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)