguity, it cannot sprout religion. The contradictions require an institution of selected experts and functionaries to question the text and to apply the old text to the new reality and the old norms to the diversity of human behavior.

One of the four original sources of Confucianism is Analects of Confucius,  from which the quotations below are taken.

Confucius was neither a retrograde nor an obscurantist.

The Master said, 'If a man keeps cherishing his old knowledge, so as continually to be acquiring new, he may be a teacher of others.'
                                           BOOK II, CHAP. XI.

He suggested the middle road in any venture but did not disapprove the venture itself.


I like to think that the conservative attitude toward life is always inspired by some kind of a shaky balance between the supply of energy and its dissipation. When large numbers of people are well today but can be on the verge of extinction tomorrow, as it happened in Chinese floods and Russian famines, not to mention the wars and revolts aggravating Chinese history, a cold conservative system has better chances of survival than a diverse and fluid structure.

The source of energy for China was not just the solar radiation but, in addition to it, the fertile river valleys that carried vast amounts of silt and, like the Nile of the pharaohs, could sustain the imperial food chain where the emperor, his officials, and his subjects depended on each other. Water does not always deliver its promise and needs a centralized power to control it, maintain the distribution of moisture over large territories, accumulate the crop, level out its consumption over time, as Joseph taught the Pharaoh, and defend the empire against the non-agricultural invaders. The less reliable the harvest, the more  authoritarian and vertically stratified the social structure. The Chinese rivers had very nasty temper, periodically throwing devastating floods.

The same could be said about the Russian climate in which a decent harvest is never to be taken for granted. The Emperor at such conditions has the true mandate of Heaven.

Confucius treasured the virtue of propriety (the following of the established order) above all. How did he manage to measure it? It seems that he understood order as modern mathematics does. He tried to order the set of moral qualities without recurring to numbers, which leaves only the tool of comparison.

Tsze-kung said, 'What do you pronounce concerning the poor man who yet does not flatter, and the rich man who is not proud?' The Master replied, 'They will do; but they are not equal to him, who, though poor, is yet cheerful, and to him, who, though rich, loves the rules of propriety.'       BOOK I, CHAP. XV. 1.

 In this story we find four combinatorial human types:

 1. The poor man who does not flatter.
 2. The rich man who is not proud.
 3. The poor man who is cheerful.
 4. The rich man who loves the rules of propriety (i.e., order).

If they are to be judged at a virtue pageant, how would they stand? I wonder how Confucius would order types 2 and 4 or the cheerful rich and the cheerful poor men.    

Obviously 3 and 4 are above 1 and 2, but what about the position within the pairs? Basing on the sole maxim, it is impossible to tell.

A set of integers, for example, from 1 to 10,  is an ordered set because for any two numbers one is more than the other. The linear sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and for that matter, any lined up objects, even identical marbles, are an ordered set because for any two objects one is farther to the right (or left) than the other. Order is any relation defined in a certain way.

In mathematics, a set is ordered by a certain relation (for example, one is  more than
the other or one is further to the right than the other),  if


* any two different members of this set always have this relation,
* the relation can never exist between two equal members, and
* the relation is transitive (i.e., if 3>2 and 5>3, then 5>2).

 In partially ordered set, some members have this relation and others do not.

The four types from the Confucian maxim form what is called partially ordered set. For some two members of the set we know the relation between them, but for others we do not.

 Let us look for the clues in the rest of Analects..

1. When the Master went to Wei, Zan Yu acted as driver of his carriage.  2. The Master observed, 'How numerous are the people!'  3. Yu said, 'Since they are thus numerous, what more shall be done for them?' 'Enrich them,' was the reply.  4. 'And when they have been enriched, what more shall be done?' The Master said, 'Teach them.'      BOOK XIII, CHAP. IX.

The Master said, 'Riches and honors are what men desire. If it cannot be obtained in the proper way, they should not be held. Poverty and meanness are what men dislike. If it cannot be avoided in the proper way, they should not be avoided.  BOOK IV, CHAP. V. 1.

The Master said, 'The mind of the superior man is conversant with righteousness; the mind of the mean man is conversant with gain.'  BOOK IV, CHAP. XVI.

This seems to put enlightenment over wealth, wealth over poverty, and enlightenment over ignorance.  But what is better, to be humble or to stay away from flattering? To be rich and not to flatter or to be rich and cheerful?

The Confucian scale of moral values is based on partial order. He consistently uses pairs to establish the superiority, but does not exhaust all possible ones.

The Master said, 'They who know the truth are not equal to those who love it, and they who love it are not equal to those who delight in it.'    BOOK 6. CHAP. XVIII

This might make reading Confucius a delight, but leaves a wide margin for guessing.

1. Tsze-kung asked which of the two, Shih or Shang, was the superior. The Master said, 'Shih goes beyond the due mean, and Shang does not come up to it.'  2. 'Then,' said Tsze-kung, 'the superiority is with Shih, I suppose.'  3. The Master said, 'To go beyond is as wrong as to fall short.'

But what is more wrong? I would ask the Master. Wasn't the purpose of Zen Buddhism, originated in China, to protect the Master from too many questions?


Although Confucius ordered some pairs, large number of moral combinations is practically impossible to order and to evaluate a man on the Confucian scale is not an easy business. If it were, Confucianism would be an obvious truth and not a deep truth (see Essay 8).

To order the combinatorial variety of real life and achieve maximal order and certainty has been a very much understandable but never attainable goal of any authoritarian government since ancient empires.

The Russia of the czars, an imperial neighbor of China,  maintained its order not through any philosophy but through the religion in which the Czar had mandate from God, like in China. Peter the Great established a very rigid hierarchy of social service. The Table of Ranks contained fourteen ranks, equivalent to the same number of ranks in the army and the navy.

        Here it is:

                                CIVIL SERVICE RANKS OF RUSSIAN EMPIRE

 I  Chancellor  VIII Collegial Assessor 
II  Real Secret Councilor IX    Titular Councilor 
III Secret Councilor  X     Collegial Secretary
IV Real State Councilor   XI    Ship Secretary
V  State Councilor XII    Provincial Secretary 
VI Collegial Councilor  XIII   Senate Registrar
VII Court Councilor XIV   Collegial Registrar


A peculiar consequence of this system was the pervasive Russian obsession with superiority, real or fake, in dealing with a stranger or an equal, or even a foreign country. This is why Russia has been fixated on self-proclaimed greatness throughout its history.

The ranks and their monetary representation are very ancient invention. In the Code of Hammurabi, the king of Babylon who lived in  the eighteenth century BC, we find:

202. If any one strike the body of a man higher in rank than he, he shall receive sixty  blows with an ox-whip in public.
203. If a free-born man strike the body of another free-born man or equal rank, he
shall pay one gold mina.
204. If a freed man strike the body of another freed man, he shall pay ten shekels in  money.

Democracy, which instead of distinction between classes tends to turn to distinctions between individuals, faces a deluge of complexity.

The respectable  US Tax Code is one of the latest repercussions of imperial bureaucracies and, paradoxically, the most complicated product of the struggle with complexity. In essence, it is a never-ending quest for lining up every droplet of the combinatorial ocean of human circumstances to the perfect linear order of the Tax Table, where the figures of income form ordered set. In a sense, it is yet another historical attempt to order a vast number of combinations, traceable to Hammurabi. This time, however, the code deals with many millions of individuals or families instead of a dozen or so social classes, estates, and casts. In 1984 it was 19,500 pages long, and in 2001 it counts 45,662 pages, no doubt, due to the fecundity of computers. Here is a sample:


  1986 - Subsec. (a). Pub. L. 99-514, Sec. 102(b), substituted
  subsec. (a) for former subsec. (a) which read as follows:
    '(1) In general. - In lieu of the tax imposed by section 1,
  there is hereby imposed for each taxable year on the tax table
  income of every individual whose tax table income for such year
  does not exceed the ceiling amount, a tax determined under
  tables, applicable to such taxable year, which shall be
  prescribed by the Secretary and which shall be in such form as he
  determines appropriate.  In the tables so prescribed, the amounts
  of tax shall be computed on the basis of the rates prescribed by
  section 1.... (etc.).

Ordered set, or, to put it differently, a gauge or a ruler, is the golden dream of any bureaucracy.  The Federal Tax Code, driven partly by the liberal intent to assist various disadvantaged groups and milk some advantaged ones, is the roster of inequality and the best proof that equality does not exist. It is the embodiment of totalitarian frame of mind: not to miss anything. Bureaucracy is the steamrolling of priorities.

Interestingly, the modern penal codes solve the problem of complexity by setting the range of punishment (unthinkable for taxes!) so that the individual combination of circumstances can be taken into account, which is an enormous progress since Hammurabi.

Democracy started as public forum and ended as a public market place where anything goes. In our time, what people buy is more important than how they vote. The policy follows the economy as the driver follows the road. The motto is: buy first and vote later. The market democracy generates enormous number of combinations that cannot be completely linearized, and money, income, and prices take advantage of this complexity by pushing out any other scale of values, impractical in the current Era of Large Numbers when money is easy on morals and heavy on litigation.

To hike over mental distances is my favorite kind of tourism and the tourist's observations are by necessity superficial.

I think about history of USA, Russia, China, Babylon, and for that matter, any nation as a precious pool of social and cultural genes, some unique and others universal, like the genes of basic biochemical metabolism are more or less similar throughout the species. I find the task of mapping the human social genome fascinating.  We could be humbled by discovering that we carry most genes, or, rather, memes (see Essay 6), common with those of very distant times and places.  In the social genetic engineering of the global future, some can be found harmful and some beneficial for the needs of the moment, but the winds could always change. Besides, the genes and memes express themselves without asking for anybody's permission.

I believe, the following tourist's observation presents an example of socio-genetic cross-pollination. In the following charts I modified the data  taken from an excellent  source of in-depth information on China.

The first chart plots the population of China from 1 AD to 2050 AD (projection).

We can see from the numbers that something dramatic happened twice, in the middle of the eighteenth century and in the middle of the twentieth century (compare with Essay 4).

The last Chinese dynasty, Qing (1644-1911), brought an unheard of peace, prosperity, and governmental efficiency to China and fell the victim of its own success because of the overpopulation and the alien pollen brought to China by  the winds from the West. If something was to blame, it was the Industrial Revolution and its political consequences plus the Western attempts to colonize China.

The second time it was the Marxist and Leninist reaction to the Industrial Revolution, also called revolution, the proletarian one. The population skyrocketed, and the authoritarian Communist government, finally, attempted to undo the numbers.

          Above: population of China between 1 AD and 2050 AD.  The detailed plots left and right of  the vertical line (1290) are presented below.  The future estimates are given in three versions.

           Source:  http://www.iiasa.ac.at/Research/LUC/ChinaFood/data/pop/pop_21_m.htm



        This is what the power of large numbers packed into limited space can do.

A big small number:

Page created: 2001                                                     Revised: 2016

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