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The goal of both science and philosophy is to explore and understand the Reality behind all existence. From the very dawn of civilization man has tried to know the nature and the universe it is embedded in. Philosophers and scientists alike have probed into the secrets of nature and the origin of the universe has been the focal point of their investigations. Philosophers have their conjectures and scientists have their theories but no one has been able to grasp the reality through intellectual efforts. According to the generally accepted theory in cosmology the universe grew impulsively from a concentrated point source of infinite energy called space-time singularity. In spite of the general acceptance the theory has many problems, one of them being the very nature of this cosmic singularity. Singularity is a mathematical concept that brings together two other mathematical concepts - zero and infinity. Although mathematics regards these as the two extremes of the number system, they have nothing in common with numbers. By very definition numbers are quantitative. They have meaning only in relation to some objects. By themselves they mean nothing; five has nothing five-like, it is only a symbol. Zero and infinity have no quantitative aspect to them, they can be referred to only qualitatively. Each is an indivisible whole. They do not depend on other objects for their meaning. In other words they are absolute not relative. Any number multiplied by zero or infinity loses its identity, it merges into the multiplier. Any number divided by zero yields infinity, but divided by infinity yields zero. This process of division is like flipping a coin having zero and infinity as the two sides. The same idea is expressed in Vedanta from a different perspective.

In Vedanta zero and infinity are regarded unknowable and therefore explaining and defining them is out of question. Strictly speaking, even in mathematics zero and infinity cannot be arrived at; a variable can only tend to either. Looking for the smallest number greater than zero or the largest number smaller than infinity is futile. For any unknowable we cannot say what it is; it can be described only in terms of what it is not. That is why in answer to the question what is God (Brahman or Brahm) the Vedas say ‘not this, not this (neti neti)’. The Sanskrit word, from which zero and all its equivalents in other languages are derived, is shunya. It is far more comprehensive in meaning than zero; it means zero only in the context of numbers. Its general meaning is void or emptiness, a concept that the mind cannot grasp unless it is itself empty (the initial premise of yoga philosophy). The same is true for infinity. In Sanskrit there is no single word for infinity. Several words like anadi (without beginning), ananta (without end) etc. are used. Upanishads characterize Brahman as ’smaller than the smallest and larger than the largest’ [1]. In the usual sense in which the words small and large are used, this statement may seem to be a contradiction in terms. This is because our concept of small and large is tied to the perception of space and time. Small and large are a pair of opposites, which according to Vedanta are two faces of the same coin. Opposites are illusion produced by language. Small is contained in the large and large is contained in the small in a latent form. As Kabir, one of the greatest Indian mystics has said: “The drop merges in the ocean. Where do you search for the drop? The ocean merges in the drop, where do you search for the ocean? The end of the search is the realization that each is in the other” [2] (translation is mine). The concept of space-time leads to that of causation - cause preceding effect. The sum total of space-time and causation is maya that obscures the Reality [3]. Being absolute, zero and infinity are two aspects of the one Ultimate Reality. In order to realize them one has to transcend space-time.

The singularity brings zero and infinity together. The cosmic singularity combines zero space-time and infinite energy. In Vedanta the primordial state of existence of the Reality is an infinite emptiness (shunyata). When the mind is completely empty and one transcends space-time in deep meditation, one is able to realize this state. However, this emptiness is not nothingness or non-existence. It is pervaded by energy and consciousness. We have arrived at the same concept coming from quantum physics; the quantum void is not empty. There are myriads of virtual particles appearing and disappearing constantly and there is an all-pervading cosmic energy. This state is the substratum of all existence and is called the ‘ground’ by mystics as well as physicists [4]. The dictionary meaning of singularity is ‘the state of being unique or only one of its kind’. In that sense we can view the Ultimate Reality also as a singularity. In cosmology a ‘naked singularity’ cannot be observed or probed because it is enveloped by the event horizon. From Vedanta’s point of view we cannot see the Reality because it is veiled by maya. The laws of physics break down at the singularity; the rational knowledge cannot penetrate through the veil of maya. Even though space-time goes to zero at the singularity, the concept is still wrapped up in space-time; it is defined as a point singularity. We cannot transcend space-time through the rational mind.

We exist in space-time. We as well as everything else in the universe are simply events in space-time. Every event has a lifetime and on cosmological time scale most of the events are transient. Anything that has name and form has a beginning and an end in space-time. How do we transcend space-time? Physically we cannot. But what one calls ‘I’ is not the physical body, ‘I’ just lives in it. Besides the gross physical body there is also a subtle body. It is the subtle body that can go beyond space-time and experience things outside the phenomenal world. When I think of an object I am already bringing in a separation, the object is something different from myself, the subject. Space-time implies separation - distance between two points or two events. When every object is viewed as an event and the object vanishes, space-time also vanishes. The vanishing of the subject-object duality essentially means that the person has merged with the universe. In this state one has transcended space-time and zero and infinity have come together. The Upanishads describe this as the state in which a person sees everything in himself and himself in everything [5].

As we have seen, zero and infinity are indivisible. The concept of undivided wholeness is basic to Vedanta and it expresses the omnipresence of God. Again the Upanishads say: ‘He moves and remains still at the same time; He is near as well as far; He is inside as well as outside’ [5]. Everything in the universe is God because it is simply a manifestation of that Ultimate Reality. The first of the four great sentences (mahavakyas) of Vedanta is ‘I am Brahm (Aham Brahmasmi)’. If I am everywhere at the same time, for me there is no space and there is no time. Still there is movement because Brahm is consciousness, which is dynamic. However, now the concept of movement is totally different; space and time are not involved in it. The concept of non-locality in modern physics has come very close to this viewpoint.

In the end we have to think about the distinction between the reality and the knowledge of reality. The uncertainty principle has ramifications far beyond the realm of physics. Any theory, scientific or philosophical, presents only a model of the reality based on the interpretation of the facts known about the reality. A model by very definition is simply a representation of the actual thing, not the thing itself. Just as in mathematics a variable can only tend to zero or infinity, the model is forever trying to get closer to the real. One can never be certain about the degree of correspondence between the knowledge and the reality. The knowledge is a bunch of ideas about reality; the ideas cannot be the reality. The situation is far more complex when it comes to knowing the Ultimate Reality. According to Vedanta God is pure consciousness, which is in every living being. So everything that we perceive or know is through Him; He is the ultimate knower. The subject-object duality precludes the possibility of being conscious of our own consciousness in the same way, as we are conscious of an object. How can the knower be known - is a famous exclamation in Vedanta [6]. Knowledge has two aspects, rational and intuitive; one comes from logic, reasoning, and intelligence, the other from insight and realization. Rational knowledge can never help one uncover the ultimate truth.

References

1. Svetashvatar Upanishad, 3-20; Kathopanishad, 1-2.20.
2. Kabir Granthavali (Complete Works of Kabir), pp. 169-170,

ed. R. K. Sharma.
3. The Complete Works of Swami Vivekananda, Vol.2, p.135.
4. J. Krishnamurti and David Bohm, The Ending of Time, p.96.
5. Ishopanishad, 5, 6.
6. Brihadaranyak Upanishad, 2-4.14

Dharmbir Rai Sharma is a retired professor with electrical engineering and physics background. He obtained his M.S. degree in physics in India and Ph.D. in electrical engineering at Cornell University. He has taught in universities here and also in Brazil, where he spent sometime. He maintains a website http://www.cosmosebooks.com devoted mainly to philosophy and science.

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In part one of this article we discussed numbers in nature and their possible philosophical significance. In this part we consider some other unusual aspects of numbers.

Zero and Infinity

Unlike other real numbers zero and infinity do not have any quantitative attribute. Strictly speaking they are not numbers although mathematicians do consider zero as a finite number. The statement that I have five dollars makes perfect sense but ‘I have zero dollars’ does not. If asked to I can show the five dollars but cannot show the zero. So zero is just a concept like infinity. The concept of a thing is not the thing itself but only an attempt to describe what it is.

Zero and infinity figure prominently in philosophy, especially in Eastern philosophies. They are considered two sides of the same coin, as are all pairs of opposites. They are interlinked like the two ends of an infinite spiral and also contained in each other. We cannot go into the details of these philosophical arguments here and simply mention that the Ultimate Reality is described as ’smaller than the smallest and larger than the largest’ (Vedanta) and ‘nothing is everything’ (Tao).

Irrational and Transcendental Numbers

Numbers that cannot be expressed as a ratio of two integers are called irrational. I suppose they are called irrational because of the belief that the only rational way of expressing a number is in terms of two other whole numbers. Their decimal expansions do not terminate nor become periodic. The most well known irrational number is the square root of two. These numbers are, by definition, indeterminate, although geometrically one can get the value of square root of two by measuring the hypotenuse of a right-angle triangle with height and base equal to one. However, as we shall see later, it brings in another problem - the error of measurement.

As one can guess from the name itself, the definition of transcendental numbers is a little complicated. (These are numbers that are not roots of integer polynomials so they are not algebraic numbers of any degree.) All transcendental numbers are irrational. Frankly I do not know what they transcend other than an easy definition! Anyway, two are most common. One is associated with circle and is denoted by the Greek letter Pi; the other is denoted by e for exponential and is associated with logarithm. Another such number is the so-called golden ratio denoted by the Greek letter Phi, which is apparently favored by nature and can be traced even in human anatomy. Architectures using this ratio are aesthetically more pleasing. It also related to Fibonacci sequence discussed in part 1 of this article.

Approximations and Errors

Since the values of irrational and transcendental numbers cannot be determined precisely, one has to resort to approximation. Any approximation has some inherent error. This means that the use of these numbers will not yield a unique result. The circumference or the area of a circle cannot be determined precisely because it involves multiplication by Pi. The exponential growth of any variable quantity cannot be calculated exactly because of the use of e. The same is true for using the golden ratio. If we use geometrical methods of determining the values of the square root of two or the circumference of a circle, the approximation error translates into measurement error which can never be completely eliminated.

This takes us back to what was mentioned about zero. No matter what we do we can never get to zero. The same holds for infinity. This bears an analogy to the spiritual goal of reaching the Ultimate Reality. If this goal is nirvana, it cannot be attained while living. One can only strive to get as close as possible and the closest possible approach is called enlightenment.

Dharmbir Rai Sharma is a retired professor with electrical engineering and physics background. He obtained his M.S. degree in physics in India and Ph.D. in electrical engineering at Cornell University. He has taught in universities here and also in Brazil, where he spent sometime. He maintains a website http://www.cosmosebooks.com devoted mainly to philosophy and science.

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In continuation of my discussion on infinity and its implications with the divine, I should mention that the concept of there existing infinities beyond imagination is quite difficult to comprehend. If you read my poem, “How Can this Be? you read in verse the proof that shows clearly that there is no such thing as one kind of infinity. (See my ezine article “How Can this Be”) The extension of this most curious fact is that there are actually an infinite number of infinities!

Occasionally I wax metaphysical in conversations with my uncle and the other evening we were discussing some points regarding the spiritual realms. In passing, I brought up the topic of infinity and I asked him his impression of it. His response, which is typical of most people, is that infinity is just thatinfinity: something that never ends. But how do we make this vague notion somewhat more concrete? I pointed my uncle’s attention to the set of natural, or counting numbers. This set comprises the familiar numbers {1, 2, 3, …}. The numbers go on and on, falling like dominoes, and never reaching a “biggest one.” This process is easy to grasp and presents no ordinary difficulty for the average person. What does become difficult to understand is why the infinity typified by this set of numbers is not unique.

Now let’s delve a little more deeply into this curious set of numbers and the topic of infinity in general. This set of counting numbers obviously never ends. If you have ever seen a chronometer counting hundredths of a second, then you have seen how fast the digits representing the hundredths of a second whiz by, not appearing for any length of time sufficient to allow recognition of the appropriate digit. And this is for hundredths of a second. Imagine the same chronometer counting off thousandths of a second. Now imagine this going on from, let us say, ten thousand years ago and continuing for another ten thousand years, starting with 1 and such that each thousandth of a second would represent the next sequential counting number. Think of how far along in the set of counting numbers you would be. We could actually compute the number but we are only interested in trying to conceptualize how large potential infinity could be.

Now that we have this huge number in hand, we could do whatever we wanted with it to project ourselves much further out in the set of counting numbers. We could multiply it by itself ten times (the mathematical way of saying we can raise the number to the tenth power); we could multiply it by itself a hundred times, a thousand times, and so on. We could then take the largest product and do the same process all over again. How big is this set?

This set is so largenever ending in factthat we should be able to use it to compare to anything else that is infinite, right? Wrong. And in a continuing article on this most fascinating subject, I will discuss how this notion of one universal infinity is completely wrong. Thus, if sets of numbers can shatter our preconceived notions of a concept like infinity, which is more or less universally accepted as something that is real, what more can we uncover by plunging into the mysteries of numbers and mathematics in general? Stay tuned……

Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Under the penname, JC Page, Joe authored Arithmetic Magic, the little classic on the ABC’s of arithmetic. Joe is also author of the charming self-help ebook, Making a Good Impression Every Time: The Secret to Instant Popularity, the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)?particularly in regard to its educcational flavor? continues to captivate readers and to earn him recognition.&

Joe propagates his teaching philosophy through his articles and books and is dedicated to helping educate children living in impoverished countries. Toward this end, he donates a portion of the proceeds from the sale of every ebook. For more information go to http://www.mathbyjoe.com

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