THE BEGINNINGS OF SCIENCE
IN EARLY GREECE
(c. 600  400 BC)
"Atlas Brings Heracles the Apples of the
Hesperides in the Presence of Athena,"
from the Temple of Zeus at Olympia,
c. 460 BC.
During this time, when gods embodied nature and interfered
in human lives, the journey from mythos to logos began.
WHAT DO WE KNOW OF THE PRESOCRATIC PHILOSOPHERS?
Not much ... not a single original work
has survived intact for us to study!
The only sources are
1. Fragments  a few quotations from Presocratic works that have survived in works written later.
2. Testimonia  comments in the writings of Plato and Aristotle on Presocratic ideas.
3. Doxography  summaries and (summaries)^{2} of Presocratic works.
Milesians 
Pythagoreans 
Eleatics 
Independent 
Atomists 



Physiologists 

Thales of Miletus 
Pythagoras of Samos 
Parmenides of Elea 
Heraclitus of Ephesus 
Democritus 
624  546 BC 
570  500 BC 
540  480 BC 
c.500 BC 
c.460  370 BC 
Water 
Number 
Eon (Being) 
Pyr and Logos (Fire and Rule) 
Atom 





Anaximander of Miletus 
Philolaus 
Zeno of Elea 
Empedocles 
Leucippus 
610  540 BC 
c.470  390 BC 
c.450 BC 
c.493  433 BC 
c.440 BC 
Apeiron (The Infinite) 


the 4 elements (Love and Strife) 






Anaximenes of Miletus 
Alcmaeon of Croton 
Melissus of Samos 
Xenophanes of Colophon 
Anaxarhos 
c.545 BC 

mid 5^{th} cent. BC 
c.570  475 BC 

Aer (Air) 


Single God 







Arhytas 

Anaxagoras 
Mitrodorus of Hios 



c.500  428 BC 




Nous (Mind) 







Hepassos 

Diogenes of Apollonia 







Hevritos 









Heketas 









Ekfantos 



adapted from http://www.forthnet.gr/presocratics/
EARLY GREEK MATHEMATICS
arithmetic: the study of "multitude," or discrete quantity
started with the commerce and trade of Phoenician merchants
handicapped by the lack of efficient numeric and algebraic notation
geometry: the study of "magnitude," or continuous quantity
geometry = "measurement of land,"
first arose in surveying practices among the ancient Egyptians
EARLY GREEK ARITHMETIC
The early Greeks proved a general rule for finding all sets of numbers
a, b, and c
such that
a^{2} + b^{2} = c^{2}
(For example, 3^{2} + 4^{2} = 5^{2})
Recipe: take any two different whole numbers p and q, both even or both odd (say p = 3, and q = 1)
a = pq = 3(1) = 3
b = (p^{2}  q^{2})/2 = (3^{2}  1^{2})/2 = 4
c = (p^{2} + q^{2})/2 = (3^{2} + 1^{2})/2 = 5
EARLY GREEK GEOMETRY
The "quadrature of the lune" was accomplished by Hippocrates of Chios (c. 440 B.C.)
Using only a compass (divider) and straightedge, Hippocrates determined how to construct a square whose area was equal to that of the lune
The early Greek philosophers sought natural, rather than supernatural, explanations for natural processes.
They believed the world is ordered and intelligible, rather than random and arbitrary. They asked the most fundamental questions:
What is the world made of?
Is the world changing or changeless and eternal?
Can something come from nothing?
How many things are there in the world?
For the concept of the universe, the whole of reality, the early Greek philosophers chose the word
"kosmos"
which derives from a word meaning
"to order" "to arrange" "to marshall"
But "kosmos" in everyday Greek also meant an adornment, as in our word "cosmetic."
The answers to their questions should not be merely ordered
and logical, but beautiful and elegant as well.
Listen to Werner Heisenberg, a founder of quantum mechanics:
"If nature leads us to mathematical forms of great simplicity and beauty, ... to forms that no one has previously encountered, we cannot help thinking that they are 'true,' that they reveal a genuine feature of nature ... You must have felt this too: the almost frightening simplicity and wholeness of the relationships which nature suddenly spreads out before us and for which none of us was in the least prepared."
 quoted in S. Chandrasekhar's Truth and Beauty
THALES OF MILETUS
624  546 BC
the first to ask, "What is the basic substance of the universe?"
water is the source or material principle (arche) of everything
earth is a flat disk floating on water
said to have predicted a solar eclipse of 584 BC (probably myth)
no difference between living and nonliving matter:
"All things are full of gods." (~ lifegerms?)
ANAXIMANDER OF MILETUS
610  540 BC
the first principle of all other things is the infinite "apeiron" (no single substance, indefinite)
from this, the heavens and the worlds in them arise
the earth is aloft, not supported by anything, because it is of equal distance from everything
earth is cylindrical in shape
ANAXIMENES OF MILETUS
fl. c. 545 BC
the infinite air is the source or material principle of everything
air differs in its substantial properties by its rarity and density
when very attenuated, fire
when more condensed, wind and then cloud
and when still more condensed water and earth and stone
the soul is like air in its nature
PYTHAGORAS OF SAMOS
570  500 BC
the Pythagoreans (~300) belonged to a semimystical scientific society (secret signs, ...)
believed the brain is the location of the soul, and that the soul cyclically migrates among living forms
forbade eating beans because they contain the souls of the dead
"All is number" (from Pythagoras's observations in music, mathematics and astronomy)
Each number has its own traits:
1: the number of reason
2: the first even or female number, the number of opinion
3: the first true male number, the number of harmony
4: the number of justice or retribution
5: marriage
6: creation
10: the perfect number, the number of the universe
discovered that vibrating strings produce harmonious
tones when the ratios of the lengths of the strings are whole numbers
HERACLITUS
fl. c. 500 BC (born in Ephesus)
all things are an exchange of fire
there are bowls in the heavens that catch fiery exhalations (stars)
change is the most important principle of the universe
"everything flows"
change generated by opposites, stability achieved by balance
universe eternal: no beginning, forever changing
Heraclitus trusted his senses, observing the world in a
state of neverending flux.
Socrates comments on Heraclitus:
"They say that Euripedes gave [Socrates] a copy of Heraclitus' book and asked him what he thought of it. He replied: 'What I understand is splendid; and I think that what I don't understand is too  but it would take a Delian diver to get to the bottom of it.'"
 Diogenes Laertius, Lives of the Philosophers
The Milesian philosophers and Heraclitus believed in a basic substance at the source of all things.
But ... how can one substance change into something else?
How many things are there in the world?
Enter the Eleatics (~ 500 BC) from the Greek colony of Elea in southern Italy.
PARMENIDES OF ELEA
540  480 BC
nothing can come out of nothing
everything in the world is eternal
rejected the possibility of actual change
our senses and observations of change are unreliable
we follow the Way of Opinion (illusion), not the Way of Truth
first to ask "Is it, or is it not?" of our
perception of the world
ZENO OF ELEA
fl. c. 450 BC
disciple of Parmenides (his "son")
sought to disprove change
believed that only one thing exists:
1. if any object is divisible into more than one thing, we can keep on dividing forever into an infinite number of smaller pieces
2. the sum of an infinite number of small (but not zero) pieces must be infinite in size  an absurdity!
3. therefore nothing can be divisible; only one thing exists
F
I
N
I
S
H
Achilles and the Tortoise
tortoise has a head start
by the time A reaches where T is, T has moved on ...
by the time A reaches where T is, T has moved on ...
this happens again and again and ...
so Achilles can never catch up to tortoise!
ZENO EXAMINED
Let's assume that Achilles runs 1 m/s, and the tortoise runs 1/2 m/s. If Achilles
has to run 1 meter, and the tortoise has to run 1/2 meter, we expect them to reach the
finish line at the same time. Follow Zeno's argument. I'll show Achilles as a blue vertical line that starts the race at 0, and the tortoise as a
green vertical line that starts the race at 1/2 meter.

  
  
 
  
  
 
0
1/2 m
1 m
A starts at 0, T starts at 1/2 meter
 
  
  
 
    
 
0
1/2 m
1 m
A travels 1/2 meter, T is 1/4 meter ahead
 
  
  
  
 
 
 
0
1/2 m
1 m
A travels 1/4 meter, T is 1/8 meter ahead
 
  
  
  
  
  
0
1/2 m
1 m
A travels 1/8 meter, T is 1/16 meter ahead
Zeno's paradox says:
Achilles travels an infinite number of steps, none of them of zero length
Therefore Achilles must travel an infinite distance to catch the tortoise: impossible!
Therefore motion (and change) is impossible!
The Roman solution:
"Solvitur ambulando"
("solved by walking") Duh!
The mathematics of infinity was not known to the Presocratic philosophers.
The mathematical solution:
the distance traveled by Achilles is
A = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...
multiply each side by 1/2:
1/2 A = 1/4 + 1/8 + 1/16 + 1/32 + ...
subtract A  1/2 A:
A = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +
...
 1/2 A = 1/4 + 1/8
+ 1/16 + 1/32 + ...
1/2 A = 1/2
multiply each side by 2 to get A = 1
thus Achilles has run A = 1 meter
Achilles does not have to run an infinite distance to catch the tortoise, just 1 meter!
Paradox solved!!!
Oh, really? Suppose Achilles takes one step each time, and starts on his right foot:
R
L
R L R..
   
  
  
  
  

0
1/2 m
1 m
Question: which foot is on the ground when Achilles has
traveled one meter and caught the tortoise?
If you can't answer this question, then what does this say about the mathematical solution to Zeno's paradox?
In Paradoxes, R. M. Sainsbury writes,
"Part of the puzzle here lies, I think, in the exact nature of the correspondence that we are setting up between mathematical series and physical space. ... In this case, it is not clear how we are to answer the question 'To what physical length does this series of mathematical points correspond?' ... The upshot is that a full response to Zeno's ... paradox would require a detailed elaboration and justification of our spatial concepts. This is the task Zeno set us  a task that each generation of philosophers of space and time rightly feels it must undertake anew."
ZENO QUESTIONS
Are space and time infinitely divisible? Or is there a smallest nonzero unit of space? Of time?
Mathematically, is there a smallest nonzero number (an infinitesimal)? Yes and no!)
In 1961 the logician Abraham Robinson (191874) showed that the notion of an infinitesimal was in fact logically consistent (but not required) and that, therefore, infinitesimals could be introduced as new kinds of numbers. This led to a novel way of presenting the calculus, called nonstandard analysis.
Back to our story ... Heraclitus says,
"everything flows" (changes)
our sensory perceptions are reliable
Parmenides says,
nothing changes
our sensory perceptions are unreliable
Empedocles realized that
insisting on a single basic substance was the problem. No single basic substance could
change into anything else and remain basic.
EMPEDOCLES
c. 492  433 BC (born in Acragas)
four primary and irreducible elements: earth, air, fire, and water
physical world is made of these unchanging eternal elements in various changing combinations
combinations of the four elements are determined by the interaction of Love (attraction) and Strife (repulsion) >>> chemistry!
today we call them positive and negative electric charge (unlike charges attract, like charges repel)
ANAXAGORAS OF CLAZOMENAE
c. 500  428 BC
first philosopher of classical Athens
the creative and driving force of the cosmos is Mind (nous)
the first to realize the moon shines by reflected light from the "redhot stone" (the sun)
the first to explain correctly the reason for eclipses of
the sun (the moon between Earth and sun) and eclipses of the moon (Earth's shadow hitting
the moon)
DEMOCRITUS
c. 460  370 BC (born in Abdera)
believed that there must be indivisible bodies ("atoms")
infinite divisibility could lead to a universe of objects of zero magnitude, a paradox since nothing can come from nothing
an infinite variety of eternal atoms combines and dissolves according to their shape, order, and position (like Legos)
the soul (not immortal) is made of round, smooth atoms
Galen (b. 129 AD) wrote, "... the primary element is qualityless, having no natural whiteness or blackness or any other color whatever, and no sweetness or bitterness or heat or cold or in general any other quality whatever. For, says Democritus,
'By convention color, by convention bitter, by convention sweet: in reality atoms and void.'
And he thinks that it is from the congregation of atoms that all the perceptive qualities came to be  they are relative to us who perceive them, and in nature there is nothing white or black or yellow or red or bitter or sweet."
HOW NOW,
BROWN COW?
Is there a color "brown?"
<
In 1666 AD, Isaac Newton passed a ray of sunlight through a prism and split the white light into a rainbow spectrum of colors.
red, orange, yellow, green, blue, indigo, violet ...
All of these colors of light waves have their own unique wavelength.
But there is no brown color in the spectrum!
no such thing as the wavelength of brown
"brown" is our internal representation of the mixture of other colors
the color brown is our invention, and begins and ends within our mind
brown (and in fact all color) is the way our mind represents the physical property of the wavelength of a light wave
The limonene molecule (C_{10}H_{16}) exists in two forms that are mirror images of each other:
llimonene
 smells piney, turpentine smell
dlimonene
 pleasing orange scent
The scents are the mind's interpretation of molecules.
Democritus believed everything we perceive is due to the movement of atoms through the void
he understood that we have come to an accommodation with the exterior world of atoms
all of our experience is with our internal interpretation of our sensory perceptions
"In reality we know nothing  the truth is in the depths."
"The Truth is Out There" (Fox Mulder of The XFiles)
At the beginning of the 21^{st} century, can something come from nothing?
Is there such a thing as "nothing?"
The closest thing to "nothing" is the vacuum.
Today cosmologists are attempting to explain the creation of everything from nothing, from the vacuum.
The vacuum is not truly "nothing;" it is a state of minimum energy where quantum fluctuations, consistent with the Heisenberg's uncertainty principle, could have led to the formation of particles in the Big Bang that created both space and time.
"Creation" implies a timeordering:
"first it is not, then it is"
Is it meaningful to say that time was created?
When was "is not?"
Is time t = 0 part of time or not?
As cosmologists push the frontiers of science, they ask Zeno questions.
At the beginning of the 21^{st} century, what is the fundamental component of the universe?
atoms?
No, they were (and continue to be) manufactured by the nuclear reactions in the centers of stars.
protons, neutrons, and electrons?
No, protons and neutrons are made of quarks and held together by other particles. All in all, there are far too many particles for any of them to be fundamental.
Best Guess: Strings!
instead of many types of pointlike elementary particles, some physicists postulate that in nature there is a single variety of stringlike object
the string is not made up of anything; rather, it is basic and other things are made up of it
as with musical strings, this basic string can vibrate, and each vibrational mode can be viewed as a pointlike elementary particle, just as the harmonics of a musical string are perceived as distinct notes!
Pythagoras would have been pleased
Some references for this presentation:
Atchity, Kenneth J., The Classical Greek Reader, Oxford University Press, 1996
Barnes, Jonathan, Early Greek Philosophy, Penguin Classics, 1987
Chandrasekhar, S., Truth and Beauty : Aesthetics and Motivations in Science, U. Chicago Press, 1991
Dunham, William, Journey Through Genius : The Great Theorems of Mathematics, Penguin, 1991
Gaarder, Jostein, Sophie's World, Berkley Books, 1996
Hertherington, Morriss S., Cosmology, Garland Publishing, 1993
Lindberg, David C., The Beginnings of Western Science, U. Chicago Press, 1992
Sacks, Oliver, The Man Who Mistook His Wife for a Hat, Harper and Row, 1987
Sainsbury, R. M., Paradoxes, Cambridge U. Press, 1995
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