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.
