Galileo's problem!

Often  we end up with exactly the opposite of what we are seeking. In the case of mathematics this is no different. Georg Cantor, was a religious man. In Halle (Province of Saxony, Eastern Germany), he often visited the Church regularly. But, the person nailed to the cross with a crown of thorns was not really his god. He was not interested in expiation of sins and redemption. Even as a child, Cantor had heard this voice, that urged him to take up Mathematics. To him, this voice was the voice of God. God to him was the one who set about the planets and the galaxies in motion in the Universe. The one who had set up immutable laws that governed the universe. And Cantor believed that it was his responsibility to seek these laws. To seek the deepest mysteries of Mathematics. What he really sought to find was certainty. But certainty does go very well with the concept of Infinity.

Probably the first modern thinker who confronted the concept of Infinity head on was the Italian scientist Galileo. Galileo came up with a simple thought experiment. He said "Consider a circle. Within draw a triangle. And then draw a square. And so on. In other words, a circle has an infinite number of sides within. You could draw an infinite number of sharp lines from the center of the circle to the infinite number of points on the circumference of the circle." Now this can be understood quite easily. There is no problem here.

However Galileo thought of another experiment that made him wish, he had not thought about it. Seeking to further understand the concept of infinity, he ran straight into a paradox. This is what perplexed Galileo. Now take a circle B. With an infinitely sharp pencil, draw an infinite number of straight lines from the center of the circle to the infinite number of points on the circumference. This is what he did earlier. Now consider a bigger circle A around the smaller circle B. Extend the infinite number of straight lines from the smaller circle to the larger circle. You observe that the lines diverge leaving gaps in circle A. This to Galileo did not make sense. The bigger circle needs an infinite number of straight lines greater than the infinite number of straight lines in the smaller circle. In other words, there is an infinity bigger than another infinity. A hierarchy if you will of infinities now exists. This was so perplexing to Galileo, that he declared that "Yes there exists infinity, however it is not for our finite minds to understand. It can only be discerned by God and God alone!"

This great scientist who was also fighting his own private battles with the papacy and the Vatican, decided not to further contemplate on this issue. He already had enough problems and was being branded a heretic. Not many scientists and mathematicians prior to dawn of renaissance  dared confront this issue.

Many years later, a mathematician from Halle decided to take up this problem. Georg Cantor, was a religious man. To him infinity meant the realm of God. It was an essential nature of the creator. He had heard voices from childhood days beckoning him to take up Mathematics. He firmly believed that he was a messenger of God, who was waiting for him to unearth the hidden secrets of realm of the heavens. It was with this intention that Georg Cantor took up the study of trans finite and infinite sets and wrote his first paper on the same. However as he delved deeper and deeper into the concept of infinity seeking solace and certainty he got exactly the opposite. This quest would finally land him in a lunatic asylum. But what he saw was so profound that it transcended mathematics and logic. In the next post, I shall describe Cantor's initial work and probably a time of his life when he was the happiest.

Infinity in many cultures

"How can I - or anyone else - ever cease being astounded
That He whom nothing can contain is contained in the heart?"
                    -- Jalaluddin Rumi

Sufi mysticism reached it's zenith in the twelfth century AD at Anatolia ( Modern Turkey) in the works of Jalaluddin Rumi. Escaping a life threatening situation in his native Balkh in Afghanistan, which was then under the invasion of the murderous Genghis Khan, Jalaluddin had settled in Anatolia as a silent Islamic preacher. That was until he met a spiritual master who was a wandering dervish called Shams of Tabrizi. Then he underwent a spiritual transformation and reached heights of spirituality, composing a great body of spiritual poetry and prose. Rumi often reached heights of divine love and Ecstasy which few Sufis could boast of. In such a state, he describes the wonder and awe he feels in the above verse. He says in wonder, how is it that, God, who is essentially formless and boundless cannot be contained by anything. He means, God is not a finite entity. His presence is not bounded. He is infinite. But Rumi wonders, that how God, who cannot be contained by anything, is contained in his heart. For Rumi is intoxicated in his love for God. There is nothing in his heart, but his beloved (God).

Pretty much like this, one can find references to infinity in the most unlikely places (As the western orthodoxy would think) like in this instance Islamic mysticism. Whatever was the case, be it religion, mythology or mysticism people more often than not encountered infinity.




For instance, in the mythology of ancient Egypt one encounters the God Heh, who was the god of Formlessness and infinity. He represented the boundless aspect of the universe. He was shown as a crouching man holding out two palm ribs in his hands, each of which terminated with a tadpole and a shen ring. The shen ring was a traditional symbol of infinity. The shen ring is at first glance a circle with a horizontal line in a tangent along its bottom edge. The circle, in most cultures, represents "eternity" or infinity as it does in this case.

The symbol for eternity or infinity in Zen Buddhism is the enso, which is verily a circle.


Coming back to Mysticism, in the philosophy of Lao Tse, Taoism one finds many references to the infinite nature of the Tao (The Tao being the one source of all).

The Xiseng Jing, the Taoist mystical book describes the Tao thus :

                                "The Tao is without shape or end" .

Thus, the mystical text goes onto say that Tao, the source of all, the essential nature of all that is, is without shape (Formless) and without end (boundless or infinite).


While in ancient Greece the earliest attestable accounts of mathematical infinity come from Zeno of Elea (ca. 490 BC? – ca. 430 BC?), a pre-Socratic Greek philosopher of southern Italy and member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".It is also known that Archimedes had an intuition for infinite quantities.

Thus mankind for a long time has encountered infinity many times over. Some have looked at it with wonder and awe. Some have been baffled. Some saw it as an attribute in their god. Some in their philosophy. I would like to end this post by quoting a beautiful mystical saying of the Tao Tse Ching which describes the infinite aspect of the Tao.


                             Look at it: nothing to see.
                                  Call it colorless.
                            Listen to it: nothing to hear.
                                 Call it soundless.
                            Reach for it: nothing to hold.
                                 Call it intangible.
                              Triply undifferentiated,
                              It merges into oneness,
                                 not bright above,
                                  not dark below.
                                  Never oh! never
                                 can it be named.

                                It reverts, it returns
                                     To unbeing.
                               Call it form of the unformed,
                                  the image of no image.
                                Call it unthinkable thought.
                                   Face it: no face.
                                 Follow it: no end.
                               Holding fast to the old Way,
                                We can live in the present.
                             Mindful of the ancient beginnings
                               we hold the thread of the Tao.
                            (Tao Te Ching, 14, emphasis added)

The nature of Infinity - The earliest conception

Om poornamadah poornamidam
Poornaat poornamudachyate
Poornasya poornamaadaya
Poornamevaavashishyate

Om shantih shantih shantih

-Ishavasyopanishad

Om (The primordial Sound), That (God) is Infinite
This (The self) is also infinite
From that infinite comes this infinite
(And how is it possible), because Infinity - Infinity
Is still Infinity!

Mathematicians, Philosophers and physicists embarked on a quest for the truth with a missionary zeal after the Renaissance. If something did not or could not be explained by reason, it was ignored. However, there was one concept which most preferred to hide under the rug. The concept of Infinity! No matter what they do, they were always confounded by the very concept of a boundless quantity. So far, it was merely something that was like an out caste. It was loathed and despised by one and all. That is, until Georg Cantor came to the scene and started working on Set theory. In his work, he inevitably started encountering infinite or transfinite (Not finite but not absolutely infinite) sets, like the set of real numbers, the set of natural numbers etc. But was this the first time ever that mankind ever stumbled upon this concept? Most modern scientists who share a disdain for all forms of mysticism  would be most baffled where this concept can be encountered. For most people in the modern world, at least in the Western society and  for those who are not acquainted with the manifold forms of Eastern mysticism the truth would be like running headlong into a speeding truck. The most unlikely place, is the ancient Vedas(The Sanskrit word véda "knowledge, wisdom" is derived from the root vid- "to know), the holy scriptures of the Indus Saraswathi Civilization, whose actual date is not established. In these ancient works, in particular in a most important Upanishad (The Sanskrit term Upaniṣad derives from upa- (nearby), ni- (at the proper place, down) and sad ("sitting down near"), implying sitting near a teacher to receive)instruction or, alternatively, "laying siege" to the teacher.) called the Isha Upanishad which appears in the Yajurveda. At the beginning of which is a Shanti mantra (A hymn invoking peace) which is what you read when you started reading this post!

I initially thought of writing on just the nature of infinity and then in passing mention the mantra, as it was one of the earliest known recorded verses containing within it this concept. Then I felt the need to explain it in detail, for each time I thought of infinity, I started contemplating on this mantra. This led me to write an entire article on the same. I hope my atheist and agnostic friends or friends not following Eastern mysticism will not declare the atheist "Fatwa" on my article and stop reading my posts. I beg them to open their mind, and look upon this from a purely historical and scientific perspective. I ask them to remove all their prejudices, and to look at the concept with an open mind. Then they are welcome to form their own conclusions.

Let us analyze the shloka. The word Poorna here is taken to mean infinity. But did the vedic seers actually mean just Infinity? Poorna actually means complete or whole. Where then is the concept of infinity here? If one were to even glance superficially at the Vedas and the Upanishads they would understand the connection. When describing the one god (Parabrahma), the Shanti Mantra and in fact the entire Isha Upanishad goes onto say, that he (Or She) is complete in itself. That is, god is complete in terms of explanation. Once this state is reached, or once one knows God, there is nothing beyond. Every conceivable thing is covered and all answers to all questions are found here. How many questions have we got about existence? How much do we know? And isn't infinity also one of the concepts or big questions that baffles us? All this is within the completeness. This completeness encompasses everything, including the concept of infinity. So the word poorna describes something that cannot be completely comprehended with any word corresponding in the English language. The closest would probably mean complete, but it is not just as ordinary as that word sounds. It  encompasses everything.

Next the shloka says, this (The self) is also Poorna (Infinite). There are several interpretations to this. One is that, the world around us, exists but in our conscious or waking state. When we are in deep dreamless sleep or when we are fully enlightened as to our true nature, we reach the same state as that of Parabrahma  (God). This philosophy was later expounded by  none other than one of the greatest philosophers in India, Shree Adi Shankaracharya as Advaita (Non-Duality). It goes on to stress the fact that Brahman (God) is the only reality and all else is an illusion. And that the self is one with Brahman (God). Hence this Upanishad says, that the self is also poorna or infinite.

The question that might come to anyone's mind is, how can one quantity which is infinite come from another quantity that is infinite. The shloka goes onto say infinity - infinity (Poornasya Poornamaadhaya) is equal to infinity. This is a mathematical fact which most high school students learn these days. But this was a fact noticed and recorded by the Vedic seers eons ago, probably around 6000-8000 years ago. The age is so estimated because, the oldest of the Vedas, the Rig Veda, describes a flowing Saraswathi river. The Saraswathi, now scientists know is an underground river, which used to flow about 6000-10000 years ago. Hence we  know how ancient the Vedas are, and therefore I conclude that this is the oldest known observation on the nature of Infinity! And since the Vedas are known to contain within them the essential truth, the one truth, the answers to all questions, they may verily contain what has been sought and  people are even now seeking. Answers to the meaning of life. With this I leave you to muse on the ancient Vedas and lying therein the concept of Infinity. I shall in the next article describe other cultures where this concept has appeared before talking about the work done by Georg Cantor.

Introduction: In search of Certainty

Man has always sought answers to questions, questions about himself, about the universe he lives in, about the past and the future. He has always sought certainty in a very very uncertain existence. The man living in the wilderness was never sure when he would be attacked by a wild beast and when his life would end. In his quest for certainty, man often went on a quest that taxed both his rational mind and his intuitive abilities. The moment he observed the heavens he noticed t that the position of stars across the night sky coincided with certain weather patterns. After astronomy, meteorology was probably one of the oldest of the sciences developed. One can find astounding astronomical feats in the brilliant calendars developed by the hoary Vedic civilization in India and the brilliant calendars of the Mayans and the other indigenous civilizations in South and Central America.

Often mankind has settled for certain theories which it thought explained everything or well, almost everything. The Greeks thought the Gods of Olympus controlled pretty much everything that happened on earth. The flash of lightning and thunder could always be explained away as the mighty thunderbolt of Zeus. The Aztecs thought that the Sun traveled in a boat around the Earth and for him to rise every morning, a human sacrifice was necessary. Different beliefs, different religions, different sects came one after the other. Every time man thought a certain belief system explained everything, a few daring people found flaws in it and went on to try to find systems which they felt explained everything. The quest has been going on and on.

The holy Roman Empire established Christianity and Holy Bible explained everything about the known universe. The papacy came to accept Ptolemy's model of the heavens (Now popularly called the Geocentric system) as the theory that explained the Universe in it's entirety. However, Nicholas Copernicus,Galileo Galeli challenged this,and proved that there were flaws in the heart of this theory. With the onset of Renaissance, the quest for knowledge from a rational perspective became the most prominent way of seeking the truth. Intuition was pushed to the background. Francis Bacon and Rene Descartes's works show a skepticism at everything. If it cannot be proven from known axioms, it cannot be held as true. With this system flowered science and maths. The western scientists, philosophers and mathematicians believed they took over from where the ancient Greeks had left off.

Thus came Newtonian mechanics. People looked upon the universe as a clockwork whose pieces can be dismantled by analysis and explained. There were immutable laws created by God, through which the universe functioned. 

People sought certainty in everything. Be it physics or politics or Maths. This was the case until Georg Cantor came on the scene.

A young mathematician who sought to find the immutable laws that god created for the universe, ran headlong into one of the monsters of Mathematics. Infinity!