[AntonioLao]

Answering the question ‘what is motion’, the calculus (differential as well as integral) as originally invented and developed by Leibniz and Newton presupposed the taming of infinity as that found in manipulating infinite series. The validity of the calculus hinges on the existence of convergences of these infinite series for determining the finite values of the infinite sums. The principle behind these convergences is the principle of directional quantizability, the smooth connective transitional direction from discrete to continuous or vice versa (at a deeper level this directive property of space and time is influenced by the principle of directional invariance). This connective transition completes Zeno’s paradox front and back. Front, when considering spacetime as discrete and back, when considering spacetime as continuous.

Taking a linear slice (straight or curved) through the spacetime continuum and representing this line by the infinite series S given as S = + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + …, without any directive property, each quantum of spacetime continuum is preceded by a plus sign. This plus sign can be used to signify outward motion with respect to each quantum, but how about all those inward motions into the quanta? For completeness, surely, these inward motions must exist in principle. The inward motions can now be represented by minus signs. Such that the series is transformed into S = - 1 – 1 – 1 – 1 – 1 – 1 – 1 – 1 – 1 – 1 - …. The mixing of pluses and minuses can be used to represent both outward and inward motions given by S = + 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + …. It is well known in the mathematical circle that the groupings of terms in the series can give at the least three different sums: ½, 1, 0, leading mathematicians to believe that there is no correct answer. Since mathematicians are mostly obsessed with finding unique answer to every mathematical problem, they have completely abandoned further study of this infinite series.