(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.


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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.


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Thales of Miletus

Pythagoras of Samos

Parmenides of Elea

Heraclitus of Ephesus


624 - 546 BC

570 - 500 BC

540 - 480 BC

c.500 BC

c.460 - 370 BC



Eon (Being)

Pyr and Logos (Fire and Rule)







Anaximander of Miletus


Zeno of Elea



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


c.545 BC


mid 5th cent. BC

c.570 - 475 BC


Aer (Air)



Single God











Mitrodorus of Hios




c.500 - 428 BC





Nous (Mind)










Diogenes of Apollonia
































adapted from


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started with the commerce and trade of Phoenician merchants

handicapped by the lack of efficient numeric and algebraic notation

geometry = "measurement of land,"

first arose in surveying practices among the ancient Egyptians


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The early Greeks proved a general rule for finding all sets of numbers

a, b, and c

such that

a2 + b2 = c2

(For example, 32 + 42 = 52)

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 = (p2 - q2)/2 = (32 - 12)/2 = 4

c = (p2 + q2)/2 = (32 + 12)/2 = 5


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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

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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


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.


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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


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624 --- 546 BC






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610 --- 540 BC



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fl. c. 545 BC


when very attenuated, fire

when more condensed, wind and then cloud

and when still more condensed water and earth and stone


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570 --- 500 BC





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


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fl. c. 500 BC (born in Ephesus)



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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


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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.










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540 --- 480 BC



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fl. c. 450 BC



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


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                                                                                               Achilles and the Tortoise


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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


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Zeno's paradox says:

The Roman solution:

"Solvitur ambulando"

("solved by walking") Duh!


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The mathematics of infinity was not known to the Presocratic philosophers.

The mathematical solution:

A = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...

1/2 A = 1/4 + 1/8 + 1/16 + 1/32 + ...

                                     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

Paradox solved!!!


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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?


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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."


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In 1961 the logician Abraham Robinson (1918-74) 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.


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Back to our story ... Heraclitus says,

Parmenides says,

Empedocles realized that insisting on a single basic substance was the problem. No single basic substance could change into anything else and remain basic.


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c. 492 --- 433 BC (born in Acragas)



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c. 500 --- 428 BC



the first to realize the moon shines by reflected light from the "red-hot stone" (the sun)


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c. 460 --- 370 BC (born in Abdera)


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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."


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Is there a color "brown?"


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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.


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The limonene molecule (C10H16) exists in two forms that are mirror images of each other:


---  smells piney, turpentine smell   






--- pleasing orange scent                          




The scents are the mind's interpretation of molecules.


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At the beginning of the 21st 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 time-ordering:

"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.


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At the beginning of the 21st century, what is the fundamental component of the universe?

No, they were (and continue to be) manufactured by the nuclear reactions in the centers of stars.

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.


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Best Guess: Strings!


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Pythagoras would have been pleased













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