A brief exploration of Infinity

By infinity, most of us would conjure up a number line or a large number to which you can add another number and make it even larger. To a mathematician, infinity is the endless extent of numbers. For others it is endless spaces and fields or even the universe; and yet others may think of the time since something happened or the long-time one will have to wait for something to happen. Therefore, when infinity is mentioned, it conjures up three images, endless numbers, space and time. There are interesting differences between the three.

If you recall school maths you will remember that this infinitely long number line is infinite in both directions – in positive as well as negative numbers. But there are many infinities within this. For example, between any two finite numbers there are an infinite number of real numbers. But if you restrict yourself to say natural numbers, there are only finitely many numbers between any two numbers, however, large!Then there are numbers and series of numbers generated by specific rules which can either diverge to infinity or converge to fixed values. But the variety of infinite series is a special branch of mathematics where mathematicians play the game of defining new ways of creating a series of numbers and then studying if it converges to a value or diverges to infinity.

There are many more aspects of the mathematics of infinity, such as number of items in a set. The concept of cardinal numbers (that measures the size of a set) can be extended to infinite sets called trans-finite cardinal numbers. For example, the set of natural numbers is a trans-finite cardinal number called aleph-null. Similarly, the cardinal number of the set of real numbers, called c (for continuum) is different from aleph-null.These are fascinating in their own right, and I would request the interested reader to read the articles by Hans Hahn  and Cheri Doucette. One of the points Hans Hahn discusses in his article is does physics work on finite world point? In other words, is the number of points in the space (uncountable) finite as some ideas of physics suggest or is it infinite? For example, if the smallest objects in the universe are not infinitely small but there is something called ‘the smallest unit of space, mass and time’, then physics is finite! While this question may seem more philosophical, at a deeper level it changes the way we formulate broader aspects of the relation between mathematics and physics. So let us look at some apparent infinities in the physical world.

There is a belief that space is infinite. To physicists, space is not infinite, just as land is not infinite to a geologist. In both cases, there is no edge or end that one can define but yet it is not infinite. To an astronomer, for example, the maximum distance between any two points is 13.7 billion light years. That is, light traveling at 3 lakh kilometres per second would take less than 14 billion years to cover the distance. This distance is increasing, but it is finite. Einstein showed that the velocity of light in vacuum is a fundamental constant and space and time must be coupled such that velocity does not change. Since it is coupled, scientists prefer to refer to space-time as a single entity rather than talk of space and time as separate entities, as we did a century ago.

In the universe, for every point there is another point that is 13.7 billion light years away as the space-time exists today. This space-time distance between points is a function of time. When the Universe was born 13.7 billion years ago, the distance between different points was zero.The image it gives is like being on an expanding sphere but the reality is more complex. For example, any sphere is a sphere in a large room in which it exists. For the universe this is not true. The expanding universe creates its own space-time.It has its own curvature and its own way of handling points at large distances.

There is an additional problem – physical space is 3 dimensional. However, there are physicists who think that in fact the universe has many more dimensions but we experience only 3, like an ant on a paper will think that the universe has only 2-dimensions. In this 3-dimensional space, we can go any way we want. But, that does not hold true for time - time flows only in one direction, past is gone, future is yet to come (and uncertain) and ‘now’ is the ever changing present.

That is what we all think. But reality is more complex.

While physically, we all know what are past, present and future, physicists are not so sure. That is because all laws of classical physics are time reversible. If you take any law of physics and put time as backwards or forwards, you get an answer that is meaningful – knowing where we are, we can find out where we came from. So it is not possible to define direction of time from purely physical principles.

The way physicists define the direction of time is to observe that everything goes from order to disorder and hence the arrow of time moves in the direction of creating maximum disorder. The definition is statistical. So if you want to create order somewhere, you have to create an even greater disorder somewhere else so that overall the disorder increases with time! This is called thermodynamic or entropic definition of time – entropy (degree of disorder) of the universe increases with time.

But planets and solid objects are, in fact, ordered. So for these to come into being, requires creating correspondingly higher disorder somewhere else by consuming a lot of energy. In the case of planets, gravity does this at the cost of binding all the mass of the planet into one strong grasp from which matter cannot come out. In case of crystals the electrical forces do the needful. A more extreme case of crystallisation is life itself. We are highly ordered and are in conflict with entropy. Nature is forever, trying to take us to disorder (from dust to dust!) and we have to continuously spend energy by eating food etc to keep the order alive. But in each case, a lot of heat must be radiated away into free space as a payment to entropy.Time therefore will become unmeasurable, when the universe is completely disordered. All stars, planets, galaxies, everything will dissolve into one huge fluff of gas, so dilute that all matter and all radiation will move without meeting anything or interacting with anything. Our recent research shows that there seems to be some ominous form of energy called dark energy which is accelerating the universe’s march towards this oblivion. The death of the universe is not going to be spectacular and there will be no restaurant at the end of the universe from where we can see the end of time.

So some of these infinities are often not infinities. To return to the question of whether physics is finite or infinite, clearly, on the large scale, space is uncountable, yet finite but for time, we are not sure.

Then there is micro infinity that asks the question,  is there an infinite amount of space between any two points, just as was the case with real numbers? We do not know and our knowledge of nature is not good enough to answer this question. For example, scientists talk of the so called Planck Length and Planck time which are the smallest units of space and time, but are they real? We also talk about black holes having zero volume and infinite density that produces an infinitely deep well in space–time. Is this infinite as physicists define it or as mathematicians understand it? The answer, I am sorry to say, does not seem to be round the corner.

But it is a fascinating subject and has kept scientists and philosophers busy. Philosophers have their own varieties of infinity, each with its own set of ideas and depth. All of these are based on instinctive ideas of space, time and numbers and some are quite ingenious.

Some ideas in Indian philosophy beautifully develop and express the concept of Infinity. The first of these sets of ideas is the infinity of time. With the concepts of yuga and mahayuga, and the repetition of these cycles, the concept of time – how long it will go on – is infinite. But this infinity is unidirectional. The beginning of the Universe is defined, for example, by Nasadiya Sukta of Rig Veda (Rig Veda X, 129) which defines the arrival of the universe from pure tapas. Panikkar, in one of the most beautiful and readable books on the Vedas titled The Vedic Experience translates it as ardor and everything, including gods and the world come afterwards.

But once born, according to conventional chronologies, the world will live infinitely long with yugas coming and going. Time therefore is essentially going to be flowing forever. There are also ideas of repetitive loops in the rendering of Ramayana. Ramanujan in his brilliant essay on Ramayana points this out. In each of these Rama tries to persuade Sita to not go to the forest with him (and fails every time) but at least in one rendering, Sita rebukes Rama by saying – look we have played out this story before and if I don’t go with you, the past stories will also have to change so just   let us get moving!

This infinity is obviously larger than the infinity of lifetimes that a soul must go through before salvation is reached. This is an individual infinity, sensitive to how one has lived the earthly life and attachments and unfulfilled desires he or she leaves behind while dying. But clearly, once the soul reaches peace he is merged into the completeness of the god (purnat purnaidam) so clearly the soul must, hopefully, have a finite life unless it remains eternally vulnerable to illusions and attachment. But even a soul, is expected to be in attendance till seven generations are born after it leaves the earth and it can attain heaven for a short period provided the life on earth was pious and the offspring perform proper ceremonies on the death, till the end of the 5 year Yuga.

The ancient philosophers also speculated about the infinity of space, where space extended in all directions from infinity to infinity. For example in the story of why Ketaki flower fell from grace there is an interesting discussion of infinities. At the beginning of time in Cosmos, Vishnu and Brahmā approached a seemingly unending column of fire – representing Shiva, and set out to find its beginning and end. Vishnu advanced to seek the end and Brahma the beginning. Taking the form of a boar, Vishnu begins digging downwards into space, while Brahma took the form of a swan and began flying upwards. However, neither could find his appointed destination. Vishnu, satisfied, came up to Shiva and bowed down to him as a swarupa of Brahman. Brahmā does not give up so easily. In his flight to the top, he sees a ketaki flower, dear to Shiva, floating down. The flower tells him that she had been placed at the top of the column of fire. Brahma's asks the flower to bear false witness and tell Vishnu that she saw Brahma at the origin. When this happens, Shiva, the all-knowing, appears and curses Brahma, that no being in the three worlds will worship him. The Ketaki flower, for bearing false witness, was cursed to be never used for the worship of Shiva. On much pleading, Shiva allowed the flower to be presented to him only 1 day a year.

As regards the infinity of numbers, the concept of unaccountably large is an old one. In Rig Veda mentions donations of 60,000 cows (RV 8.4.20) and numbers as large as 100,000 (RV.II.14.6) are mentioned.

The philosophical concept of Infinity is different. There is infinity of inner mind. In meditation, the inner world is essentially infinitely deep and takes one further and further into understanding the outer world and the Great Creation. It is an example where an internal infinity connects to outer infinity. Then there is infinity of knowledge, ideas and words, the cacophony of which keeps us busy and occupied. But behind all this vastness and infinity, there is a sense of something incomprehensible but beautiful.

The Nobel Laureate Astronomer George Gamow expressed his awe at infinity as follows:

There was a young fellow from Trinity,
Who took the square root of infinity.
But the number of digits, Gave him the fidgets;
He dropped Math and took up Divinity.

But the only infinity that worries me is infinity of human capacity for stupidity which cannot be surpassed by any of these.

Dr Mayank Vahia is a scientist working at the Tata Institute of Fundamental Research since 1979. His main fields of interest are high-energy astrophysics, mainly Cosmic Rays, X-rays and Gamma Rays. He is currently looking at the area of archeo-astronomy and learning about the way the our ancestors saw the stars, and thereby developed intellectually.   He has, in particular, been working on the Indus Valley Civilisation and taking a deeper look at their script.