The Beginnings
We will come back to episteme, but we need to start where science started—with the
useful bits. There were a few things that were universally important to early civilizations.
The first was food. If one were a nomadic herder, it was important to be able to navigate
to where the food and water were. That required knowledge of the land and perhaps a
knowledge of the sky.
First, people left Africa. Second, they settled in the three great river valleys which were
chosen primarily because the climate was tolerable and the land replenished itself with
the annual floods. There was some development of science in the Ganges Valley and in
China, especially in arithmetic and all three acquired detailed knowledge of mathematics
and astronomy. The knowledge of Chinese science really came with Joseph Needham’s
24 volume Science and Civilization in China.
The Man who loved China - Joseph Needham
Biography tells secrets of Joseph Needham's China love
(Xinhua)
Updated: 2008-09-24 13:50 Comments(0) Print Mail LONDON -- In China, Li Yuese,
the Chinese name for an English intellectual Joseph Needham, is at least a household
name among the well-educated -- his Science and Civilization in China, a twenty-fourvolume masterpiece, is known as the most important books telling the west what Chinese
have contributed to the world.
The 17th-century philosopher-statesman Francis Bacon declared that nothing had
changed the world more profoundly than three great inventions: gunpowder, printing and
the compass. But what the philosopher didn't know was that all the three had already
been conceived of and successfully employed by a single people -- the Chinese.
And it was not until over 300 years later, that one young man in Cambridge gave these
people the credit they rightly deserved. The man was Joseph Needham, or better known
in China, Dr. Li Yuese.
"Needham was the first bridge builder between China and the rest of the world," Simon
Winchester, writer of the biography on Needham said on Tuesday in London.
In his book titled Bomb, Book & Compass -- Joseph Needham and the Great Secrets of
China, Simon is trying as he said "to bring the human side of the great man to the world,
and let the world know better what Needham was as a human being."
As a charismatic young biochemist, working towards a glittering career at Cambridge, he
fell in love with a young Chinese student and his passion for his mistress, Lu Gwei-djen,
led quickly to a fascination with her country's language and history and soon he
developed an astonishing reputation as a self-taught, albeit eccentric, scholar of Chinese
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culture, Simon said.
When in 1943, the British government sent him on a diplomatic mission to help save
China's universities from the occupying Japanese forces, Needham began the research
that would occupy him for the rest of his life and which ended up to create the greatest
work on China ever created in the Western world, the biography said.
None have succeeded in finding out what the first ever Chinese characters Needham
learnt under Lu Gwei-djen, Simon said on the launching of his biography in central
London's British Library, revealing that "cigarette" was first Chinese words the heavy
smoker learned under the Chinese lady.
Lu Gwei-djen, a brilliant biochemist from Nanjing, married Needham in 1989, more than
half a century after they first met.
Needham's 24-volume Science and Civilization in China remains an unrivaled account of
the nation's astonishing history of invention and technology, and Simon's book on the
Cambridge scholar is a story of the man and the "extraordinary rise of the Chinese nation
that continues to this day."
So, here’s the question. The Chinese were the first to record observations of comets (they
catalogued 29 of them observed over 300 years in 400 BC), solar eclipses (2137 BC), and
supernovae. They had a version of the sundial 4000 years ago and invented the abacus
between 1000 and 500 BCE. They invented gunpowder (300 AD) and the crossbow.
They used movable type in 700AD. They had a magnetic compass in 1086. They even
learned to produce coke from soft coal to replace charcoal and prevent deforestation.
The images are Yang Hui Triangle (Pascal’s Triangle) from a book by Zhu Shijie, 1303
AD and from the 1726 The Sea Island Mathematical Manual written by Liu Hui in the 3rd
C. The next two slides show a few of the many curiosities of Pascal’s triangle. Note that
the triangle is formed by adding the two squares immediately above each with 1’s down
the sides. The first curiosity is the counting and triangular counts. Then the rows go in
powers of two. Then going up diagonally gives a Fibonacci series. The triangle is side to
side symmetrical. And it does the odds. There is a whole list of others.
Part of the problem is that it was almost entirely engineering. They never developed a
cohesive and comprehensive body of theory behind the techne. For all their ability, they
learned that the earth was round from Italian Jesuit Matteo Ricci in the 1584 working
from atlases he took with him (and whose 12.5 foot folding screen map of the world from
1602 is the oldest copy introduced the Chinese to the Western Hemisphere and sold for
over a million dollars in January 2010). Ricci was a Jesuit priest who was the first
westerner admitted to the hidden city and who lived in Peking from 1582 until his death
in 1610. The map (along with the 1506 Waldsemuller map that has the first use of the
word America) was bought by the University of Minnesota.
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They had a real tendency to fall back on explanations by religion or invoking
explanations outside the natural system. An example is the idea flow of Chi to various
points in the skin as justification for acupuncture. No experiment can verify that (at least
yet).
Jared Diamond had another theory. He suggested that China, lacking geographical
barriers, could be molded into a large governmental entity under a single ruler who could
suppress technology or ideas on a whim. In Europe, the political entities were separated
by mountains and rivers and were small. If a ruler suppressed technological advances, he
would fail in competition with close by neighbors. Of course that does not explain the
fact that the US and USSR succeeded so well or that modern China is succeeding. It is
more likely that they ossified because they had a society hostile to scientific theory and
they managed to remain isolated until the 17C. Since they have opened up, they are more
than competitive.
More to the point, in the early civilizations who had just learned to grow crops in soil
annually replenished by flooding rivers, that meant having a calendar that was accurate
enough to allow one to predict the floods and control the crop output. Pure techne.
Once you could grow enough food that everyone was not involved every minute in food
production, part of the population could turn its attention to more esoteric things—like
building, art, and medicine. We will come back to the last two, but science in general
and mathematics in particular were vital to calendar making and building.
Of course, the most important scientific fact for the Egyptians was when the Nile would
go into flood. For that, they built a system of calculating the days based on astronomy.
The Egyptian year was divided into three seasons (Inundation/Autumn, Growing/Winter
and Harvest/Summer). Each season was divided into four months of each thirty days
making a total of 120 days in each season. Each month was further divided into three
decans to become three weeks of ten days each. At the end of the year, five extra days
were thrown in to compensate for the year being 365 days. This occurred in antiquity late
June, due to the progression of the stars nowadays late July, when the inundation of the
Nile began. As the inundation never occurred on a fixed date, the rising of the Dog Star
(Sirius), which had a fixed date, was used as a sign that the inundation was near.
But so far, we have used astronomy to measure time in the large scale. People being
what they are, at some point the Egyptians wanted to look at smaller units than months
and years. By 1500 BCE, they also had a device to measure hours based on the
movement of the sun. They had a stick with a cross bar sticking out. On the crossbar
were names for the six hours. In the morning, you faced the bar tip to the west and the
sun’s shadow gradually moved in toward the vertical piece. At noon, you turned it
around and the shadow gradually moved out toward the end. Of course, at dawn and
dusk, the shadow became infinitely long and wasn’t useful. It did nothing for you when
it was cloudy or night. Also, the length of the day is constant only at the equator. The
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farther you get from the equator, the more the length of days varies during the year, and
the Egyptians had no ability to account for that. Also, to correct local solar time to a
constant, you have to correct for longitude—witness time zones around GMT. The word
sundial, by the way, comes from the Latin dies or day. The device was also useless for
measuring anything less than an hour.
But remember that the length of the day varied during the year, so the hour was not a
constant length of time. In the summer, it was long during the day and short during the
night and the reverse in the winter. There would not be a constant hour until the Greeks
when Hipparchus proposed measuring the hour at the equinox and using that period of
time for the rest of the year as well. But that required a way of telling time other than
using the sun.
Within five hundred years of the sun stick from the reign of Thutmose III (who ruled
from 1479 to 1425 BCE (22 years of which he shared the rule with his stepmother—and
aunt—Hatshepsut and two of which he shared with his son Amenhotep II) the Egyptians
had figured out another way to measure time. They reckoned that water flowed from a
hole in a pot at a constant and measurable rate. The sun, after all, was useless in
measuring time at night. Flowing water could work in the dark. Interestingly, water
clocks were in the realm of Thoth who was also god of the night, god of learning, god of
writing, and god of measurement. That says what else time was tied up with.
The oldest water clock of which there is physical evidence dates to c. 1417-1379 BC,
during the reign of Amenhotep III where it was used in the Temple of Amen-Re at
Karnack. The oldest documentation of the water clock is the tomb inscription of the 16th
century BC Egyptian court official Amenemhet, which identifies him as its inventor.
These simple water clocks, which were of the outflow type, were stone vessels with
sloping sides that allowed water to drip at a nearly constant rate from a small hole near
the bottom. The sides were sloped because the pressure against the hole decreased as the
column got shorter so the diameter was also decreased to compensate and keep the water
flowing at a constant rate.
There were twelve separate columns with consistently spaced markings on the inside to
measure the passage of "hours" as the water level reached them. This was the original
division of daylight into twelve segments. The columns were for each of the twelve
months to allow for the variations of the seasonal hours. These clocks were used by
priests to determine the time at night so that the temple rites and sacrifices could be
performed at the correct hour and may have been used in daylight as well.
The water clock was refined by the Greek inventor Ctesibius who lived in Alexandria,
Egypt in the second century BCE. The Greeks, in fact, used water clocks for a number of
things. Athenian courts used them to limit the length of legal arguments. Court clocks
that have been recovered ran for six minutes. Demosthenes actually asked that the hole
in the clock be plugged during time he spent reading laws into the record as part of his
arguments.
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The pragmatic use was to measure time at night so the length of a “watch” could be
measured. The night was divided into three equal parts for that use. While there are no
surviving water clocks from the Mesopotamian region, most evidence of their existence
comes from writings on clay tablets. Two collections of tablets, for example, are the
Enuma-Anu-Enlil (1600–1200 BC) and the MUL.APIN (7th century BC). The latter is a
Babylonia star chart that is in the British Museum in London that relates the rising and
setting of stars to the 30 day month calendar. In these tablets, water clocks are used in
reference to payment of the night and day watches (guards).
In Babylon, water clocks were of the outflow type and were cylindrical in shape. Use of
the water clock as an aid to astronomical calculations dates back to the Old Babylonian
period (c. 2000 BC–c. 1600 BC. They did have one interesting adaptation. They did not
measure the water height with sticks or marks on the side of the bowl. Instead, they
measured the weight of the water they put it and then how long it took to come out. That
takes away the problem of a varying pressure as the water comes down. They understood
that hours were different in different parts of the year and they adjusted the amount of
water they put in the clock depending on the month. To define the length of a 'night
watch' at the summer solstice, one had to pour two mana of water into a cylindrical
clepsydra; its emptying indicated the end of the watch. One-sixth of a mana had to be
added each succeeding half-month. At equinox, three mana had to be emptied in order to
correspond to one watch, and four mana were emptied for each watch of the winter
solstitial night.
At Nalanda, a Buddhist university, four hours a day and four hours at night were
measured by a water clock, which consisted of a copper bowl holding two large floats in
a larger bowl filled with water. The bowl was filled with water from a small hole at its
bottom; it sank when completely filled and was marked by the beating of a drum at
daytime. The amount of water added varied with the seasons and this clock was operated
by the students of the university.
The water-powered mechanism of Su Song's astronomical clock tower, featuring a
clepsydra tank, waterwheel, escapement mechanism, and chain drive to power an
armillary sphere and 113 striking clock jacks to sound the hours and to display
informative plaques
In China, as well as throughout eastern Asia, water clocks were very important in the
study of astronomy and astrology. The oldest reference dates the use of the water-clock in
China to the 6th century BC. From about 200 BC onwards, the outflow clepsydra (from
the Greek Klepto and Hydra or water thief) was replaced almost everywhere in China by
the inflow type with an indicator-rod borne on a float.
Huan Tan (40 BC–AD 30), a Secretary at the Court in charge of clepsydrae, wrote that he
had to compare clepsydrae with sundials because of how temperature and humidity
affected their accuracy, demonstrating that the effects of evaporation, as well as of
temperature on the speed at which water flows, were known at this time.[17] In 976, Zhang
Sixun addressed the problem of the water in clepsydrae freezing in cold weather by using
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liquid mercury instead. Again, instead of using water, the early Ming Dynasty engineer
Zhan Xiyuan (c. 1360-1380) created a sand-driven wheel clock, improved upon by Zhou
Shuxue (c. 1530-1558).
Romans still had only sundials and water clocks, but they made a few ingenious
adaptations. They had a one and a half inch sundial that served as a pocket watch. At the
other extreme, they had the giant obelisk on the Campus Martius that also served as a
sundial. It was built by Augustus and is 525 feet high surrounded by bronze lines in the
pavement that mark the hours. It stood until the 8th century when it fell and was buried. It
was excavated and re-erected in 1789. The Roman courts also measured arguments from
a water clock and Martial actually suggested that one particularly long winded lawyer
might ease the judges boredom if, when he need a glass of water to ease his voice, he
would take it from the bowl of the water clock.
The Central American civilizations were different—perhaps in part because their
agriculture was not tied to a river flood. Although their agriculture was somewhat tied to
the seasons, remember that they were close to the equator where the temperature did not
vary a great deal through the year and where it rained in most every season. They did,
however, use the calendar to tie their lives to their history. Pacal, the great king of
Palenque calculated the exact date of the 80th revolution of the Long Calendar after his
accession to the throne. It would, just for your information, be on our October 15, 4772.
They also counted forward from the beginning of time which, converted to our calendar,
they reckoned to have been August 11, 3114 BCE. They also used a calendar to quite
accurately track and predict the phases of the moon and the relations of Venus and the
other planets to one another.
More useful, but more complicated were the great round calendars that were basically
gears within gears. The smallest wheel marked 13 sections that rotated against twenty
day signs so there were 260 combinations of a number and a day sign. These rotated
against a large wheel that was divided into 18 months of twenty days each (360 days).
The left over days were held as a five day “rest month” and basically ignored. A given
date had a name from the month cycle that was a number and a day sign. There was
another combination of a number and a sign from the large wheel. Together they defined
a date and a cycle that regularly repeated every 18,980 days or 52 years.
Of course, water froze in winter and was unsuitable for northern climates. On the other
hand, anything that flowed could be used to measure time and it just took talented
glassmakers to create a clock where sand ran from one place to another. Legend says the
sand hour glass was invented by a monk at Chartres in the eighth century although there
is quite good evidence that the Greeks had them and even had ones using mercury instead
of sand. The sand had to be perfectly dry and then perfectly sealed to keep its flow
constant. One early description details using fine black marble dust soaked nine times in
wine and dried after each soaking before being sealed in the glass. The problem was size.
A sand glass that would work for 12 hours (and Charlemagne actually had one built) was
huge. Columbus carried one that had to be turned every thirty minutes around the clock.
Of course, he didn’t use it to navigate but rather to keep track of the canonical hours for
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worship. They were also used to measure the length of sermons and an English law in
1483 required that the sermon glass be placed over the pulpit so the congregation could
keep track of how much of the sermon was left. The House of Commons used a sand
glass to measure the length of speeches (and still use it to mark the time to a vote or
division) and masons and other workers used them to measure the length of the work day.
They were also used to measure the length of lectures and one Oxford don threatened to
bring a two hour glass to class if the students did not start doing their work.
By the sixteenth century, hour glasses were used to measure the speed of a ship. A rope
with one knot tied every seven fathoms (six feet) with a log chip on the end of the rope.
The rope was thrown over the side and the hour glass started. How many knots paid out
in half a minute was measured. If five knots paid out, the ship was doing five nautical
miles an hour or five knots. Heaving the log remained the primary way to measure a
ships speed almost to the end of the 1800s.
The giant hour glass in Red Square was built in 2008 and at 40 tons is the world’s largest
chronograph. The time wheel in Budapest will run for an entire year and takes four men
45 minutes to turn with a series of cables.
Even though the constant hour dated to the Greeks, there was no reliable mechanical
timepiece until 14th century Europe. The original abbey clocks did not have hands or a
face at all. They merely rang bells for matins, nones, complines, and so forth. In fact the
word clock is from the German glock or bell. There was one of those in St. Paul’s in
London in 1286. The first with a face came in the 1340s (one in 1340 by Peter Lightfoot
a monk in Glastonbury, one in 1344 by Giacomo Dondi in Padua, and one built by Henry
de Vick for Charles V of France in Paris at what is now the Palais de Justice. It was
finished in 1379 and it is still there and was working until 1850.) It had an hour hand
only. The word clock comes from the Latin clocca or bell or possibly German glocke
which means the same thing. De Vick’s clock also worked with an escapement
mechanism.
The basis for the mechanical clock is escapement or controlled release of power.
Galileo had the idea of using the constant period of an oscillating pendulum to tell time
and the first pendulum clock appeared in 1656. The story (probably apocryphal) is that
he measured the swing of the lamp in the Duomo at Pisa using his pulse while a student
there. The period of a pendulum’s swing (more or less) varies not with the width of the
swing but with the length of the pendulum. A 39.13 inch (99.38 cm) length pendulum has
a period of exactly one second. Two of the earliest uses of the pendulum were to measure
pulse (devised by a physician friend of Galileo) and to keep time in music (metronome).
He let the clock idea go until 1641 when he was 77, blind, and under house arrest. His
friend Vincenzo Viviani wrote
“One day in 1641, while I was living with him at his villa in Arcetri, I remember that the
idea occurred to him that the pendulum could be adapted to clocks with weights or
springs, serving in place of the usual tempo, he hoping that the very even and natural
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motions of the pendulum would correct all the defects in the art of clocks. But because
his being deprived of sight prevented his making drawings and models to the desired
effect, and his son Vincenzio coming one day from Florence to Arcetri, Galileo told him
his idea and several discussions followed. Finally they decided on the scheme shown in
the accompanying drawing, to be put in practice to learn the fact of those difficulties in
machines which are usually not foreseen in simple theorizing.”
The mechanical pendulum clock has five parts:
A power source which is either a weight on a cord that turns a pulley or a spring
A gear train that increases the speed of the power so that a pendulum can be kept moving
An escapement that gives precisely timed pulses of power to the pendulum and that also
releases gear train wheels to move forward a specific amount with each power release
and to ultimately make hands rotate
A pendulum which is a weight on a rod—these were long so that the period of the
pendulum could be small and would approach isochronism.
An indicator or dial that keeps track of how often the escapement rotates. Typically that
is the clock face.
The magic was in devising the verge and foliot as an escapement mechanism. Verge
comes from the Latin virga or stick. The mechanism was probably incorporated in water
clocks as early as that of Villard de Honnecourt in 1287.
For the slide: The main wheel (1) is pulled in a circle by a hanging weight but is held in
check by the toothed wheel (2). This is controlled by the vertical verge (3) which has
blades set to catch on the teeth of the wheel alternating between the top and the bottom.
The weight that keeps the wheel from spinning freely comes from the inertia of the T
shaped foliot which exerts just enough pressure to hold the blade of the foliot in place for
a moment. It then releases and the bottom blade catches. Tick tock. Consider that now
time did not flow constantly but was divided up into small packets. By moving the
weights on the foliot, you can change the momentum and the rate at which the wheel is
allowed to turn. The same mechanism could trigger an alarm which allowed monks
charged with waking each other for prayers every four hours to sleep.
Time essentially changed. Around 1330, the hour became constant. Prior to that, it had
been temporal or temporary since it varied with the season. Now it was constant whether
winter or summer and whether day or night.
The earliest clocks did not have hands—only bells since their job was to note the times of
prayer (matins, compline, nones, etc). But they rapidly spread past churches. Large cities
got clocks governed by the king’s time. Smaller cities got them in their cathedrals and
churches and they determined when the city’s gates would be opened and closed and
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when fires had to be out at night to prevent fires. They marked the beginning and end of
curfew that protected citizens from attacks at night. They set the times for watchmen and
most of all determined when people started and stopped work.
The power initially was from the controlled fall of weights. The control came from a
verge and folio mechanism. The earliest spring clock was that given to Peter the Good,
Duke of Burgundy around 1430 and introduced the fusee. In the 1400s in Nuremberg, an
unknown metalworker (perhaps a locksmith) figured out that the pendulum could be
replaced by a spring and invented the fusee. The fact that a spring exerts less tension as it
unwinds was solved by running it through a cone shaped spindle so that it did the most
work on the narrow part of the cone when the spring was tight and progressively less as it
moved to the wider part of the cone. Now there could be pocket watches (by mid 1500s).
The first minute hand was in 1475 and the second hand appeared in 1560.
As they became smaller, they became status symbols. The astarium Giovanni di Dondi
made for the Viscount of Pavia had 297 moving parts and told hours, minutes, rising and
setting of the sun, Church feasts, days of the month, trajectories of the five known
planets, phases of the moon, an Easter calendar (the last movable feast since it is
determined by a lunar calendar), star time, and an annual calendar.
The problem with pendulums was that they required a stable platform and could not be
used on ships. Star charts were done at a fixed position and changed as one moved east
or west. The charts said where in the sky the sun, the moon, or the stars should be at a
particular time but they were only good for finding how far north or south of the equator
one was on a single meridian. For instance, since the day has 24 hours and the
circumference of the Earth 360 degrees, every 15 degrees west one goes, the sun rises
one hour earlier. The earth goes through 360 degrees in 24 hours (1440 minutes). The
circumference of the Earth is 24,901.55 miles at the equator. So, if the earth rotates 360
degrees in 24 hours, it rotates 360/1440 degrees per minute or ¼ degree per minutes. A
point on the equator travels 25,000 miles in 360 degrees or 70 miles per degree. 70 miles
per degree time ¼ degrees per minute equals 17 miles per minute of clock error. Every
minute a clock is off means the position east or west is off by 17 miles. Not acceptable.
Springs were too variable to be that accurate since the metal varied and the spring tension
changed with temperature.
Absent a clock, one could not determine longitude since that required knowing sunrise
time in exact relation to a fixed geographical point (Greenwich). In the wake of a
1707disaster in which an entire British fleet sank when they hit the Scilly Isles—a well
known group of 140 islands off the British coast. Sir Clowdesly Shovell was one of
Britain’s best admirals. He was washed ashore at Porthellick and was said to have been
injured but alive until a local woman murdered him to steal his emerald ring. (The
woman confessed thirty years later on her deathbed and the ring was returned to the
family). It was a national disaster and shame that the vaunted British navy had lost
hundreds of men and one of its best admirals from inaccurate determination of longitude.
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In 1714, Parliament offered a prize of £20,000 pounds (some millions currently) for
anyone who could solve the problem.
The first step came from Benjamin Huntsman who adopted glass making techniques and
the use of high temperature coke to make “crucible steel” that had a very high tensile
strength in 1740. John Harrison used the steel and, in 1761, built his H5clock that was
accurate to within 5 seconds over ten weeks.
Although the first of the great mathematicians that we know by name were Greek, they
readily acknowledged their debt to the Mesopotamians and particularly to the Egyptians.
The Egyptians had a numerical with signs for 1, 10, and so forth to 100,000 that they
(like the Romans) marked by repeating the sign for each one. They did multiplication
and division as a series of additions and subtractions. They were clumsy in their handling
of fractions, reducing them to a series of proper fractions with a numerator of 1. For
instance, 2/29 was written as 1/24+1/58+1/174+1/232. Difficult but correct to four digits.
They knew that the squares of 3 and 4 added up to the square of five and that a triangle of
sides 3, 4, and 5 was, of necessity, a right triangle and they used that fact in their
building. They had a special case of the Pythagorean Theorem worked out but not the
generality. This knowledge allowed the pyramid builders to calculate the angle at which
facing stone had to be cut.
The Babylonians were ahead of the Egyptians in time and somewhat more sophisticated
in their mathematics. First, they used a place value system. The Egyptians, like the
Romans later, did not. For them, 64 was a sign for forty plus four unit signs. The
Babylonians would have used a sign for the sixty followed by a sign for 4 with the place
location determining the value: they used the base ten for numbers less than 50 but then
converted to an ingenious base six system in which the value of the symbols was based
not on how many times they were written but rather on their order. So the first position
was units, the second was units x 60, the third units x 602 and so forth. 4,096 was written
1,8,16 or 1 x 602 + 8 x 60 + 16. Why 60? Possibly because it is divisible by 2, 3, 4, 5, 6,
10, 12, 15, 20, 30, and 60. That makes dealing with fractions considerably easier. Of
course, that got us to 60 seconds in a minute and sixty minutes in an hour.
They also needed a place holder for something with no value so they had (in that limited
sense) a zero. The idea of using position to indicate value would not appear in Greek
mathematics or in any of Western Europe until it was imported from India in the 8th
century.
They also built tables of multiplication and division, of all the squares up to 60, of cubes
up to 16, and of square and even cube roots. That kind of multiplication and division was
really only feasible in placeholder numbers. The Egyptians and the rest of the west right
to the Middle Ages were reduced to manipulating complex fractions.
They recognized that there was a relation between the sides and hypotenuse of a right
triangle and they made empirical efforts to arrive at a generalized solution but never quite
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got there. They also tried to calculate Π and got as close as 3 1/8. And all of that was in
the third millennium BCE (2600 BCE in Sumer).
Greek science was different. Here we get to the real questions of episteme or ontological
(metaphysical) knowledge. You have to give Greek science a certain amount of credit
considering that they were doing it entirely without instruments.
From Jowett, Timaeus.
That science (at least what we know of it) can be said to have originated in the city of
Miletus on the coast of Asia Minor in the 6th century BCE with three men who are
traditionally said to have been students one of the next: Thales, Anaximander, and
Anaximenes. Aristotle credits Thales as the first of the Greek scientists (remember that
that is an anachronism since the word scientist did not come to currency until the 1800s.
From the OED and quoting the Quarterly Review of 1834, “Science loses all traces of
unity. A curious illustration of this result may be observed in the want of any name by
which we can designate the students of the knowledge of the material world collectively.
We are informed that this difficulty was felt very oppressively by the members of the
British Association for the Advancement of Science, at their meetings in the last three
summers. Philosophers was felt to be too wide and too lofty a term; savans was rather
assuming; some ingenious gentleman proposed that, by analogy with artist, they might
form scientist, and added that there could be no scruple in making free with this
termination when we have such words as sciolist, economist, and atheist—but this was
not generally palatable.”
A sciolist is a superficial pretender to knowledge.
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It is no surprise that the origins of Greek science came from the Asian side of the
Mediterranean where they could benefit from what had been going on in Mesopotamia
for over a thousand years. Most of eastern “science” was actually technology—making
pottery, weaving cloth (up to 160 threads per inch in Egypt), extracting metal from ores,
and agriculture. In the basic sciences, there were advances in astronomy and
mathematics which were closely related. These were the basis from which the Milesians
worked. Medicine was there as well but will come later.
Thales (c 624-546 BCE) (whom Aristotle thought the first Greek philosopher which
brings up the question as to whether these men were actually scientists or philosophers)
was the first to try for a cosmology that did not require an outside motive force.
There are two prime characteristics of Thales’s view of the world that, as far as
we can tell, differ from everyone who preceded him. First, he thought that nature was
completely material—that is there was nothing supernatural or magical in its
composition. Second, whatever happens in nature is caused by nature and there is no
supernatural intervention. This is a remarkable leap from a world in which the gods are
constantly present and constantly acting. This may be a function of the fact that the
Greeks were at the edge of literacy and their brains may be working differently a la Julian
Jaynes. For Homer and Hesiod, everything that happened was unique and under direct
influence of the gods. For Thales, natural events could be generalized and were caused
by nature itself.
He hypothesized that everything could ultimately be reduced to water. Moreover,
he posited that the earth floated on water and that the heavens (which arose as mist off of
the waters) also floated on an arching sea above the earth. "Thales", says Cicero, "assures
that water is the principle of all things; and that God is that Mind which shaped and
created all things from water.”He explained earthquakes by supposing the earth was
floating and the quakes were caused by waves in that water. Anaximenes (ca. 585-528
BCE) would change that to everything being composed of air.
It is likely that Thales wrote nothing. Anaximenes and Anaximander on the other hand
clearly wrote things that the later Greeks had access to.
In geometry, Thales' theorem (named after Thales of Miletus) states that if A, B and C
are points on a circle where the line AC is a diameter of the circle, then the angle ABC is
a right angle. Thales' theorem is a special case of the inscribed angle theorem.
Thales’ Theorem is stated in another article. (Actually there are two theorems called
Theorem of Thales, one having to do with a triangle inscribed in a circle and having the
circle's diameter as one leg, the other theorem being also called the intercept theorem.) In
addition Eudemus attributed to him the discovery that a circle is bisected by its diameter,
that the base angles of an isosceles triangle are equal and that vertical angles are equal. It
would be hard to imagine civilization without these theorems.
Anaximander (ca. 610-546 BCE) went farther and some of his ideas are strikingly
modern. He realized that having the earth float on a giant sea meant the sea had to be on
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something so he dispensed with it and hypothesized that the earth was merely floating in
space unsupported.
Anaximander added fire to earth, air, and water and was the first to imagine that
the world was composed of four elements.
He also guessed that land animals had come from the sea (“Man in the beginning
resembled another animal, to wit, a fish.”) And, noting that fossils of marine creatures
could be found high in the hills, that the sea had once been much deeper than it was in his
time. Some of his ideas were less impressive: thunder came from wind and lightning was
the result of clouds splitting in half. On the other hand, the Milesians deserve credit for
replacing supernatural with natural explanations for phenomena.
The man that Bertrand Russell called the most influential philosopher of the Western
world was born about 580 BCE in the beautiful Greek island of Samos. The island is
only a mile from the coast of Asia Minor, a fact that may have some bearing on the fact
that, although he is at the heart of western science and philosophy, Pythagoras was more
than slightly a mystic who spent his life exploring and trying to define the borderlands
between science and religion. Also, Samos was the center of the Greek worship of Hera,
the wife of Zeus, and that may partially account for the fact that he always treated men
and women as equals and equally capable of learning. In fact, his daughter was a student
and became quite a famous mathematician in her own right.
We are hampered in studying what he did by the fact that not a single one of his works
survives except as oral tradition handed down by his unusually devoted followers.
Pythagoras’s mother was from Samos and his father was a Phoenician merchant. The
young man’s time on Samos was cut short when he fled the island to escape the tyrant
Polycrates who had assumed control in a coup with his two brothers. Polycrates had then
executed one of the brothers and expelled the others and taken sole control. Members of
the old aristocracy, including Pythagoras, left the island. Parenthetically, the residents of
Samos who stayed actually liked Polycrates although he was ultimately overthrown by
the Persians and the island was given to his surviving brother.
Pythagoras went to study in Memphis under the Egyptians and later to the libraries at
Tyre and Byblos before settling in Croton in Calabria. Legend also has it that he was
taken as a captive to Babylon where he learned Mesopotamian mathematics. In Croton,
he formed his own secret cult which may have been largely composed of slaves or former
slaves. He took males and females without differentiation and put them in a life of
religious study, exercise, philosophy, and common meals. His close followers
(mathematikoi) were bound by a rule of silence and the penalty for breaking the rule was
death. The reason for this was his belief that one should only speak when there was no
doubt of the truth of his words and that was unlikely ever to happen. They were ascetic
and vegetarian and were the first recorded example of a monastic lifestyle. The lifestyle
must have worked since he died of old age at 90. The Pythagorean ideal keeps cropping
up in things like the Templar legends and Masonic Lodges.
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He and his followers also believed in transmigration of the soul. He thought the soul
survived death and was reincarnated into humans, animals, and plants. He personally
claimed to remember four former lives and to be able to recognize the voice of a
deceased friend in the bark of a dog. Part of the justification for the life of personal
deprivation and contemplation was an attempt to move the souls of him and his followers
up to the gods. The ultimate goal of reincarnation was to become so moral that the cycle
could stop. The better the person’s behavior, the higher position he was likely to have in
reincarnation including position with the gods at the end. Interestingly, and unlike his
contemporaries, Pythagoras was quite sure that the brain and not the heart was the seat of
thought and knowledge.
Besides the mathematikoi, Pythagoras allowed people who were not part of the cult or
bound by its rules to listen to him teach (although they could only hear him since he
lectured from behind a screen). These akousmatikoi were taught abbreviated versions of
Pythagorean knowledge (often in short aphorisms such as “Don’t eat beans” which made
sense in a culture that had endemic favism—the most common human enzyme defect
characterized by hemolytic anemia after exposure to broad beans or some drugs) and they
were told the great secrets behind the cult’s philosophy.
At the core of Pythagoras’s philosophy was his belief in the spiritual primacy of numbers.
Arithmetic had been around since long before there was writing. Hunter gatherers were
undoubtedly capable of figuring out that two groups of two wildebeests was a four animal
herd. But the jump from arithmetic to abstract mathematics was a long one and one that
Pythagoras gets credit for making. See the “Fish, fish” article. He believed that pure
knowledge was the ultimate purification of the soul and the purest knowledge and purest
reality lay in numbers. Iamlichus of Chalcis quoted him as saying, “number is the ruler
of forms and ideas and the cause of gods and daemons.” Where Thales and his followers
believed reality was built of separate elements, the Pythagoreans actually believed it was
built on numbers.
They also believed that nature operated as it does because it is in the nature of the objects
in nature to behave as they do and not because some external force (the gods) made them
do things.
To understand Pythagoras’s influence, one needs to understand his theorem, his theory of
music, and his theories of astronomy.
People had been aware of right angles long before there was writing. Arguably, the two
greatest sensory influences are gravity which everyone experiences every day of their life
and the horizon which dominated the views of coastal and desert civilizations. Gravity is
always vertical and the horizon is (cleverly) always horizontal and where they intersect is
a right angle.
Ancient Egyptian builders had completely understood the uniqueness of the right triangle.
Their tool kits included a set square with sides in a 3:4:5 proportion. The Babylonians
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understood that, in a right triangle, the square of the hypotenuse is equal to the sum of the
squares of the other two sides and had hundreds of formulae based on that understanding
as early as 2000 BCE. Attributing the formula to Pythagoras was probably a bit of
propaganda spread by Plato to enhance his importance. What Pythagoras did do,
however, was prove that an empiric observation was a general truth. He linked geometry
to numbers and, two hundred years later, Euclid would take that knowledge to Alexandria
and put it in his Elements which, until recently, was the most reprinted book after the
Bible in the world.
Proof in Euclid's Elements
In Euclid's Elements, Proposition 47 of Book 1, the Pythagorean Theorem is proved by
an argument along the following lines. Let A, B, C be the vertices of a right triangle, with
a right angle at A. Drop a perpendicular from A to the side opposite the hypotenuse in the
square on the hypotenuse. That line divides the square on the hypotenuse into two
rectangles, each having the same area as one of the two squares on the legs.
He showed that a fundamental property of the space we occupy reduces to a very simple
set of numbers. As an offshoot, Pythagoras discovered square numbers. He also
believed that numbers had inherent properties, e.g. odd numbers were masculine and
even ones feminine. His student Hippasos wondered what happened when the right
triangle had two short sides of length (1). That would make the hypotenuse the square
root of (2), an irrational number. Pythagoras was so distressed at the violation of
symmetry that he had Hippasos executed.
Pythagoras also discovered that harmony in music is based on proportional intervals of
the numbers one through four. Parenthetically, the Greeks used base ten probably not
because of fingers and toes but rather because 10=1+2+3+4. If you divide a vibrating
string into an exact number of parts, the sounds are in harmony. Anything else is
discordant.
The legend was that Pythagoras made the discovery while walking by a blacksmith shop.
He noticed that the tones from the blacksmiths striking anvils sounded like music and
then noticed that notes coming from anvils of proportional size were in harmony. An
anvil half to size of its neighbor was one octave higher. One a third the size was in
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harmony but not the same note. This took him to strings, probably since the lyre was the
common instrument of the day.
The whole string is the base note. Half is one octave up. 1/3 of the way is a fifth and
one fourth of the way is another octave. A fifth of the way would be a third, but
Pythagoras did not get that far.
You could do the same thing by using strings of different length. Take four strings of the
same diameter and substance and make them 6, 8, 9, and 12 inches long. The 6 and 12
are an octave apart. The 8 and 12 are separated by a fifth. The 9 and 12 are separated by
a fourth, and the 8 and 9 are one note apart. Music and numbers are the same thing.
Chords come from exact divisions of the string into whole numbers. The base plus a
fifth plus a third makes a major chord. If it is three and then four, you have a minor
chord.
The fact that something as elemental as harmony came as the result of exact intervals
from one to four was so stunning as to be almost magical. It could just as well have been
2.7 or 6.8 or Π. This was such a striking finding that the extension was that numbers must
be at the base of everything. Numbers must be how the gods designed the earth. The
only way to truth was through mathematics and one could, by extension, know and
understand everything if one just knew the formulae.
Perhaps the most impressive thing is the idea that complex phenomena can be broken into
simple ones. This is probably the most important idea in the development of science and
is at the heart of Newtonian and Einsteinian physics.
The next logical extension was to take it to the stars. Why astronomy and not biology or
medicine? After all, what happened to one’s body had more immediate impact than what
was going on in the night sky. Perhaps it was because astronomy actually looked like
something that could be reduced to numbers. Biology has only entered that realm in the
last couple of decades.
Pythagoras divided the heavens into the sublunar sphere or Uranos, the outer,
perfect sphere where the gods lived (Olympos), and the sphere of moving bodies (the
stars) which lay between the other two—the Cosmos.
Greeks believed that the various heavenly bodies were layered onto a nested set of crystal
spheres that moved around one another. The Pythagoreans figured those spheres, like
music, must be based on whole numbers and must be in harmony. It was the idea of the
“music of the spheres.” Perhaps the earliest definable case of a logical fallacy in which a
discovery which is true in one area and is beautiful is projected onto another because
logic says anything that beautiful must apply everywhere. We will see that again….and
again.
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It was obvious to the Greeks that the moon revolved around the earth. By extension, it
was only reasonable to assume that all the other heavenly bodies did so as well. Also,
since the circle was (they believed) the most perfect of shapes (no corners, no end,
perfectly symmetrical), it was only reasonable to assume that the gods would make the
heavenly bodies rotate in circles. Thus the crystal spheres. That would remain an article
of faith (literally) in the West for the next 1400 years—until a Polish astronomer would
have the temerity to challenge it.
He probably thought the earth was a sphere mostly because the sphere was the perfect
shape than from observation. He was the first to recognize that the evening star and the
morning star were both Venus.
Pythagoras and his followers looked at the earth as well and had an active interest in
geography. They understood that the earth was round and rotated on an axis and believed
that the planets did as well. The also believed that the planets orbited a single point.
Initially they thought that was the earth but could not make the observations fit that, so
they hypothesized a “central fire” for them to orbit but did not believe that was the Sun.
Christians (and the Jews that preceded them) were committed numerologists. God used
six days to create the world because six is the product of 1x2x3. Seven is magical and
there are seven heavenly spheres because 7 is the sum of 3 (the Trinity) and 4 (the
number of elements—earth, air, fire, water). Also 3x4 equals the number of apostles.
The primacy of numbers had to do with the fact that the Greeks had no real ability to do
experiment because they lacked the tools. No fancy lab glassware, no accurate balance,
no microscope, and no telescope so they were restricted to what came from their minds.
The following is from the Timaeus and commentary by Benjamin Jowett:
Plato was heavily influenced by Pythagoras and believed that geometry was the essence
of reality. Over his door was printed “Let no one destitute of geometry enter my doors.”
Like the Pythagoreans, Platonists believed that numbers were the only way to get close to
god. Like Pythagoras, Plato believed the soul had a place in the material world. In fact,
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the Platonic ideal state in his Republic, bears striking resemblance to the Pythagorean
community at Croton.
Parenthetically, both the Rosicrucians and the Freemasons consider themselves direct
descendants of the Pythagoreans.
It is not surprising that the Greeks concentrated on geometry. Their system of
numbers used letters to represent numbers (as in Latin) and they did not have a place
holder system so any arithmetic beyond simple addition and subtraction was virtually
impossible. Also, they had no concept of using abstractions for numbers—no algebra.
The Ionians concentrated on defining a material structure for the world and the
Pythagoreans on mathematical and geometric relationships in nature but that left of the
whole matter of change. Movement, growth, decay, and mental activity have no material
representation . Heraclitus of Ephesus (c. 550BC-475 BC) believed that all nature was in
a constant state of flux. The paradigmatic statement proven that reality was in constant
flux was that you could never stand in the same river twice.
Parmenides of Elea (~480 BC) took the exact opposite point of view contending
that all change was just an illusion. In fact, Parmenides took the point of view that any
“knowledge” obtained from the senses was inherently flawed and not to be trusted as a
representation of reality. Is basic argument was that, if change were to occur, at some
point something would have to come from nothing and go back to nothing and that was
logically impossible. Zeno’s paradox, which purported to show that movement was
impossible since you could never reach a destination when each step of half way was
followed by another of half way, was archetypically Pythagorean. Only logic and not
sensation could get you to the actual truth.
Archimedes in his book The Sand Reckoner wrote:
“You (King Gelon) are aware the 'universe' is the name given by most astronomers to the
sphere the center of which is the center of the Earth, while its radius is equal to the
straight line between the center of the Sun and the center of the Earth. This is the
common account as you have heard from astronomers. But Aristarchus has brought out a
book consisting of certain hypotheses, wherein it appears, as a consequence of the
assumptions made, that the universe is many times greater than the 'universe' just
mentioned. His hypotheses are that the fixed stars and the Sun remain unmoved, that the
Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle
of the Floor, and that the sphere of the fixed stars, situated about the same center as the
Sun, is so great that the circle in which he supposes the Earth to revolve bears such a
proportion to the distance of the fixed stars as the center of the sphere bears to its
surface.”
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He also recognized that the lack of parallax meant that the stars were very far away. The
contemporary response was that he should be tried for impiety.
But what about the whole “knowledge” question? Actually, that is at the heart of Plato’s
wrestling with philosophy.
Socrates (470BC-399BC) had no interest in looking at the world as a series of mechanical
and predictable events. The only sentence in all the writings of the physicists that he
found appealing was the opening of Anaxagoras’s Nature of Things which said, “In the
beginning everything was in confusion, the Mind came and reduced them to order.” Of
course Anaxagoras never said another word about the concept of mind, but that is what
captured Socrates and he set out looking for evidence of use in how things were—for the
teleological rather than the mechanistic. And his change in emphasis would last to
Newton. Of course, the paradox in that is that the Socratic method of breaking a complex
problem into manageable parts is the exact description of what we call the Scientific
Method. A series of hypotheses are proposed, tested, and discarded until one that stands
up best to questioning remains. Nonetheless, Socrates thought any experimentation was
fruitless since by the very nature of designing an experiment, the observer had to
manipulate nature and then you were studying a manipulation and not nature itself. This,
of course, will resurface with Heisenberg and quantum physics.
The true philosopher will not waste his time on observations based on the senses
but will instead concentrate on using logic and his brain to seek out the true essences of
things that are intrinsic to consciousness. Pursuit of knowledge of the ideal is the goal of
the philosopher.
For Socrates, perception was variable from person to person and of no quantitative
usefulness. What was useful was the mind or the soul behind it. And knowledge came
not from observation but from divine inspiration. The one science that was interesting to
Socrates was mathematics since mathematics was entirely an intellectual and not a
perceptual exercise.
Socrates’ relation to writing is interesting. Egyptian hieroglyphics (the word
comes from Greek for sacred carvings) had been present since something like 4000 BC
but fully syllabic writing only dated to 1500 BC and alphabetic writing to only about 500
BC so it was quite new when Socrates lived and he deeply mistrusted it contending that
writing would make memory unnecessary and memory skill would die out because of
writing. On the other hand, anything written down can be carefully examined by a
number of people and can be challenged. It is not like having what you said handed down
by Plato as stories. That may be another reason Socrates was not a fan. As we have
previously said, writing probably fundamentally changed how brains work. Of course,
writing also made it possible to accumulate information over time. The great example of
that in ancient times was the accumulation of astronomical observations that allowed
models of the universe to be created.
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Since most of what we know of Socratic philosophy comes directly from Plato, it is quite
difficult to differentiate the two.
Plato (427BC-347BC) was born either in Athens or Aegina and it is rumored that his
family, on his mother’s side, was descended from the poet and legislator Solon and to
two of the Thirty Tyrants who ruled Athens after the First Peloponnesian War. He may
have been related to the kings of Athens and Messina on his father’s side. His actual
name was Aristocles (for his grandfather) but it is said that his wrestling coach gave him
the nickname Platon (broad) because of the size of his fingers. It is also said that he got
the nickname from the size of his forehead or, more of a compliment, from the breadth of
his knowledge. At any rate, his family could afford to give him an excellent education
and he was said to have been a good enough athlete to wrestle in the Isthmian Games.
After traveling (perhaps as far as Egypt and Sicily) he came back to Athens at age 40 and
opened a school in an olive grove that had once been owned by Academus. The school
would last until it was closed as pagan by Justinian I of Byzantium in 529—although
probably not continuously. It was probably closed a couple of hundred years after Plato’s
death and re-opened around 300 AD.
Plato is best known to us as a philosopher and teacher (from his Academy) but he was
also a creditable mathematician and his view of science would remain influential for over
a millennium. His Academy (named for the owner of the olive grove where the
peripatetic lectures were held) functioned although probably intermittently until 529 AD
when it was closed by Justinian I of Byzantium.
Although the words are put in Socrates’ mouth, they have been given the name
Platonism. "Platonism" is a term coined by scholars to refer to the intellectual
consequences of denying, as Socrates often does, the reality of the material world. In
several dialogues, most notably the Republic, Socrates inverts the common man's
intuition about what is knowable and what is real. While most people take the objects of
their senses to be real if anything is, Socrates is contemptuous of people who think that
something has to be graspable in the hands to be real. In the Theaetetus, he says such
people are "eu a-mousoi", an expression that means literally, "happily without the muses"
(Theaetetus 156a). In other words, such people live without the divine inspiration that
gives him, and people like him, access to higher insights about reality. Socrates’ idea that
reality is unavailable to those who use their senses is what puts him at odds with the
common man, and with common sense. Socrates says that he who sees with his eyes is
blind, and this idea is most famously captured in his allegory of the cave, and more
explicitly in his description of the divided line. The allegory of the cave (begins Republic
7.514a) is a paradoxical analogy wherein Socrates argues that the invisible world is the
most intelligible ("noeton") and that the visible world ("(h)oraton") is the least knowable,
and the most obscure.
According to Socrates, physical objects and physical events are "shadows" of their ideal
or perfect forms, and exist only to the extent that they instantiate the perfect versions of
themselves. Just as shadows are temporary, inconsequential epiphenomena produced by
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physical objects, physical objects are themselves fleeting phenomena caused by more
substantial causes, the ideals of which they are mere instances. More explicitly, Plato
himself argues in the Timaeus that knowledge is always proportionate to the realm from
which it is gained. In other words, if one derives one's account of something
experientially, because the world of sense is in flux, the views therein attained will be
mere opinions. And opinions are characterized by a lack of necessity and stability. On the
other hand, if one derives one's account of something by way of the non-sensible forms,
because these forms are unchanging, so too is the account derived from them. It is only in
this sense that Plato uses the term "knowledge.”
This coincides with the allegory of the divided line describing the levels of knowledge
which also comes from the Republic.
The idea is that a perfect circle or a perfect triangle does not exist in nature and, since
mathematics can conceive of it with no experience, that conception must be inborn.
There must be a perfect knowledge in the soul that is polluted by the senses and
experience and that will only emerge in an afterlife when the purified soul is all that
remains. That is the idea that persisted and led to the persecution of Galileo.
Plato's influence has been especially strong in mathematics and the sciences. He helped
to distinguish between pure and applied mathematics by widening the gap between
"arithmetic", now called Number Theory and "logistic", now called arithmetic. He
regarded logistic as appropriate for business men and men of war who "must learn the art
of numbers or he will not know how to array his troops," while arithmetic was
appropriate for philosophers "because he has to arise out of the sea of change and lay
hold of true being." Plato's resurgence further inspired some of the greatest advances in
logic since Aristotle, primarily through Gottlob Frege and his followers Kurt Gödel,
Alonzo Church, and Alfred Tarski; the last of these summarized his approach by
reversing the customary paraphrase of Aristotle's famous declaration of sedition from the
Academy (Nicomachean Ethics 1096a15), from Amicus Plato sed magis amica veritas
("Plato is a friend, but truth is a greater friend") to Inimicus Plato sed magis inimica
falsitas ("Plato is an enemy, but falsehood is a greater enemy"). Albert Einstein drew on
Plato's understanding of an immutable reality that underlies the flux of appearances for
his objections to the probabilistic picture of the physical universe propounded by Niels
Bohr in his interpretation of quantum mechanics. Of course, Einstein would like Plato
since he thought that the only worthwhile experiments were ones carried out in the mind.
The duality is still present. Classical scientists who believe that the physical universe has
its own existence and that life emerged from it and the Platonists or mystics who believe
that the universe is only the interpretation of experience by living things and that life
comes first and inanimate nature after. Well, actually, the second is not unlike modern
physics (Schrödinger’s Cat) which holds that nothing exists until it is observed.
Schrödinger and Einstein had exchanged letters about Einstein's EPR article, in the
course of which Einstein had pointed out that the quantum superposition of an unstable
keg of gunpowder will, after a while, contain both exploded and unexploded components.
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To further illustrate the putative incompleteness of quantum mechanics, Schrödinger
applied quantum mechanics to a living entity that may or may not be conscious. In
Schrödinger’s original thought experiment, he describes how one could, in principle,
transform a superposition inside an atom to a large-scale superposition of a live and dead
cat by coupling cat and atom with the help of a "diabolical mechanism". He proposed a
scenario with a cat in a sealed box, wherein the cat's life or death was dependent on the
state of a subatomic particle. According to Schrödinger, the Copenhagen interpretation
implies that the cat remains both alive and dead (to the universe outside the box) until
the box is opened.
For Plato, true knowledge had to be universal, certain, and necessary. The problem with
any knowledge from experience is that it is by definition particular—that is unique to the
observer and subject to the vagaries of perception. But he also believed that there were
universal truths that were independent of experience. In these truths was universality and
certainty. In The Sophist, he argues vigorously against the group that held that there was
no certain knowledge, only more or less probable beliefs and opinions.
The Sophists came to Athens around the middle of the 5th century BC and brought
a fundamental change to Greek education. It had previously been oriented to athletics
and the arts and was really only elementary. The sophists were itinerant and offered
more advanced education on a contract basis. They concentrated on training young
Athenians to be better citizens and more politically aware and effective.
Plato was not entirely without the supernatural. He viewed the cosmos as the
work of the Demiurge, a benevolent craftsman who built the universe but who was
restricted in what he had to work with so that his imitations of the perfect form of each
object was not quite perfect and the cosmos as well was not perfect. The Demiurge was
also a mathematician and his creation was built on sound principles of geometry.
Also note that his thinking is deep but his actual ability to arrive at an approximate
description of nature that is useful was very limited by the fact that he had no useful tools
to measure nature and mathematical ability which was essentially limited to geometry.
The elements are a good example and are typical for being constructed out of his
imagination rather than out of observation or experiment.
Here are a few things to think about that came from this intellectual approach to science.
First, what constitutes a good scientific theory? We said earlier that it allows one to
explain, predict and control nature. Let me add a couple of other things to that. First, it
needs to be elegant. That is to say, it needs to be impressive in its simplicity and its
“beauty.” Think how impressive it is to say that the same rules that make a musical
chord pleasing also govern the movement of the stars and planets. Second, the theory
must have as few “fudge factors” as possible. Modern physics has a paucity of
unexplainable “constants” without which the universe would not work (c, λ, Avogadro’s
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number, Einstein’s cosmological constant). When the number of fudge factors starts to
grow, it is probably time to look for another theory.
Finally, when one does come up with a series of “laws of nature” that make the universe
run (and that even God has to follow if you are Newtonian or after), you need to ask three
questions: 1) Where did the laws come from 2) Can the laws ever change, and 3) Are
there other possible laws that would explain nature equally well (cf Ptolemy and
Copernicus).
Where Plato was deductive and studied nature from inside his mind, his student Aristotle
was almost an encyclopedist. He set out to collect as many observations about nature as
possible and to categorize what he collected. His primary approach was a teleological
one—everything in nature had a will and a desire and acted to fulfill that desire. Rocks
fell because they wanted to be closer to the center of the earth. All things did what they
had been designed to want to do. This would ultimately fit very well with medieval
Christian orthodoxy.
Aristotle, unlike Plato, collected evidence and organized it and tried to define specific
patterns and explanations—induction.
Aristotle (384BC-322BC) was Plato’s student and Alexander the Great’s teacher. His
influence would last almost 2000 years until he was finally supplanted by Newton in
physics and Linnaeus and Darwin in biology. It is likely that we still have only about 1/3
of what he wrote in his lifetime.
He was born a bit east of modern day Thessaloniki in Chalcidice. His father was
the personal physician to the king of Macedonia and he was raised as an aristocrat in a
relatively wealthy family. He spent 20 years studying at Plato’s Academy before the
latter’s death and his being supplanted as successor by Plato’s nephew Speussipus. He
ultimately came back to Athens and established the Lyceum. Besides anatomy,
embryology, botany, zoology, astronomy, geography, and physics, he wrote on ethics,
economics, and politics. Basically, he created a coherent system that attempted to
organize and explain all of nature and would last until the edge of the Renaissance.
He was also the source of a series of perpetuated errors: the earth is the center of the
universe, heavy objects fall faster than light ones, and males have more teeth than
females. He believed there were five elements: earth, air, fire, water, and aether and that
the universe was a series of 55 concentric crystal spheres.
Aristotle differed from Plato in that he valued observation and took the point of
view that the senses might be imperfect but they were all we had. Intelligence could only
be productively applied to what we knew or at least observed of the world around us.
Aristotle’s system attempted to do two things: 1) catalogue and organize all natural
objects and 2) verify knowledge using proof of truth that would stand up in a litigious
society. The only extra-natural component of Aristotle’s system was the Unmoved
Mover since all motion had to come from something and there had to be an initial
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something. Later Christianity was quite happy to identify the Unmoved Mover as God
and to accept most of Aristotle’s system branching out from that.
The system had four qualities: hot, cold, wet, dry. These came in combinations
and acted on the four elements—earth, air, fire, water—that made up an object. The
elements were the acted on by the qualities. These are the material and formal “causes”
of nature. In general, you cannot separate the material from the formal. Then there is the
efficient cause or the agency which created the object. The final cause was the purpose
of the object. This is the tough one and is entirely teleological. Birds were created as
they were because they were supposed to fly and act like birds. Fish had fins and scales
and lived without breathing because they were created to be fish. Teleology is ultimately
circular reasoning, but it fit well with Aristotle’s lack of better knowledge and really fit
Church doctrine that held that everything was exactly as it had been originally created.
Another example would be a block of marble (the matter) shaped into a statue (the form)
by a sculptor (the efficient cause) for the beautification of Athens (the final cause).
Aristotelian motion was a function of what the object was made of. Each object
moved to a center defined by its components. A stone, being mostly earth, tried to seek
its natural position at the center of the earth and always fell in that direction unless
diverted. Fire sought the Sun and rose. Trees were composite. The roots sought the
center of the earth and the leaves sought the sky.
Motion was caused by a force and would gradually diminish if the force was not
constantly applied. Note that this is the exact reverse of Newton’s first law of motion. It
also required that there be a prime mover to get things started. So there was a natural
motion which was the objects tendency to go toward its perfect location (a stone to the
center of the earth) and an imposed of “violent” motion that deviates the object from
where it would naturally go. The violent motion will dissipate if not constantly applied.
Aristotelian biology is categorical. He divided animals into two major groups:
blooded (i.e. red blooded) and bloodless. The blooded are further divided into oviparous
and viviparous quadrupeds, marine mammals, birds, and fish which are in turn divided
into molluscs (such as octopus), crustacean, testacea (snails and oysters), and insects.
These were in turn put in a hierarchy based on what Aristotle assumed to be their vital
heat.
Life forms are also hierarchically categorized based on the type of soul they
possess. Plants have a nutritive soul that allows them to seek nourishment, grow, and
reproduce. Animals have, in addition to the nutritive, a sensitive soul that allows them to
feel and move. Humans are the only life forms with a rational soul that allows them to
reason.
Aristotelian logic was based on the syllogism: 1) a major premise derived from
either an axiom or some previously proven statement. “All men are mortal.” 2) A minor
premise which is the condition being investigated. “Socrates is a man.” 3) A conclusion
that inevitably results from the major and minor premise. “Therefore Socrates is mortal.”
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The errors lie in the premises. For instance, “All dogs have three legs. Lassie has four
legs. Therefore Lassie is not a dog.” The axioms have to be true and Aristotelians spent
a great deal of time proving the truth of their axioms. Unfortunately, the Greeks had
neither the inclination nor the ability to do much experimentation so the only way to
arrive at truth of axioms was reason and thought experiments.
There were two other Athenian schools at roughly the same time. The followers
of Epicurus (342-271 BC) adopted Democritus’s idea that everything was composed of
innumerable (but not infinite) numbers of indivisible and indestructible particles that
combined and recombined in various ways. The Epicureans met in a garden just outside
the walls of the Academy and took a rigid view that only the senses could define reality.
Because of their emphasis on the senses, they got a reputation for being both “sensual”
hence Epicurean and were viewed as atheistic by Jews, Moslems, and Christians since
they excluded any supernatural influences.
The other group met near a set of painted columns (the Stoa poikile) in a corner of
Athens’ agora and was called the Stoics. The school was founded in 312 by Zeno of
Citium.
The Lyceum lasted until about 86 BC when Athens was sacked by Sulla and the
Romans but Marcus Aurelius endowed chairs of study for each of the four schools.
After Alexander the Great and Aristotle died, the center of Greek intellectual
activity moved to Alexandria which was ruled by one of Alexander’s generals—the
empire had been divided among several of them. Ptolemy I began the great library and
the temple of the Muses (museum). Euclid (325-265 BC) worked at the Museum as did
Claudius Ptolemy.
As we said, Euclid lived in Alexandria in about 300 BCE during the reign of
Ptolemy I. Virtually nothing is known of his life and the earliest written references to him
were 800 years after his death in spite of the fact that Elements was the second most
reprinted book after the Bible until very recently and was the main text for teaching
mathematics until the early 20th century. Over 1,000 editions were printed after 1482.
It is built entirely on deductive reasoning. Take a set of initial definitions and
assumptions. If the assumptions are true, then a deduction that must be true comes from
them. The truth of the deduction is universal, necessary, and certain (provided the
assumptions are correct). This would be Plato’s favorite method of thought.
His proofs were clear, concise, and built one on the other. The Elements begins with
definitions and five postulates. The first three postulates are postulates of construction,
for example the first postulate states that it is possible to draw a straight line between any
two points. These postulates also implicitly assume the existence of points, lines and
circles and then the existence of other geometric objects are deduced from the fact that
these exist. There are other assumptions in the postulates which are not explicit. For
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example it is assumed that there is a unique line joining any two points. Similarly
postulates two and three, on producing straight lines and drawing circles, respectively,
assume the uniqueness of the objects the possibility of whose construction is being
postulated.
The fourth and fifth postulates are of a different nature. Postulate four states that all right
angles are equal. This may seem "obvious" but it actually assumes that space is
homogeneous - by this we mean that a figure will be independent of the position in space
in which it is placed.
The famous fifth, or parallel, postulate states that one and only one line can be drawn
through a point parallel to a given line. Euclid's decision to make this a postulate led to
Euclidean geometry. This one has always been problematic and there have been repeated
attempts to use the other four postulates to derive it so it would not have to be an
assumption. It was not until the 19th century that this postulate was dropped and nonEuclidean geometries were studied.
There are also axioms which Euclid calls 'common notions'. These are not specific
geometrical properties but rather general assumptions which allow mathematics to
proceed as a deductive science. For example:Things which are equal to the same thing are equal to each other.
Zeno of Sidon, about 250 years after Euclid wrote the Elements, seems to have been the
first to show that Euclid's propositions were not deduced from the postulates and axioms
alone, and Euclid does make other subtle assumptions.
The Elements is divided into 13 books. Books one to six deal with plane geometry. In
particular books one and two set out basic properties of triangles, parallels,
parallelograms, rectangles and squares. Book three studies properties of the circle while
book four deals with problems about circles and are thought largely to set out work of the
followers of Pythagoras. Book five lays out the work of Eudoxus on proportion applied to
commensurable and incommensurable magnitudes. Heath says [9]:Greek mathematics can boast no finer discovery than this theory, which put on a sound
footing so much of geometry as depended on the use of proportion.
Book six looks at applications of the results of book five to plane geometry.
Books seven to nine deal with number theory. In particular book seven is a self-contained
introduction to number theory and contains the Euclidean algorithm for finding the
greatest common divisor of two numbers. Book eight looks at numbers in geometrical
progression but van der Waerden writes in [2] that it contains:-
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... cumbersome enunciations, needless repetitions, and even logical fallacies. Apparently
Euclid's exposition excelled only in those parts in which he had excellent sources at his
disposal.
Book ten deals with the theory of irrational numbers and is mainly the work of
Theaetetus. Euclid changed the proofs of several theorems in this book so that they fitted
the new definition of proportion given by Eudoxus.
Books eleven to thirteen deal with three-dimensional geometry. In book eleven the basic
definitions needed for the three books together are given. The theorems then follow a
fairly similar pattern to the two-dimensional analogues previously given in books one and
four. The main results of book twelve are that circles are to one another as the squares of
their diameters and that spheres are to each other as the cubes of their diameters. These
results are certainly due to Eudoxus. Euclid proves these theorems using the "method of
exhaustion" as invented by Eudoxus. The Elements ends with book thirteen which
discusses the properties of the five regular polyhedra and gives a proof that there are
precisely five. This book appears to be based largely on an earlier treatise by Theaetetus.
It is likely that Claudius Ptolemy was heavily influenced by Euclid. Ptolemy was an
Alexandrine of either Greek or Egyptian parentage. His Egypt was under Roman rule
and he lived roughly from 90-168 AD and spent his working life in Alexandria. His great
works included the Almagest, his Geography, and the astrological test known as the
Tetrabiblos in which he tried to unite Aristotelian philosophy to astrology.
The fact that Ptolemy also carried the Roman Claudius proves his Roman citizenship.
Ptolemy was a common name among the Macedonian upper classes. So his name was
half Greek and half Roman suggesting that he was a Roman citizen of Greek ancestry
living in Egypt.
His Almagest (The Greatest Compilation or al majisti in Arabic) is the only surviving
comprehensive ancient astronomy text and would remain the accepted authority until
Copernicus’s De Revolutionibus in 1543. It shares the distinction of being the scientific
text in the longest use with Elements. It is worth noting that Ptolemy never contended
that his system was the actual way things worked. He only held that it was a model that
fit observations quite well. Actually, it held up for 1,300 years and is still used in small
craft navigation. The same can hardly be said for any other scientific theory in the past
couple of centuries.
In it, he claimed that he had used astronomical observations dating back 800 years to
create models of motion for the heavenly bodies. From the observations, he created
tables that could be used to predict or postdict positions of the planets. The tables also
allowed accurate predictions of eclipses of the Sun and Moon.
In the Ptolemaic system of astronomy, the epicycle (literally: on the circle in Greek) was
a geometric model used to explain the variations in speed and direction of the apparent
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motion of the Moon, Sun, and planets. It was first proposed by Apollonius of Perga at the
end of the 3rd century BC and formalized by Ptolemy of Alexander in his 2nd-century
AD astronomical treatise the Almagest. In particular it explained the retrograde motion of
the five planets known at the time. Secondarily, it also explained changes in the apparent
distances of the planets from Earth.
In the Ptolemaic system, the planets are assumed to move in a small circle called an
epicycle, which in turn moves along a larger circle called a deferent. Both circles rotate
eastward and are roughly parallel to the plane of the Sun's orbit (ecliptic). Despite the fact
that the Ptolemaic system is considered geocentric, the planets' motion was not actually
centered on the Earth. Instead, the deferent was centered on a point halfway between the
Earth and another point called the equant. The epicycle, meanwhile, rotated and revolved
along the deferent with uniform motion. The rate at which the planet moved on the
epicycle was fixed such that the angle between the center of the epicycle and the planet
was the same as the angle between the earth and the sun.
There is also a star catalogue that he mostly stole from Hipparchus. It includes 48
constellations but they do not match the modern system and include only the sky visible
from the eastern Mediterranean. Still, it was considered definitive through the Middle
Ages. His star almanac predicted appearance and disappearance of stars over the course
of the year.
We have gotten the Great Treatise and the name Almagest because the originals were
destroyed and we have copies passed down through Arabic libraries. It was translated
back into Latin from Arabic in the 12th century (once in Sicily and once in Spain).
Like Pythagoras, Ptolemy thought of the universe as a set of nested spheres. He thought
the sphere containing the Sun was 1210 Earth radii away and that of the fixed stars
20,000 radii away.
The final five books of the Almagest discuss planetary theory. This must be Ptolemy's
greatest achievement in terms of an original contribution, since there does not appear to
have been any satisfactory theoretical model to explain the rather complicated motions of
the five planets before the Almagest. Ptolemy combined the epicycle and eccentric
methods to give his model for the motions of the planets. The path of a planet P therefore
consisted of circular motion on an epicycle, the centre C of the epicycle moving round a
circle whose centre was offset from the earth. Ptolemy's really clever innovation here was
to make the motion of C uniform not about the centre of the circle around which it
moves, but around a point called the equant which is symmetrically placed on the
opposite side of the centre from the earth.
He also compiled substantially all that was known of the Earth’s geography at the time.
He included work from earlier Romans and from the Persians. He divided the earth into
a grid with lines of latitude beginning at the equator. Rather than defining degrees of
latitude as parts of the arc of a circle, however, he defined the levels of latitude by the
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length of the year’s longest day at that latitude (12 hours at the equator and 24 hours at
the poles). His latitudes are typically off by an average of about 2 degrees even though
some of his contemporaries using star sightings were accurate within a minute.
He did use degrees to define longitude and made the most western place he knew
(probably the Cape Verde Islands) as 0. His world went 180 degrees from the Blessed
Isles to the middle of China and from the Shetlands in the north to the east coast of Africa
in the south. He knew that his knowledge only encompassed about a quarter of the actual
globe.
Our oldest versions of his maps date only to about 1300 and what we have of the text
may have been considerably altered since he wrote it.
He viewed his astrology as the second part of a study that began with the Almagest and
emphasized the heating, cooling, moistening, and drying effects of the planets on the
“sub-lunar sphere.”
Like Pythagoras, he also wrote on music theory and argued for basing musical intervals
on mathematical ratios. His other major work was Optics which only survives in an
incomplete Arabic translation translated back into Latin in about 1154. He understood
that reflection, refraction, and color were properties of light.
In fairness, we must remember that Ptolemy was not the only Greek with a theory of how
the heavens worked. There was (four hundred years earlier—310-230 BCE) another idea
and it came from the same small island off Asia Minor that gave us Pythagoras.
Aristarchus took the Pythagorean central fire and moved it to the Sun.
One of the cleverer Ptolemaic experiments was Eratosthenes of Cyrene’s (c. 273192 BC) estimate of the diameter of the Earth. He knew that, at the summer solstice, a
vertical rod in the Egyptian city of Syene (now Aswan) cast no shadow. That is because
the city is on the Tropic of Cancer. He had actually been told that a person leaning over a
deep well at noon would block out the reflection of the Sun. He used a rod (gnomon) at
Alexandria the same time and day and found that it cast a shadow 7°12’ south of the rod.
From that he reasoned that the distance from Alexandria to Syene must be 1/50 of the
diameter of the Earth (7°12’/360°). He knew that distance to be 5,000 stadia which
translated to 700 stadia per degree or a diameter of 252,000 stadia. If the Egyptian stade
was 600 Roman feet (about 11.5 inches), that would be a circumference of 24,000 miles
or an error from the actual value of about 1.6%. Columbus knew about the estimate but
opted not to believe it. Good thing.
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