Poetry, Parable and Proof
Thomas J. McFarlane
16 June 1998
www.integralscience.org
We once thought the Earth was real,
And the air and fire and water real.
Until they were found to change.
The real was then the elements.
Those atoms, those indivisibles.
Then the indivisible was divided.
The real was the electrons and nucleons.
But not even they were real for long.
Matter itself wasn't even real.
The real was energy and spacetime
Whose origin is the timeless spaceless Mystery
Before the first instant of the Big Bang.
That Mystery is the ground of all other realities:
Space, time, energy, matter, atoms, fire, air, water, earth.
And yes, the ground of you.
Once thought identified with the body,
Thinking the eyes of the body saw the world.
Until the eyes were seen. So who sees the eyes?
Then thought identified with emotions and thoughts.
Until they, too, were seen. So who saw them?
Then the habits of contracted attention were noticed,
And the sense of identification with these subtle processes.
Until Consciousness recognized itself,
The ground and substance of all experience.
And, yes, the ground of you.
Life's energy goes around, life living on death.
Cycles within cycles, powered by the sun.
Our human bodies, fueled by food of animals and plants and bacteria.
All fueled by the radiance of the sun.
The candle burning at the romantic dinner.
The grapes that made the wine.
The graceful movement of the dancers.
The energy that draws their every breath.
All derives from that single solar source.
Outside the winds blow, the rains fall.
Thunder and lightning, tornadoes and floods.
Streams merge into rushing rivers and fill the oceans.
All are only sun in diverse form.
Every movement of life and nature
Is ultimately driven by that one radiant source.
It is all sun energy transformed.
Life energy.
Nature energy.
Even the energy from the split atom
Derives from an ancient star
Which, in its last gasp of life,
Exploded its energy, creating that atom.
Thus all creation's energy derives from the energy of stars.
All life is made of star matter, and animated by star energy.
And powering these stellar sources
Is the universal mystery of cosmic attraction,
Pulling creation back to the uncreated.
If you have scissors and tape, you can create your own Möbius strip. If not, you can easily imagine doing this. Or maybe you can remember having done it before, in grade school perhaps.
Here's how to do it: Cut off an edge of this sheet of paper so you get a strip of paper. Wrap it into a ring. Now put a half-twist in it, then join the ends. Tape or staple the ends together, and you have created a Möbius strip.
This object has some paradoxical properties. For example, if you consider just a small part of the strip, it obviously has two sides. But if you follow one side all the way around the loop, you end up on the "other" side of the paper without having crossed to the other side. The two sides are actually just one side, even though they appear to be two.
There is only one edge as well. Try it: pick one edge and follow it around the loop. You will end up on the "other" edge after going around once. It is the same single edge, even though it looks like it has two.
This remarkable object is proof that two things as opposite and distinct as the two sides of a piece of paper can, in fact, be identical and not two at all.
There is no real opposition or conflict between inner work and outer work.
Even to work in both dimensions, as if they were separate spheres, is subtle conflict.
Whenever we act, we are working in both spheres simultaneously:
Every effort to foster a worldwide community
Has very real and positive effects on our inner development.
Every step of inner development that blesses us
Will inevitably manifest in helping create a positive community around us.
It is a beautiful truth that the two are inseparable.
Inner work and outer work are just different sides of the same coin
We help ourselves most by helping the world,
We help the world most by helping ourselves.
How else could it possibly be
since we and the world are one?
A prime number is a number that is evenly divisible only by itself and one. Some examples of prime numbers are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41.
The word "theorem" means "something to behold" in the Greek...
Theorem (Euclid): The number of prime numbers is not finite.
Proof: Suppose as a hypothesis that the number of prime numbers is finite, and let the finite list of all the prime numbers be P1, P2, ..., PN. Now multiply these N prime numbers together and add one. Call this number P. So, P = (P1 P2 PN ) + 1. Now either P is prime or not prime. If P is not prime, then P must be evenly divisible by at least one of the N prime numbers. Let this number be Pi. But notice what happens when we actually divide P by Pi:
Because this is not a whole number, but has a fractional remainder, P is not evenly divisible by any prime number. Therefore, P is actually prime. But if P is prime, then it must be equal to one of the N prime numbers. It is easy to see, however, that P is larger than every prime number in the list, so it cannot be equal to any prime number. So P cannot be prime. This contradiction means that our hypothesis was false. Thus, the number of primes is not, in fact, finite. Q. E. D.
The number of prime numbers is not finite -- this is an eternal truth. It will not be later qualified or refined. No postmodern critique will show that the number of primes is actually finite. This theorem will not be disproved when the next paradigm shift comes along. Theories of physical reality may come and go. But mathematics is forever. This theorem will always be true. There is, was, and always will be no doubt about it. Think about it.
Ages ago, before humans knew counting, there was a man who preached that there were universal principles of number that permeated the world, and that anything could be counted. He showed people how to add, subtract, and even multiply any kind of thing at all, whether rock, tree, animal, or mountain. Perhaps most remarkably, he showed people how to count without even having anything to count. Eventually, he moved on, wandering off into the mountains. Several years later, a cult grew up around his doctrine and people spent their Sundays counting rocks, communing with the divine order of things.
One day some cult members got word of another strange cult across the mountains that also had similar doctrines of counting. Some members made the journey and heard the doctrine of this other cult. To their amazement, these people were counting wrong! This whole cult was engaging in totally different rituals and had very different rules about adding and subtracting and multiplying their numbers. It was blasphemy!
They began to argue with these people, showing them their "true" science of counting. But these other people insisted that their counting was the true doctrine. Anger grew, until finally fighting broke out between them. They threw rocks at each other. It was a holy war. They were even counting the rocks as they threw them. The one group was throwing them, shouting 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,... as the rocks flew. The other group was throwing their rocks shouting I, II, III, IV, V, VI, VII, VIII, IX, X,...
Appalled by such insanity, many onlookers declared counting irrational. Because one group counted one way, while the other group counted another way, counting was not based on universal principles after all, it seemed to them. There were two conflicting doctrines, so counting must be based on faith alone. So counting was dismissed as unscientific dogma because there was no way to establish the true method of counting.
Meanwhile, the visionary wandering counter, whose teachings had become the doctrines of these cults, was relaxing under a tree in a far off land, enjoying the summer breeze, and marveling over the beautiful unity of counting, and the wonderful diversity of manifestations it takes.
The first numbers were the so-called "whole" numbers we use for counting: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on. There are also the fractions, like 1/2, 2/3, 3/4. These are called the rational numbers because they can be represented as the ratios of whole numbers. Rational numbers can also be defined as those numbers that can be represented by a repeating decimal expansion. For example,
The first three examples have a single-digit repeating, while the fourth example has a four-digit sequence of repeating digits. All rational numbers have a decimal expansion that ultimately ends up repeating a fixed sequence of digits.
An irrational number is a number that cannot be expressed as a ratio of two whole numbers. Equivalently, an irrational number is a number whose decimal expansion has no repeating pattern. How can we express or talk about such numbers? We cannot write them as ratios. And because their decimal expansions never repeat, we would have to write down an infinite sequence of digits in order to express an irrational number. One might even wonder whether such numbers exist at all.
It is said that Pythagoras first discovered that irrational numbers exist. His theorem, you may recall, states that the sum of the squares of the sides of a right triangle equals the square of the hypotenuse:
12+12=c2
Now, if the two sides of a particular triangle have sides equal to 1 (a=1 and b=1), then the square of the hypotenuse is equal to two (c2=2).
Thus, the number c is the number which, when multiplied by itself, gives two. We call this the square root of two, and write c=2.
Here is another theorem to behold:
Theorem: Let c be a number such that c2=2 (i.e. c=2). Then c is not rational.
Proof: Suppose, as a hypothesis, that c is rational. Then c is equal to the ratio of two whole numbers. Call these numbers a and b. We can thus write c = a/b. If a and b are both even, then we could simplify the fraction until at least one of them is odd. So we can pick a and b so that at least one of them is odd. Now square both sides of this equation to get c2=(a/b)2. But c2=2, so (a/b)2=2. Rearranging this, we get a2=2b2. Now, because a2 is two times another number, it must be even. Thus, a itself is even (if a were odd, a2 would be odd also since an odd number times an odd number is still odd). Because a is even, we can write it as a=2a. Now substitute this into a2=2b2 and we get (2a)2=2b2. Simplifying this, we get 4(a)2=2b2, or 2(a)2=b2. Now, because b2 is two times another number, it must be even. So b itself is even, too. But both a and b are not even because the fraction a/b was simplified. This contradiction implies that our original hypothesis was not correct. Thus c is not rational. Q. E. D.
Like the theorem that there are an infinite number of primes, this theorem is eternal. Noone will discover tomorrow two integers whose ratio squared is equal to 2. This number 2 is, and always will be, irrational. Did you know that your mind could know complete certainty about something? Moreover, did you know that you can have complete certainty about something that you cannot ever express explicitly?
Concepts and symbols are incomplete only if we think they can completely represent more than they do, e.g. if we think concepts can capture the whole Truth. But if we recognize that the conceptual representations are limited, then they are suddenly completely true. For example, if I think that a finite decimal expansion is a complete representation of an irrational number like 2, then I'm wrong and it is incomplete. But if I recognize that it is only an approximation, then it is completely true as such. Moreover, you could write down an approximation in base 10, and I in base 2, and they could both be equally accurate and true even though they look very different.
Although we can develop very general descriptions of Truth, they aren't going to be the Truth any more than a very precise decimal approximation to 2 will actually be 2. This isn't to say that we can't have an immediate recognition of Truth, though. It's just that we can't represent it completely in a conceptual system. Just because we can't write down the complete decimal expansion for 2 doesn't mean we can't have a perfect understanding of 2 through insight. To know 2 is to leap beyond not only all rational numbers, but to leap beyond all concepts of rational numbers, too. Similarly, to Know the Truth is to leap beyond all conceptual theories of Truth into an unknown and infinite realm of Reality.
The mystery of existence is so vast and infinite that it cannot be contained in or reduced to any strictly conceptual explanation or understanding. In that sense, the mystery will never be solved. To deeply understand this irreducibility of the mystery of existence, however, liberates us from our desire to reduce the mysterious to the unmysterious. Because the mystery is then embraced fully for what it is, there is no longer any need to solve the mystery. The mystery is then forever solved because it is finally recognized that it is forever unsolvable.
Although from the human perspective we experience Truth through our human psychological structures, insofar as we are a spiritual being who is not limited to a human perspective, Truth can be known directly and immediately without the distortion of mediating perceptive or conceptual faculties.
When we try to represent the Unlimited in a limited way (e.g. with words) we are not actually capturing all of it. And if we think we are, then we run into logical contradictions. The contradiction vanishes, though, once we realize that the representation is incomplete. Not holding onto statements as absolutes, the mind is then open and accepting. Here is where Love and Wisdom abide together in perfect harmony.
Conception cannot capture the One any more than a circle can be drawn that captures all of space within it. But a circle not only divides the one space into an inside and an outside, but it also connects the inside and outside to form one space. This suggests that, in addition to its mode of dividing, conception can also operate to bind and unify opposites. This is the mode of conception that can be used to catapult us beyond conception.
