The Golden Ratio

When I studied Philosophy at A-Level we were given a talk by a composer and he brought the golden ratio to my attention and it is one of those things, like pi for instance, that just blow your mind away. The ratio, 1:1.61803, is not just a mathematical constant but is a ratio that can be found in nature and art with such frequency that the Romans titled it 'sectio divina' or 'the divine section' and it is one of the supposed proofs put forwards for the teleological argument for the existence of God (otherwise known as the argument from design).

The golden ratio, also known as phi after the Ancient Greek mathematician who seems to have discovered it and applied it in the construction of the Parthenon, can be found everywhere. Take your fingers for instance, the ratio from the largest bone to the middle bone is phi, so is the ratio between the middle bone and the smallest bone. You also find phi in the arrangement of branches on the stem of a plant, in the growing points in a plant (the distance the shoot grows before it is strong enough to support another branch), in the replication of patterns in leaves, it has been found in the proportions of chemical compounds in crystals. It has even been found as the proportion of drones to the population of bees in a beehive.

The reason it was a composer who lectured on this subject was because phi has been found in Mozart. His piano sonatas which are often conveniently split into two parts can be found to contain phi in the proportion from from one movement to the other. Phi can also be found in the ratio between key changes in Debussy's 'Image, Reflections in Water'. More recently Shostakovich applied the golden ration to his music though from memory it seems to have not produced that pleasant a piece. Leonardo da Vinci incorporated phi into his work, the Mona Lisa being being a prime example of its exercise. The golden ratio, for many, is the key to beauty. Leon Battista Alberti, the fifteenth century Italian architect, believed that beauty was a matter of proportion and that if a body was divided up into 600 parts beauty would be ‘a Harmony of all the Parts, in whatsoever Subject it appears, fitted together with such proportion and connection, that nothing could be added, diminished or altered, but for the worse’. The proportion which he believed would secure a harmony of all the parts was phi.

We humans have an incredible ability to see patterns everywhere, phi really could just be another example of this talent at making order of chaos or it could be one of the keys to unlocking our understanding of the universe. As an aside, Pythagoras was evidently spooked by the discovery of the golden ratio because he worked to keep it secret, its discovery was punishable by death.

Pythagoras

Like millions of children the world over I was forced into studying Pythagoras and his familiar theorem of Euclidean geometry which states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two opposite sides. Pythagoras' influence in the field of mathematics is undoubted and given my ineptitude in that particular field I shall say little more about it, what interests me about this ancient philosopher is how he inspired a form of mysticism as bizarre and influential as the Homeric poems. Quite little is known of his early life other than to say that he was born on the island of Samos around 532 B.C. and lived under the despotic rule of the tyrant Polycrates. He was a genuinely odd chap, Bertrand Russell described him as 'a combination of Einstein and Mrs Eddy'.

Pythagoras wrote on a number of fields from Mathematics and logic through to metaphysics and religion. It is important to remember that he came a couple of hundred years before Aristotle, the man who was responsible for the categorisation of different spheres of thought, physics, metaphysics and politics. Therefore the early, pre-Socratic philosophers wrote on anything and everything, they truly lived up to the etymology of the word; philo sophia, the love of knowledge.

Pythagoras was the St. Francis of his time in that he preached to the animals which is quite understandable when you understand that one of the central tenets of his religion (yes he began a religion) was the concept of the transmigration of souls, or metempsychosis for those of you who are familiar with Joyce's Ulysses. In his time Pythagoras' religion exerted considerable authority and became responsible for unusual rules, for instance:

• One must always abstain from beans (not even Aristotle could understand this one, he mused that perhaps the reason Pythagoras ban their eating was that they looked like genitals.
• One must not break bread or eat from a whole loaf.
• One must not let Swallows share one's roof.
• One must never look in a mirror besides a light (probably a sign of the general mathematicians fear of the concept of infinity).
  
• On rising from bed one must smooth out the imprint that the body has left, etc...

There were some more progressive sides to his religion, for instance in his society men and women were considered equals. Property was held in common (Plato wrote a book on Pythagoras, now lost, perhaps he was inspiration for 'the Republic') and the advances made by that society were considered as a result of collective rather than individual achievement, I like him all the more already. Mathematics provide axiomatic truths and form the basis of our understanding of ourselves and the external world. There are elements in Newton's Principa mathematica which can be directly traced back to Pythagoras so he is one of the giants upon whose shoulders modern science stands, the fact that he was something of a loon makes him all the more endearing.

Mind Experiments

It is something quintessentially characteristic of me that I desire to talk about or indeed write about the things that I happen to learn and the lateral thoughts which sprout, this is going to be one of those posts. I make no claim at interestingness so feel free to jump ship, as it were, at this stage with due impunity.

John Rawls in his book 'A Theory of Justice' conducts what can only be described as a mind experiment. He wanted to ascertain what were the basic principles that people would agree on if they were completely unaware of their status. In this hypothetical forum people would be under a 'veil of ignorance' without knowledge of themselves, anything that would lead them to distort their principles so that they would not work only to serve their own ends. Ignorance would extend to their age, sex, class, colour, religion, where they lived or the status of their society, ignorant even of the degree of their own intelligence. Rawls reasoned that with all these restraints on their knowledge that the agreements they would come to, their conceptions of justice if you will, would protect the least advantaged in society because under that veil of ignorance, one could never know if that were a position in society reserved for oneself. This was Rawls' 'basic position' and one could sit and pick holes both in his method and his conclusion but that's for another time, it got me to thinking about the other 'mind experiments' philosophers are prone to.


Descartes, in his 'Meditations on First Philosophy', conducts what he calls a 'project of pure enquiry' and invokes the notion of a malign being, the devil if you are of a Judeo-Christian bent, who could be tricking him into believing in his own existence. Following on with the theme of philosophical scepticism Bertrand Russell asks whether we are not a brain in a jar in some mad scientist's experiment with the 'knowledge' we have of ourselves and the world around us, beamed directly into our brain.

Bertrand Russell had other unusual mind experiments which he used as philosophical tools, one was a teapot. Russell is what you can call a teapot atheist, he noted that many people believed in God because they had not been shown enough evidence to refute his existence. Russell argued that there was perhaps a teapot orbiting the earth, a small teapot, too small to be picked up by the most powerful of telescopes. Whilst you cannot prove the existence of the teapot you also cannot prove that it doesn't exist. There has been an updating of the teapot argument and that is the Flying Spaghetti Monster, the argument works just the same but the imagery is that bit funnier. When I studied philosophy at a-level my tutor used an argument along these lines but she argued for the existence of a perfect pizza chef, it was an ontological argument of sorts. a) I have an idea of a perfect pizza chef in my head, b) it is more perfect to exist than not exist, c)for my pizza chef to be perfect he must exist, c) my pizza chef is perfect therefore exists. The argument is flawed for all the same reasons the ontological argument is flawed but it made me laugh.

When you challenge the dictates of common-sense as philosophers are prone to do, things like belief in one's own existence, you place yourself in the awkward position of coming up with other ideas and solutions. The result is a body of work rich enough mental imagery to compete with the most abstract fantasy novel. I'm not saying that if you dig manga then you should go pick up Aristotle's Nicomachean Ethics because the chances are you will be disappointed. But if you enjoy journeying within your mind; philosophy could be a discipline for you -- consult your nearest philosopher for advice.

How to be alone

The title of my blog comes from a quote from the fantastic philosopher Michel de Montaigne so it is only right that I devote an entry to him but before I go on with the topic I must give my two favourite quotes of his.

"Upon the highest throne in the world, we are seated, still, upon our arses"

The other quote deals with a pretty similar theme:

"Kings and philosophers shit: and so do ladies"

Montaigne was a man who believed that you can live a virtuous life even if you speak no ancient Greek, fart and not know the ancient philosophers, but as long as you strive towards wisdom (even if you never stray too far from folly). He spent a large amount of his life in isolation, locked up in the most wonderful library atop a tower in his estate in Perigord, France. He was not only surrounded by his amassed book collection but by sixty or so maxims from ancient Greek and Roman scholars (including the likes of Cicero, Seneca, Virgil and Socrates) carved into the wooden beams.

As you can imagine from Montaigne's style of life, he was not afraid to live alone, he was in fact keen to make the most of it:

"Now since we are undertaking to live, without companions, by ourselves, let us make our happiness depend on ourselves; let us loose ourselves from the bonds which tie us to others; let us gain power over ourselves to live really and truly alone -- and of doing so in contentment"

His first piece of advice seems almost intuitive, to keep occupied though to tailor that occupation the best way to suit one's humour. Unusquisque sua noverit ire via -- let each man choose the road he should take. But, nothing should be done to excess:

"Whether we are running our homes or studying or hunting or following any sport, we should go to the very boundaries of pleasure but take good care not be involved beyond the point where it begins to be mingled with pain".

Montaigne and I share a method of occupation and he singled it out for a special mention:

"Books give pleasure: but if frequenting them eventually leads to loss of our finest accomplishments, joy and health, then give up your books."

His second piece of advice is not to expect too much from your time alone. He baulks at the ideas of Pliny the Younger and of Cicero, ideas of attaining glory for 'ambition is the humour most contrary to seclusion'.

"We must do like the beasts and scuff out our tracks at the entrance to our lairs...withdraw into yourself, but first prepare yourself to welcome you there"

Okay, I'd be the first person to admit that Montaigne's ideas for how to be alone are, well, pretty naff and if you asked Montaigne himself whether one should be alone he would in all probability say no but learning to be alone will always be an important thing to learn because it is an inevitability in life that we will spend time with only ourselves as companions.

"We should have wives, children, property and, above all good health...if we can: but we should not become so attached to them that our happiness depends on them...so that when the occasion arises that we must lose them it should not be a new experience to do without them"

So how should we be alone? Montaigne seems to say that's really up to you, find out what works and then do it, just not to excess. It's not a way of living that he would suggest we chose, but one we learn to accommodate. Life can throw up the most unexpected incidents, we may lose our families, be thrown in jail, in essence we are a slave to fortune. The most telling quote from his essay 'on solitude' comes from the founder of the school of cynical philosophy, Antisthenes, 'man ought to provide himself with unsinkable goods, which could float out of a shipwreck with him'. Learning how to be alone is Montaigne's unsinkable good.