From A Random World to a Rational Universe
Randomness, Luck, Astragali and Dice
In the time before the mathematical idea of randomness was
discovered, people thought that everything that happened was part
of the will of supernatural beings, the gods, who looked down upon
human affairs and decided to 'tip the balance' one way or another
to influence events. Hence, sacrifices were made and rituals
performed to discover the 'will of the gods' or to try to influence
human affairs. This idea still prevails, and many people all over
the world use lucky charms, engage in superstitious practices, use
horoscopes, and still have some kind of belief that there are such
ways of influencing their lives. The gods may be dead, but 'Lady
Luck' still survives.
The astragalus is a small bone, about an inch cube, found in the
heel of hoofed mammals. Astragali have six sides but are not
symmetrical, so there is no way of knowing which way they will
eventually come to rest. For many ancient civilizations, astragali
were used by priests to discover the opinions of their gods. It was
customary in divination rites to roll, or cast, five astragali.
Typically, each possible configuration was associated with the name
of a god and carried with it the sought-after advice.
[Insert Astragali 4]
Showing the four positions of rest. The small one in the foreground
is made from pottery. Astragali found in excavations typically have
their sides numbered or engraved. They were also used in board
games in the First Dynasty in Egypt, c 3500 BCE; archaeological
evidence consists of boards, counters, and astragali for various
games, including one similar to Snakes and Ladders, still popular
today.
[Insert Game of Hounds and Jackals]
The astragali have been used from classical times for gambling, and
similar stones are still in use today for games like 'fivestones'
or 'jacks'.
Gradually, over thousands of years, astragali were replace by dice,
and pottery dice have been found in Egyptian tombs. The earliest
die known was made from pottery and excavated in Northern Iraq
dating from about 3,000 BCE. It has dots arranged as in (Die a)
[Insert Early Die a]
By about 1400 BCE a die from Egypt shows that the markings became
the familiar 2 - partitions of 7 (i.e. 1 opposite 6, 2 opposite 5,
and 3 opposite 4) (Die b).
[Insert Early Die b]
Dice with other markings like the names or portraits of gods have
been found, probably used for special games or rituals, and others
where some numbers are repeated, or 'loaded', for special purposes
or possibly for cheating. (Die c)
[Insert Early Die c]
Once the Greeks had worked out the geometry of the polyhedra, dice
of other shapes began to be constructed. However, whether cube or
polyhedral, the shapes were not entirely regular and were therefore
biased.
Over time, gamblers would get used to using the same dice, and have
an intuitive idea of how they would fall, but given another set of
dice, the odds would be different. Later, as the manufacture of
dice became more exact, some ideas of the possible combinations of
number began to emerge.
The Earth and The Cosmos
There were many other forms of rituals hoping to overcome the
randomness of nature and man's condition. A few of these which
became of particular mathematical interest are geomancy, the nine
square grid or magic square, and temple designs, the ancestors of
board games.
Geomancy
Geomancy means divination
of or by the earth
, and is a system of 16 mathematically related arrangements of
stones, beans or other available small objects used to make
decisions, answer questions, or foretell the future. The stones are
cast upon the ground and the pattern formed is interpreted. The
symbols represent a series of binary 'opposites' like good and
evil, male or female, sadness and happiness, etc. Combinations of
these opposites can be used to represent odd and even numbers.
[Insert 16 Figures of Geomancy]
As in all methods of divination, each of these figures has a number
of interpretations depending on its relation to other figures
shown, and many other circumstances like the time of day, the
weather, and the kind of person who is asking the question.
The Grid of Nine Squares
The Nine square grid is said to come from an ancient system for the
division of land, probably from feudal India. In China the
nine-square configuration was supposed to be an ideal arrangement,
with eight farmers' fields surrounding a central well. The grid of
nine squares, or a circle divided into nine sections by straight
lines often appears as a central form in Tibetan sacred diagrams.
In Scotland, the pattern was used at Beltane (the eve of May) where
eight squares were cut out from the turf, and a bonfire lit on the
central square.
In this way, from practical beginnings in different cultures, the
nine-square grid acquired mystic importance and symbolised divine
order, and the representation of control by the gods.
Magic Squares are directly
related to the Sacred Grid, supposedly being the numerical mystery
which underlies their physical form. The simplest magic square is
the square of nine, ascribed to Saturn, where each row and column
adds up to 15; the total of the rows and the columns is 45, and the
diagonals 30. The 4x4 square with row and column numbers 34 is
assigned to Jupiter, the 5x5 with row or column numbers 65 to Mars,
and so on for the Sun, Venus, Mercury and the 9x9 square with row
or column numbers 369, to the Moon [link to Magic Squares articles
on the site].
As with other devices, these magic squares are all said to have
correspondences to different numbers, various deities, days of the
week, natural objects, different qualities, and so on. In the Hindu
Temple Yantra you can see the nine squares, the 'sacred space', or
source of energy, in the centre.
[Insert Hindu Temple Yantra]
Board Games are clearly
linked with divination, astrology and sacred geometry, and the
designs of the boards can show their sacred or occult origins. The
popular game of 'snakes and ladders' is controlled by the throw of
dice, and the ladders and snakes originally referring to good and
bad fortune, now refer to good and bad 'luck' in the progress of
the game. In some cases the designs of the boards are the same as
the plans of temples and holy cities with a 'sacred space' in the
centre.
[Insert Korean Game Board] [Insert Nine Men's Morris] [Insert
Robert Burton Horoscope]
Mathematics and Magic
In ancient times, few people could understand even the simplest
arithmetic and geometry, and the confusion of mathematics with
magic has a long history.
People who had knowledge of the regular movements of the heavens
were able to predict the position of planets, and the particular
the times when astronomical events appeared in certain sections of
the sky. In ancient civilisations these were highly skilled
technicians, called 'priests', and their activities were partly
scientific, and partly religious. In Europe, after the arrival of
Christianity, the religious aspect of these practices was condemned
as superstition. Because numbers were used in these processes,
anyone who used numbers was regarded with considerable suspicion.
In this way genuine mathematicians were looked upon with suspicion
by the ignorant, and the titles of Astrologer, Mathematician and
Conjurer were virtually synonymous.
An early Bishop of the Church, St. Augustine of Hippo (354-430 CE)
once said:
"The good Christian should beware
of mathematicians and all those who make empty prophecies. The
danger already exists that mathematicians have made a covenant with
the devil to darken the spirit and confine man in the bonds of
Hell."
Augustine was arguing that belief in astrology denies the freedom
of the will.
Roger Bacon (1214 - 1292), often called England's first Scientist,
had a reputation as a 'great necromancer' because of his ingenious
experiments and John Dee (1527 - 1609) probably one of the foremost
mathematicians in Europe of his time, gained a reputation as a
'Conjuror' while he was at Oxford becuase he was respnsible for
developing a simple mechanical device for by which an actor
appeared to fly, and people claimed he was in league with the
devil.
[Insert Portraits of Bacon, Dee, Recorde ?]
During the sixteenth century in England, mathematicians like Robert
Recorde (1510-1558) and Thomas Digges (1546-1595) published many
works showing the everyday practical usefulness of mathematical
knowledge for ordinary people clearly showing that mathematics was
not an occult practice.
Following the foundation of the Oxford chairs in mathematics and
astronomy in 1619, some parents kept their sons away from the
university in fear of them becoming contaminated by the 'Black
Art'.
As the predictive power of astronomy and other practical uses of
mathematics became apparent, mathematicians were able to dispel the
idea that many events were not controlled by the goddess Fortuna,
but could be explained in a rational way.
[Insert Castle of Knowledge Title page]
The Beginnings of Probability
Since dice were used in gambling, in religious ceremonies and for
divination, it is believed that those who used the dice had a good
intuitive idea of the likely frequency of various number
combinations. The first printed document showing the possibilities
with three dice was the Latin poem De Vetula, which shows all the
combinations for the fall of three dice, and is believed to have
been written in the early 13th century. The idea of using binomial
coefficients [link or explanation] to calculate the possibilities
appears in the poem, but is not taken up until much later.
[Insert De Vetula]
Since the Christian Church was against gaming, and there was much
superstition about divination, it is not surprising that a theory
of probability did not begin to appear until the 16th century.
Cardano, writing with considerable personal knowledge of gambling,
recognised that if the die was honest, each face would have an
equal chance of appearing. His manuscript, Liber De Ludo Aleae , was written
about 1526 but only found after his death, and not published until
1663. He gave tables of the results for one, two and three dice,
but these are not all correct. However, Cardano is credited with
recognising that the abstraction of the 'honest die' is the key to
a theory of probability based on mathematical principles.
[Insert Cardano Portrait]
Tartaglia (1500 - 1557) and others discuss various versions of the
division of the stakes when a gambling game is stopped, called the
'problem of points', and this shows that Cardano's ideas were
likely to be common knowledge among scholars of the later 16th and
early 17th century.
Galileo (1564 - 1642) wrote on probability but his work was not
published until1718. He stated that with three dice there can only
be one way of obtaining a 3 (1,1,1) and an 18 (6,6,6) but there are
three ways of obtaining a 6, and four ways for a 7. However,
although 9 and 12 could be made up in the same number of ways as 10
and 11, from their experience, gamblers agreed that the occurrence
of 10 and 11 were more likely! Galileo showed that the total number
of possible throws with three dice are 216, and he gave a table of
the number of possible throws for a total of 10, 9, 8, 7, 6, 5, 4
and 3, showing that the throws for 11 to 18 were symmetrical with
these. In this way he showed that there were 27 possible throws to
obtain a 10, and 25 for a 9.
[Insert Galileo's Table]
His work showed that by this time there was no doubt about the
general method for calculating chances with a die, and it was clear
that the mathematical concepts of the equal probability of the
throw of a die, and the procedures to analyse the results were well
known.
Pascal's Triangle
By the mid 16th century the theory of probability became
established on a rigorous basis with the work of Pascal and Fermat.
However, as we have seen, the idea of the application of 'Pascal's
Triangle' had been suggested as early as the 13th century but
forgotten for some 200 years. The triangle itself was known and
published before, by Stifel (Arithmetica Integra 1543) Tartaglia
(Trattato 1556) Stevin (Arithmetic 1625) Pierre Herigone (Cours
Mathematique 1634), and we also know it was known to the Chinese
and the Arabs by the mid 13th century, but Pascal was the first to
apply it to probability.
Notes
References
Web
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