“I protest…against the use of infinite magnitude as if it were something finished; this use is not something admissible. The infinite is only a facon de parler, one has in mind limits approached by certain ratios as closely as desirable while other ratios may increase indefinitely.” Carl Friedrich Gauss, presumably the foremost mathematician of the 19th century, expressed this view in 1831 in reply to an idea of Schumacher and in the process uttered a horror infiniti which up to almost the end of the century was the common attitude of mathematicians and seemed unassailable considering the authority of Gauss. Mathematics should deal with finite magnitudes and finite numbers only while the treatment of the actual infinity, whether infinitely great or infinitely small, might be left to philosophy.
It was the mathematician Georg Cantor (1845-1918) who dared to fight this attitude and in the opinion of the majority of 20th century mathematicians, has succeeded in the task of bestowing legitimacy upon infinitely great magnitude. Besides the creative intuition and the artistic power of production, which guided Cantor in his work, an enormous amount of energy and perseverance was required to carry through the new ideas, which for two decades were rejected by his contemporaries with the arguments that they were meaningless or false or “brought into the world of mathematics a hundred years too early”. Not only Gauss and other outstanding mathematicians were quoted in evidence against the actual infinity but also leading philosophers such as Aristotle, Descartes, Spinoza and modern logicians. Set theory was even charged with violating the principles of religion, an accusation rejected by Cantor with particular vigour and minuteness. Only in the last years of the nineteenth century, when Cantor had ceased engaging in mathematical research, did set theory begin to infiltrate many branches of mathematics.
Cantor has shown how definite and distinctly great magnitudes can be handled — another evidence of the free creation which is characteristic of mathematics to a higher extent than that of the other sciences. It is no mere accident that at the birth of the set theory, the slogan was coined: the very essence of mathematics is its freedom.
Ref: Abstract Set Theory by Abraham A. Fraenkel.
Shared by Nalin Pithwa …(a piece that I enjoyed in the beginning of the book)