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ContentsNumber: the Language of Science
Infinity: the Elegant Idea
Blaise Pascal, a French mathematician and philosopher, once stated, “Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.” These notions of nothingness and infinity have remained fodder for the ravenous minds of philosophers and logicians for centuries. The number zero arose from the need to denote nothingness and represents perhaps the most important part of the number line. However, nothingness, though not physically existent, seems far easier to comprehend and utilize than its exceedingly abstract counterpoint, infinity. Though philosophers may revel in contemplating the infinite, mathematicians view the concept as fundamentally essential; infinity remains one of the most important concepts in the discovery and advancement of mathematics. The power of the idea lies in the impossibility to imagine infinity juxtaposed with the necessity to conceive this indispensable mathematical abstraction.
Throughout history, the notion of infinity seemed intrinsically tied to omniscience, power, and religion—math philosophers have suggested it exists as the embodiment of God. In his novel, ^ , Tobias Dantzig quotes, “There is a last number, but it is not in the province of man to reach it, for it belongs to the gods” (Dantzig 64). Descartes once defined imagination and conception through an experiment in visualizing a one thousand-sided polygon. Though entirely possible to conceive and utilize the shape, it is impossible to imagine it. He proceeded to suggest “the greatest obstacle to true faith…is the attempt to imagine God when we can only conceive God” (Suber). This scenario embodies the basic frustration and fascination with the concept of infinity. Descartes proves that the unimaginable nature of infinity does not prevent its practical and functional use. Nevertheless, the concept itself seems analogous to the idea of faith, which is perhaps the reason for infinity’s recurring ties to religion.
Historically, the concept of infinity has been regarded with a pervading sense of unease and anxiety. The struggle to grasp the idea has led many to revoke its essential nature; nevertheless, “the concept of infinity, though not imposed upon us either by logic or by experience, is a mathematical necessity” (Dantzig 77). Cantor’s ingenious method of proving the one-to-one correspondence of infinite sets provided a counter-intuitive, but mathematically enlightening understanding of infinity. The concept of infinity also remains an essential part of the fundamental theorem of calculus and the simple existence of the idea allows the creation of entirely new domains of mathematics. The elementary axioms of math dictate that infinity must exist, but it is this very stipulation that creates the frustrating struggle to comprehend an unending and boundless set. Nevertheless, this cognitive battle is necessary in an attempt to fully understand the number line. Not only is infinity crucial for math, but it exists as a concept so utterly abstract that its applicability reveals the beauty of the idea.
It has been postulated that “the deep explorers of the infinite, even in its strictly mathematical forms, recurrently find it to be sublime” (Suber). The impossibility to imagine infinity, but the capacity to conceptualize it embodies the centuries-old human struggle to understand the nature of the intangible. The ability to conceive of such an abstract concept in practical terms epitomizes the appeal of infinity. It is this awe-inspiring power which correlates the notion of infinity to religion and to that of God Himself. Perhaps it is the elegant idea of infinity in mathematics that links so strongly to the universal design. Though unreachable, human nature will cause the continued pursuit of the infinite, which is imperative for understanding the “sublime” nature of mathematics.
Dantzig, Tobias. “Number: the Language of Science.” New York: Penguin Group, 2005.
Suber, Peter. “A Crash Course in the Mathematics of Infinite Sets.” 1998.
Suber, Peter. “Infinite Reflections.” 1998. < http://www.earlham.edu/~peters/writing/infinity.htm>.