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Systematic Studies: The Infinity Problem in Modern Mathematics
Edited by Yi Lin
1 Calculus and theory of limits: a brief history recall
During the seventeenth and eighteenth century, because of the appearance of calculus and its wide range applications in various areas of learning, the theory of calculus was developed rapidly. However, during this time period, the theory of calculus was established on the ambiguous and vague concept of infinitesimals. Since, the foundation was not solid, the theory of calculus was criticized severely from many different angles. The most central criticism is the famous Berkeley paradox. To illustrate this paradox, let us use the example of computing the instantaneous speed of a free falling object at the time moment t0 . The well-known distance formula for a free falling object is given by S =(1/2)gt2 . When t =t0 , the distance the object has fallen through is S0 =(1/2)gt20 . When t =t0 +h , the distance of falling is S0 +L =(1/2)g (t0 +h )2 . This end implies that during the h seconds, the object has fallen through the distance L , as shown in Figure 1 [Figure omitted. See Article Image.], that is given by: Equation 1 [Figure omitted. See Article Image.] So, within the h seconds, the average speed of the falling object is: Equation 2 [Figure omitted. See Article Image.] Evidently, the smaller the time interval h is, the closer the average speed is to the momentary speed at t =t0 . However, no matter how small h is, as long as h ≠0, the average speed is not the same as the speed at t =t0 . When h =0, there is no change in the falling distance. So, V =L /h =gt0 +(1/2)gh becomes the meaningless 0/0. So, it becomes impossible to compute the instantaneous speed of the falling object at t =t0 .
Both Newton and Leibniz once provided several explanations to get rid of this difficulty:
- Assume that h is an infinitesimal. So, h ≠0 and the ratio L /h =((1/2)g (2