The science of Chaos was born out of the persistent attempts of a determined group of people from a dozen different scientific disciplines who, often working alone and in direct opposition to orthodox scientific beliefs, strove to make sense of what appeared essentially senseless and chaotic.
Working in a holistic manner that sought to look at the effect they were studying as a whole rather than isolate a single one of its parts, they came to grips with nothing less than the boiling broth of the magician: creating order out of anarchy...on a cosmic scale!
Their holistic approach paid off.
They found that there is a correspondence of scaling that is constant in everything, from the way leaves on a tree are shaped to the shape assumed by the dancing flame of a candle.
They realised what magicians have known all along: To make things work you do not require a minute knowledge of the working of every possible part of the connection you are studying, but a visual approximation of the generality governing it. Much like visualising the effect rather than the entire complex sequence of events that must lead to it.
Nature does the rest.
The man credited with laying down the foundations of the new science is physicist Mitchell J. Feigenbaum. His playground of speculation was a by now familiar numerical territory known as a complex plane that is composed of ‘real’ and ‘imaginary’ numbers, now recognised as arbitrary as both sorts can be as real or imaginary as any other sort.
What is remarkable is that the modern day definition of the complex plane sounds like an updated reiteration of the astral plane concept propagated by Eliphas Levi in 19th century Paris as a way of recovering the microcosm and macrocosm which science had rejected. As we’ve already seen the complex plane consists of two axis where numbers co-exist with their opposites. Levi said that the astral plane gives a picture of the “ultimate reality comprised by a unity compounded of opposites”!
Working within the context provided by the complex plane, Feigenbaum discovered that he could visualise shapes that corresponded not only to static things but also to motion.
Eliphas Levi in The Key to the Mysteries, wrote that the astral plane is filled with astral light “which is the fluidic and living gold of alchemy” and to control it is to master all things, “To direct the magnetic forces is then to destroy or create forms; to produce to all appearance or to destroy bodies; it is to exercise the almighty powers of Nature”!
The procedures and trappings of ritual magic then became -like the studying of the scientist- a means of directing and shaping the Will, making it part of the network of correspondences. Links between the universe and the human mind, which obtained in miniature all factors existing in the world outside it.
Even more to the point, the astral plane is the realm in which thoughts, imaginings and desires have an independent reality. In modern magical theory it is there that the traditional ascent of the spheres mentioned in The Sefer Yetsirah or Book of Creation takes place, enabling man to ascend to the point where he can see the nature of God.
Studying the pictures of the fractal shapes provided by mathematical manipulations in the complex plane, scientists can at last see a simple, beautiful, unifying order underlying the fabric of chaotic processes. An order that is encountered from the microscopic to the macrocosmic. An order that may reveal, according to Australian physicist Paul Davies, not just the nature of God but how God thinks!
Tuesday, June 5, 2007
Chaos in all directions
It wasn’t like that always however. Mathematics was one of the less glamorous of the sciences and mathematicians, in the public eye, were no more exciting than accountants.
This all changed one wintry afternoon in 1975 when an IBM mathematician by the name of Benoit Mandelbrot, preparing his first major book-length work for publication, thumbed through a Latin dictionary looking for a word that would describe some peculiar-looking shapes his calculations had come up with.
He came across the adjective fractus from the verb frangere - to break. And, with a little playing around, the word fractals, was born.
Fractals, as the name suggests, are a family of jugged, tangled, twisted, splintered and fractured shapes which seem to underlie the very fabric of nature. They are present not only in static shapes like serrated edges and coastlines but also in the shapes formed by charting the scaling of motion (in the way a butterfly beats its wings or a pendulum swings, for example) or the rise and fall of cotton prices in the Stock Exchange and the way molecular cells transmit data.
Many of us will have probably seen the twisted, colourful designs which fractals today make, but what is truly beautiful about them is that their forms, as complicated as the mind of God according to those who study them, require but the most basic of instructions to recreate.
Using powerful IBM computers to analyse the formation of these structures, Mandelbrot discovered that they are but a variation of a single fractal shape which has since become known as the Mandelbrot set.
The essense of the discovery illustrates the sensitivity of any event to initial conditions and became the foundation of the inter-disciplinary science of Chaos.
Like new magicians, the Chaoticians that sprang up in the wake of Mandelbrot’s discovery, did not look upon the world in the reductionist, analytical manner of their predecessors but sought to gain an understanding of the underlying simplicity governing the behaviour of complex systems.
What Chaos teaches is that nothing can ever be predicted with accuracy because even the tiniest change in initial conditions can be responsible for tremendously large effects.In a statement of almost metaphysical quality chaoticians say that “the gentle beating of the wings of a butterfly in China is responsible for the tornadoes in Texas”!
If you think that’s overstating matters a little, or even edging away from the cold, precise logic of science, more shocks are in store, for the language used by today’s Chaoticians to describe the nature of fractals and the physical events they affect parallels that used by the historian and theoritician of magic, Eliphas Levi, and subsequently the controversial, self-styled ‘Great Beast’ Alistair Crowley.
This all changed one wintry afternoon in 1975 when an IBM mathematician by the name of Benoit Mandelbrot, preparing his first major book-length work for publication, thumbed through a Latin dictionary looking for a word that would describe some peculiar-looking shapes his calculations had come up with.
He came across the adjective fractus from the verb frangere - to break. And, with a little playing around, the word fractals, was born.
Fractals, as the name suggests, are a family of jugged, tangled, twisted, splintered and fractured shapes which seem to underlie the very fabric of nature. They are present not only in static shapes like serrated edges and coastlines but also in the shapes formed by charting the scaling of motion (in the way a butterfly beats its wings or a pendulum swings, for example) or the rise and fall of cotton prices in the Stock Exchange and the way molecular cells transmit data.
Many of us will have probably seen the twisted, colourful designs which fractals today make, but what is truly beautiful about them is that their forms, as complicated as the mind of God according to those who study them, require but the most basic of instructions to recreate.
Using powerful IBM computers to analyse the formation of these structures, Mandelbrot discovered that they are but a variation of a single fractal shape which has since become known as the Mandelbrot set.
The essense of the discovery illustrates the sensitivity of any event to initial conditions and became the foundation of the inter-disciplinary science of Chaos.
Like new magicians, the Chaoticians that sprang up in the wake of Mandelbrot’s discovery, did not look upon the world in the reductionist, analytical manner of their predecessors but sought to gain an understanding of the underlying simplicity governing the behaviour of complex systems.
What Chaos teaches is that nothing can ever be predicted with accuracy because even the tiniest change in initial conditions can be responsible for tremendously large effects.In a statement of almost metaphysical quality chaoticians say that “the gentle beating of the wings of a butterfly in China is responsible for the tornadoes in Texas”!
If you think that’s overstating matters a little, or even edging away from the cold, precise logic of science, more shocks are in store, for the language used by today’s Chaoticians to describe the nature of fractals and the physical events they affect parallels that used by the historian and theoritician of magic, Eliphas Levi, and subsequently the controversial, self-styled ‘Great Beast’ Alistair Crowley.
The Complex Plane
It would be all too easy, at this stage, to make a great deal about the traditional, age-long connection of mathematics with magic.
I won’t because it will neither lead anywhere nor provide any insights beyond the fact that at every stage of human history, there is a body of knowledge that is fiercely guarded from outsiders and may acquire semi-mystical status. And if you think that we’re beyond this stage now, just spend some time in the company of any self-styled computer ‘expert’ and you’ll see what I mean.
What I will make a point of mentioning however is that from the very first moment we looked at our world, mathematics has provided a key to both representing it and understanding its evolution.
Galileo Galilei rightly said, way ahead of his time, that “Mathematics is the language in which God wrote the Universe.” Today complex algorithms enable us, if we wish, to numerically codify everything we see around us, including our own selves and represent it in mathematical terms.
The capacity to do that has proved a double-edged knife. On the one hand it gave us greater understanding and it has given us the almost magical power to make predictions. From forecasting the weather to predicting the potential power of an explosion.
On the other hand, it armed Newtonians with the emasculating belief that everything was predictable in a clockwork universe in which we functioned as simple cogs in a machine of truly universal proportions. This single belief, more than anything else, stunted scientific thought for a great many years and proved almost the death of every magical concept.
Whatever the drawbacks however, the advantages of representing everything in terms of numbers formed the basis of ‘formalism’ in mathematics. This allowed mathematics to become more and more the formal manipulation of symbols and it gave rise to the place where mathematicians can dream while awake and the manipulation of both real and imaginary numbers can affect the world. Today, this is known as the Complex Plane of numbers. In very simple terms the complex plane is comprised of two axis at right angles to each other.
One axis goes from east to west and goes onto infinity, and on it is every positive and negative number we know (which are called ‘real’ numbers). The other axis, goes from north to south. It too is also infinite and on it, like familiar shadows, are the so called imaginary numbers. Each one representing the imaginary doppleganger of a real number.
The reason for this arrangement is that instead of looking at numbers on a single line which would give a one, dimensional and thus rather limiting, view of the world, we can now look at them in two dimensions.
Every number we want now can be expressed as a combination of its real and imaginary coordinates. This means that it does not exist on its individual line but rather somewhere in one of the four quadrants of the complex plane.
This simple technique allows mathematicians to examine not only the solidity of objects (say like a cube or a square) but also to study normally invisible quantities, like the shapes of motion, and the formations of temperature differences!
Their sojourns through the Complex Plane have made confirmed Neoplatonists out of some of the best mathematicians, who in turn have found themselves, unwittingly reiterating some of Plato’s most profound statements regarding the state of reality.
The Princeton mathematician Kurt Gödel, who formulated the dictum (known as Gödel’s theorum) which states that no matter how precise mathematical computations become there will always be truths whose existence cannot be mathematically proved, envisaged these truths as already existing in a Platonic domain somewhere “out there”, beyond our limited, mortal ken. More recently, the Oxford mathematician Roger Penrose said that “Mathematical truth is something that goes beyond formalism,” and went on to write that “There often does appear to be some profound reality about these mathematical concepts, going quite beyond the deliberations of any particular mathematician. It is as though human thought is, instead being guided towards some external truth - a truth which has a reality of its own, and which is revealed only partially to anyone of us.”And discussing the implications of the system of complex numbers to be found in the complex plane Penrose rather apocryphically said that he felt it had “a profound and timeless reality”!
As if the examples of the overt similarities between the complex plane of mathematics and the Astral plane of High magic were not enough, the apparent ability to enter the complex plane (limited naturally only to those who’re adept at mathematics,) and toy around in it has given rise to discoveries and assertions which in turn have drastically altered the way we look at, and relate to the world around us (any of it sounds familiar so far?).
I won’t because it will neither lead anywhere nor provide any insights beyond the fact that at every stage of human history, there is a body of knowledge that is fiercely guarded from outsiders and may acquire semi-mystical status. And if you think that we’re beyond this stage now, just spend some time in the company of any self-styled computer ‘expert’ and you’ll see what I mean.
What I will make a point of mentioning however is that from the very first moment we looked at our world, mathematics has provided a key to both representing it and understanding its evolution.
Galileo Galilei rightly said, way ahead of his time, that “Mathematics is the language in which God wrote the Universe.” Today complex algorithms enable us, if we wish, to numerically codify everything we see around us, including our own selves and represent it in mathematical terms.
The capacity to do that has proved a double-edged knife. On the one hand it gave us greater understanding and it has given us the almost magical power to make predictions. From forecasting the weather to predicting the potential power of an explosion.
On the other hand, it armed Newtonians with the emasculating belief that everything was predictable in a clockwork universe in which we functioned as simple cogs in a machine of truly universal proportions. This single belief, more than anything else, stunted scientific thought for a great many years and proved almost the death of every magical concept.
Whatever the drawbacks however, the advantages of representing everything in terms of numbers formed the basis of ‘formalism’ in mathematics. This allowed mathematics to become more and more the formal manipulation of symbols and it gave rise to the place where mathematicians can dream while awake and the manipulation of both real and imaginary numbers can affect the world. Today, this is known as the Complex Plane of numbers. In very simple terms the complex plane is comprised of two axis at right angles to each other.
One axis goes from east to west and goes onto infinity, and on it is every positive and negative number we know (which are called ‘real’ numbers). The other axis, goes from north to south. It too is also infinite and on it, like familiar shadows, are the so called imaginary numbers. Each one representing the imaginary doppleganger of a real number.
The reason for this arrangement is that instead of looking at numbers on a single line which would give a one, dimensional and thus rather limiting, view of the world, we can now look at them in two dimensions.
Every number we want now can be expressed as a combination of its real and imaginary coordinates. This means that it does not exist on its individual line but rather somewhere in one of the four quadrants of the complex plane.
This simple technique allows mathematicians to examine not only the solidity of objects (say like a cube or a square) but also to study normally invisible quantities, like the shapes of motion, and the formations of temperature differences!
Their sojourns through the Complex Plane have made confirmed Neoplatonists out of some of the best mathematicians, who in turn have found themselves, unwittingly reiterating some of Plato’s most profound statements regarding the state of reality.
The Princeton mathematician Kurt Gödel, who formulated the dictum (known as Gödel’s theorum) which states that no matter how precise mathematical computations become there will always be truths whose existence cannot be mathematically proved, envisaged these truths as already existing in a Platonic domain somewhere “out there”, beyond our limited, mortal ken. More recently, the Oxford mathematician Roger Penrose said that “Mathematical truth is something that goes beyond formalism,” and went on to write that “There often does appear to be some profound reality about these mathematical concepts, going quite beyond the deliberations of any particular mathematician. It is as though human thought is, instead being guided towards some external truth - a truth which has a reality of its own, and which is revealed only partially to anyone of us.”And discussing the implications of the system of complex numbers to be found in the complex plane Penrose rather apocryphically said that he felt it had “a profound and timeless reality”!
As if the examples of the overt similarities between the complex plane of mathematics and the Astral plane of High magic were not enough, the apparent ability to enter the complex plane (limited naturally only to those who’re adept at mathematics,) and toy around in it has given rise to discoveries and assertions which in turn have drastically altered the way we look at, and relate to the world around us (any of it sounds familiar so far?).
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