How the Mandelbrot and Julia Set mathematically proves St. Thomas Aquinas’ statement that God is the starting point of every effect.

By:Karen Quinto

Written for Dr. Gary Toop (PHL 201 “Problems in Philosophy”) on July 31, 2010


I am sure it is not a unique thought that we as human beings should never know what created the universe because we ourselves are in it and are limited by it.  Because of the limitations, seeking the truth is not an easy task.   As a human being, I am often overwhelmed where to begin.  Questions seem to be the very nature of the human consciousness, the answers to which seems to formulate even more complex questions, and those questions answered creating infinitely more.

As humans, we persist in the curiosity of the metaphysical, and through reason alone, we have developed the study of natural philosophy to better understand God’s effects. This is the God that which we understand—if we observe motion present in the universe—to be analogous to the unmoved mover and the uncaused cause of every effect thereafter.  This is derived from the cosmological argument of St. Thomas Aquinas who, with the second law of thermodynamics in mind, thought of God as being the “potential energy” of the universe that created everything.  The argument of motion states that through God, the universe was created from potential into actual.  Aquinas proposes that we can better know God through his effects; after all, a potential motion can be measured through the actual motion, just as we are able to know how high a rubber ball can bounce if we knew how high it was dropped.  God’s effects, the realm of actuality, is everything that can be observed in nature and described through abstractions in mathematics and other branches of the sciences.  We see parallels of the cosmological argument most popularly in the big bang, a theory that proposes that a spontaneous explosion created the universe, expanding continuously through time.  This has been proven through the Doppler effect that describes the moving of celestial objects away from each other inferred by the wavelengths of lights.

The continuous expansion of the universe can also be described by the second law of thermodynamics known as Entropy, which states that everything in the world moves towards disorder, and towards chaos in an irreversible process, as in the case of time. Perhaps chaos is simply an abstraction of order.  It is simply a form of order that our mind cannot, because of the limits of the human intelligence, decipher.   After all, many shapes in nature have extreme complexity; shapes that exist, yet cannot be defined within simple geometry, although geometry is certainly a part of it.

I see signs of God in science where patterns emerge as soon as I make sense of them, like finding meaning where there didn’t used to be.  Berkeley put it succinctly that the world is dependent upon our mind for its existence.  When something is from a great distance, it is beyond our perception and thus cannot exist by way of our senses. Upon entering our periphery, we may sense it a little, perhaps like a dot to acknowledge its physical manifestation, but for the most part we rely heavily upon our reason, intuition and imagination to infer the nature of the object.  It seems to me that God is that which resides in the periphery, constantly within the boundary of knowing, and yet, not fully knowing.  Many philosophers have tried to explain the existence of God through mathematics, and perhaps it is because it is the only language available to humans that can best describe all the abstractions of complexity.   In order to derive meaning behind a phenomenon, we must be able to find the pattern; first, by being able to describe them, so that we can pull things that seem very far and very complex into the nearer realm of order, which are the things that we are able to describe through our senses and reason.

Reasoning is the faculty which sets humans apart from animals, and the progress we have made in the sciences to better understand ourselves is a testimony to the power of our will to reason.  Only humans can take into account the order we see in nature—patterns which we strive to discover in order to uncover more complex patterns that can be explained through the study of the sciences.  Yet nature is the part of the whole of which science is only a part, just like order is only a small part of chaos.  By language of mathematics, humans take into account the order that we see in nature, make sense of it and understand it because nature is meaningful, and so it must necessarily follow that God, who is the cause of nature must also be meaningful.

In my understanding of God, He must necessarily exist for anything in science to have meaning.  And even more evidence I see of God is the sheer complexity of things that humans cannot understand now, but may be able to decipher their pattern in due time.  He is the necessary being that remains the same throughout time and space, being unbound by it. The understanding of God as a Whom or What may vary across different interpretations (and even religions, or lack thereof), yet God is essentially that same point in the periphery.  So that in my understanding of Him, God is the potential that transforms into the actual and is simultaneously both, in the same way that light is both particles and wave as a duality (perhaps that is why many religions associate God with light).

Contingency is Aquinas’ argument that humans and the universe are “contingent” things that do not possess a necessary existence, so that we could not exist if God did not will it to be so.  This contingency argument is personified in a sub-plot of the novel Memnoch the Devil by Anne Rice.  God wanted to know who He was, so He made the universe in His image to better understand Himself.  In effect: I am willed to existence by God so that I am able to conceive of Him as He conceives of me.  This iterative dynamic system of God feeding into me feeding into God feeding into me in a loop, is what I immediately knew by reason and intuition when I was introduced to the iterative dynamical system of the Mandelbrot and Julia set of complex planes.

The formula, Z —> Z2 + C , is a beautiful one, akin to a mathematical story of creation.  In the beginning, there was nothing except an initial point.  The purpose or goal, was to evolve the single point into an interconnection of things that a repeating “loop” system could potentially, given the right conditions make something out of nothing.  To illustrate this in science, biological life did form from the initial combinations of nucleic acids that could potentially transform itself into DNA under the condition that it could replicate itself. This is the argument of causation of St. Aquinas, who describes God to be the starting point of every effect; the “nucleic acid” of existence.   Just as things did not appear out of nothing– because nothing can come out of nothing–God is the necessary cause of our existence.  Mandelbrot himself remarks that the interconnectedness of the Mandelbrot and Julia set is an ‘act of God’.  This is the notion of the everlasting, continuously returning something that originates from the initial point, and through the re-iteration of the system creates an infinitely connected shape that continues to feed itself.

Depending on the constant in the formula, C, the Julia Set may transform into different shapes called fractals.  For example, if the constant is zero (Z —> Z2 + Cthe shape of the Julia set is a perfect circle.  Other numbers conjure up shapes that seem like lightning, like clouds, like a snowflake–yet pick a different number and just as easily it turns into dust.  Mandelbrot explains that the constant, C, must be a number within the Mandelbrot Set in order for the Julia Set to be connected.  Any number outside of the Mandelbrot set will only return fractal dust, the nothingness that results when the Julia Set is disconnected.  Thus we must necessarily be inside God to be interconnected, as the Julia set is contingent upon the Mandelbrot Set to be connected.  And we are indeed an interconnected web of life emerging amidst other unconnected ‘dusts’ in the universe.   This is not such a difficult idea to explore, for we are on a living Earth surrounded by the void and vacuum of outer space.

The patterns occurring in the natural world is an overwhelming proof that Mandelbrot discovered the mathematical design of nature–the shapes of the Julia set that can only exist because the Mandelbrot set exists.  With only the iterative dynamic proof of God in mathematics, we arrive at the argument that the interconnectedness of life has a necessary origin and is contingent upon something in order to exist.  Without God, how can everything come from nothing?