# THE PLATONIC SOLIDS

The properties of solid figures have kept mathematicians occupied for centuries. Regular polyhedra are formed from regular polygons such as squares or triangles and mathematicians have failed to find any more than five of them.

Although they were defined by Pythagoras two hundred years before Plato was born, they are known collectively as the platonic solids, named in honour of PLATO by the geometer Euclid.

“THE PLATONIC SOLIDS – The regular polyhedron is defined as a three-dimensional solid comprising regular polygons for its surfaces – and with all its surfaces, edges and vertices identical. The five regular polyhedra are the tetrahedron (four triangular faces), the cube (six square faces), the octahedron (eight triangular faces), the dodecahedron (twelve pentagonal faces) and the icosahedron (twenty triangular faces).”

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# PLATO (c.427 – c.347 BCE)

387 BCE – Athens

‘The safest general characterisation of the European philosophical tradition is that it consists of a series of footnotes to Plato’

So said the English mathematician and philosopher Alfred North Whitehead (1861-1947)

A pupil of Socrates, Plato was introduced to the notion of ‘reality’ being distorted by human perceptions, which became important in his approach to science and to metaphysics. Socrates taught a method of thinking that elucidated truth though a series of questions and answers. The Socratic method was to ask for definitions of familiar concepts like ‘justice’ and ‘courage’, and then probe the definition by asking a series of questions. His intention was to lead people to start to contradict themselves and in so doing uncover any weaknesses in the initial definition. Socratic dialogue is thus often better at revealing ignorance than producing answers. Socrates was convinced that the wisest people are those who are aware of how little they know.
Socrates fell foul of a newly elected democratic government and was put on trial for allegedly corrupting the youth of Athens with his rebellious ideas. He was sentenced to death and elected to drink hemlock rather than argue in favour of a fine or accepting the offer of help to escape. His rational for his obstinacy was that he believed that doing harm damaged one’s soul. As the soul survives death then it would be better to die.

## PHILOSOPHY

Plato determined to reveal a world of certainty that existed beyond the world of change and decay. The physical world we see is merely the world of ‘becoming’ – a poor copy of the ‘real’ world of the Forms which can only ever be grasped through thought.
After PYTHAGORAS and HERACLITUS, most Greek philosophers believed that knowledge had to be as stable and fixed as the certainties of mathematics, kept safe from Heraclitan change and from sceptical relativism.
Knowledge could only come through thought and although observation was useful, it was an inferior and misleading way of understanding the world and the place of human beings within it. Such a view helps to explain why it is that the ancient Greeks invented extremely sophisticated mathematics, astronomy and philosophy but little in the way of technology.

Plato produced nearly all the central questions for philosophy in epistemology, metaphysics, ethics, politics and aesthetics.
Central to Plato’s thinking is that people should seek virtue studying what he called the Good, a non-physical absolute concept that never changes. If you know Good, you will live well because your thoughts and desires will automatically be shaped by that knowledge.

During the decade of his travels after the execution of Socrates, Plato wrote his first group of ‘dialogues’ – which include the ‘Euthyphro‘, ‘Apology’, the ‘Crito’, ‘Phaedo‘ – concerning the trial and death of Socrates.
Further accounts of Socrates debates with friends on various subjects are found in other works; ‘Charmides’ (temperance); ‘Laches’ (courage); ‘Lysis’ (friendship); ‘Hippas Minor’, ‘Hippas Major’, ‘Gorgias’, ‘Ion‘, and ‘Protagoras’ (ethics and education). In his plays, Plato used Socrates as a character, bringing his mentor back from the grave and throwing light on his concepts. In Gorgias, Plato portrays Socrates confronting Polus, or the sophist Callicles, who holds that immoral acts can bring the greatest amount of pleasure, measuring actions in terms of their immediate material outcome. Socrates disagrees. Whatever the immediate pleasure, he says, immorality will damage the soul.

Opposing DEMOCRITUS, Plato believed that all substances are composed of one kind of matter, possessing the qualities of form and spirit. He accepted the Greek notion, first suggested by EMPEDOCLES in the fifth century BCE, that matter was made up of mixtures of the four elements – earth, water, air or fire. Because these four are only fundamental forms of the single type of matter, they cannot be related to any idea of ‘elements’ as understood by modern science – they could be transmuted into each other. Different substances, although composed of matter would have different properties due to the differing amounts of the qualities of form and spirit. Thus a lump of lead is made of the same type of matter (fundamental form) as a lump of gold, but has a different aggregation of constituents. Neither lead nor gold would contain much spirit – not as much as air, say, and certainly not as much as God, who is purely spiritual.

399 BCE on the execution of SOCRATES, Plato leaves Athens in disgust.

387 BCE Returns to Athens. Plato founds his academy (‘ Let no one enter here who is ignorant of geometry ‘) – a bastion of intellectual achievement until its closure on the orders of the emperor Justinian in CE 529.

Plato’s ‘Theory of Forms’ consisted of the argument that Nature, as seen through human eyes, is merely a flawed version of true ‘reality’, or ‘forms’.

Plato argued that everything we see and call beautiful in some way resembles the form of Beauty. Two people independently come to the conclusion that a person or an object is beautiful because they both recognise the form of Beauty. In a similar way, everything that we see as ‘Just’ resembles the form of ‘Justice’. Disputes about the rightness of actions then depend on how well the outcome will conform to the form of Good. For a person to act justly requires that while they seek the form of Good, they keep the three parts of their personality in balance. The person needs wisdom, which comes from reason; courage, which comes from the spirited part of man; and self-control, which rules the passions.

In ‘The Republic’ Plato expands the idea that if you educate a person so that he can see that a particular action is not good for them, then they will not perform that action. This knowledge will enable them to make good decisions and to rule wisely, hence the idea of a philosopher king who has mastered the discipline of ‘dialectic’ and studied the hierarchy of Forms. The society is organised into a rigid hierarchy of workers, soldiers and rulers who all know their relative positions and there is a communism of property and family. The rulers have totalitarian powers and a harmonious communal life can only be achieved at the expense of individual freedoms.
Plato’s educational syllabus in ‘The Republic’ is based on Spartan methods – selfless dedication to the welfare of the State is essential.

## SCIENCE

Plato encountered the Pythagoreans in Croton, who became a major influence. For Plato, there had always existed an eternal, underlying mathematical form and order to the universe, and what humans saw were merely imperfect glimpses of it, usually corrupted by their own irrational perceptions and prejudices about the way things ‘are’. Consequently, for Plato, the only valid approach to science was a rational mathematical one, which sought to establish universal truths irrespective of the human condition. This has strongly impacted on modern science; for example, arithmetic calculations suggesting that future discoveries would have particular properties has led to the naming of unknown elements in DMITRI MENDELEEV‘s first periodic table.

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# ARISTOTLE (c.384 – c.322 BCE)

335 BCE – Athens

ARISTOTLE

384 BCE – Born in the Greek colony of Stagira. The son of Nicomachus, court physician to the king of Macedonia
367 BCE – Enters Plato’s Academy in Athens
347 BCE – On Plato’s death Speusippus succeeds Plato as head of the Academy. Aristotle leaves the Academy for Lesbos
342 BCE – Becomes tutor to the young Alexander (the Great), son of Phillip of Macedon
335 BCE – Returns to Athens and founds the Lyceum
321 BCE – Accused of impiety, returns to Chalcis where he dies a year later

Aristotle reinforced the view espoused by PYTHAGORAS that the earth is spherical. The arc shaped shadow of the earth cast upon the moon during a lunar eclipse is consistent with this view. He also noted that when traveling north or south, stars ‘move’ on the horizon until some gradually disappear from view.

Proposing that there was no infinity and no void he accepted the notion of the earth at the centre of the universe, with the moon, planets, sun and stars all orbiting around it in perfect circles.
The universe existed as beautiful spheres surrounding the Earth, placed at the centre of the cosmos. This system was later refined by the Alexandrian astronomer Ptolemy and become the dominant philosophy in the Western world.

Explaining why the heavens rotate in perfect, uniform order, with none of the disturbances associated with earthly elements; he described the fifth element added to the traditional four, ‘Aether‘, as having a naturally circular motion. Everything beyond the moon was regulated by aether, explaining both its perfect movement and stability, while everything below it was subject to the laws of the four other elements.

Aristotle rejected the ideas of zero and infinity, hence he had explained away Zeno’s paradoxes – Achilles runs smoothly past the tortoise because the infinite points are simply a figment of Zeno’s imagination; infinity was just a construct of the human mind.
By rejecting zero and infinity, Aristotle denied the atomists’ idea of matter existing in an infinite vacuum, infinity and zero wrapped into one.
In contrast to the theory of atoms, like Plato, Aristotle believed that matter is composed of four elements ( Ignis, Aqua, Aer and Terra ) with differing qualities ( hot, wet, cold, dry )

[ Fire – hot + dry ; Water – cold + wet ; Air – hot + wet ; Earth – cold + dry ]

He believed that the qualities of heat, cold, wetness and dryness were the keys to transformation, each element being converted into another by changing one of these two qualities to its opposite.

Agreeing that things were composed of a single, primal substance (prote hyle) that was too remote and unknowable, he accepted EMPEDOCLES elements as intermediaries between the imponderable and the tangible world, concealing the complications behind a philosophy of matter.

The four elements always sought to return to their ‘natural place’. Thus a rock, for example, would drop to the earth as soon as any obstacles preventing it from doing so were removed – because ‘earth’ elements, being denser and heavier, would naturally seek to move downwards towards the centre of the planet. Water elements would float around the surface, air would rise above that and fire would seek to rise above them all, explaining the leaping, upward direction of flames.

Although the Aristotelian view of matter has been undermined as experiments proved that neither air nor water are indivisible; today, scientists define matter as existing in four phases, solid, liquid, gas and plasma.

### MATTER (hyle); FORM (morphe); CAUSE; PURPOSE;

The place where his ideas converge with Plato’s is that for Aristotle, the pinnacle of the tower of superiority is the Good. According to Aristotle, all aims eventually lead to the Good, not necessarily of the individual but of humankind. Humans by nature are social and moral and everyone is part of a group, a family, village, town or city-state. There is no place for individualism or freethinkers, as without the happiness of the group then the individual cannot be happy.
The consequence of this emphasis on the community as opposed to the individual is hierarchy and subordination and as a result slavery was a very normal part of a well-ordered society.

• Matter is itself only one component of the world – others being form and spirit. There are different sorts of living being in the world.
Human beings possess immortal souls.
He believed that there is in living creatures a fundamental vital principle, a ‘life force’, which distinguishes them from non-living material. The gods breathed this vital principle into living things, and thereby gave them their life – ( nous – spontaneous generation ).

The soul is governed by reason, spirit and appetite.
‘All human actions have one or more of these seven causes: chance, nature, compulsion, habit, reason, passion, and desire’ ( source )

• Forms are incorporated in individual particulars as potentiality.
All particular acorns possess the form of the potential oak tree.

Although Aristotle was a pupil at Plato’s Academy for almost twenty years, the two great thinkers were diametrically opposed on a number of subjects; he criticised Platonic forms for being impossibly transcendent and mystical.

Aristotle pursued his ideas unrestricted by Socratic theories that non-physical forms such as Truth and Beauty were the keys to understanding.

• Four Causes – efficient, formal, material, final – (agent, form, matter, goal). – The ‘Timaeus’ – ( Plato’s work in which the chief speaker is encouraged to provide his account of the origins of the universe.)

• ‘Action exists not in the agent but in the patient’
To study a situation, or an action, Aristotle would categorise it into a series of subordinate and superior aims.

### MOTION

• Motion of Place – A to B

• Motion of Quantity – change in amount

• Motion of Quality – green apples turning red or from sour to sweet

Aristotle could explain why a rock, when thrown, would travel upwards through the air first before heading downwards, rather than straight down towards the earth. This was because the air, seeking to close the gap made by the invasion of the rock, would propel it along until it lost its horizontal speed and it tumbled to the ground.

Such notions made a lasting impact for the next two thousand years, if only by slowing down progress due to their unchallenged acceptance.

Some of Aristotle’s biology was faulty, such as defining the heart, not the brain as the seat of the mind.

Aristotle’s model of ‘the hydrologic cycle’ is uncannily close to the ideas we have today. The Sun’s heat changes water into air ( as defined as ‘elements’ by EMPEDOCLES ). Heat rises, so the heat in this air pulls the air up to the skies ( modern explanations of the nature of heat give a fuller understanding of the mechanisms involved ). The heat then leaves the vapour, which thus becomes progressively more watery again, and this process is marked by the formation of a cloud. The positive feedback of the increased ‘wateriness’ of the mixture in the cloud driving away its opposite ( the ‘heat’ ) and causing the cloud to become colder and shrink results in restoration of the true wateriness of the water, which falls as rain or, if the cloud is now cold enough, as hail or snow.

Aristotle was one of the first to attempt a methodical classification of animals; in ‘Generation of Animals’ he used means of reproduction to differentiate between those animals which give birth to live young and those which lay eggs, a system which is the forerunner of modern taxonomy. He noted that dolphins give birth to live young who were attached to their mothers by umbilical cords and so he classified dolphins as mammals.

Based on the Pythagorean universe, the Aristotelian cosmos had the planets moving in crystalline orbs.
Since there is no infinity, there cannot be an endless number of spheres; there must be a last one. There was no such thing as ‘beyond’ the final sphere and the universe ended with the outermost layer.
With no infinite and no void, the universe was contained within the sphere of fixed stars. The cosmos was finite in extent and entirely filled with matter.
The consequence of this line of reasoning, accounting for Aristotle’s philosophy enduring for two millennia was that this system proved the existence of God.

The heavenly spheres are slowly spinning in their places, making a divine music that suffuses the cosmos. The stationary earth cannot be the cause of that motion, so the innermost sphere must be moved by the next sphere out, which, in its turn must be moved by by its larger neighbour, and on and on. With a finite number of spheres, something must be the ultimate cause of motion of the final sphere of fixed stars. This is the Prime Mover.
Christianity came to rely on Aristotle’s view of the universe and this proof of God’s existence.
Atomism became associated with atheism.

The ideas of Aristotle were picked up by the twelfth century Andalusian philosopher Abu al-Walid Muhammad ibn Ahmed ibn Rushd (AVERROES) and were later adopted by the medieval philosopher THOMAS AQUINAS in the thirteenth century; whose concept of Natural Law is the basis of much thinking in the Christian world.

Aristotle had greater influence on medieval scholastic thought than Plato, whose rediscovery in the Italian renaissance influenced Petrarch, Erasmus, Thomas More and other scholars to question the dogmas of scholasticism.

Aristotle’s work in physics and cosmology dominated Western thought until the time of GALILEO and NEWTON, when much of it was subsequently refuted, though his work still underpins both Christian and Islāmic philosophy. His importance lies as much in his analytical method as in the conclusions he reached.

Aristotle expanded Plato’s concept of ‘virtue’ by dividing virtues into two groups, the 12 ‘moral’ and 9 ‘intellectual’ virtues, believing that each lay between the non-virtuous extremes of excess and deficiency.

Deficiency Virtue Excess
Cowardice Courage Rashness
Licentiousness (disregarding convention, unrestrained) Temperance (restraint or moderation) Insensibility (indifference)
Illiberality (meanness) Liberality (generosity) Prodigality (wasteful, extravagant)
Pettiness Magnificence Vulgarity
Humble-mindedness High-mindedness Vanity
Lack of ambition Proper ambition Over ambition
Irascibility (easily angered) Patience Lack of spirit
Understatement Truthfulness Boastfulness
Boorishness Wittiness Buffoonery
Cantankerousness Friendliness Obsequiousness
Shamelessness Modesty Shyness
Malicious enjoyment Righteous indignation Envy/spitefulness

His intellectual virtues consisted of :

 art    scientific knowledge    prudence    intelligence    wisdom    resourcefulness    understanding    judgment    cleverness

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# CHRISTIAN THEOLOGY & WESTERN SCIENCE

The speculative Greek philosophers, considering the great overarching principles that controlled the Cosmos, were handicapped by a reluctance to test their speculations by experimentation.
At the other end of the spectrum were the craftsmen who fired and glazed pottery, who forged weapons out of bronze and iron. They in turn were hindered by their reluctance to speculate about the principles that governed their craft.

WESTERN SCIENCE is often credited with discoveries and inventions that have been observed in other cultures in earlier centuries.
This can be due to a lack of reliable records, difficulty in discerning fact from legend, problems in pinning down a finding to an individual or group or simple ignorance.

The Romans were technologists and made little contribution to pure science and then from the fall of Rome to the Renaissance science regressed. Through this time, science and technology evolved independently and to a large extent one could have science without technology and technology without science.

Later, there developed a movement to ‘Christianise Platonism’ (Thierry of Chartres).

Platonism at its simplest is the study and debate of the various arguments put forward by the Greek philosopher PLATO (428/7-348/7 BCE).
The philosopher Plotinus is attributed with having founded neo-Platonism, linking Christian and Gnostic beliefs to debate various arguments within their doctrines. One strand of thought linked together three intellectual states of being: the Good (or the One), the Intelligence and the Soul. The neo-Platonic Academy in Greece was closed by the Emperor Justinian (CE 483-565) in CE 529.
During the early years of the Renaissance, texts on neo-platonism began to be reconsidered, translated and discoursed.

Aristotle’s four causes from the ‘Timaeus’ were attributed to the Christian God, who works through secondary causes (such as angels).

Efficient Cause – Creator – God the Father

Formal Cause – Secondary agent – God the Son

Material Cause – The four elements: earth, air, fire & water.
Because these four are only fundamental forms of the single type of matter, they cannot be related to any idea of ‘elements’ as understood by modern science – they could be transmuted into each other. Different substances, although composed of matter would have different properties due to the differing amounts of the qualities of form and spirit. Thus a lump of lead is made of the same type of matter (fundamental form) as a lump of gold, but has a different aggregation of constituents. Neither lead nor gold would contain much spirit – not as much as air, say, and certainly not as much as God, who is purely spiritual. ( ALCHEMY )

Final Cause – Holy Spirit

All other is ‘natural’ – underwritten by God in maintaining the laws of nature without recourse to the supernatural.
Science was the method for investigating the world. It involved carrying out careful experiments, with nature as the ultimate arbiter of which theories were right and which were wrong.

Robert Grosseteste (1168-1253) Bishop of Lincoln (Robert ‘Bighead’) – neo-platonic reading of Genesis – emanation of God’s goodness, like light, begins creation. Light is thus a vehicle of creation and likewise knowledge (hence ‘illumination’), a dimensionless point of matter with a dimensionless point of light superimposed upon it (dimensions are created by God). Spherical radiation of light carries matter with it until it is dissipated. Led to studies of optical phenomena (rainbow, refraction, reflection).

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# ROGER BACON (1214- 94)

#### (Doctor Mirabilis) ‘The Marvelous Doctor’

(Franciscan friar) Oxford – 1257

‘Mathematics (The first of the sciences, the alphabet of philosophy, door & key to the sciences), not Logic, should be the basis of all study’

Converted from Aristotelian to a neo-Platonist.

ROGER BACON

The Multiplication of Species; the means of causation (change) radiate from one object to another like the propagation of light.

‘An agent directs its effect to making the recipient similar to itself because the recipient is always potentially what the agent is in actuality.’

Thus heat radiating from a fire causes water placed near the fire,
but not in it, to become like the fire (hot). The quality of fire is multiplied in the water (multiplication of species).

All change may be analysed mathematically. Every multiplication is according to line, angles or figures. This thinking comes from the ninth century al-Kinde and his thoughts on rays and leads to a mathematical investigation into light.

Fear of the Mongols, Muslims and the Anti-Christ motivated the Franciscans. Franciscan neo-Platonism was based on Augustinian thought with a mathematical, Pythagorean, approach to nature. Bacon subscribed to this apocalyptical view, suffered trial and was imprisoned.
The Dominicans chose Aristotle – with a qualitative, non-mathematical approach to the world.

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# THOMAS AQUINAS (1225- 74)

#### (Doctor Angelicus, Doctor Communis, Doctor Universalis)

THOMAS AQUINAS

‘A theological need to explain a cause becomes the basis for a specific scientific explanation of the world’

Established by Christians and Muslims in order to confound the dualist philosophies coming out of Persia.

Thomas Aquinas was a Dominican priest, theologian, and philosopher. Called the Doctor Angelicus (the Angelic Doctor,) Aquinas is considered one the greatest Christian philosophers to have ever lived. Two of his most famous works, the Summa Theologiae and the Summa Contra Gentiles, are the finest examples of Christian philosophy.

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# FRANCIS BACON (1561-1626)

1620 – England

‘Scientific laws must be based on observations and experiments’

Bacon rejected ARISTOTLE‘s deductive or a priori, approach to reasoning and suggested his own, inductive, or a posteriori, approach. Bacon developed the scientific method – but he did not make any significant scientific discovery.

‘I shall content myself to awake better spirits like a bell-ringer, who is first up to call others to church’

FRANCIS BACON

Bacon, a philosopher, advocated a new method of enquiry, completely different from the philosophical methods of the ancient Greeks, in his book Novum Organum – which has influenced scientists since its publication in 1620.

The text proposed the sentiment of ‘The Advancement of Learning’ (1605) signaling dissatisfaction with the limits of, and approaches to, knowledge to date and foresaw a future where the ancient masters would be far surpassed – Aristotle had written a text called Organum or ‘Logical Works’ and Bacon’s ‘new’ approach suggested an alternative direction to scientific study.

Bacon strongly criticised Aristotle’s deductive method of science, which involved formulating abstract ideas and ‘logically’ building upon them step-by-step to find ‘truths’, without thorough consideration of whether the theoretical foundation in itself was ever valid.

Rather than rely on superstition or accept unquestioningly the flawed solutions of the ancient academics as had largely been the case for two thousand years, Bacon’s alternative was to argue for ‘inductive’ reason, where the only ‘certain’ statements that should ever be made were based on observation and proof collected from the natural world. The essence of his method is to collect masses of data by observations and experiments, analyse facts by drawing up tables of negative, affirmative and variable instances of the phenomenon ( ‘Tables of Comparative Instances’ ), draw (induce) hypotheses from the evidence, then to collect further evidence to proceed towards a more general theory. The most important aspect of this method was the idea of drawing up tentative hypotheses from available data and then verifying them by further investigations.

‘A true and fruitful natural philosophy has a double scale or ladder ascendant or descendant, ascending from experiments to axioms and descending from axioms to the invention of new experiments’, he wrote in Novum Organum.

Bacon cautioned those trying to practice his new method, urging them to repudiate four kinds of intellectual idol

• Perceptual Illusions – ‘idols of the tribe’

• Personal biases – ‘idols of the cave’

• Linguistic confusions – ‘idols of the market place’

• Dogmatic philosophical systems – ‘idols of the theatre’

more

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