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?).