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

64 the mathematical infinityThis kind of reckoning reminds us very much of Archimedes’ method: a nonrigorousapproach to problems, leading to a correct line of thought, that can be perfected tofully rigorous treatment at a later stage. For the example at hand, it is quite easy totransform it into a rigorous statement, working in Aristotelian fashion by using onlyfinite quantities. The precise result can be stated as∑p≤X1= log log X + C + ε(X),pwhere C is a certain numerical constant, X ≥ 3, and ε(X) → 0 when X increaseswithout limit. We see here an advantage of the finite method: whenever it works, itleads to quantitative statements, that is, information and knowledge that may be lostby going to the limit.Euler was somewhat of an exception in his ability to handle the infinity correctly andwith ease, even if not in a rigorous way. Others were not so successful, as witnessed byendless and at times acrimonious debates about the value of the infinite sum 1 − 1 +1 − 1 +···A century later, with the work of Cesàro laying the foundation of a theorygiving precise meaning to such processes without a limit in the traditional sense (thetheory of summability), it became possible to disentangle the heuristic methods of thepast, separating meaningful ideas from paradoxical calculations.2.6 Rigorous ProofsStarting with Cauchy at the beginning of the nineteenth century, the finitistic view,avoiding the question of actually defining infinity or infinitesimals as precise mathematicalentities, became generally accepted. The current way of teaching calculus, withthe traditional epsilons and deltas indicating arbitrarily small numbers, is indeed thesimplest example of this general trend. Thus, infinity took a precise meaning, especiallyin analysis, and specific rules and conditions to be met for handling infinite sums andintegrals were formulated, thereby adding a new effective weapon in the armory ofanalysis.Should we then say that the Aristotelian point of view, regarding the mathematicalinfinity as a convenient shorthand for potentially unlimited quantities or constructions,has won the day at last? Is infinity nothing other than a mathematical convention? Notso. Infinity has come back into mathematics in a far more powerful way, as well as inmany different forms, to coexist with the finitistic approach of Aristotle. There is noneed to coerce every proof into a finite argument.In the view of many mathematicians, including myself, the intellectual contortionsneeded to remain within the realm of the finite (if possible at all) indicate that a wholesalerejection of infinity in mathematics is not a good thing. What really matters is thefinal understanding, coupled with a good foundation. Thus, the modern mathematicianapproaches the foundational question of the nature of infinity in a pragmatic way: whatmatters first is knowledge. How it is obtained is also very important, but that takessecond stage.