Discussions surrounding the nature of the infinite in mathematics have been under way for two millennia. Mathematicians, philosophers, and the- ologians have all taken part. The basic question has been whether the infinite exists only in potential or whether it exists in actuality. Only at the end of the 19th century a set theory was created that works with the actual infinite. Ini- tially, this theory was rejected by other mathematicians. The creator behind the theory, the German mathematician Georg Cantor, felt all the more the need to challenge the long tradition that only recognised potential infinite.The issue turns surprisingly theological, with St. Augustine, St. Thomas Aquinas, and the Pope on the side of Cantor.
Cantor was convinced that the knowledge of infinite numbers had been revealed to him by God, who guided his steps from pure mathematics to an interest in theol- ogy and philosophy so that he could improved a proper understanding of God and nature.Cantor's papers from 1883 to 1895 were met with skepticism, but by 1897 Mathematicians were on board with his theory of infinities.
Xkcd just had a cartoon about large numbers, explained here.
Cantor really did violate the laws of classical logic, and you'll hear as much from the likes of a Graham Priest. Look at the measure theory paradox of Mycielski and Sierpiński. It cannot be eliminated by relaxing the axiom of choice or assuming all sets of reals to be Lebesgue measurable. Ultrafinitism was essentially proved with the introduction of computer algebra systems. Bogus analysis proofs disguise a finite algebra underneath, like the Risch algorithm. Even incompleteness assumes an impredicative infinite concatenation. Cantor's pseudomathematics was rightly criticized by those such as Kronecker, Poincare, Solomon Feferman, Doron Zeilberger, etc. Even our neurons are well-approximated with fairly simple GLMs, if you're familiar with computational neurology. Sets, homotopy types, categories, etc. are arbitrary metaphors and only a form of redundant metaphysics. Logicism has won but all kinds of crackpot literature says otherwise.
ReplyDeleteInfinity has no business being treated either as a quantity or a number, or being considered in a ratio of any kind, as it can not be measured or determined in reality other than with a lot of vague hand waving and a 'because I said it is so'.
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