Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. It is usually assumed, based on Plato's Parmenides (128a–d), that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides' view. Thus Plato has Zeno say the purpose of the paradoxes "is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one." Plato has Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point. Some of Zeno's nine surviving paradoxes (preserved in Aristotle's Physics and Simplicius's commentary thereon) are essentially equivalent to one another. Aristotle offered a refutation of some of them. Three of the strongest and most famous—that of Achilles and the tortoise, the Dichotomy argument, and that of an arrow in flight—are presented in detail below. Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum, also known as proof by contradiction. They are also credited as a source of the dialectic method used by Socrates. Some mathematicians and historians, such as Carl Boyer, hold that Zeno's paradoxes are simply mathematical problems, for which modern calculus provides a mathematical solution. Some philosophers, however, say that Zeno's paradoxes and their variations (see Thomson's lamp) remain relevant metaphysical problems. The origins of the paradoxes are somewhat unclear. Diogenes Laërtius, a fourth source for information about Zeno and his teachings, citing Favorinus, says that Zeno's teacher Parmenides was the first to introduce the paradox of Achilles and the tortoise. But in a later passage, Laërtius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees. Paradoxes of motion Dichotomy paradox Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on. The resulting sequence can be represented as: This description requires one to complete an infinite number of tasks, which Zeno maintains is an