PROTAGORAS of ABDERA (c.480 – c.411 BCE)



‘Of all things the measure is Man, of the things that are, that they are, and of the things that are not, that they are not’
‘About the gods, I am not able to know whether they exist or do not exist, nor what they are like in form; for the factors preventing knowledge are many: the obscurity of the subject, and the shortness of human life’

A contemporary of Socrates
In 415 BCE he was forced to flee Athens because his works were condemned for impiety

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ZENO of ELEA (c.490 – after 445 BCE)

Fifth Century BCE – Greece

‘A number of paradoxes; some of which seemed to prove the impossibility of motion’



Zeno was a member of the Eleatic school of thought founded by Parmenides. The School held the belief that the underlying nature of the Universe was unvarying and immobile.

In support of Parmenides’ argument, Zeno’s puzzles appeared to show that change and motion were paradoxical and that everything is one – and changeless.

Zeno’s paradoxes were based on the false assumption that space & time are infinitely divisible: that is, the sum of an infinite number of numbers is always infinite. Though they were based on fallacies, the paradoxes remained unsolved for two millennia.

Without the concepts of zero, infinity and the idea of limits, Greek philosophy and mathematics were not equipped to solve the puzzles.

The race between Achilles and the tortoise is not made up of a number of tiny distances but it is continuous until the end.

In the seventeenth century CE the Scottish mathematician James Gregory showed that it is sometimes possible to add infinite terms together to get a finite result – but to do so the terms being added together must approach zero. (This is a necessary, but not sufficient, condition – if the terms go to zero too slowly, then the sum of the terms does not converge to a finite number).
Such a series of numbers is called the convergence series, which occurs when the difference between each number and the one following it becomes smaller throughout the sequence.

The numbers 1, 1/2, 1/4, 1/8, 1/16… are approaching zero as their limit.

When you add up the distance that Achilles runs, with the terms becoming smaller and smaller, each term becomes closer to zero; each term is a step along a journey where the destination is zero.

Since the Greeks rejected the number zero they could not understand that this journey could ever have an end. To them, the terms aren’t approaching anything; the destination does not exist. Instead the Greeks saw the terms as simply getting smaller and smaller.

Using modern definitions, we know that the terms have a limit. The journey has a destination. We now may ask how far away is that destination and how long it will take to get there?

We sum up the distances that Achilles runs:
1 + 1/2 + 1/4 + 1/8 + 1/16… + 1/2n + …
As each step that Achilles takes gets smaller and smaller, it gets closer and closer to zero and the sum of these steps gets closer and closer to 2. How do we know this?
Starting with 2, we subtract the terms of the sum, one by one:

2-1=1 ; 1-1/2=1/2 ; 1/2-1/4=1/4 ; 1/4-1/8=1/8 ; 1/8-1/16=1/16…

We already know that series has a limit of zero, thus as we subtract the terms from 2 we have nothing left and the limit of the sum 1 + 1/2 + 1/4 + 1/8 + … is 2

The source of the trouble is infinity – Zeno had taken continuous motion and divided it into an infinite number of steps, assuming that the race would continue for ever as the steps get smaller and smaller. The race would never finish in finite time, but with the concept of zero we find a key to solving the puzzle.

Of course, Achilles wins the race.
Achilles takes two steps in catching up with the tortoise, even though he does it in an infinite number of increments.

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EMPEDOCLES of AGRIGENTUM (Sicily) (c.494 – c.434 BCE)

‘The four roots of all things are: AIR, WATER, FIRE and EARTH’

Two forces exist – LOVE and STRIFE.

The view of Empedocles developed the monists’ ideas that all substances are derived from a single source, into the concept of objects consisting of different compositions of these four basic elements.
The materials of the natural world being wrought from different blends of the four elemental principles, brought about through the eternal conflict between Love and Strife; their waxing and waning applied to cause mixing when Love is dominant, or separation by Strife.
Empedocles argued that this was the cause of transformation not just of the elements but also of the lives of people and cultures.

Popular mythology described how Aphrodite fashioned the human eye out of the four elements, held together by Love. She kindled the fire of the eye at the hearth fire of the universe, so that it would act like a lantern, transmitting the fire of the eye out into the world and making sight possible.
Empedocles realised that there must be more to sight than this, and that the darkness of night is caused by the body of the Earth getting in the way of light from the sun.

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ANAXAGORAS (c.500 – c.428 BCE)

‘The notion of the indivisible particle’

Anaxagoras came from Ionia but settled in Athens, where he remained for thirty years and taught both Pericles and Euripides. Charged with impiety because of his theory that the Sun is a red-hot stone (such an explanation, denying the role of Helios the sun-god, was enough to warrant prosecution) he fled Athens before the trial and settled in Asia Minor.
What we know about Anaxagoras is based on references to him by later writers.

In the cosmology of Anaxagoras, the Universe began as a homogenous sea of identical basic particles. Nous gave this sea a stir, in the knowledge that in time the particles would so combine to arrange themselves such that everything would be as it is today.

Bust said to be of ANAXAGORAS ©


Picture of a document showing a seal with a likeness of Anaxagoras ©

Nous was a vital principle akin to the life force of vitalism – the nearest English words being ‘mind’ or ‘intellect’.
The range of the word ‘nous’ is vastly greater, however, as it refers to the combination of insight and intuition which permits the apprehension of the fundamental principles of the cosmos – the concept is closer to the oriental idea of ‘seeing’ than the occidental notion of intelligence founded upon EUCLIDEAN LOGIC.

At the same time, Nous could be the creative, motive intelligence behind the cosmos, almost indistinguishable from the Christian concept of the will of God.

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THALES of MILETUS (624 – 545 BCE)

bust of Thales


‘WATER is the basis of all matter’

Thales is remembered because he disregarded the classical, mythical understanding of life and the universe in favour of a more physical understanding. Thales and his followers worked with the conviction that there exist natural laws governing the behaviour of natural processes and that future events could be predicted by understanding these laws. One did not have to examine chicken entrails in order to understand the vagaries and whims of the gods who ultimately determined these events.
The basic principle that you can understand nature by studying these natural laws and what things are made of forms the foundation of future philosophy and science.

Born in Miletus on the Aegean coast, Thales engaged in disciplines as varied as engineering and statesmanship.
He had a belief that there must be a fundamental substance or building block from which everything else is made. His conclusion was that this was water. Water was essential for life, drink it and you will grow, take it away and everything dies. Water can exist in a solid form as ice or snow; a bowl of water left exposed will vanish into the air. Water, he thought, could readily be made very fine, in which case it becomes air; alternatively it could be compacted into a slime that becomes earth. Not only was everything made of this elemental substance, but also all life was supported on it, literally. The Earth, said Thales, floated on water.
There were two compelling strands of evidence. First, it couldn’t be supported by air because air was incapable of supporting anything, but water could hold up large objects like logs or ships. Secondly, you could observe the effects of the Earth floating on water in that on occasions it rocked suddenly – an earthquake. Quite obviously this was caused by the water’s movement.

Legend has it that Thales, while traveling in Egypt observed that the ratio of the length of a shadow and the height of an object is the same for all objects if measured at the same time of day, hence when his shadow was the same length as his height, then at the same moment the height of the pyramid would be equal to the length of its shadow.

Nothing remains of Thales’ original work, and all knowledge of him is derived from later philosophers. His way of thinking marks the beginning of a way of investigating the hidden nature of natural things by investigation and systematisation.





ANAXIMANDER of MILETUS (c.611 – c.547 BCE)

‘Apeiron is the basis of all matter’

mosaic of Anaximander


A pupil of THALES of MILETUS. As with Thales, little is known about Anaximander’s life and most of what we know comes from later Greeks, notably Aristotle and Theophrastus.
Anaximander conceptualised the Earth as suspended completely unsupported at the centre of the universe. It had been assumed by other thinkers that the Earth was a flat disc held in place by water, pillars or some other physical structure. Although without a notion of gravity, Anaximander supported his argument by supposing that the Earth, being at the centre of the universe, at equal distances from the extremes,

‘has no inclination to move up rather than down or sideways; and since it is impossible to move in opposite directions at the same time, it necessarily stays where it is’ – ( ARISTOTLE explaining Anaximander’s theory).

Moreover, because the Earth was suspended freely, it allowed Anaximander to propose the idea that the sun, moon and stars orbited in full circles around the Earth.

Anaximander proposed the idea of space or a universe with depth. Rather than the view of the Earth caged in a planetarium style ‘celestial vault’, he argued the celestial bodies (the sun, moon and stars) were different distances away from the Earth, with space or air between them.

Picture of head of statue said to depict ANAXIMENES ©


ANAXIMENES of MILETUS (c.585 – c.528 BCE)

‘AIR is the basis of all matter’



HERACLEITUS of EPHESUS (c.535 – c.475 BCE)

‘FIRE is the basis of all matter’

Thales’ pupil Anaximander avoided the issue of ‘prote hyle‘, or first matter by contending that all materials were composed of apeiron, the indefinite and unknowable first substance. Wanting to understand the transformations observed daily in the world, Anaximander believed that change came about through the agency of contending opposite qualities; hot and cold, and dry and moist.





These pre-Socratics survive only in enigmatic fragments.

Pittacus of Mitylene (c.650 – c.570 BCE)  ‘Know thine opportunity’

Solon of Athens (638 – 559 BCE)  ‘Know Thyself’

Thales of Miletus (c.642 – c.545 BCE) ‘Who hateth suretyship is sure’

Periander of Corinth (died 585 BCE)  ‘Nothing is impossible to industry’

Chilon of Sparta (sixth century BCE)  ‘Consider the end’

Bias of Prinene (sixth century BCE)  ‘Most men are bad’

Cleobus of Lindoe (sixth century BCE)  ‘The golden mean’ or ‘Avoid extremes’

In Athens of the fourth century BCE, empirical science held little appeal. Socrates had an aversion to natural philosophy.

PLATO celebrated mathematics but opposed any form of experimentation.
ARISTOTLE was more receptive to natural philosophy, but practiced little experimentation or even observation.