The Paradoxes in Modern Times
Infinite processes remained theoretically troublesome in mathematics until the late 19th century. The epsilon-delta version of Weierstrass and Cauchy developed a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes.
While mathematics can be used to calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Brown and Moorcroft claim that mathematics does not address the central point in Zeno's argument, and that solving the mathematical issues does not solve every issue the paradoxes raise.
Zeno's arguments are often misrepresented in the popular literature. That is, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite–with the result that not only the time, but also the distance to be travelled, become infinite. However, none of the original ancient sources has Zeno discussing the sum of any infinite series. Simplicius has Zeno saying "it is impossible to traverse an infinite number of things in a finite time". This presents Zeno's problem not with finding the sum, but rather with finishing a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event"?
Today there is still a debate on the question of whether or not Zeno's paradoxes have been resolved. In The History of Mathematics, Burton writes, "Although Zeno's argument confounded his contemporaries, a satisfactory explanation incorporates a now-familiar idea, the notion of a 'convergent infinite series.'" Bertrand Russell offered a "solution" to the paradoxes based on modern physics, but Brown concludes "Given the history of 'final resolutions', from Aristotle onwards, it's probably foolhardy to think we've reached the end. It may be that Zeno's arguments on motion, because of their simplicity and universality, will always serve as a kind of 'Rorschach image' onto which people can project their most fundamental phenomenological concerns (if they have any)."
Read more about this topic: Zeno's Paradoxes
Famous quotes containing the words paradoxes, modern and/or times:
“The way of paradoxes is the way of truth. To test Reality we must see it on the tight-rope. When the Verities become acrobats we can judge them.”
—Oscar Wilde (1854–1900)
“Men must speak English who can write Sanskrit; they must speak a modern language who write, perchance, an ancient and universal one.”
—Henry David Thoreau (1817–1862)
“After years of vain familiarity, some distant gesture or unconscious behavior, which we remember, speaks to us with more emphasis than the wisest or kindest words. We are sometimes made aware of a kindness long passed, and realize that there have been times when our Friends’ thoughts of us were of so pure and lofty a character that they passed over us like the winds of heaven unnoticed; when they treated us not as what we were, but as what we aspired to be.”
—Henry David Thoreau (1817–1862)