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the UNIVERSITY of GREENWICH

BA ECONOMICS

MATHEMATICS FOR ECONOMISTS1999-2000

MODULE 1: THE ECONOMICS OF ADDING UP

WHY MATHEMATICS?

Economics deals with quantities:

Prices Output Profits Wages The interest rate

It deals with relationships between quantities:

Consumption and production Income and spending Supply and Demand Investment and Output

? What other economic quantities can you think of?? What relationships can you think of between the quantities above?

Mathematics is the technology of thought. Economists use it to explain relationships between quantities

rapidly. Like all technologies, it saves labour.

HOW DOES MATHEMATICS SAVE LABOUR?

? Add CXXIV and XLVII.? Now add 124 and 47. You are using a mental technology which took 10,000 years to develop.? How do you think addition and multiplication were done before the modern number system?Mathematics makes thinking quicker. The oldest technology in existence, it has been with humans since

the dawn of history. We shall use three of its branches


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Mathematics for Economists1999-2000 Module 1: The Economics of Adding Up

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Arithmetic

the technology of measurement

Geometry

the technology of imagination

Algebra

the technology of symbolism

MATHEMATICS

A UNIVERSAL LANGUAGE

If you write a number inArabic-Hindu Numerals (1,2,3,4,5,6,7,8,9,0)it will be understood by everyone in

the world who has been taught how to read, no matter what language they speak. (Ironically, Arabic uses

different numerals; it kept the original numerals that the rest of the world modified to what we use today;

but Arabic speakers can use either Latin or Arabic-script numbers. So they get the best of both worlds)

ARITHMETIC THE TECHNOLOGY OF MEASUREMENT

Measuring is the oldest science. The earliest known number system was invented in Central or Western

Africa around 8,000 years ago. By the year 4,000 BC the peoples of ancient Egypt had evolved a highlysophisticated system and built pyramids accurate to within 0.07. Our number system, descended from the

Egyptian, was developed by mediaeval Indian and Arabic thinkers. This is why we speak of 'Arabic'

numberals (1,2,3,...) as opposed to 'Roman' (I, II, III,...)

All societies measure. They measure land, food, beasts, people, money, time, water the list is endless.

Once you measure, you must add, subtract, multiply and divide. But you can't remember every possible

combination of numbers to add or multiply.Arithmetic evolved

Written signs to express quantitiesnumbers Rules for combining the signsthe rules of arithmetic (tables, and so on). Calculating aidsbeads, abacusses, paper, and now calculatorsARITHMETIC AS FORWARD PLANNING

When you add two numbers say, 41325 and 62917 you take a leap in the dark. You do not know

from your direct experience what they add up to. You have never taken 41325, put it together with

62917 and counted it. Maybe no-one has ever done it. But such is your faith in the system of numbers,

based on 10,000 years of human experience, that you are absolutely confident that, were you ever to

possess these vast sums and put them together, you would indeed find 1004242 burning a hole in yourpocket.

Every day you use this to plan your life. You count your change and decide what you can buy. You calculate

the time of your journey and decide when to start it.

This property of mathematics its capacity to predict and plan is the foundation of its social usefulness.

Mathematical skills acquired a premium and became the exclusive property of castes such as priests and

scribes who predicted natural events and organised the distribution of land, water, or food and the

collection of taxes. In the hands of master builders they became the main mental instrument for

constructing buildings: houses, monuments, temples.


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GEOMETRY THE TECHNOLOGY OF IMAGINATION

Pictures and drawings are the oldest evidence of thinking, of any kind, which we know of. Cave paintings

date back 50,000 years. Graphics are an aid to the imagination.They let you visualise relationships

spatially. This brings to bear many of your senses which relate to the experience of space: not just vision

but movement, touch, balance, direction. Pictures also speed communication. Like the rules of arithmetic,they save the time of thinking

? From the table below, is the share of government spending in GNP is rising or falling?Year 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989

Government spending 70 71 71 72 73 74 74 75 76 77 77

GNP 327 328 324 327 335 342 354 368 386 401 410

? The graph on the next page gives the same information. What can you tell from it?

Graphs Convey

Information

Fast

UK Government Spending as a

share of GNP

0.18

0.2

0.22

0.24

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

Year

Share

ofGNP

PICTURES AS PLANS

Like the number system, pictures let uspredict the result of an action before it happensand hence to

plan. Probably the earliest geometers were builders, who possessed mysterious abilities possessed by fewmortal beings, such as putting shelves up straight. Even today the symbol of the master builder is the

pyramid, the set square, and the 'eye of illuminatus' the symbol of divine illumination. These can be

found on the back of any one-dollar bill, surrounded by the words (in Latin) 'New World Order'. They are

also the symbols of numerous Masonic Orders.


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A

B

X

X

X

X X X X X

X

X

X

X

How to build a pyramid

Tie knots in a piece of rope, an equal distance apart, as

shown by the points marked 'X'. Put three pegs in the

ground where you see a , so that the rope makes a

triangle of sides 3,4 and 5 knots, and so that the linemarked 'A' lies along the first side of the pyramid. Line

'B' will show you the second side of the pyramid.

ALGEBRATHE TECHNOLOGY OF SYMBOLS

Every human being who learns a language has conquered the basic device of algebra, which is to use the

names of things symbolsin place of the things themselves. Formulae express relations between

quantities using names that we give them, for example:

Profit is equal to receipts minus expenditure

This describes a rule which you follow to calculate profits. Another example is

Receipts are equal to price times quantity

Using these two rules we can deducea new rule which conveys new knowledge:

Profit is equal to price times quantity, minus expenditure

The idea of combining rules expressed in symbols to produce new knowledge is attributed to the Athenians

and particularly Aristotle. The Athenians seem to have been very preoccupied with winning arguments, and

Aristotle codified the laws of argument in two works on what he called 'Analytics'. By a strange quirk of

fate, the same laws which helped win arguments sometimes made it easier to arrive at truth. Aristotle's

methods were codified in what is now known as Logic, or sometimes 'Symbolic Logic', the study of

reasoning.

In Logic, symbols or names usually stand for objects rather than numbers. In the hands of mediaeval Arabic

thinkers an entirely new use of symbolic reasoning was made, allied to arithmetic, and designed to solve

problems in which symbols stood above all for quantities. The new science is known by a derivative of its

Arabic name Algebra.

ALGEBRA AND LOGIC COMPARED

"When I considered what people generally want

in calculating, I found that it always is a

number"

Al-Khwarizmi, inventor of the algorithm

"First we have to state the subject-matter of our

enquiry: it is demonstration, i.e. demonstrative science"

Aristotle, author of 'Analytics'

THE ALGORITHM,OR HOW TO PROGRAMME YOUR BRAIN

The great Eastern thinker Al-Khwarizmi gave his name to the algorithm, which word means a set of

mechanical rules for calculating something.


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When you learn an algorithm, you programme your brain. You learn instructions which must be followed in

order to produce a result. Provided you can memorise the instructions and carry them out methodically,

the answer comes out at the end, just like cranking the handle on a mincer.

'Computers' were originally people. They computed things so they were called computers.

The word Computer was added to the Latin language in the 6th

Century AD when the Christian church

wanted all Christians to celebrate Easter at the same time. The church had to reconcile the lunar and

solar calendars. The skill of planning, adding, and enumerating was referred to as Computus. The Year

2000 bug was created.

All the more complex operations of algebra boil down to applying rules like these, methodically and

systematically. The important thing to understand is that this procedure is in the main completely

mechanical. It can, and indeed is, performed by computers. In order to do it, you do not need to be a

genius. You do need to be methodical. Programming your brain is like acquiring any other skill it comes

with practice.

As an example consider theformulafor profit which we gave above. mathematics uses names in place of

quantities so copiously that it is convenient to use short, abbreviated symbols. Instead of words, therefore,

we use letters:

Let Pbe profit Let Rbe receipts Let Ebe expenditure Let pbe price Let Qbe quantity.

The two rules we just stated can be rewritten by replacing each word by the letter it stands for it:

Profit is equal to receipts minus expenditure

P = R E

Receipts are equal to price times quantity

R = p Q

Now suppose a company sells 50 videos at 12 each at a cost (expenditure) of 400. How can we work out

profits? The starting point of the symbolic method is substitution. This rule says that if a symbol is equal to

some expression or other (for example, here, Ris equal topQ), then we can replace the symbol by the

expression. In the formula

P= RE

you can replace the letter Rby the expressionp Qto give a new formula

P=p QE


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Butpstands for 12, Qstands for 50 and Estands for 400. Replacing the letters by the numbers gives

P= 12 50 400

and so P= 600400 = 200.

THE GRAPH,OR HOW TO DEPROGRAMME YOUR BRAIN

The picture is an old idea but the graph is only three hundred years old. It was constructed by the French

philosopher Descartes, best known for his catchphrase 'I think, therefore I am' which he deduced while

sitting in a stove.

I think, therefore I use pictures: graphs combine geometry and algebra

The graph combines algebra and geometry, giving rise to the uninspired name of algebraic geometry. It

represents pairs of numbers by a point in space, providing a mapof the relation between them. Thus the

point marked X on the graph corresponds to the two numbers given by Q= 2.5, = 290.25. These

numbers are known as the Cartesian co-ordinatesof the point, in honour of Descartes.

Nearly every problem in economics can be looked at in two ways: graphically, or algebraically. This is very

fortunate, because it allows you to use both halves of the brain, one of which it is claimed by some

psychologists is adapted to logical, symbolic and rule-driven thinking (typical of algabra), while the other

half is receptive to spatial, intuitive thinking, typical of geometry.

WHY DO ECONOMISTS USE GRAPHS?

No profession seems to use graphs as much as economists. Open the Financial Timesor The Economistat

any page and you'll find a chart. Open any textbook: little pictures leap out. There are four reasons for this

obsession.

Marginalist ideas are

easy to demonstrate

with graphs

Graphs are quick Graphs are vague Graphs give you power over

your fellow-beings

MARGINALISM

The marginalist school, now dominant, uses two main constructs:

the maximisation of profit and consumer satisfaction the equalisation of supply to demand.

A company's profits are found through experiment to obey the formula

= 300(Q-4)23Q


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wherestands for profits and Qstands for the quantity of goods. How can the compay make the greatest

profit?

The algebra of maximisation (called calculus) is quite sophisticated and requires a certain amount of study.

But graphs can present these ideas in a simple way, freeing the mind from the clutter of the algebra. In

short, graphs are used to visualise complicated relationships. Consider, for example, the following problem:

The algebra needed to solve this exactly will not be covered in this course (They are covered in Advanced

Mathematics and Computing). But a graph rapidly yields the answer: profit is largest when the graph is

highest, at about Q= 2.5.

Profit as a function of output

Output

278

280

282

284

286

288

290

292

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

X

X is the maximum where Q=2.5

Supply and demand are even easier to demonstrate using pictures. During this course you will extensively

two basic graphs which are used time and again in economics. It is essential that you understand how

these graphs work. They are a graph showingpartial microeconomic equilibriumin the demand for a single

good; and the 'Keynesian Cross' diagram, used in so-called 'IS-LM' analysis, which shows macroeconomic

equilibrium in the goods market, between the total ('aggregate') supply of traded goods and the total

demand for them. Some of the mathematics of these two graphs was discussed in the Introductory Skills

Test. The graphs are reproduced below to remind you.

Estimated aggregate demand (Yd) and aggregate

supply (Ys) for various levels of national income (Y)

Y

Yd,Ys(bn)

300

310

320

330

340

350

360

370

380

390

400

300

310

320

330

340

350

360

370

380

390

400

Aggregate Supply (Ys)

Aggregate Demand Yd

Yd=Ys (Equilibrium national Income)

when Y=341bn

AB=Excess Demand of 5bn

when Y=310bn

A

B

Supply and Demand for Oil

Price per barrel in $

50

52

54

56

58

60

62

64

21 22 23 24 25

Supply Demand

Equilibrium Price = $23.4

Excess Supply at $24.5 = 10 bbl/day

Graphs of supply and demand


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GRAPHS ARE QUICK,GRAPHS ARE VAGUE

The advantage of graphs is that they convey information rapidly. The disadvantage is that the information

is not accurate. This is for two reasons:

a graph, unlike other mathematical representations, is scaled-down a graph, like any mathematical representation, is a simplification

Look again at the graph above that you used to maximise profits. What are the profits? Are they 291?

292? In fact the formula tells us that because Q= 2.5, must be 290.25. But you cannot see this

accurately from the graph.

Using algebra profits can be found to any desired accuracy by using more decimal places. The only way to

make the graph more accurate is to make it bigger.

The Argentinian writer Juan Luis Borges tells a short story which alleges that in ancient times a map of

China was made on a scale of 1:1 as big as China itself. Any pebble or rock could be found, as accurately

as if you came upon the rock yourself. But there was nowhere to put the map. Eventually it broke up into

fragments carried by the wind, and parts of it still turn up in remote places. This is the problem with

graphs: they are impractical for the accuracy needed in real life.

MODELS AND THEIR LIMITS

The fact that graphs are vaguer than algebra has given rise to a kind of myth, which is that algebra or

calculus are 'more exact' than graphs. This is not quite true.Allmathematical representations are only

'models' of the real thing. Like a video, they tell a story which takes elements from reality, elements fromfiction, and only lasts a short time because the tape runs out.

Mathematics has given humans great power over nature. As Zaslavsky (see quotations at the end) points

out, this makes it tempting for those humans who do it to make it a mystery and keep it secret. Essential to

this endeavour is to inculcate into non-mathematicians a sense of awe, and foster the belief that

mathematics is infallible. Modern economics has become very mathematical, partly because it is studying

difficult issues but perhaps also because it helps cover up the fact that most of the time, economists get

the wrong results.

But if the economics is wrong, the mathematics cannot put it right. At the end of the day it is a device for

condensing, automating and speeding up thinking. If the thinking is garbage, then you will get from maths

is automated garbage.

MATHEMATICS AS RELIGION

Testimony to the power of symbols, numbers and pictures is the universal role they play in magic and

legend. Any follower of fantasy novels knows that the most powerful hold you can have over someone is

to know their true name. The golem, forerunner of the robot, was built of clay by the sixteenth-century

master Elijah of Chelm, and came to life when he wrote the secret name of God on the figure's forehead.

Rabbi Judah Low ben Belazel of Prague repeated this feat and, frightened by the monster, erased thesecret word, turning the man back to clay.


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Magic numbers are the raw material of sorcery and religion alike: think of the pentangle, the trinity, the

seven-branched candelabrum, the number of the beast. The famous Tower of Babel, built in Babylon, was

a a magical monument with seven rising stages, each dedicated to a planet. Its angles symbolised the four

corners of the world. "The old tradition of a fourfold world was reconciled with the seven heavens of later

times." says Seligman (p38) "For the first time in history numbers expressed the world order".

As for pictures and images, which play a central role in magic, their prohibition is a recurring feature of

many modern religionsso much so that one of the commandments expressly forbids making 'graven

images'. A simple if gruesome test demonstrates the mental power of a picture: pick up a newspaper and

see if you can stick a pen into a photo of someone's eye without wincing.

The divine or supernatural power which these traditions give to numbers, names and images is a pale

reflection of the real power which they confer in society today. There is practically nothing you eat, wear,

live in, enjoy or fear that has not been assembled from parts that were named, measured, and depicted.

From the earliest times, those who have power over the representation of reality have been beset by two

contradictory urges: to keep it secret, and to pass it on to others. The ancient priesthoods almost certainly

held a monopoly over the mathematics of survival: above all they could predict floods, seasons and tides.

In later times, mediaeval mathematicians who discovered how to solve cubic and quartic equations used to

guard their secrets jealously and hold public showdowns in front of the crowned heads of Europe, where

each would challenge the other to solve an equation until one of them failed.

WHY MATHEMATICS APPEARS TO BE DIFFICULT

It is much easier to blind with science than to illuminate. Teachers are not immune to this. Mathematics is

powerful and never ceases to impart awe and wonder. It is hard to avoid showing it respect. But it is easyto confuse respect with worship. All mathematics teachers are prone to turn their own sense of inferiority,

faced with the unfathomable mystery of their subject, against their unfortunate pupils.

This is the foundation of the notion of mathematics as a mysteryas something which only a small (and

privileged) elite can handle. This gives rise to feeling among many probably most people that

mathematics is an innate ability which they cannot hope to understand.

Economists are more than normally prone to the worship of mathematics. Perhaps they believe that, since

they get so few things right, the power of the symbols, numbers and little pictures will bring magical

assistance.

HOW CAN YOU LEARN MATHEMATICS?IS IT HARD?

Modern schooling has created a modern disease: Mathophobia, the fear of mathematics. It is the subject

of learned discourses in the Times Educational Supplementand at teachers' conferences.

One cause of this disease is the worship of mathematics by its teachers. You cannot do anything about this.

You can control your own learning process.

More than any other subject, every step you take in mathematics depends on the previous steps. Miss a

step, and you fall. If you missed out or failed to absorb an early part of the mathematics you were taught

at school, then you found yourself unable to progress any farther because you could no longer understand


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what was being taught. You slowly began to earn your teachers' contempt, you lost your self-respect, and

before long you were on the slippery slope to ruin.

Things are made worse by the idea that mathematical ability is innate and cannot be taught. Pupils who fall

by the wayside find themselves in a foreign country; they feel stupid and incompetent, ashamed at their

apparent ignorance, and try to compensate by guessing or memorising the answers.

What then happens is a mental block. You find that there are certain types of problems which you always

get stuck on and can never seem to cope with. The block is caused by a combination of factors which

re-enforce each other:

fear of seeming stupid using techniques which you don't understand repeatedly making the same mistake at an early part of the calculation

ONE WAY OUT:LEARNING MATHEMATICS AS A SKILL

Each person will find a different way of overcoming these blocks. A crucial starting point, however, is to

understand that, deep down, mathematics like every technology is a skill learned bypractice.

Provided you learn the correct practice, your understanding will develop as you use your skill.

You can use mathematics without understanding it, just as you can drive a car without knowing how it

works. This means you can learn mathematics by practice.

? Explain why a bicycle doesn't fall over.? If you can't explain, does that stop you riding it?However, after youve learnt to do mathematics, if you want to be an effective economist, it helps to

understand it also. Unfortunately, many highly-paid economists dont seem to grasp this, which is why a

healthy mistrust of mathematical economics is always useful. Doing mathematics is not a substitute for

understanding mathematical concepts; but if your mathematical skills are second nature so that you don't

have to worry about them, then it will free your mind to concentrate on the meaning of what you are

doing. Moreover by analysing the skills you already possess, by studying what you actually do when you

solve a problem, you will gain insight into the concepts behind them.

MATHEMATICS:ANOTHER FOREIGN LANGUAGE

Is it difficult to learn a foreign language? Many people think so. But one group of people never fail to learn

language: children. If children can do it, why can't adults? The answer is that they canif they are forced

to use it. If you live in a country where a new language is spoken, you learn it because you have to use it.

Maths is the language no-one learns as a child. Like any other language, you can learn it only by using it.

You are an immigrant in the world of mathematics: it will be foreign until you converse in its language. If

you want to be an economist, you will learn to speak mathematics because you have to. This course will


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teach you to converse about economics in the language of mathematics. For this the most important thing

of all is practice.

The seminars will be used for working through examples. The coursework will consist mainly of practical exercises. Computer tutorials will allow you to drill in basic mathematical skills

At the end of the day, however, you are here to learn economics. Mathematics will help you do this better.

But it is a tool for solving problems, not a substitute. No matter how sophisticated or impressive an

economic model looks, if it is based on wrong theories then it will not work.

MATHEMATICS AS A WAY OF UNDERSTANDING ECONOMIC THEORIES

Whether or not mathematics is necessary for economics, there is one very substantial advantage to

learning it: it provides a model of the way the theory works, free from the clutter of words. When youwrite down a theory in mathematical form (calledformalising it), you condense the wordy and vague

formulations of the economists into a succinct set of formulae and rules. They may not tell you how the

economy works: but they certainly do explain how the theory works. And if the formalisation has been

done properly, it will also expose for examination the weaknesses and contradictions of the theory.

WHAT MATHEMATICAL SKILLS DO YOU NEED TO DO ECONOMICS?

Modern economics uses mathematics extensively; but very rarely effectively. Here is a list of the skills you

will need to do economics. There are four basic skills which you must have:

You must be able to solve linear equations (equations whose graph is a straight line) You must be able to draw and use graphs You must understand the relation between the two You must be able to handle the mathematics of growth and change, for which you need to calculate

compound interest, and understand powers and logarithms

To do these you will need the following basic arithmeticalskills.

Adding, subtracting, multiplying and dividing, negative and positive numbers and fractions Using a calculator, including calculating powers of a number Percentages and proportions Converting between decimals and fractions


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You will need the following algebraicskills

Expressing economic relations with symbols and formulae. Substituting for a symbol in a formula. Removing brackets from expressions Manipulating equations by moving numbers or symbols from one side of an equation to another Collecting terms and simplifying them Solving simultaneous equations by elimination, and by substitution


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SELECTED READINGS

THE ISHANGO BONE AND THEAFRICAN ORIGINS OF ARITHMETIC [VAN SERTIMA P111]

During the later stone age, hunting and fishing societies developed in the Nile Valley, in West Africa, and in

East Africa. From the mathematical point of view the most interesting find is a carved bone discovered at

the fishing site of Ishango on Lake Edward, in Zaire (Democratic Republic of the Congo). It is a bone tool

handle having notches arranged in definite patterns and a bit of quartz fixed in a narrow cavity in its head.

It dates back to the period between 9000 BC and 6500 BC. The discoverer of the artifact, Dr Jean de

Heinzelin, suggests that it may have been used for engraving or writing.

He is particularly intrigued by the markings on the bone. There are three separate columns, each consisting

of sets of notches arranged in distinct patterns. One column has four groups composed of eleven, thirteen,

seventeen and nineteen notches; these are the prime numbers between ten and twenty. In another

column the groups consist of eleven, twenty-one, nineteen and nine notches, in that order. The pattern

here may be 10+1, 20+1, 20-1 and 10-1. The third column has the notches arranged in eight groups, in the

following order: 3,6,4,8,10,5,5,7. The 3 and the 6 are close together, followed by a space, then the 4 and

the 8, also close together, then another space, followed by 10 and the two 5s. This arrangement seems to

be related to the operation of doubling. De Heinzelin concludes that the bone may have been the artifact

of a people who used a number system based on ten, and who were also familiar with prime numbers and

the operation of duplication....

The man who herded the cattle and the farmer who cultivated the fields had to observe the passage of the

days and the seasons. But it was a separate class of priests who abstracted the practical knowledge into

both a scientific study and a religion. Their observations over a period of many centuries enabled them to

foretell the behaviour of the seasons and the appearance of the heavenly bodies. This knowledge they

kept secret. They were agents through whom the people propitiated their gods to ensure the coming of

the rains. the appearance of the new moon the survival and proseperity of the society. Through religious

observances the priests exerted their influence over the populace. Knowledge of natural events enabled

the priests to predict and claim credit for their occurrence. Frequently they also held the power to divide

the land, to demand tribute from the people, to organize public works, and to build vast monuments to the

glory of the gods and the kings. Only a stable society, one that had passed the bare subsistence level, could

afford to maintain such a superstructure of unproductive rulers and priests.

In ancient Egypt the flooding of the Nile River necessitated annual redivision of the land. Private ownership

of the land and the ability to produce a surplus of commodities enabled the owners to exchange their

products for their private gain or to store them for future use. Thus arose the need for a system of weights

and measures. Mathematical operations of addition, subtraction, multiplication, division and the use of

fractions are recorded in Egyptian payri in connection with the practical problems of the society.

Their methods of doubling and halving, called 'duplication and mediation' were still considered separate

operations in mediaeval Europe. This is how the method is used for multiplication. Let us find the product

of 27 and 11. The process consists of successively doubling one factor and halving the other:

11* 27

5* 54


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2 108

1* 216

Now we find the sum of everything in the second column (in bold) where there is an odd number (marked

with an asterisk) in the first column: 27 + 54 + 216 = 297, the desired product.

GREEK WRITERS ON EGYPTIAN MATHEMATICS [FAUVEL AND GRAY,P 21]

Herodotus (mid fifth Century BC)

The king moreover (so they say) divided the country among all the Egyptians by giving each an equal

square parcel of land, and made this his source of revenue, appointing the payment of a yearly tax. And

any man who was robbed by the river of a part of his land would come to Sesotris and declare what had

befgallen him; then the king would send men to look into it and measure the space by which the land was

diminished, so that thereafter it should pay in proportion to the tax originally imposed. From this, to my

thinking, the Greeks learned the art of geometry.

Plato (early fourth Century BC)

SOCRATES: I have heard that at Naucratis, in Egypt, there was one of the ancient gods of that country, to

whom was consecrated the bird, which they call Ibis; but the name of the deity was Theuth. That he was

the first to invent numbers and arithmetic, and geometry and astronomy, and moreover draughts and dice,

and especially letters, at the time when Thaumus was king of all Egypt, and dwelt in the great city of the

upper region which the Greeks call Egyptian Thebes, but the god they call Ammon; to him Theuth went and

showed him his arts, and told him that they ought to be distributed among the rest of the Egyptians.

Aristotle (mid fourth Century BC)

Hence it was after all such inventions [the practical arts] were already established that those of the

sciences which are not directed to the attainment of pleasure or the necessities of life were discovered;

and this happened in the place where men had leisure. This is why the mathematical arts were first set up

in Egypt; for there the priestly caste were allowed to enjoy leisure.

Proclus (fifth Century AD)

According to most accounts geometry was first discovered among the Egyptians, taking its origin from the

measurement of areas. For they found it necessary by reason of the rising of the Nile, which wiped out

everybody's proper boundaries. Nor is there anything surprising in that the discovery both of this and of

the other sciences should have had its origin in a practical need, since everything which is in the process of

becoming progresses from the imperfect to the perfect. Thus the transition from perception to reasoning

and from reasoning to understanding is natural. Just as exact knowledge of numbers received its origin

among the Phoenicians by reason of trade and contracts, even so geometry was discovered amont the

Egyptians for the aforesaid reason.


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DOES GOD DO ARITHMETIC?MATHEMATICS AND CLASS[FARRINGDON,P29]

Florus (according to Plutarch in Dinner-Table Discussions)

Lycurgus is said to have banished the study of arithmetic from Sparta, as being democratic and popular in

its effect, and to have introduced geometry, as being better suited to a sober oligarchy and constitutionalmonarchy. For arithmetic, by its employment of number, distributes things equally; geometry, by the

employment of proportion, distributes things according to merit. Geometry is therefore not a source of

confusion in the state, but has in it a notable principle of distinction between good men and bad, who are

awarded their portions not by weight or lot, but by the difference between vice and virtue. This, the

geometrical, is the system of proportion which God applies to affairs. This it is, dear Tyndares, which is

called by the names of Dike and Nemesis, and which teaches is that we ought to regard justice as equality,

but not equality as justice. For what the many aim at is the greatest of all injustices, and God has removed

it out of the world as being unattainable; but he protects and maintains the distribution of things according

to merit, determining it geometrically, that is in accordance with proportion and law"

KNOWLEDGE AND POWER:Francis Bacon in The Advancement of Learning, I.VII.2

We see the dignity of the commandment is according to the dignity of the commanded: to have

commandment over beasts, as herdmen have, is a thing contemptible; to have commandment over

children, as schoolmasters have, is a matter of small honour; to have commandment over galley-slaves is a

disparagement rather than an honour. Neither is the commandment of tyrants much better, over people

which have put off the generosity of their minds...But the commandment of knowledge is yet higher than

the commandment over will; for it is a commandment over the reason, belief, and understanding of man,

which is the highest part of the mind, and giveth law to the will itself. For there is no power on earth which

setteth up a throne or a chair of state in the spirits and souls of men, and in their cogitations, imaginations,

opinions and believes, but knowledge and learning.

THEARABIC ORIGINS OFARITHMETIC (SWETZ P27)

The words 'algorism' and 'algorithm' owe their etymological origin to the name of the Muslim scholar and

author Abu Jafar Muhammed ibn Musa al-Khwarizmi (Muhammed, the father of Jafar and the sone of

Musa, the Khwarismian, c825). Among his writings, al-Khwarizmi produced an arithmetic on the Hindu

numerals and their computational schemes. This arithmetic found its way to Spain where, in the twelfth

century, it was translated into Latin by an Englishman, Robert of Chester, and bore the titleAlgoritme denumero indorum...Thus the Latinised name of al-Khwarizmi used in [a] typical Islamic salutation became

associated with the new numerals and methods of computation.

THE ORIGIN OF THE MULTIPLICATION SIGN (SWETZ P204)

Crocetta, or as in the Treviso[an early Italian treatise on arithmetic], per croxetta simplice, was a very

common method of multiplication in old Italian arithmetics. Perhaps the best way to understand this

algorism is by use of a diagram enumerating the distinct steps taken:

1 2 1. First multiply the units together,writing down the unit result, and hold


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1 2

the ten's result, if there is one

4

1 2

1 2

2. Multiply the units and tens together.

Add to this the tens obtained in step 1

and write down the resulting number of

tens. If any hundreds are obtained they

are held for the next step.

4 4

1 2

1 2

3. Multiply the tens terms together, add

any tens carried over from the previous

step, write down the result.

1 4

Some early works did have such illustrative diagrams. In accomodating these diagrams, printers often

placed an between the multiplicand and the multiplier:

3 2

2 5

This clearly distinguished the problem as a multiplication exercise and the evolved as the symbol for

the operation.

ADAND BC:COMPUTERS,CALENDARS AND CATHOLICISM

The abbot Dionigi Esiguo was charged in 525 by the Pope with calculating the date of Easter for the

following year. Until then this was the preoccupation of Alexandrian scholars, whose Greek texts were

translated into Latin. They spoke of the sancte pasche compotum[sacred computation of Easter] with the

same solemnity as if the calculation of time remained, as in Caesars epoch, an occult science, the exclusive

concern of high priests and experts. Dionigio rejected this Hellenistic haughtiness. He made a clear

separation between the Lords Easter, dominicum pascha, and the calculation of lunar orbits, lunae

computus; the rules for calculating Easter were not of a terrestrial origin, but rather came from the

illumination of the Holy Spirit. The Christians sense of time was only oriented secondarily to natural signs

and acquired procedures. With equal vigour, Dionigi intervened on the social determination of terrestrial


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dates. He condemned the political practice of dating the years of the calendar on the basis of the reigns of

the Roman Emperors, in particular that of Diocletian, who had enthusiastically persecuted the Christians. In

opposition to this, he referred his table of Easwter to the initial date, ab incarnatione domini nostri Jesu

Christi[from the incarnation of our Jesus Christ]. This festival, which recurred each year, recalled precisely

the Lords re-incarnation, the event of our redemption and the origin of our hopes.

If Jesus Christ was the Lord of time, the Christians were authorized to re-insert his incomparable terrestrial

existence into the spiral of time. Dionigi not only calculated Easter Sunday for five successive lunar cycles

of nineteen years, from 532 to 626, but with the aid of approximative rules he backdated the principal

Christian festival to the birth of Christ, 525 years earlier, dispensing moreover with the oriental lunar year

and its characteristics, and with the roman solar year with its intercalendary daysfrom then on, it would

be enough to place the ecclesiastical calendar in a single table without calculating it every year, as had

been the practice.

Computus: Tempo e numero nella storia dEuropa. Borst, Arno(1990),

References and further reading

Bacon, Francis. The Advancement of Learning. Dent & Dutton (Everyman)

Borst, Arno(1990), Computus: Tempo e numero nella storia dEuropa. Genova: il melangolo (in Italian)

Farringdon, B. Science and Politics in the Ancient World. Allen and Unwin

Fauvel, John and Gray, Jeremy, The History of Mathematics a Reader. Open University Press, 16.50

Seligman, Kurt. Magic, Supernaturalism and Religion. Paladin

Swetz, Frank J.Capitalism and Arithmetic: the New Math of the 15th Century. La Salle, Illinois: Open Court

van Sertima, Ivan, Blacks In Science, Ancient and ModernNew Brunswick and London:Transaction Books

Zaslavsky, Claudia.Africa Counts: Number and Pattern in African CulturePub: Prindle, Weber and Schmidt

AND FINALLY...

Michael Fish on economics:

If the economists got it right as often as the weather forecasters, they'd be well pleased

01-the economics of adding up

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