Zeno’s Paradox and its Contributions to The Notion of Infinity
Name: Dejvi Dashi
School: King’s-Edgehill School
IB nr: 000147-0006
Mathematics Exploration
May 2014
Date: March 31st, 2014
Word Count: 2681
Achilles and the Tortoise is one of the many mathematical and philosophical paradoxes that were expressed by Zeno of Elea. His purpose was to present the idea that motion is nothing but an illusion. Many solutions have been offered as an explanation to these paradoxes for many years now. Some of these solutions include the factor of time, arguing that a mathematical result can be obtained when a certain amount of time is set for the race. However, many others have resulted in the fact that solutions, which include a set time, have simply missed the point of Zeno’s Paradoxes. There is also a philosophical reach that many mathematicians have had to carry out in order to expand the net of solutions to these problems. Mathematicians like Weierstrass and Cauchy propose ways that are achieved due to a fusion of mathematical ability and reasoning.
Zeno’s Paradoxes have led to many contributions in math and calculus through the attempts that have been made to understand them. Therefore my goal is to analyze how mathematical solutions have contributed in a better understanding of the philosophy behind Zeno’s problems.
1. Achilles and the Tortoise
Achilles is to run a race against the tortoise and the tortoise is given a head start due to the idea that the tortoise is weaker. Zeno argues that Achilles will never be able to reach the tortoise no matter how fast he runs. For Achilles to reach the tortoise he has to run the distance the tortoise has been given at first. As he does so, the tortoise will ha...
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...trass saw these problems form a purely mathematical point of view and that helped them redefine the mathematical concept of a limit. Others have thought of these paradoxes as a way of feeding our skepticism and doubting the deficiency of what we presume.
To me is seems that we may profit form these problems either now. By looking at the solutions provided over the years, I think that the quest to find a “proper” solution shows an interest in finding out if Zeno’s ideas can be outsmarted rather than knowing what his goal really was. A paradox is meant to juggle with the value of truth in a statement and I believe Zeno was well aware of that when he proposed them. Therefore, though it has had massive impacts in many areas, Zeno’s paradoxes were an invitation to open skepticism to our presumptions and consequently a more keen yet broad perception of mathematics.
Achilles, son of Thetis, also had divine blood flowing through his veins. He, however, was well aware of his mortality, as he chose a shortened lifespan full of glory over a longer, non-glorious life. "Alas, that you should be at once short of life and long of sorrow above your peers," exclaims his mother. (Butler, I). Despite accepting his mortality, Achilles, like Gilgamesh, was blessed with unequaled strength and skill as t...
In this paper, I offer a reconstruction of Aristotle’s argument from Physics Book 2, chapter 8, 199a9. Aristotle in this chapter tries to make an analogy between nature and action to establish that both, nature and action, have an end.
In order to understand the concept of Moore’s Paradox, we must first assess and understand the behavior of logical and performative contradictions. Credited for devising and examining this paradox, George Edward Moore, a British philosopher who taught at the University of Cambridge and studied ethics, epistemology, and metaphysics describes the paradox in its omissive and commissive forms in which we will discuss thoroughly. I will then express my standpoint on which solution is the most optimal choice for Moore’s Paradox in order to analyze and explain why I believe my solution is superior to other solutions. I will also discuss any issues that arise
Phoenix’s paradigm narrative fails to persuade Achilles to rejoin the war because the specifics of that narrative fail to align with Achilles’ specific concerns. In particular, Phoenix neglects the pernicious effects of Agamemnon 's actions on Achilles’ notions of honor and pride.
In his Meditations, Rene Descartes attempts to uncover certain truths about existence. In his Third Meditation, he establishes his "special causal principle" (SCP). Descartes uses this principle to explore the origin of ideas, and to prove the existence of God. I agree that there is much logic to be found in the SCP, but I disagree with Descartes method of proving God's existence, and in this essay I will explain why. I will begin by explaining the SCP, and will then demonstrate how Descartes applies this principle to prove that God exists. I will then present my critique of the SCP, and expose the flaws in both of Descartes proofs with regards to the principle. A conclusion will then follow.
The Greeks placed great importance on personal honor. Why is this? Is it because to them man I nothing without honor. Or is it that the honor is more important than the man? "Honor to the Greeks is something that is won by a man's prowess, his ability to fight and be victorious on the battle field"(Schein 62). This is just one example of how honor is obtained. A second method of gaining honor is to be a great orator, one must posses the ability to speak in the assembly and express his ideas eloquently, and persuasively to the gathered body. A third way of achieving personal honor is to demonstrate athletic ability.
The paradox arises due to a number of assumptions concerning knowledge, inquiry and definition made by both Socrates and Meno. The assumptions of Socrates are:
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This book was at times difficult to grasp its principles when reading the calculus, but with its inclusion of geometry the material becomes accessible to most educated readers. Because it has the feel of a textbook for spreading the understanding of the new philosophy this book should be recommended to anyone studying the history of science, philosophy, or any of the various influential philosophers who contributed to understanding and truth through experimentation.
Homer's two central heroes, Odysseus and Achilles, are in many ways differing manifestations of the same themes. While Achilles' character is almost utterly consistent in his rage, pride, and near divinity, Odysseus' character is difficult to pin down to a single moral; though perhaps more human than Achilles, he remains more difficult to understand. Nevertheless, both heroes are defined not by their appearances, nor by the impressions they leave upon the minds of those around them, nor even so much by the words they speak, but almost entirely by their actions. Action is what drives the plot of both the Iliad and the Odyssey, and action is what holds the characters together. In this respect, the theme of humanity is revealed in both Odysseus and Achilles: man is a combination of his will, his actions, and his relationship to the divine. This blend allows Homer to divulge all that is human in his characters, and all that is a vehicle for the idyllic aspects of ancient Greek society. Accordingly, the apparent inconsistencies in the characterization of Odysseus can be accounted for by his spiritual distance from the god-like Achilles; Achilles is more coherent because he is the son of a god. This is not to say that Achilles is not at times petty or unimaginative, but that his standards of action are merely more continuous through time. Nevertheless, both of Homer's heroes embody important and admirable facets of ancient Greek culture, though they fracture in the ways they are represented.
Infinity has long been an idea surrounded with mystery and confusion. Aristotle ridiculed the idea, Galileo threw aside in disgust, and Newton tried to step-side the issue completely. However, Georg Cantor changed what mathematicians thought about infinity in a series of radical ideas. While you really should read my full report if you want to learn about infinity, this paper is simply gets your toes wet in Cantor’s concepts.
was the accepted way of life in the Bronze Age, which is when Achilles lived.
Wigner, Eugene P. 1960. The Unreasonable Effectiveness of Mathematics. Communications on Pure and Applied Mathematics 13: 1-14.
The Iliad, the Greek epic documented by Homer that describes the battles and events of the ten year siege on Troy by the Greek army. Both Trojans and Greeks had their fair share of heroes and warriors, but none could match the skill and strength of the swift runner, Achilles. Achilles had the attributes of a perfect warrior with his god-like speed and combat abilities. However, even though he was Greek’s greatest warrior, he still possessed several flaws that made him fit the role of the Tragic Hero impeccably. Defined by Aristotle, a Tragic Hero is someone who possesses a high status of nobility and greatness, but must have imperfections so that mere mortals cannot relate to the hero. Lastly, the Tragic Hero’s downfall must be partially their own fault through personal choice rather than by an evil act, while also appearing to be not entirely deserved of their unfortunate fate. Achilles is a true Tragic Hero because he withholds all of these traits. Achilles proves to be a good man that puts his loved ones first, reveals his tragic flaws of pride and anger, shows dynamic qualities as a character when his flaws are challenged, and has a moment of clarity at the end of his rage. Achilles truly exemplifies the qualities of a Tragic Hero.
Prime numbers have been of interest to mathematicians for centuries, and we owe much of our existing knowledge on the subject to thinkers who lived well before the Common Era––such as Euclid who demonstrated that there are infinitely many prime numbers around 300 BCE. Yet, for as long as primes have been an element of the mathematician’s lexicon, many questions about prime numbers remain unreso...