Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

CHAPTER 2The Mathematical InfinityEnrico Bombieri2.1 Early HistoryIn my youth I read a popular book, written by the famous physicist Gamow (1947),aimed at guiding the reader to a glimpse of modern science, with special emphasison the microcosm of the atom, the macrocosm of the galaxies all the way back to theBig Bang, and Einstein’s theory of relativity along the way. It was a fascinating bookindeed. The title was One,Two,Three...Infinity, in reference to how counting mayhave started in primitive tribes as “One, two, three ...many.” 1We may smile, thinking proudly of how far ahead we have gone in our understandingof counting, but in a certain sense we have not made much progress beyond this. Studieshave shown that the average person, when shown a multitude of objects, is not ableto recollect more than seven of them with accuracy, if not even less. So our innerway of counting may still be today, “One, two, three, . . . seven ...many.”Ontheotherhand, there are ways of understanding the very large and the incredibly small, andmathematics provides the tools to do so.What is infinity? Is it the inaccessible, the uncountable, the unmeasurable? Or shouldwe consider infinity as the ultimate, complete, perfect entity? Can mathematics, thescience of measuring, deal with infinity? Is infinity a number, or can it be treated as such?The concept of infinity plays a fundamental, positive role in today’s mathematics, butit was not always so positive in antiquity. The Greek mathematicians and philosopherstook, at times, a negative view of infinity.For Pythagoras, the Eleatic school, and the philosophers Parmenides and Plato,infinity was accepted as a negative concept: it could not be reached or described infinite terms; it was the irrational; it was formless because it could not be increased ordecreased. Simply put, it was inaccessible. In arithmetic and geometry, it meant thatunending mathematical constructions were not allowed.Zeno based his proofs of the impossibility of movement on such ideas (Achilles willnever reach the tortoise, because in the time he reaches the distance that separates him1 This is not an exaggeration. The Pirahã Amazonian tribe actually counts “One, two, many.”55