Hippasus
1,9,45,165,495,1287,
The paradox: Suppose a runner is supposed to cover a distance of one unit. First she has to get halfway across the unit. Then she has to get halfway across the remaining expanse of the unit. Then she has to get halfway across the new remainder. Ad infinitum. So the runner does not traverse the unit after all and change is unreal. Being and logic are real though (Zeno of Elea was a Parmenidean student).
The modern response: The sigma notation of Riemann sums suggests a summation of discrete terms coinciding with the interval [0,1] through a limiting process. We suppose the discrete summation of terms to morph into the continuity and magnitude constituted by the interval [0,1]. We logically require the notion of infinity, that is, a hypothetical abstract limiting process where an infinite number of arithmetical operations is permissible. This is a convergent geometric series where the individual terms are summed as the summation coincides with the interval [0,1]. The first term of this series is (1/2) and the common ratio of the series is (1/2). In expanded form the series would look as follows: [1/2 + 1/4 + 1/8 + . . . 1/2^n].
This is the general form of how this series is defined:
Riemann sum:
(Upper-case sigma), index: n=1,2,3, . . . infinity
(1/2)^n
This is saying the series equals the number one:
= (1/2)/(1-(1/2)) = interval [0,1]
We are only able to say this because the common ration of 1/2 is less than 1. Having a common ratio less than 1 makes the series convergent. Often series diverge to infinity. This one does not, although it involves an infinite number of operations.
The modern response: The sigma notation of Riemann sums suggests a summation of discrete terms coinciding with the interval [0,1] through a limiting process. We suppose the discrete summation of terms to morph into the continuity and magnitude constituted by the interval [0,1]. We logically require the notion of infinity, that is, a hypothetical abstract limiting process where an infinite number of arithmetical operations is permissible. This is a convergent geometric series where the individual terms are summed as the summation coincides with the interval [0,1]. The first term of this series is (1/2) and the common ratio of the series is (1/2). In expanded form the series would look as follows: [1/2 + 1/4 + 1/8 + . . . 1/2^n].
This is the general form of how this series is defined:
Riemann sum:
(Upper-case sigma), index: n=1,2,3, . . . infinity
(1/2)^n
This is saying the series equals the number one:
= (1/2)/(1-(1/2)) = interval [0,1]
We are only able to say this because the common ration of 1/2 is less than 1. Having a common ratio less than 1 makes the series convergent. Often series diverge to infinity. This one does not, although it involves an infinite number of operations.
Last edited: