A Mathematical Explanation of Spiritual Enlightenment
by: Frank N. Wiley
Copyright © , 1996.
Copyright © , 1996.
In The Perennial Philosophy, author Aldous Huxley discusses the power of various languages to convey meaning. He notes how the musical scale captures the essence of all music, and how the symbols of mathematics have become the vocabulary of science. He laments that there is no "calculus of the Spirit", no language which can express the mystical awareness that transports us "beyond words and thoughts."
While it's true that we cannot actually describe the indescribable, perhaps we can still use language to identify the means of achieving that transcendent awareness. To do this, this article will explore a rather unusual path: the language of mathematics.
Mathematics, as used in modern science, has proven to be an extremely powerful language. In physics, it has reached the point where scientists do not believe a particular theory has any value unless it can be captured in an equation. Often, a theory evolves simply because the mathematics of it is "elegant", and then later, experiments discover that the equations were right. In other words, the math comes first, and the verification, in actual experimental reality, follows. Such was the case with Einstein's famous theories of relativity. So, obviously, there exists some deep connection between mathematics and reality.
What, then, if Huxley was wrong, and we could uncover a "spiritual calculus.". What if we were able to describe aspects of the spiritual experience via mathematics? Would we then also discover the same predictive power in the equations that physicists sees? Might not the math help point us in the right direction for our own spiritual exploration?
Before we continue, it is important that we distinguish clearly between an explanation and an experience. What will be presented here is an explanation of enlightenment; an equation that demonstrates how the awakening process works. However, this is certainly not the same as having the experience of enlightenment. That can only come through dedication to a spiritual practice. Hopefully, though, this explanation will allow us to understand why certain spiritual practices work better than others, and point us in the right direction for our own exploration, just as the physics equations help direct the experiments in the labs.
The Nothing That Is Everything
What, then is the "mystical experience"? All religions, despite their external differences, share a common core of truth, an esoteric theme that runs beneath them all like an underground river. The essence of that common truth can be summarized in the Hindu expression tat tvam asi, "Thou art that." Mystics of all traditions perceive God, or Brahman, or Buddha-Mind as an underlying Ground of Being out of which springs the reality we perceive. Within each of us lies a spark of that divinity; the "kingdom within" of Christianity, the "Atman" of Hinduism or Buddha-nature. Ultimately, upon achieving enlightenment, the mystic realizes that her inner self, her Atman, is exactly the same as the outer Ground, or Brahman. Atman and Brahman are one -- "thou art that."
How does one come to this mystic awareness, this enlightenment experience? It is here we begin to describe the process which later we will capture in a mathematical equation. However, we must first understand that the mystical tradition defines God, or the Ground of Being, in terms far different than traditional Judeo-Christian beliefs. The God of the mystics lies beyond rational definition. As Lao Tzu states in the Tao Te Ching, "The Tao which can be named is not the eternal Tao." In Eastern traditions particularly, this mystic Ground is often described by terms such as "the void", "emptiness", or "nothingness". And for the mystic to achieve union with this Ground, he too must become nothing.
The mystic injunction could be summarized as: "Free yourself of everything." You must let go of your ego, of desire, of thought, of all sense of an individual self. If you can do this -- if you can truly become nothing -- then an amazing thing happens. You suddenly, in that moment of enlightenment, realize that you are everything! This experience is proven time and again in the writings of the great mystics, who talk of their union with all of life, of becoming one with the entire universe.
So, then, to simplify, the mystic formula appears to be: become nothing and you will become everything. It sounds contradictory and paradoxical. If we put this formula into an equation, it looks even more ludicrous. The mystics seem to be saying that if you can become nothing, or zero, you will also realize that you are everything, or infinity. So, the equation would read:
Yet, how can nothing be the same as everything? Yet, we will see there are a number of proofs of this hypothesis to be found in mathematics, cosmology, physics and finally, spirituality. Hopefully, this exploration will give you a better understanding of the mystical experience, and point you in the right direction for your own spiritual practice.
The Mathematical Proof
Up until the last one hundred years or so, the concept of infinity was considered an abomination, not only by the Western church, but by mathematicians too. Only in the last century have we begun to explore infinity in any systematic way, and have even looked at the fact that different infinities might exist.
For example, if we take the positive numbers starting at one, and begin to count, we get a string like this:
1, 2, 3, 4, 5, ...
and so on, and so on, up to infinity. But what if we only take every other number, like this:
2, 4, 6, 8, 10...
This progression also will continue on to infinity. And, surprisingly, this infinity is exactly the same size as the first group, even though it appears we've only included half the numbers. And what about adding these two infinite groups together? What would be the result? According to the tenets of fundamental arithmetic, if you add anything to infinity, the answer is still infinity. So, if you add two infinities together, the answer seems rather obvious:
Obviously, if you add two infinities together, you're still going to get infinity. Or, will you?
To answer that questions, let's take a look now at the opposite end of our spectrum, the number zero (0). If you look it up in the dictionary, you find the following definition: "The figure or symbol 0, which in the Arabic notation for numbers stands for the absence of quantity; naught; nothing." About what you would expect. It's the next definitions that proves very interesting: "The origin of any kind of measurement, positive or negative, on a scale."
First, consider the beginning of that definition, "the origin of any kind of measurement." It begins to hint at a nothing that is very important to everything. But, let's investigate it further. If we draw a mathematical scale, it looks like this:
Taking into account both positive and negative numbers, we see a balance exists that pivots around the number 0. For every positive number, there exists its negative equivalent: +1 has its -1; -154 is balanced by +154. And, again using simple arithmetic, if you add a positive number to its negative counterpart, the result is 0. So, +1 added to -1 makes zero, just as -154 and +154 equal 0.
Let's go back and explore our infinities again, as we discussed them above, only lets look at both sides of this scale. Just as there is a positive infinity of whole numbers (1,2,3...), there is a corresponding negative infinity of whole numbers (-1,-2,-3...). Only this time, when we add them together, they cancel each other out. So, in this case, the sum of one infinity and another is not infinity, but zero. We can do this for any other infinity along this scale: all the positive and negative powers of two; all the positive and negative fractions. In each case, the sum of these infinities is equal to zero! Mathematically, that would be expressed in this way:
This equation, known as Sigma Notation, merely means the sum, starting at negative infinity and going to positive infinity, of ALL infinities is zero. Hopefully, based on the above description, this now makes some sense to you.
However, as is true of any math expression, if A equals B, then B must also equal A. So, if we turn the above expression around, it appears this way:
What this clearly demonstrates is that zero equals the sum of every number! When you add it all up, when you take the totality of all numbers, what you get is...nothing. It is a Nothing that is Everything.
The Scientific Proof
Well, you may be asking, so what? Certainly it's a rather interesting equation, and not one you typically think of, but that's just pure math. What does it have to do with the real world? As discussed above, a lot.
Since the dawn of modern science at the end of the middle ages, researchers have been in awe of the apparent connection between mathematical equations and the reality they have been exploring. Newton revolutionized our understanding of the universe when he discovered a universal equation for gravity. He showed, with one mathematical expression, the way that gravity functions whether you're standing on the surface of the Earth, or looking at the attraction of two galaxies to each other. So powerful was this connection that it lead Galileo to state: "The language of nature is mathematics."
For three hundred years, this relationship has continued. As we learned more about nature, such as electricity and magnetism, for instance, we eventually found equations that could be used to explain the behavior of what we saw. In one sense, you might think there is nothing unusual about that. We were just using math, like words, to describe what we uncovered in the lab. While the close ties between the two might be surprising, still it was just a descriptive language.
That view began to change with the dawn of quantum physics at the beginning of this century. As we began to learn more and more about the atom, the world of sub-atomic particles became stranger and stranger. It lead Werner Heisenburg, one of the fathers of quantum physics, to lament, "we could not believe that nature was as strange as our experiments showed. Still, the results were there." And despite how strange reality got, going beyond our ability to frame it in words, we still found we could provide math equations that matched the results.
Moreover, what scientists discovered was that their math equations began to predict reality, not just describe it. From time to time, a series of equations designed to explain a certain behavior will introduce certain factors or new elements not yet known in the lab. In this case, a scientist will predict that a certain particle must exist, even though no one has found it yet, or else the equations won't work. Time and time again, these statements have proven to be true. Within a few months or years, more powerful experiments find the predicted particle and reality thereby conforms itself to the mathematics.
Based on this connection between mathematics and reality, our spiritual equation would therefore seem to predict that nothing is exactly the same as everything, not just in the realm of mathematics, but in the world itself. Outlandish, to be sure, yet there are several current theories in science that would indicate it may in fact be true.
First, in cosmology, recent evidence indicates our universe came into being in a single instant, known as the Big Bang. Based on astronomical observations, our universe is expanding in all directions, as though the galaxies rested on the surface of a balloon which was inflating at a fantastic rate. According to the Big Bang, this expansion must have started at some point in the past. If you roll back the "film" of history, drawing the galaxies closer and closer, eventually they all converge into an infinitesimally small point known as a "singularity", a mathematical point, in fact a nothing, out of which the everything of the galaxies somehow emerged.
Still, taking that as given, the mathematics of our equation only makes sense in the subsequent history of the universe if there is a balance of positive and negative, pivoting around the zero point. Otherwise, the sum of infinities would not cancel, and the result would be infinity, not nothing. Does this positive/negative balance have its equivalent in the world of matter? To be sure, it does.
With the birth of the universe at the Big Bang, the creation of matter followed a dualistic path. For every particle created, there existed an anti-particle as its negative reflection. Electrons had their counterpart in positrons; protons matched with anti-protons. And if you combined a positive particle with its negative equivalent, the result was total annihilation, a burst of energy, then nothingness. Just like +1 and -1, these particle pairs add together to make zero.
In fact, if you look at all of reality from a mystic viewpoint, this duality exists all the way up and all the way down. For every hot there is a cold; for every up, a down. Even at the human level, we balance love with hate, joy with sadness. Science puts forward a number of conservation laws, such as the conservation of momentum or the conservation of energy. Perhaps there is another, based on this discussion: The Conservation of Nothingness. This appears to be the way creation unfolds. From the void, the nothingness, reality springs forth in pairs -- a plus and a minus -- evenly balanced. Add them back together and the result returns to nothing, the ground of being.
This conservation of nothingness clearly demonstrates itself in wave mechanics, where you can frequently add together any number of waves and achieve a result of zero. Waves are energy patterns composed of crests and troughs which travel through some medium, like air or water. An interesting aspect of waves is how they combine. If two crests meet, they add together and the resultant crest is twice the size. Likewise, if two troughs meet, they also combine and double in size. However, if a crest and a trough of equal size combine, they cancel each other out and the result is zero. So, if you took two wave patterns of equal dimension but 180 degrees out of phase, like this:
and added them together, the result would be zero. The waves still exist, its just that their combined energy equals nothing. So, in theory, we could take an infinite number of waves, add them to an equally infinite number of counter-phased waves and the result would end up being zero. Still, while scientifically possible, this appears to be only a thought experiment. Does it have any corresponding verification in experimental reality?
Physicists such as John Wheeler hypothesize that the "empty" vacuum of space may actually be quite full, in terms of energy. If you add together all the wavelengths of energy available in any point of space, the result turns out to be nearly infinite. This means, in any given cubic-centimeter of "empty" space, you can find an almost limitless amount of energy in the form of waves. According to Wheeler's calculations, the amount of energy in the vacuum of space is equal to 1094g/cm3 . This is a one followed by 94 zeros. By comparison, the amount of energy available in solid matter (based on Einstein's famous E=MC2) is only 1014g/cm3 , a difference of 80 zeros! So, in fact, the nothingness of empty space is far more full than the somethingness of matter -- by a long shot! Yet, somehow, these waves cancel themselves out (via the crest/trough mathematics described above) and the result, in terms of measurement, is zero. So, once again, we have a scientific verification of the fact that nothing does equal everything, that zero can be the sum total of infinity.
A new theory in cosmology demonstrates this "conservation of nothingness" with a vengeance. In the May, 1996, issue of Astronomy magazine, theoretical physicist Michio Kaku discusses quantum cosmology, a theory seeking to combine Einstein's relativity with the findings of quantum physics. According to this new theory, our universe may be just one of an infinite number of universes which bubble forth from an endless, eternal ocean of nothingness.
According to Kaku:
"Creating universes out of nothing may seen to violate cherished conservation principles, until we realize that it takes no energy to create a universe. If the universe is closed like a bubble, then the energy content of its matter is positive, while the energy of its gravity is negative. The sum is exactly zero...Thus it takes no net energy to create new bubbles, which are constantly being created in the sea of nothing." (Italics are mine.)Science & Spirit
However, it must be clearly understood that these discussions of science are not an attempt to prove God exists through quantum physics. There have been many books that have taken such a tack, starting with Frijof Capra's The Tao of Physics in the mid-70's, and continuing unabated to this day. Dozens of authors, typically scientists, have discovered the parallels between the results of their research and the insights of mystics throughout history. They go on to show how the quantum soup is scientific proof of the Ground of Being, and therefore claim that science has found God.
While the parallels between our modern discoveries and the mystical insights surely exist, the former do not prove the latter. That these parallels occur should not really surprise us. Using very different tools, the mystic and the scientist explore the same thing --ultimate reality-- and so a commonality of results is to be expected. However, a problem arises from the reductionistic approach science attempts to take.
There is an obvious hierarchy in physical reality which extends from sub-atomic particles through atoms, to molecules, cells, organs and living creatures. In parallel to this external, matter-bound hierarchy, we have an internal hierarchy of the psyche, what philosopher Ken Wilber calls a "spectrum of consciousness" which extends from the non-awareness of rocks, through sensory awareness of one celled animals, to the rudimentary consciousness of lower life forms, to self awareness in human beings. At its highest level, this spectrum reaches the trans-personal revelations of the mystics.
Physicists tend to focus their research on the basic building blocks of matter, sub-atomic particles and energy fields. They then attempt to explain how everything resting on top of these building blocks in the hierarchy can be explained by the properties of the particles themselves. They attempt to show how all aspects of the upper levels of the hierarchy derive their properties from the lowest levels; everything, from protons to God, can be reduced to the laws of physics.
Mysticism approaches reality from the opposite end of the spectrum. Through awakening, mystics experience the highest reality available to human beings. And from this reality, they then are able to understand the lower as limited expressions of the ultimate ground of being. So, the alpha and omega of divinity, the yin and yang, find their counter part in the dualities of physical reality. The physical does not explain the spiritual, rather it reflects it. Just so, the scientific verifications of our equation do not prove that this relationship must exist at the highest level. Rather, they are like reflections of the moon in pools of water, which would indicate that the moon exists, but are not, in fact, the moon itself.
So, let us now turn our attention to the "moon" of our discussion, the enlightenment experience itself, and see what insights we might derive from our mathematical formulation.
The Mystic Viewpoint
As discussed in the introduction, the mystic view of God differs dramatically from the traditional Judeo-Christian concept. Gone is the mythic figure, sitting on his throne in heaven, to be replaced by a nebulous, amorphous field of divinity, the Ground of Being. This Ground cannot be grasped intellectually, but can only be hinted at, through concepts such as "the void" or "emptiness." To the mystics, God is an eternal, infinite Nothing, that at the same time is the source and foundation of Everything.
An analogy might help explain this trans-rational concept. Consider this Void or Emptiness like the silence just before the start of a symphony. In that breathless instant before the conductor waves his baton, anything is possible. The orchestra might launch into the majesty of Beethoven's Ninth Symphony, or play a humorous rendition of "Happy Birthday." The potential of everything rests in that moment of nothing. Yet, once the first note is played, the options diminish dramatically, and each succeeding note moves you further into a single track -- Beethoven or birthday.
And, so, the mystics seek to align themselves with this Divine Void, to free themselves of any concept of a little self or ego, and merge with the Emptiness which lies at the heart of creation. Think of it like a wave and the ocean. Each wave (or person) gains its being from the underlying source of the ocean (or God). Yet, as a wave, it is possible to believe yourself to be independent, a unique wave, different from all of the other waves around you. You may then spend your time trying to be the best wave, or the biggest wave. You might worry whether the wave next to you has a more frothy crest, or a more appealing arch. You can spend your entire life trying to keep up with the Joneses of other waves.
Or, like the mystics, you can let go. You can stop your ceaseless struggle to prove your unique identity as a wave, and sink back into your source, there to discover that you've been the Ocean all along. When you become Nothing as a wave, you recapture your truth as the Infinity of Ocean-ness. So, once again, nothing and everything prove to be one.
But, why nothingness? Why should we seek emptiness? Many spiritual writings talk about unity, about merging with the One. This certainly seems more appealing. Yet, ultimately it is a limited perspective, based on the arguments presented in this article. First, from the mathematical perspective, one is not the sum of all numbers; that, we have demonstrated, can only be zero. Second, all mystics warn that God lies beyond our rational comprehension, and cannot be conceived in terms the mind can understand. Yet, we can grasp the concept of One; and we might be tempted to put an image to it, to possibly return to the God of our fathers. But, zero, nothingness, that is a concept that defies our rational ability to grasp. When we reach for it, it slips through our fingers, like smoke, and confounds all our attempts to describe it. Third, scientifically, we see that emptiness lies at the heart of matter. Even our bodies, which appear so solid, are in fact 99.9999% empty space. Yet, that void is not "empty" in the sense we usually think about it, but infinitely full. Recall that every cubic centimeter of space contains more energy that all of the matter in the entire universe!
So, the choice is ours: Do we focus on some outer manifestation, even something as all comprehensive as the One; or do we turn inward toward the infinite/eternal nothingness that rests beneath all things, as the ocean rests gently beneath the waves?
A final analogy might prove beneficial. The mystic attempts to understand the Divine by merging with it, by grasping it whole as an experience that lies outside of rational thought. In much the same way, we attempt to grasp the beauty of a work of art by experiencing it in its totality. Who is not moved by the grandeur of Michealangelo's masterwork, the David? Standing before such beauty, the only proper response is awe. Yet, how moved would you be if all you could see was one square inch of this statue, say a small rectangle taken from his knee cap. Not much. Now, let's continue this analogy to its logical extreme. Remember the difference in energy, in manifestation, between "empty" space and matter; emptiness outweighed fullness by a factor of 1080. Coming back to our statue, this would be like trying to experience the beauty of the David when all you could look at was a single atom, or even more, a sub-atomic particle! To focus on anything less than the Void is to miss the entire picture.
The Mystic Path
Mysticism, like physics, is a science; it puts forth a proposition, or hypothesis (Atman equals Brahman), and then provides a series of experiments to prove that hypothesis. These experiments, the spiritual practices of the mystic tradition, offer proven and tested methods for you to perform the experiment yourself. And, the results of your experiment can be verified, just like an experiment in quantum physics, by a group of experts in the field. So, when a Zen student believes he has achieved enlightenment through focus on his koan, he goes before the master and others of his temple, and has his experience verified or refuted. So, too, Tibetan monks are grilled by a panel of Masters before give the title of Awakened.
Many of us, however, do not have the opportunity or inclination to study with a Master or guru. We may make a conscious choice to pursue our own spiritual awakening as a solitary path, without the involvement or guidance of a guru. In this case, the mathematical proof presented here can give you an enormous insight into the best way to continue your own growth. By recognizing the validity of the spiritual injunction, nothing equals everything, we can see clearly see what the goal of our quest should be. Mahatma Gandhi summed it up in this way: "I am the most ambitious man in the world. I want to become zero."
Such a statement, which once may have seemed esoteric or meaningless, now takes on a deep and spiritual significance. "Free yourself of everything" the mystics tell us. And where once such a requirement seemed too demanding, too ascetic, we can now understand that it is the only way to become Infinite.
And with that rational understanding, that mathematical proof, we give ourselves permission to begin the experiment. Because the explanation is not the experience. We may now understand why the mystic path works, why we can achieve unity with all things by becoming no thing, but we have not yet had the experience. That requires work. Mystical literature is filled with techniques to achieve this awakening, from meditation to mantras, from centering prayer to sufi dancing. Such a profusion of options can often be confusing. However, now, as you consider your path, you hopefully have a clearer understanding of the final goal: To become nothing. And, in this way, you will most assuredly become everything!
email address: FNWiley@aol.com
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