Mathematics and Reality

The question is sometimes raised as to how it is that mathematics, which is a creation of the human mind, without any empirical reference to external reality, should match reality so well. When we make the distinction between the reality we experience and the underlying reality, the correlation between mathematics and reality is not so surprising.

Science takes our observations of the external world and seeks to understand how they occur and to discover underlying patterns and principles. In doing so, it inevitably draws upon experience When atoms were first imagined, they were thought of as small solid balls of matter–a model clearly drawn from everyday experience. Then, as physics realized that atoms were composed of more elementary particles (even the word "particle" contains an implicit assumption as to their nature), the model shifted to one of a central nucleus surrounded by orbiting electrons–again based on experience at the human level. Now, as we try to interpret quantum theory, we inevitably draw upon other concepts derived from our perception of reality. We interpret them as waves or bundles of energy, possessing "spin" and mass. Yet every model we come up with, fails in some way or another to capture the essence of the underlying reality.

At first we might find it surprising that the conclusions of modern physics are so far removed from our experience or reality. But it is not actually that surprising at all. All scientific models and theories have their roots in human experience. They are all based on the way the human mind interprets the incoming sensory, which is itself based on our particular, and partial, perception of the world around. What would be far more surprising would be to find that the image of reality created in the human mind was indeed a faithful representation of the thing-in-itself.

Mathematics on the other hand is purely a creation of the mind. Mathematics is that body of knowledge that is arrived at by pure reason, and does not rely upon any observations of the phenomenal world. It is free from the limitations imposed by the particular way human minds create their experience of the underlying. As such it probably the closest the human mind can come to understanding the thing-in-itself.

The only thing that pure mathematics depends upon anything is the notion of distinction. If I experience two apples I am experiencing two phenomena that can distinguished one from the other; I can eat one and keep the other. I can distinguish between the black ink and the white paper of this page. Even in the underlying reality there is distinction; we may not know what the thing-in-itself is really like, but we can measure its separation in the spacetime interval from another thing-in-itself. If there was no distinction in the cosmos, there would be no difference of any kind. No experience whatsoever. The existence of distinction is as undeniable as the existence of experience itself

If there are distinctions, we can count them. The base of the counting may vary. We use ten (probably because we have ten fingers), computers use two, the Babylonians used sixty (which is why we count sixty seconds in a minute and sixty minutes in an hour), other cultures have used five, twelve or twenty as their base.

From counting comes the concept of number, and all the integers. We can add numbers together, leading to multiplication of numbers, and the their opposites, subtraction and division. From this simple arithmetic come the concept of nothing, zero; and beyond zero, the negative numbers–not part of our direct experience, but a concept we readily accept and quite happy to work with. In between the integers we discover fractional numbers–numbers such as a half, or two thirds, which can be expressed as the ratio of two integers. Hence their name, the rational numbers.

Counting all the numbers we arrive at the notion of infinity. And between the rational numbers we discover an infinity of irrational or transcendental numbers that can be expressed as the ration of two integers. Numbers such as "pi", the ratio of the circumference of a circle to its diameter, or "e", the base of natural logarithms. They can be defined, but never written down exactly as a number for they go on forever, to an infinite number of decimal places. All this from the notion of distinction.

And there is more. Any positive number has a square root, the number that when multiplied by itself produces that number. The square root of one is one; of four it is two; and of eight it is 2.828... (another irrational number that goes on forever). But what, asked mathematicians of negative numbers, what multiplied by itself gives minus one? Nothing in the range so far discovered–any number, positive or negative when multiplied by itself results in a positive number. So they defined the square root of minus one to be a totally new number, an "imaginary" number, not part of the range or "real" numbers, and gave it the symbol "i". From this arose a new and even larger set of numbers, the so-called "complex" numbers, that were a combination of real and imaginary numbers. And these, it turned out were invaluable in helping mathematicians solve equations that had no solution in the realm of real numbers. Moreover the solutions applied to the real world.

Out of this panoply of numbers a most remarkable and intriguing relationship appeared. The irrational number "pi", the irrational number "e", and the imaginary number "i", come together in one of simplest equations ever; "e to the power of i times pi = -1".

Many mathematicians have eulogized over the significance and beauty of this equation. Out on the very edge of number theory a relationship is discovered that seems to show it is all in some way pre-ordained. Little wonder that some mathematicians feel that God is to be found in the beauty and perfection of mathematics.

That these three seemingly unconnected numbers should be related in such a simple way was startling enough; but even more was in store. This simple equation is the basic equation of any wave motion. Every wave from a wave on water, the air waves coming from a violin string, to light waves, can be expressed as a combination of simple equations of this form. It also expresses the orbits of the planets, the swing of a pendulum and the oscillation of an atom. In fact, every single motion in the cosmos can ultimately be reduced to an equation of this form. The whole of quantum physics depends upon it. If mathematicians had not discovered this most remarkable relationship, the strange story of the quantum would never have been told.

And all of this without a single empirical observation. No wonder then, that in the end all science comes down to mathematics. The very fact that it is not based upon phenomena, is why it is probably the best approximation to the underlying reality we have.



When we are under the illusion that my image of the world is the real world, we may assume that many of the qualities of the phenomena we observe are independent realities existing in the noumena.

For a long time it was assumed that space and time were fundamental to the underlying reality. Einstein's Theory of Special Relativity came as a great shock to this assumption. He showed that what we observe as space and what we observe as time are but two aspects of a more fundamental spacetime continuum. What this continuum is like we never know -- in this respect it sounds very much like Kant's noumena. All we ever know of it are the ways in which it manifests as the two very different qualities of space and time. Moreover, how much of the continuum manifests as space and how much manifests as time varies according to the relative motion of the observer.

Space and time have fallen from their absolute status. They are both created through the act of perception, and so belong to the relative world of experience. This is not to imply that they are not fundamental to our experience; they are the dimensional framework within which we structure our mental image of the world. But we deceive ourselves when we assume that they are also fundamental to the underlying reality.


The Reality of Light

In proposing his theory Einstein postulated that the speed of light was a universal constant. However fast you may be moving relative to a light beam, you will always measure the speed of light to be the same -- 186,000 miles per second. Even if you are moving at 99% the speed of light, a light will still appear to travel past you at 186,000 miles per second.

Although this is totally counter-intuitive, experiments show that it does indeed seem to be the case. This raises two difficult questions: How come the speed is always the same? And why is light so special? When we distinguish the image of reality from the underlying reality, the apparent constancy of the speed of light takes on a very different nature.

According to Einstein's equations, as an observer's speed increases, time slows down, and space (in the direction of motion) contracts. At the speed of light, time has slowed to a standstill and space contracted to a point. Although no object with mass can ever attain the speed of light (Einstein's equations predict that it would then have an infinite mass), light itself does (by definition) travel at the speed of light. From light's point of view it has traveled no distance, and has taken no time to do so.

This reflects a unique property of light. In the spacetime continuum there is no separation between the emission of a light ray and its absorption. What Einstein called the "spacetime interval" bewtween the two ends of a light ray is always zero.

How should we understand this? The answer is that we probably should not even try to understand it. Any attempt to do so would once again fall into the mistake of applying concepts derived from our image of reality to the underlying reality. All we need to recognize is that from light's perspective it traverses no spacetime interval.

However, when we perceive the world from our human frame of reference we do indeed observe a separation between the two ends of the light beam -- the exact amount of separation depending upon our speed. We could say the act of perception "stretches out" the zero interval, and divides it into a certain amount of space and a certain amount of time. Since the total interval remains zero, the amount of space created exactly balances the amount of time created. For every 186,000 miles of space, we create 1 second of time.

What we conceive of as the speed of light is actually something completely different. From light's point of view -- and this after all must be the most appropriate perspective from which to consider the nature of light, not our matter-bound mode of experience -- light travels no distance in no time, and therefore has no need of speed. What we take to be the speed of light is actually the ratio in which space and time are created in our image of reality. It is this ratio that is fixed -- and this is why in the phenomenal world the apparent "speed" of light is fixed.


Wave-Particle Duality

When we recognize that in the real world light does not travel across space or time a difficult conundrum in quantum physics becomes much easier to understand. In our image of reality we observe energy traveling from one end of a light ray to the other. It is only natural to ask how the energy travels: Is it a wave? Or is it a particle? (Two models both drawn from our image of reality.)

The answer, it appears, is both. In some situations light behaves as a continuous wave spreading out in space -- but a wave without a medium. In other situations it behaves as a particle traveling through space -- but a particle without mass. Physicists have accommodated these two strange and seemingly paradoxical conclusions by deciding that light is a "wave-particle." In certain circumstances it appears as a wave; in others as a particle.

But if we look at things from light's point of view, it is neither. Since it did not travel through space and time, it needed no vehicle or mechanism of travel -- it has no need to be either a wave or a particle. As far as light itself is concerned, there is no duality, no paradox.

The physicist's conundrum appears only when we mistake our image of reality with the "thing in itself", and try to visualize light in concepts and terms appropriate to our image of reality -- i.e., waves and particles.


The Material World

A second conclusion of Einstein's Special Theory of Relativity is that matter and energy are related to each other in a similar way as are space and time. Atomic physics had already shown that solid matter did not really exist, our experience of solid substance being an appearance generated in the mind. Einstein's work went further, showing that matter does not exist in the real world as an independent substance. What appears to us as matter and energy are bound together in his famous equation e=mc2.

More fundamental than both matter and energy is action. Planck laid the foundations of quantum physics with his realization that the indivisible unit in the physical world, the "quantum" as he called it, was action -- .

When we speak of the material world we usually think we are referring to the underlying reality -- the world that we are perceiving "out there". In fact we are only describing our image of reality. The materiality we experience, the solidness we feel, the whole of the "real world" that we know are all aspects of the image created in the mind; they are part of our interpretation of reality. Paradoxical as it may sound, matter is something created in the mind.

When we realize that everything we know, including the whole material world that we experience "out there" is part of the phenomenon, the image constructed in consciousness, we find the truth is a complete reversal of our everyday view. Matter, as we know it, is a creation of consciousness. Not the other way around as contemporary science presumes.

Thus the ultimate nature of reality -- the reality we experience that is, not the reality of of the noumena, of whose nature we have no knowledge -- is consciousness. Space, time, matter, energy -- the whole substantial world built up from our sense perceptions -- is created within consciousness. The essence of this whole phenomenal world is not matter but consciousness.

The Fabric of Reality

Everything we know is part of the picture of reality arising in consciousness. This is true not only of the objects we experience in the world around; thoughts, feelings and ideas are likewise manifestations within consciousness, and so are the theories we construct about the nature of the world around. Everything we know is structured in consciousness.

Consciousness is the fabric of reality. It is the medium from which every aspect of our experience manifests. Color, sound, taste, smell, space, time, matter -- every quality we ever experience in the world is a form or quality within consciousness. Our entire image of reality is generated in and from consciousness.

Similar claims have sometimes been made by spiritual teachings -- probably most coherently by the Vedantic philosophy of ancient India. The Western scientific mind has usually dismissed such suggestions, since they seem to make no sense whatsoever. But they are only nonsensical if we confuse the two realities and think these ancient philosophers were speaking of the underlying reality (of which we cannot, of necessity, know or speak). If we consider the reality we experience, then we have to accept that in the final analysis they are correct: Consciousness is the essence of everything -- everything in the known universe.

The Hard Question

When we look at the world, we do not see consciousness "out there". All we see are the various forms and qualities that consciousness has taken on. To us the "material world" appears to be devoid of consciousness.

The reason we do not find consciousness in the world we observe is because consciousness is not part of the picture generated in our minds. It is the canvas on which the picture is painted. But when we mistakenly assume that the picture of reality painted in our mind, is the underlying reality, we find ourselves presented with a very difficult question regarding consciousness: How does conscious experience arise or emerge from matter? This is the so-called "hard question" to which many scientists and philosophers are currently devoting considerable time and attention.

The hard question that these people think they are asking is: "How does the noumena give rise to consciousness?" But knowing very little of the noumena, we are not really in any position even to ask this question.

The question these people are actually asking has more to do with our image of reality than the fundamental reality. They are asking how it is that a complex network of neurons can give rise to conscious experience. How does something as immaterial as consciousness arise from something as unconscious as the material world? Is it a result of the complex patterning of data across the neural net? Is it due to quantum coherence effects in microtubules within the neurons? Or is it something else?

What all these approaches have in common is that they are trying to explain consciousness in terms of phenomena that belong to our image of reality, which is itself a manifestation within consciousness.

The so-called "hard question" is actually a mistaken question. When we distinguish between the two realities, the question disappears to be replaced by its opposite: How is it that matter, space, time, color, sound, form, and all the other qualities we experience emerge in consciousness? What is the process of manifestation within the mind?


Locating Consciousness

Another recurrent question concerns the seat of consciousness. Where is the self, our sense of "I-ness", located? Is it in the brain? If so, where? Despite much thought and discussion, no one has yet come up with clear answers to such questions.

As with some of the other problematic issues we have looked at, this one too stems from confusing the two realities. The question that is actually being asked is "Where is consciousness located in our image of reality?"

There are two answers to this question.

On the one hand, consciousness is not located anywhere within the world; the whole world -- our entire image of reality, including our bodies and brains -- is itself a manifestation within consciousness. Consciousness is the container of our world; it is not contained within it.

On the other hand, we do clearly experience ourselves to be located somewhere within that image. We have created this image of reality and have quite naturally put ourselves at the center of this image. The whole world we have constructed is constructed around a central point, the center of our perception.

The central point of most of our sensory experience is somewhere in the middle of the head. We see ourselves to be somewhere behind the eyes, and hear ourselves to be somewhere between the ears. This is where we quite naturally place ourselves within our image of reality. Since the brain is also located in the middle of the head, it is easy to assume that consciousness is somehow located in the brain. But this need not necessarily be so at all.

Imagine your brain being located in your pelvis. This would not change your experience of being somewhere behind the eyes and between the ears.

In short, consciousness is not located anywhere within the world, it is that within which the world is located. But we create a sense of location for ourselves within our image of the world by placing ourselves at the center of our perceived world.



We are now in a better position to understand two recurrent themes in spiritual experience. Throughout human history there have been mystics of one kind or another who have proclaimed that "I am God", or words to that effect. To the ears of established religion this has often sounded like heresy; "How can this lowly individual claim that he (or sometimes she) is the almighty, eternal creator?" Heresy enough to get one imprisoned, tortured, or even burned at the stake.

Such people are not necessarily deluded zealots; they are usually people who have spent considerable time exploring the depths of human consciousness, and their realizations are not to be lightly dismissed. If we look more closely at their statements, what they seem to be saying is that the "I", that innermost essence of ourselves, that pure consciousness that lies at our core, is a universal essence. Whatever we may be conscious of, the faculty of consciousness is something we all share.

This consciousness is the one Truth we cannot deny. It is the absolute certainty of our existence. It is eternal in that it is always there whatever the contents of our experience. It is the essence of everything we know. It is the creator of our world. This is the "God" that we intuitively knew existed, but never quite found.


A second recurrent theme in mystical literature is the knowledge of being one with all things, the realization that "I am the Universe", that all is me, and all is in me. As before, these are not necessarily the ravings of a deranged mind. In most spiritual traditions they signify a high state of consciousness, and generally come from adepts with many years of inner exploration. It is far more likely that they represent people who have experienced first-hand that the entire universe -- everything we know from the cells in our bodies to the distant twinkling stars -- exists within the mind, not the other way around.

Far from suffering from an illusion, a person in this state is knowing the phenomenal reality for what it is. It is we who are under an illusion when we believe that the world we see around us is actually around us, not within us. These inner explorers have discovered that it truly is "all in the mind".


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