When Pythagoras came to Croton, he founded a philosophical school called 'semicircle'. The school had many followers, but the inner circle of the school had smaller number of philosophers/mathematicians who were called mathematikoi . They lived as if they were in a monastery, having no private property and observing a number of strict rules. Among other things, they believed that
- Nature and the whole reality has an underlying mathematical structure
- That philosophy should be used for spiritual purification
- That the soul which is pure can rise to experience the union with the divine
- That certain symbols (including mathematical) have a mystical significance.
Some remarks on the video:
The supernatural numbers are already called hypernatural numbers. They are "unlimited" numbers and not finite.
Cantor wanted to believe in an actual infinite existing in his set theory. He wanted there to be a highest level of infinity but Russell dispelled that notion with Russell's paradox. Since then, Quine and others have developed set theories with a maximally, absolutely infinite set.
The Ultimate court: "if I am sane, I will tell the truth, only the truth, however not all of the truth." (<--from the video) Sanity is often an issue with mathematicians. This whole post was inspired by a remark I made to greeney2 about what happens to some people who let God all the way in.
The number of provable theorems is countably infinite. The number of proofs is countably infinite. The number of true statements that lack proof is also countably infinite.
The vid also mentions pantheism briefly.
United with his fellow-men by the strongest of all ties, the tie of a common doom, the free man finds that a new vision is with him always, shedding over every daily task the light of love. The life of Man is a long march through the night, surrounded by invisible foes, tortured by weariness and pain, towards a goal that few can hope to reach, and where none may tarry long. One by one, as they march, our comrades vanish from our sight, seized by the silent orders of omnipotent Death. Very brief is the time in which we can help them, in which their happiness or misery is decided. Be it ours to shed sunshine on their path, to lighten their sorrows by the balm of sympathy, to give them the pure joy of a never-tiring affection, to strengthen failing courage, to instil faith in hours of despair. Let us not weigh in grudging scales their merits and demerits, but let us think only of their need -- of the sorrows, the difficulties, perhaps the blindnesses, that make the misery of their lives; let us remember that they are fellow-sufferers in the same darkness, actors in the same tragedy as ourselves. And so, when their day is over, when their good and their evil have become eternal by the immortality of the past, be it ours to feel that, where they suffered, where they failed, no deed of ours was the cause; but wherever a spark of the divine fire kindled in their hearts, we were ready with encouragement, with sympathy, with brave words in which high courage glowed.
-- Bertrand Russell,
Russell was the one who may have convinced Cantor that the God he had envisioned as the linchpin in his set theory does not exist. For Cantor, his study of the infinite was theology.
Thinking of math as theology or as a way to actually understand reality is not at all new. These particular forms of pantheism are at least as old as the Pythagoreans in the Semicircle society.