Magnificent Pythagorean Paradoxes In Ancient Greece

The ancient Indians had a fair amount of mathematical accomplishments and discoveries, all of which we still use today. The main, most broad accomplishment that they contributed to modern civilization is the naming of the numbers. We are

The creations of Pythagoras were very powerful during the era in which he lived in. He created a community of followers (known as the Pythagoreans) who believed that mathematics was fundamental and ‘at the heart of reality’ (source 1). The people in the society were all proficient mathematicians took mathematics very seriously, to the extent that it was similar to a religion (source 1).

The Abbasids were the first ones to study and translate important Greek and Indian mathematical book like Euclid's geometry text the Elements. They adopted a very Greek approach to mathematics of formulating theorems precisely and proving them formally in Euclid's ways.

When he was given freedom, he formed a school in Samos, called "The Semicircle," but soon left to travel to Italy. He traveled to southern Italy, to the town of Croton, where he founded a religion based school. He also developed a small group of his top followers called the Mathematikoi. In this school, Pythagoras made great discoveries. Some achievements of Pythagoras include, classifying numbers into even and odd, classifying perfect numbers, and classifying triangle numbers. His biggest discovery is most definitely the Pythagorean Theorem. This property's equation states that a^2+ b^2 = c^2, with the variables a, and b acting as the two legs of a right triangle, and c acting as the hypotenuse. The Pythagorean Theorem was the start of basic trigonometry, and geometry. When you hear the phrase Pythagorean Theorem, most people revert to saying that Pythagoras invented it. Well... no. Basically, Pythagoras heard the idea proposed in Babylon, so he stole it for himself and refined it a bit. Even so, the little bit of refining he did was something that the Persians probably couldn't have done. Pythagoras was proud of his achievement, but then one of his own students turned on

The Pythagorean School, for example, contributed many ideas to the mathematic community, among them, studies of geometry and the theory of proof.5 Euclid also lived in the time of ancient Greece and became a prominent mathematician, as well as author of a book about geometry called, The Elements, considered the second best-selling book of all time.6 The works of Pythagoras and Euclid have become fundamental building blocks for any person with an eighth-grade understanding mathematics. With these advancements, as well as innovations from Archimedes and Apollonius of Perga, Islamic scholars translated their works and contributed even more, growing the worldwide understanding of mathematics.7 Greek progressions in geometry as well as the theory of proof contributed greatly to our ultimate understanding of contemporary mathematics, without which, our modern society would be

Ancient egyptians made some of the earliest forms of technology, mathematics was one of the most important form of technology as it

The author of Journey through Genius, William Dunham, begins this chapter by depicting how mathematics was spurred and developed in early civilizations. Dunham focuses primarily on the works’ and achievements’ of early Egypt, Mesopotamia, and Greece in this section. These ancient societies, as they developed, produced mathematicians such as; Thales, Pythagoras, and Hippocrates, who turned a basic human intuition for space and quantity into applicable everyday mathematics. The primary influences driving the development of early mathematics were the issues of growing civilizations, most notably counting commodities, taxation, and the division of land equally, rather than a pure desire for understanding that is seen in mathematics today. These influences culminated in the development of early arithmetic and geometry.

Zeno’s four paradoxes of motion collectively attempt to demonstrate Parmenides’ assertion that there is no motion. For the sake of simplicity and convenience, we will demonstrate Zeno’s paradoxes of motion using the first three paradoxes. We begin with the first and most well-known of the quartet: the bisection paradox. According to the bisection paradox, in order to walk across a room and reach the opposite side, one must first walk halfway across the room. Once an individual has reached the halfway point across the room, they must then reach the halfway point between their current position and the remainder of the distance to the room. The individual must complete a perpetual sequence of halfway points before ever reaching the room, making the act of doing so (theoretically) impossible. No matter how smaller the distances between each halfway points become, it never reduces to a value of zero with any given units of measurement. The second paradox is the Achilles paradox, which involves a race between Achilles and a tortoise. In this scenario, the tortoise is given a head start since it is much slower than Achilles. Once Achilles begins running after the tortoise and reaches the its starting point, the tortoise is already ahead of Achilles by a few meters. By the time Achilles reaches that same spot a few meters ahead, the tortoise still maintains its lead (albeit slightly reduced). Zeno’s second paradox attempts to demonstrate that despite Achilles’s being

The world of chaos and disorder has become the world of order and harmony. Ever since the ancient Greek times, chaos has been in existence. This led people to start thinking outside of the box. Many people started questioning why somethings were the way they were. As a result many philosophers emerged in the pre-Socratic times.

The concept of arrow paradox claims that when objects occupy an equal space at rest and locomotion is constant at the moment; then a flying arrow will be declared to be motionless. Zeno claims that at any instant an object is motionless because it cannot move to a place that it is not due to the fact that time has not elapsed. On the other hand, an object cannot move to where the object already is because that would create a paradox, seeing as the object is already there. It is impossible to observe any form of motion when the time is paused. Furthermore, if space is occupied by the objects then also the arrow should be dormant. Thus, the entire elaboration of this notion creates a fallacy. One can conclude form the arrow paradox that motion is impossible because everything is motionless at each instant, and time in essence is the sum of many instances.

Pythagoras started to persuade people of his mathematics in the 7th century, he was able to convince people to follow him and his mathematics. Pythagoras was fair and kind, which answers the driving question “How did Islamic peoples transmit knowledge from the ancient world to influence modern intellectualism?”. He invited women to his lectures in which he spread his intellectualism.

I will now discuss Zeno’s paradox of motion. Zeno argues that motion does not exist through this argument: 1) there is an object at point A that is moving to a point B; 2) in order to reach point B, the object must pass the halfway point of points A and B; 3) we continue halving the remain distance and point B, all the way up to infinity; 4) this means that the object is taking an infinite distance to cross, and therefore, motion cannot exist, as an object cannot move an infinite

As with Achilles and a tortoise this one even further stresses the idea that motion is just an illusion. In this example Zeno assumes that time, in his definition, is only made up of a series of ‘’moments’’. In this paradox one arrow, just a regular arrow, has been shot or is just simply in motion. Zeno himself states that with his paradox, no motion is occurring while the arrow is in motion. This is because he assumes that for each instance of time, which can be broken up into small intervals as defined earlier by him, no movement occurs. Again explaining this more in detail, he is saying that because of all these ‘’moments’’ occurring and being able to divide these up into smaller intervals which then ultimately lead to the point where the arrow is not moving at all because time has moments and no movement occurs in that moment of time. This is why Zeno claims that the arrow is not moving at

Mathematics has greatly advanced since the time of Zeno and Aristotle and this has brought to light another way to counter Zeno’s paradoxes. With the creation of calculus, Zeno’s paradox has been more or less resolved. The calculus solution to this paradox considers the infinite sequence of ½, ¼, 1/8, …, 1/2^n, which is an infinite geometric sequence. Geometric sequences have a formula of ∑_(n=1)^∞▒〖(〖 1/2〗^i )〗=1 were the number of half steps are balanced out by the increasingly short amount of time needed to journey through all those distances. However, this is a convergence of an infinite series, which means that the sum of this can be approximated very closely to one if you add up enough terms. This mathematical explanation seemed to suffice for a lot of people, since the sum of all of those terms can be approximated closely enough to a doable number to justify motion being an actual thing. However, if we take a closer look at Zeno’s paradox, the problem was not that the runner wouldn’t get close to the final point, point B. The problem was that the runner would never actually manage to reach that final point because no matter how small the distance between the runner and the final point, the distance can still be divisible by half and therefore there would never be a final point were the runner would confidentially reach point B. Because the calculus equation can only approximate very closely to the final point, but never the actual point, this solution is still not the correct

Omar was also a poet, philosopher, and astronomer. Omar’s works were translated in 1851, which was research on Euclid’s axioms. In the medieval period, he expanded on Khwarizmi’s and the Greeks mathematic works. He only worked with cubic equations only and focused on geometric and algebraic solutions of equations. In 1145AD, Al-Khwarizmi’s book was translated by Robert Chester, which made it possible for algebra to be introduced to Europe. After algebra was introduced in Europe, European mathematicians developed and expanded on algebra concepts. Even though algebra began in the Arabic countries, once European mathematicians obtained the information of algebra, they became the leaders of mathematical discoveries in the world (“Mathematics”).