**Contents:**

# CHAPTER NINE.

MATHEMATICS BEFORE THE GREEKS

## NATURE'S MISTAKE

WE HAVE ALREADY mentioned the process of counting as an example
of using a model of reality that is not contained in the brain but
is created at the level of language. This is a very clear example.
Counting is based on the ability to divide the surrounding world
up into distinct objects. This ability emerged quite far back in
the course of evolution; the higher vertebrates appear to have it
in the same degree as humans do. It is plain that a living being
capable of distinguishing separate objects would find it useful
in the struggle for existence if it could also count them (for example,
this would help one become oriented in an unfamiliar area). Description
by means of numbers is a natural, integrated complement to differential
description by recognition of distinct objects. Yet the cybernetic
apparatus for recognizing numbers, for counting, can be extremely
simple. This task is much easier than distinguishing among separate
objects. Therefore one would expect that, within limits imposed
by the organization of the organs of sight, recognition of numbers
would have appeared in the course of evolution. The human eye can
distinguish tens and hundreds of distinct objects at once. We might
expect that human beings would be able to tell a group of 200 objects
from a group of 201 just as easily as we tell two objects from three.

But nature did not wish or was unable to give us this capability.
The numbers which are immediately recognizable are ridiculously
few, usually four or five. Through training certain progress can
be made, but this is done by mentally breaking up into groups or
by memorizing pictures as whole units and then counting them in
the mind. The limitation on direct discrimination remains. It is
in no way related to the organization of the organs of sight and
apparently results from some more deep-seated characteristics of
brain structure. We do not yet know what they are. One fact forces
us to ponder and suggests some hypotheses:

In addition to spatial discrimination of numbers there is temporal
discrimination. You never confuse two knocks at a door with three
or one. But eight or ten knocks is already, no doubt, ''many'' and
we can only distinguish such sounds by their total length (this
corresponds to the total area occupied by homogeneous objects in
spatial discrimination). The limit which restricts both types of
discrimination is the same. Is this a chance coincidence? It is
possible that direct discrimination of numbers always has a temporal
nature and that the capacity of the instantaneous memory limits
the number of situations it can distinguish. In this case the limitation
on spatial discrimination is explained by the hypothesis that the
visual image is scanned into a time sequence (and there is a rapid
switching of the eye's attention from object to object, which was
discussed above) and is fed to the very same apparatus for analysis.

Be that as it may, nature has left an unfortunate gap in our mental
device; therefore human beings begin work to create a "continuation
of the brain" by correcting nature's mistake--humans learn to count,
and thus mathematics begins.

## COUNTING AND MEASUREMENT

FACTS TESTIFY clearly that counting emerges before the names of
the numbers. In other words, the initial linguistic objects for
constructing a model are not words but distinct, uniform objects:
fingers, stones, knots, and lines. That is natural. During the emergence
of language, words refer only to those concepts which already exist,
which is to say, those which are recognized. The words ''one," ''two,''
and possibly ''three'' appear independently of counting (taking
"counting" to mean a procedure which is prolonged in time and recognized
as such) because they rely on the corresponding neuronal concepts.
There is as yet nowhere from which to take the words for large numbers.
To convey the size of some group of objects, the human being uses
standard objects, establishing a one-to-one correspondence among
them, one after the other. This is counting. When counting becomes
a widespread and customary matter, word designations begin to emerge
for the most frequently encountered (in other words small) groups
of standard objects. Traces of their origin have remained in certain
numbers. For example, the Russian word for five, *pyat,'* is
suspiciously similar to the old Slavic *pyad,' *which means
hand (five fingers).

There are primitive peoples who have only ''one,'' ''two,'' and
''three''; everything else is "many.'' But this in no way excludes
the ability to count by using standard objects, or to convey the
idea of size by breaking down objects into groups of two or three,
or by using as yet unreduced expressions, such as ''as many as the
digits on two hands, one foot, plus one.'' The need for counting
is simply not yet great enough to establish special words. The sequence
''one, two, three, many'' does not reflect an inability to count
to four and beyond, as is sometimes thought, but rather a distinction
the human mind makes between the first three numbers and all the
rest. For we can only unconsciously--and without exertion--distinguish
the numbers to three. To recognize a group of four we must concentrate
especially. Thus it is true for us as well as for savages that everything
which is more than three is ''many.''

To convey large numbers people began to count in "large units'':
fives, tens, and twenties. In all the counting systems known to
us large units are divisible by five, which indicates that the first
counting tool was always the fingers. Still larger units arose from
combinations of large units. Separate hieroglyphs depicting numbers
up to ten million are found in Ancient Egyptian papyruses.

The beginning of measurement, just as with counting, goes back
to ancient times. Measurement is already found among the primitive
peoples. Measurement assumes an ability to count, and additionally
it demands the introduction of a unit of measure and a measurement
procedure that involves comparing what is being measured against
a unit of measurement. The most ancient measures refer to the human
body: pace, cubit [lit. ''elbow''; the unit was the length of the
forearm], and foot.

With the emergence of civilization the need for counting and for
the ability to perform mathematical operations increases greatly.
In developed social production, the regulation of relations among
people (exchange, division of property, imposition of taxes) demands
a knowledge of arithmetic and the elements of geometry. And we find
this knowledge in the most ancient civilizations known to us: Babylon
and Egypt.

## NUMBER NOTATION

THE WRITING OF NUMBERS in ancient times demonstrates graphically
the attitude toward the number as a direct model of reality. Let
us take the Egyptian system for example. It was based on the decimal
principle and contained hieroglyphs for the one (vertical line)
and ''large ones.'' To depict a number it was necessary to repeat
the hieroglyph as many times as it occurred in the number. Numbers
were written in a similar way by other ancient peoples.

Figure 9.1. Number notation by different ancient
peoples. Adapted from G. I. Gleizer *Istoriya matematiki v shkole*
(The History of Mathematics in School) Prosveshchenie Publishing
House Moscow, 1964.

The Roman system is close to this very form of notation. It differs
only in that when a smaller unit stands to the left of a larger
it must be subtracted. This minor refinement (together with introduction
of the intermediate units V, L, and D) eliminated the necessity
of writing out a series of many identical symbols, giving the Roman
system such vitality that it continues to exist to the present day.

An even more radical method of avoiding the cumbersome repetition
of symbols is to designate key numbers (less than 10, then even
tens, hundreds, and so on) by successive letters of the alphabet.
This is precisely what the Greeks did in the eighth century B.C.
Their alphabet was large enough for ones, tens, and hundreds; numbers
larger than 1,000 were depicted by letters with a small slash mark
to the left and beneath. Thus, [beta] signified two, [eta] signified
20, and /[beta] signified 2,000. Many peoples (such as the Armenians,
the Jews, the Slavs) borrowed this system from the Greeks. With
alphabetic nummeration the model form of the number completely disappears;
it becomes merely a symbol. Simplification (for purposes of rapid
writing) of characters which initially had model form leads to the
same result.

"Arabic'' numerals are believed to be of Indian origin, although
not all specialists agree with this hypothesis. Numbers are first
encountered in Indian writings in the third century B.C. At this
time two forms of writing were used, Kharoshti and Brahmi, and each
one had its own numerals. The Kharoshti system is interesting because
the number four was selected as the intermediate stage between 1
and 10. It is likely that the oblique cross (x) used as a 4 tempted
the creators of the Kharoshti numbers by its simplicity of writing
while still preserving the modeling quality in full (four rays).
The Brahmi numerals are more economical. It is believed that the
first nine Brahmi characters finally gave rise to our modern numerals.

Figure 9.2. Kharoshti numerals.

Figure 9.3. Brahmi numerals

Figure 9.4. The genealogy of modern numerals (according
to Menninger, Zohlwort and Ziffer)

The loss of the model form in numbers was more than compensated
for in the ancient world by the use of the abacus, a counting board
with parallel grooves along which pebbles were moved. The different
grooves corresponded to units of different worth. The abacus was
probably invented by the Babylonians. It was used for all four arithmetic
operations. Greek merchants used the abacus extensively and the
same kind of counting board was common among the Romans. The Latin
word for pebble, ''calculus,'' began also to mean "computation.''
And the Romans conceived the idea of putting the counting pebbles
on rods, which is how the abacus still in use today originated.
These very simple counting devices were enormously important and
only gave way to computations on slateboards or paper after the
positional system of notation had completely formed.

## THE PLACE-VALUE SYSTEM

THE BABYLONIANS laid the foundations of the place-value system
the number system they borrowed from the Sumerians, we see two basic
''large ones'': ten and sixty, from the most ancient clay tablets
which have come down to us, dating to the beginning of the third
millennium B.C. We can only guess where the number sixty was taken
from. The well-known historian of mathematics O. Neigebauer believes
that it originated in the relation between the basic monetary units
in circulation in Mesopotamia: one *mana* (in Greek *mina*
was sixty *shekels)*. Such an explanation does not satisfy
our curiosity because the question immediately arises: why are there
sixty shekels in a mana? Isn't it precisely because a system based
on sixty was used? After all, we don't count by tens and hundreds
because there are 100 kopecks in a ruble! F. Thureau-Dangin, an
Assyriologist, gives linguistic arguments to show that the number
system was the primary phenomenon and the system of measures came
second. Selection of the number sixty was apparently a historical
accident, but one can hardly doubt that this accident was promoted
by an important characteristic of the number sixty, namely that
it has an extraordinarily large number of divisors 2, 3, 4, 5, 6,
10, 12, 15, 20, and 30). This is a very useful feature both for
a monetary unit (since its existent money has been evenly subdivided)
and for establishing a system of counting (if we assume that some
wise man introduced it, guided considerations of convenience in
calculation).

The mathematical culture of the Babylonians is known to us from
texts dating from the Ancient Babylonian (1800-1600 B.C.) and the
Seleucidae epoch (305-64 B.C.). A comparison of these texts shows
that no radical changes took place in the mathematics of the Babylonians
during this time.

The Babylonians depicted 1 by a narrow vertical wedge --, while
10 was a wide horizontal wedge.

The number 35 looked like this

All numbers up to 59 were represented analogously. But 60 was depicted
once again by a narrow vertical wedge, just the same as 1. In the
most ancient tablets it can be seen that the wedge representing
60 is larger than the wedge for 1. thus the number 60 was not only
understood as a "larger one" but was so represented. "Large tens"
appeared correspondingly for large units multiplied by 10. Later,
the difference between large and small wedges was lost and they
began to be distinguished by their postion. In this way the positional
system arose. A Babylonian would write the number 747 = 12 X 60
+ 27 in the form The
third 60-base position corresponds to the number 60^{2}
= 3,600 and so on. But the most remarkable thing is that the Babylonians
also represented fractions in this way. In a number following the
number of ones, each unit signified 1/_{60}, in the next
number each unit was 1/_{3,600}, and so on. In modern decimal
notation we separate the whole part from the fraction part by a
period or comma. The Babylonians did not. The numbercould
signify 1.5 or 90 with equal success. This same uncertainty occured
in writing whole numbers: the numbers *n, n* X 60, *n*
X 60^{2}, and so on were indistinguishable. Multipliers
and divisors divisible by 60 had to be added according to the sense.
Because 60 is a fairly large number, this did not cause any particular
problems.

When we compare the Babylonian positional system with our modern
one we see that the uncertainty in the multiplier 60 is a result
of the absence of a character for zero, which we would add the necessary
number of times at the end of a whole number or the beginning of
a fraction. Another result of the absence of the zero is an even
more serious uncertainty in interpreting a numerical notation that
in our system requires a zero in an intermediate position. In the
Babylonian notation, how can the number 3,601 = 1 . 60^{2}
+ 0 . 60 + 1 be distinguished from the number 61 = 1. 60 + 1 ? Both
of these numbers are represented by two units (ones). Sometimes
this kind of uncertainty was eliminated by separating the numbers,
leaving an empty place for the missing position. But this method
was not used systematically and in many cases a large gap between
numbers did not mean any thing. In the astronomical tables of the
Seleucidae epoch one finds the missing position designated by means
of a character resembling our period. We do not find anything of
the sort in the Ancient Babylonian epoch. But how were the ancient
Babylonians able to avoid confusion?

The solution to the riddle is believed to consist in the following.[1]
The early Babylonian mathematical texts which have come down to
us are collections of problems and their solutions, unquestionably
created as learning aids. Their purpose was to teach practical methods
of solving problems. But not one of the texts describes how to perform
arithmetic operations, in particular the operations, complex for
that time, of multiplication and division. Therefore, we assume
that the students knew how to do them. It is improbable that they
performed the computations in their head: they probably used some
abacus-like calculating device. On the abacus, numbers appear in
their natural, spontaneously positioned form and no special character
for the zero is needed; the groove corresponding to an empty position
simply remains without pebbles. Representation of a number on the
abacus was the basic form of assigning a number, and there was no
uncertainty in this representation. The numbers given in cuneiform
mathematical texts serve as answers to stage-by-stage calculations,
so that they could be used to check correctness during the solution.
The student made the calculations on the abacus and checked them
against the clay tablet. Clearly the absence of a character for
empty positions did not hinder such checking at all. When voluminous
astronomical tables became widespread and were no longer used for
checks but rather as the sole source of data, a separation sign
began to be used to represent the empty positions. But the Babylonians
never put their own ''zero'' at the end of a word; it is obvious
that they perceived it only as a separator, and not as a number.

Having familiarized themselves with both the Egyptian and the Babylonian
systems of writing fractions and performing operations on them,
the Greeks selected the Babylonian system for astronomical calculations
because it was incomparably better, but they preserved their own
alphabetic system for writing whole numbers. Thus the Greek system
used in astronomy was a mixed one: the whole part of the number
was represented in the decimal nonpositional system while the fractional
part was in the 60-base positional system--not a very logical solution
by the creators of logic! Following their happy example we continue
today to count hours and degrees (angular) in tens and hundreds,
but we divide them into minutes and seconds.

The Greeks did introduce the modern character for zero into the
positional system, deriving it--in the opinion of a majority of
specialists--from the first letter of the wo doudeg, which means
''nothing." In writing whole numbers (except for the number 0) this
character was not used, naturally, because the alphabetic system
which the Greeks used was not a positional one.

The modern number system was invented by the Indians at the beginning
of the sixth century A.D. They applied the Babylonian positional
principle and the Greek character for zero to designate an empty
place to a base of 10, not 60. The system proved to be consistent,
economical, not in contradiction with tradition, and extremely convenient
for computations.

The Indians passed their system on to the Arabs. The positional
number system appeared in Europe in the twelfth century with translations
of al-Kwarizmi's famous Arab arithmetic. It came into bitter conflict
with the traditional Roman system and in the end won out. As late
as the sixteenth century, however, an arithmetic textbook was published
in Germany and went through many editions using exclusively ''German,''
which is to say Roman, numerals. It would be better to say ''numbers,''
because at that time the word ''numerals" was used only for the
characters of the Indian system. In the preface of this textbook
the author writes: ''I have presented this arithmetic in conventional
German numbers for the benefit and use of the uneducated reader
(who will find it difficult to learn numerals at the same time).''
Decimal fractions began to be used in Europe with Simon Stevin (1548-1620).

##

## APPLIED ARITHMETIC

THE MAIN LINE to modern science lies through the culture of Ancient
Greece, which inherited the achievements of the ancient Egyptians
and Babylonians. The other influences and relations (in particular
the transmission function carried out by the Arabs) were of greater
or lesser importance but, evidently, not crucial. The sources of
the Egyptian and Sumerian-Babylonian civilizations are lost in the
dark of primitive cultures. In our review of the history of science,
therefore, we shall limit ourselves to the Egyptians, Babylonians,
and Greeks .

We have already discussed the number notation of the Egyptians
and Babylonians. All we need now is to add a few words about how
the Egyptians wrote fractions. From a modern point of view their
system was very original, and very inconvenient. The Egyptians had
a special form of notation used only for so-called 'basic'' fractions.
that is, those obtained by dividing one by a whole number; in addition
they used two simple fractions which had had special hieroglyphs
from ancient times: 2/3 and 3/4. In the very latest papyruses, however,
the special designation for 3/4 disappeared. To write a basic fraction
the symbol

which meant "part," had to be placed above the conventional number
(the denominator). Thus

The Egyptions expanded the other fractions into the sum of several
basic fractions. For example, 3/8 was written as 1/4 + 1/8, and
2/7 was written as 1/4 + 1/ 28. For the result of dividing 2 by
29 an Egytian table gave the following expansion:

2/29 = 1/29 + 1/58 = 1/174 + 1/232.

We are not going to dwell on the computational techniques of the
Egyptians and Babylonians. It is enough to say that they both were
able to perform the four arithmetic operations on all numbers (whole,
fractional, or mixed) which they met in practice. They used auxiliary
mathematical tables for operations with fractions; these were tables
of inverse numbers among the Babylonians and tables of the doubling
of basic fractions among the Egyptians. The Egyptians wrote intermediate
results on papyrus, whereas the Babylonians apparently performed
their operations on an abacus and thus the details of their technique
remain unknown.

What did the ancient mathematicians calculate? One fragment of
an Egyptian papyrus from the times of the New Empire (1500-500 B.C.)
describes the activity of the pharaoh's scribes very colorfully
and with a large dose of humor; for this reason it is invariably
cited in we shall not be an exception. Here is the excerpt, somewhat
shortened:

I will cause you to know how matters stand with you, when you
say "I am the scribe who issues commands to the army. ". . . I
will cause you to be abashed when I disclose to you a command
of your lord, you, who are his Royal Scribe.... the clever scribe
who is at the head of the troops. A building ramp is to be constructed,
730 cubits long, 55 cubits wide, containing 120 compartments,
and filled with reeds and beams; 60 cubits high at its summit,
30 cubits in the middle, with a batter of twice 15 cubits and
its pavement 5 cubits. The quantity of bricks needed for it is
asked of the generals, and the scribes are all asked together,
without one of them knowing anything. They all put their trust
in you and say "You are the clever scribe. my friend! Decide for
us quickly!" Behold your name is famous.... Answer us how many
bricks are needed for it?[2]

Despite its popularity, this text is not too intelligible. But
it nevertheless does give an idea of the problems Egyptian scribes
had to solve.

Specifically, we see that they were supposed to be able to calculate
areas and volumes (how accurately is another question). And in fact,
the Egyptians possessed a certain knowledge of geometry. According
to the very sound opinion of the Ancient Greeks, this knowledge
arose in Egypt itself. One of the philosophers of Aristotle's school
begins his treatise with the words:

Because we must survey the beginning of the sciences and arts
here we will state that, according to the testimony of many, geometry
was discovered by the Egyptians and originated during the measurement
of land. This measurement was necessary because the flooding of
the Nile River constantly washed away the boundaries. There is
nothing surprising in the fact that this science, like others,
arose from human need. Every emerging knowledge passes from incomplete
to complete. Originating through sensory perception it increasingly
becomes an object of our consideration and is finally mastered
by our reason. [3]

The division of knowledge into incomplete and complete and a certain
apologetic tone concerning the "low'' origin of the science are,
of course, from the Greek philosopher. Neither the Babylonians nor
the Egyptians had such ideas. For them knowledge was something completely
homogeneous. They were able to make geometric constructions and
knew the formulas for the area of a triangle and circle just as
they knew how to shoot bows and knew the properties of medicinal
plants and the dates of the Nile's floods. They did not know geometry
as the art of deriving "true" formulas; among them it existed, as
B. Van der Waerden expressed it, only as a division of *applied
arithmetic*. It is obvious that they employed certain guiding
considerations in obtaining the formulas, but these considerations
were of little interest to them. They did not affect their attitude
toward the formula.

##

## THE ANCIENTS' KNOWLEDGE OF GEOMETRY

WHAT GEOMETRY did the Egyptians know?--the correct formulas for
the area of a triangle, a rectangle, and a trapezium. The area of
an irregular quadrangle, to judge by the one remaining document,
was calculated as follows: half the sum of two opposite sides was
multiplied by half the sum of the other two opposite sides. This
formula is grossly wrong (except where the quadrangle is rectangular,
in which case the formula is unnecessary). There is no reasonable
sense in which it can even be called approximate. It appears that
this is the first historically recorded example of a proposition
which is derived from "general considerations," not from a comparison
with the data of experience. The Egyptians calculated the area of
a circle by squaring 8/9, of its diameter, a difference of about
1 percent from the value of pi.

They calculated the volumes of parallelepipeds and cylinders by
multiplying the area of the base by the height. The most sophisticated
achievement of Egyptian geometry known to us is correct computation
of the volume of a truncated pyramid with a square base (the Moscow
papyrus). It follows the formula

*V*=( a^{2} +ab +b^{2}) *h*/3

where *h* is the height, *a* and *b* are the sides
of the upper and lower bases.

We have only fragmentary information on the Ancient Babylonians'
knowledge of mathematics, but we can still from a general idea of
it. It is completely certain that the Babylonians were aware of
what came to be called the ''Pythagorean theorem''--the sum of the
squares of the sides of a right triangle is equal to the square
of the hypotenuse. Like the Egyptians they computed the areas of
triangles and trapeziums correctly. They computed the circumference
and area of a circle using a value of [pi]=3, which is much worse
than the Egyptian approximation. The Babylonians calculated the
volume of a truncated pyramid or cone by multiplying half of the
sum of the areas of the bases by the height (an incorrect formula).

##

## A BIRD'S EYE VIEW OF ARITHMETIC

THE SITUATIONS and representations in the human nervous system
model the succession of states of the environment. Linguistic objects
model the succession of situations and representations. As a result
a theory is a ''two-story'' linguistic model of reality.

Figure 9.5. Diagram of the use of a linguistic
model of reality

This diagram shows the use of a theory. The situation *S*_{1}
is encoded by linguistic object L1. This object may, of course,
consist of a set of other objects and may have a very highly complex
structure. Object *L*_{1} is the name for *S*_{1}.
Sometime later situation *S*_{1} changes into situation
*S*_{2}. A certain linguistic activity is performed
and we convert *L*_{1} into an other object *L*_{2};
if our model is correct *L*_{2} is the name of *S*_{2}.
Then, without knowing the real situation *S*_{2}, we
can get an idea of it by *decoding* linguistic object *L*_{2}.
The linguistic model is plainly determined by both the semantics
of objects *L*_{1} (the "material part'' according
to Russian military terminology) and the type of linguistic activity
which converts *L*_{1} into *L*_{2}.

Notice that we have not said anything about "isolating the essential
aspects of the phenomenon," ''the cause-effect relation," or other
such things which are usually set in places of honor when describing
the essence of scientific modeling. And in our presentation, situation
*S*_{1} does not "generate" situation *S*_{2}
but only ''changes into'' it. Of course, this is no accident. The
diagram we have drawn logically precedes the above-mentioned philosophical
concepts. If we have a linguistic model (and only to the extent
that we do have one) we can talk about the essential aspect of a
phenomenon, idealization, the cause-effect relation, and the like.
Although they appear to be conditions for the creation of a linguistic
model, all of these concepts are in fact nothing but description
in general terms (although very important and necessary ones) of
already existing models. Although these concepts appear to ''explain''
why a linguistic model can exist in general, in reality they are
elements of a linguistic model of the next level of the control
hierarchy and, of course, appear later in history than the primary
linguistic models (for example arithmetic ones). Before using these
concepts, therefore, we must ascertain that linguistic models exist
in general. And on this level of description we need not add anything
to the diagram shown in figure 9.5.

But theories are created and developed by the trial and error method.
If there is a starting point, then beginning from it a person tries
to build linguistic constructions and test the results. The phases
of building and testing are constantly alternating: construction
gives rise to testing and testing gives rise to new constructions.

The starting point of arithmetic is the concept of the whole number.
The aspect of reality this concept reflects is the following: the
relation of the whole to its parts, the procedure for breaking the
whole down into parts. The same thought can be expressed the other
way around: a number is a procedure for joining parts into a whole,
that is, into a certain set. Two numbers are considered identical
if their parts (set elements) can be placed in a one-to-one correspondence;
establishing this correspondence is counting. It is obvious, however,
that numbers are not enough for a theory; we must also have operations
with them. These are the elements of the model's functioning, the
conversions *L*_{1} -> *L*_{2}. Let
us take two numbers n and m and represent them schematically as
two modes of breaking a whole down into parts (figure 9.6 a).

Figure 9.6. Operations on whole numbers

How can we from these two numbers obtain a third--that is, a third
mode of breaking down the whole into parts? Two modes come to mind
immediately. They can be called parallel and sequential joining
of breakdowns. In the parallel mode both wholes form parts of a
new whole (figure 9.6 b). This breakdown (number) we call the *sum*
of the two numbers. With the sequential mode we take one of the
breakdowns and break down each of its parts in accordance with the
other breakdown (figure 9.6 c). The new number is called the *product*.
It does not depend on the order of the generating numbers. This
can be seen very well if we interpret the actions with the numbers
not as joining breakdowns but as forming a new set. The sum is obviously
the result of merging the two sets into one (their union). The prototype
of the product is the set of combinations of any element of the
first set with any element of the second (in mathematics such a
set is called the *direct product *of sets). The connection
between this definition and the preceding one can be traced as follows.
Suppose the first breakdown divides whole number *A* into parts
*a*_{1}, *a*_{2}, . . ., and the second
divides *B* into parts *b*_{1}, *b*_{2},
. . . , *b*_{m} . After performing the first breakdown
we mark the parts obtained with the letters *a*_{i}.
Breaking down each part into parts *b*_{j}we keep the
first letter and add a second. This means that in each part of the
result there will be an *a*_{j} *ab*_{j},
and all these combinations will be different.

The approaches from the whole to the part and from the part to
the whole complement one another. It is also easy to see from figure
9.6 c that multiplication can be reduced to repeated addition. Of
course the ancients who were creating arithmetic were far from this
reasoning. But then again, the frog did not know that its nervous
system had to be organized on the hierarchical principle either!
What is important is that we know this.

Having linguistic objects that depict numbers and being able to
perform addition and multiplication with them we receive a theory
that gives us working models of reality. Let us figure a very simple
example, which clarifies the diagram in figure 9.6.

Suppose a certain farmer has planted wheat in a field 60 paces
long and 25 paces wide. We shall assume that the farmer expects
a yield of one bushel of wheat per square pace. Before beginning
the harvest he wants to know how many bushels of wheat he will get.
In this case *S*_{1} is the situation before the wheat
harvest, specifically including the result of measuring the width
and length of the field in paces and the expected yield; *S*_{2}
is the situation after the harvest, specifically including the result
of measuring the amount of wheat in bushels; *L*_{1}
is the linguistic object 60 x 25 (the multiplication sign is a reflection
of situation *S*_{1} just as the numbers 60 and 25
are; it reflects the structure of the set of square paces on the
plane as a direct product of the sets of linear paces for length
and width); *S*_{2} is the linguistic object 1.500.

Note that by *theory* we mean simply a linguistic model of
reality which gives something new in comparison with neuronal models.
This definition does not take into account that theories may form
a control hierarchy; this fact is difficult to reflect without introducing
mathematical apparatus. More general models can generate more particular
ones. We shall consider the terms *theory* and *linguistic
model* to be synonymous, but nonetheless when we are speaking
of one model generating another, we shall call the more general
one a theory and the more particular one a model.

##

## REVERSE MOVEMENT IN A MODEL

A THEORY THAT has just been created must first be tested comprehensively.
It must be compared with experience and searched for flaws. If the
theory is valid, an attempt must then be made to give the model
''reverse movement,'' to determine specific characteristics of *L*_{1}
on the basis of a given *L*_{2}. This procedure is
by no means without practical importance. A person uses a model
for planning purposeful activity and wants to know what he must
do to obtain the required result--what *L*_{1} should
be in order to obtain a given *L*_{2}. In our example
with the farmer, the question can be put as follows: given the width
of a certain field, what should the length be to obtain a given
amount of wheat?

But studying the reverse movement of a model is not always dictated
by practical needs of the moment. Often it is done for pure curiosity--to
"see what happens.'' Nonetheless, the result of such activity will
be a better understanding of the organization and characteristics
of the model and the creation of new constructions and models which,
in the last analysis, will lead to greatly enlarged practical usefulness.
This is the supreme wisdom of nature, which gave human beings ''pure''
curiosity.

In arithmetic the reverse movement of a model leads to the concept
of the *equation*. The simplest equations generate the operations
of subtraction and division. Using modern algebraic language. we
define the difference *b--a* as the solution of the equation
*a + x = b*-- in which *x* is the number that makes this
equality true. The quotient from dividing *b* by a is determined
analogously. The operation of division generates a new construction:
the fraction. Repeated multiplication of a number by itself generates
the construction of exponential degree, and the reverse movement
generates the operation of extracting the root. This completes the
list of arithmetic constructions in use among the Ancient Egyptians
and Babylonians.

##

## SOLVING EQUATIONS

WITH THE DEVELOPMENT of techniques of computation, and with the
development of civilization in general, increasingly complex equations
began to appear and to be solved. The ancients, of course, did not
know modern algebraic language. They expressed equations in ordinary
conversational language as is done in our grammar school arithmetic
textbooks. Nevertheless, their substance was not changed. The ancients
(and today's school children) were solving equations.

The Egyptians called the quantity subject to determination the
*akha* which is translated as ''certain quantity'' or ''bulk.''
Here is an example of the wording of a problem from an Egyptian
papyrus: "A quantity and its fourth part together give 15.'' In
modern mathematical terminology this is the problem of ''parts,''
and in algebraic language it corresponds to the equation *x*
+ 1/4 *x* = 15.

Let us give an example of a more complex problem from Egyptian
times. ''A square and another square whose side is 1/2 + 1/4 of
a side of the first square together have an area of 100. Calculate
this for me.'' The solution in modern notation is as follows:

Here is the description of the solution in a papyrus:

"Take a square with side I and take 1/2 + 1/4 of 1, that is, 1/2
+ 1/4, as the side of the second area. Multiply 1/2 + 1/4 by itself;
this gives 1/2 + 1/l6. Because the side of the first area is taken
as 1 and the second as 1/2 + 1/4, add both areas together; this
gives 1 + 1/2 + 1/16. Take the root from this: this will be 1 +
1/4. Take the root of the given 100: this will be 10. How many times
does 1 +1/4 go into 10? It goes eight times.''

The rest of the text has not been preserved, but the conclusion
is obvious: 8 ^{.} 1 = 8 is the side of the first square
and 8 ^{.} (1/2 + 1/ 4) = 6 is the second.

The Egyptians were able to solve only linear and very simple quadratic
equations with one unknown. The Babylonians went much further. Here
is an example of a problem from the Babylonian texts:

''I added the areas of my two squares: 25 ^{.} 25/60. The
side of the second square is 2/3 of the side of the first and five
more." This is followed by a completely correct solution of the
problem. This problem is equivalent to a system of equations with
two unknowns:

*x*^{2} + *y*^{2} = 25 ^{.}
25/60

*y* = 2/3 *x* + 5

The Babylonians were able to solve a full quadratic equation:

*x*^{2} +- *ax = b*,

cubic equations:

*x*^{3} *= a*

*x*^{2} (*x* + 1 ) = *a*

and other systems of equations similar to those given above as
well as ones of the type

*x +- y = a, Xy = b*

In addition to this they used formulas

(*a + b*)^{2} = *a*^{2} + 2*ab*
+ *b*^{2}

*(a + b)(a-b)* = (*a*^{2} - *b*^{2})

were able to sum arithmetic progressions, knew the sums of certain
number series, and knew the numbers which later came to be called
Pythagorian (such whole numbers *x, y*, and *z* that *x*^{2}
+ *y*^{2} = *z*^{2} ).

##

## THE FORMULA

THE PLACE of Ancient Egypt and Babylon in the history of mathematics
can be defined as follows: the *formula* first appeared in
these cultures. By formula we mean not only the alphanumeric expression
of modern algebraic language but in (general any linguistic object
which is an exact (formal) prescription for how to make the conversion*
L*_{1} *-> L*_{2}, or any auxiliary conversions
within the framework of language. Formulas are a most important
part of any elaborated theory although, of course, they do not exhaust
it because a theory also includes the meanings of linguistic objects
*L*_{i}. The assertion that there is a relation between
the magnitudes of the sides in a right triangle. which is contained
in the Pythagorian theorem, is a formula even though it is expressed
by words rather than letters. A typical problem with a description
of the process of solution ("Do it this way!") and with a note that
the numbers may be arbitrary (this may not be expressed but rather
assumed) is also a formula. It is precisely such formulas which
have come down to us in the Egyptian papyruses and the Babylonian
clay tablets.

[1]
See B. L. van der Waerden's book *Ontwakende Wetenschap*, in
English: *Science Awakening*, New York: Oxford University Press,
1969).

[2]
van der Waerden, *Science Awakening*, p. 17.

[3]
This fragment has come down to us through Procul (fitth century
B.C.), a commentator on Euclid.