“I call this the improper infinite”
From Kant to Hilbert (1996)
Context: As for the mathematical infinite, to the extent that it has found a justified application in science and contributed to its usefulness, it seems to me that it has hitherto appeared principally in the role of a variable quantity, which either grows beyond all bounds or diminishes to any desired minuteness, but always remains finite. I call this the improper infinite [das Uneigentlich-unendliche].
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Georg Cantor 27
mathematician, inventor of set theory 1845–1918Related quotes
As quoted by Sir William Osler in his introduction to The Life of Pasteur (1907) by Rene Vallery-Radot, as translated by R .L. Devonshire (1923)
Discours de réception de Louis Pasteur (1882)
Context: He who proclaims the existence of the Infinite, and none can avoid it — accumulates in that affirmation more of the supernatural than is to be found in all the miracles of all the religions; for the notion of the Infinite presents that double character that forces itself upon us and yet is incomprehensible. When this notion seizes upon our understanding we can but kneel... I see everywhere the inevitable expression of the Infinite in the world; through it the supernatural is at the bottom of every heart. The idea of God is a form of the idea of the Infinite. As long as the mystery of the infinite weighs on human thought, temples will be erected for the worship of the Infinite, whether God is called Brahma, Allah, Jehovah, or Jesus; and on the pavement of these temples, men will be seen kneeling, prostrated, annihilated by the thought of the Infinite.
“The Tao is called the Great Mother:
empty yet inexhaustible,
it gives birth to infinite worlds.”
Source: Tao Te Ching, Ch. 6, as interpreted by Stephen Mitchell (1992)
“I will prove that there are infinite worlds in an infinite world.”
The Other World (1657)
Context: I will prove that there are infinite worlds in an infinite world. Imagine the universe as a great animal, and the stars as worlds like other animals inside it. These stars serve in turn as worlds for other organisms, such as ourselves, horses and elephants. We in our turn are worlds for even smaller organisms such as cankers, lice, worms and mites. And they are earths for other, imperceptible beings.
Just as we appear to be a huge world to these little organisms, perhaps our flesh, blood and bodily fluids are nothing more than a connected tissue of little animals that move and cause us to move. Even as they let themselves be led blindly by our will, which serves them as a vehicle, they animate us and combine to produce this action we call life.
I. Asimov: A Memoir (1994)
Context: If I were not an atheist, I would believe in a God who would choose to save people on the basis of the totality of their lives and not the pattern of their words. I think he would prefer an honest and righteous atheist to a TV preacher whose every word is God, God, God, and whose every deed is foul, foul, foul.
I would also want a God who would not allow a Hell. Infinite torture can only be a punishment for infinite evil, and I don't believe that infinite evil can be said to exist even in the case of Hitler. Besides, if most human governments are civilized enough to try to eliminate torture and outlaw cruel and unusual punishments, can we expect anything less of an all-merciful God?
I feel that if there were an afterlife, punishment for evil would be reasonable and of a fixed term. And I feel that the longest and worst punishment should be reserved for those who slandered God by inventing Hell.
The Perks of Being a Wallflower
As quoted in Mind Tools: The Five Levels of Mathematical Reality (1988) by Rudy Rucker. ~ ISBN 0395468108