Aristotle's Wheel Paradox

The paradox of Aristotle's wheel is mentioned in ancient Greek text Mechanica. The problem has hunted some of the greatest mathematicians for centuries. Consider a small wheel, diagrammed as two concentric circles. A one-to-one correspondence exists between points on the larger circle and those on the smaller  circle; thats is, for each point in the large circle, there is exactly one point on the small circle and vice versa. Thus, the wheel assembly might be expected to travel the same horizontal distance regardless of whether it is rolled on a rod that touches the smaller wheel or rolled along the bottom wheel that touches the rod. But how can this be? After all, we know that the two circumferences of the circles are different.
Today, mathematicians know that a one-to-one correspondence of points doesn't mean that two curves must have the same length. Georg Cantor (1845-1918) showed that the number, or cardinality, of points on line segment of any length is the sam. He called this Transfinite Number of points the "continuum." For example, all the points in a segment from zero to one can even be put in a one-to-one correspondence with all points of an infinite line. Of course, before the work of cantor, mathematicians had quite a difficult time with this problem. Also note that, from a physical standpoint, if the large wheel did roll along the road, the smaller wheel would skip and be dragged along the lines that touches its surface.
The precise date and authorship of Mechanica may be forever be shrouded in mystery. Although often attributed as the work of Aristotle, many scholars doubt that Mechanica, the oldest-known textbook in engineering, was actually written by Aristotle. Another possible candidate for authorship is Aristotle's student Stratus of Lampsacus (also known as Strato Physicus), who died around 270 B.C.

Aristotle's Organon

Aristotle was a Greek philosopher and scientist, a pupil of Plato, and teacher of Alexander the Great. The Organon (Instrument) refers to a collection of six of Aristotle's work in logic: Categories, Prior Analytics, De Interpretatione, Posterior Analytics, Sophistical Refutations, and Topics. Andronicus of Rhodes (c. 428-348 B.C.) determined the ordering of the six works around 40 B.C. Although Plato (c. 428-348 B.C.) and Socrates (c. 470-399 B.C.) delved into logical themes, Aristotle actually systematized the study of logic, which dominated scientific reasoning in the Western world for 2,000 years.
The goal of Organon is not to tell readers what it is true, but rather to give approaches to how to investigate the truth and how to make sense of the world. The primary tool in Aristotle's tool kit is the syllogism, a three-step argument, such as "All women are mortal; Cleopatra is a woman; therefore, Cleopatra is mortal." If the two premises are true, we know that the conclusion must be true. Aristotle also made a distinction between particulars and universals (general categories). Cleopatra is a particular term. Woman and mortal are universal terms. When universal are used, they are preceded by "all", "some", or "no." Aristotle analyzed many possible kinds of syllogisms and showed which of them are valid.
Aristotle also extended his analysis to syllogisms that involved modal logic - that is, statements containing words "possibly" or "necessarily." Modern mathematical logic can depart from Aristotle's methodologies or extend his work into other kinds of sentences structures, including ones that express more complex relationships or ones that involve more than one quantifier, such as "No women like all women who dislike some women." Nevertheless, Aristotle's systematic attempt at developing logic is considered to be one of humankind's greatest achievements, providing an early impetus for fields of mathematics that are close in partnership with logic and even influencing theologians in their quest to understand reality.  

Platonic Solids

A Platonic Solid is a convex multifaceted 3-D object whose faces are all identical polygons, with sides of equal length and angles of equal degrees. A Platonic solid also has the same number of faces meeting at every vertex. The best-known example of a Platonic solid is the cube, whose faces are six identical squares.
The ancient Greeks recognized and proved that only five Platonic solids can be constructed: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. For example, the icosahedron has 20 faces, all in a shape of equilateral triangles.
Plato described the five Platonic solids in Timaeus in around 350 B.C. He was not only awestruck by their beauty and symmetry, but he also believed that the shapes described the structures of four basic elements thought to compose the cosmos. In particular, the tetrahedron was the shape that represented the fire, perhaps because of the polyhedron's sharp edges. The octahedron was air. Water was made up of icosahedra, which are smoother than the other Platonic solids. Earth consisted of cubes, which look sturdy and solid. Plato decided that God used the dodecahedron for arranging the constellations in the heavens.
Pythagoras of Samos - the famous mathematician and mystic who lived in the time of Buddha and Confucius, around 550 B.C. - probably knew of three of the five Platonic solids (the cube, tetrahedron, and dodecahedron). Slightly rounded versions of the Platonic solids made of stone have been discovered in areas inhabited by the late Neolithic people of Scotland at least 1,000 years before Plato. The German astronomer Johannes Kepler (1571-1630) constructed models of Platonic solids nested within one another in an attempt to describe the orbits of planets about the sun. Although Kepler's theories were wrong, he was one of the first to insist on a geometrical explanation for celestial phenomena.

Quadrature of the Lune

Ancient Greek mathematicians were enchanted by the beauty, symmetry, and order of geometry.Succumbing to this passion, Greek mathematician Hippocrates of Chios demonstrated how to construct a square with an area equal to a particular lune. A lune is a crescent-shaped area, bounded by two concave circular arcs, and this Quadrature of the Lune is one of the earliest-known proofs in mathematics In other words, Hippocrates demonstrated that the areas of these lunes could be expressed exactly asa rectilinear area, or "quadrature."
For the ancient Greeks, finding the quadrature meant using a straightedge and compass to construct a square equal in area to a given shape. If such construction is possible, the shape is said to be "quadrable" ( or "squarable"). The Greeks had accomplished the quadrature of polygons, but curved forms were more difficult. In fact, it must have seemed unlikely, at first, that curved objects could be quadrable at all.
Hippocrates is also famous for compiling the first-known organized work on geometry, nearly a century before Euclid. Euclid may have used some of Hippocrates' ideas in his own work, Elements. Hippocrates' writings were significant because they provided a common framework upon which other mathematicians could build.
Hippocrates' lune quest was actually part of a research effort to achieve the "quadrature of the circle" - that is, to construct a square with the same area as the circle. Mathematicians have tried to solve the problem of "squaring the circle" for more than 2,000 years, until Ferdinand von Lindemman in 1882 proved that it is impossible. Today, we know that only five types of lunes are quadrable. Three of these were discovered by Hippocrates, and two more were found in the mid-1700s.

Zeno's Paradox

For more than a thousand years, philosophers and mathematician have tried to understand Zeno's paradoxes, a set of riddles that suggest that motion should be impossible or that it is an illusion. Zeno was a pre-Socratic Greek philosopher from southern Italy. His most famous paradox involves the Greek hero Alchilles and a slow tortoise that Achilles can never overtake during a race once the tortoise is given a head start. In fact, the paradox seems to imply that you can never leave the room you are in.  In order to reach the door, you must first travel half the distance there. You;ll also need to continue to half the remaining distance, and half again, and so on. You won't reach the door in a finite number of jumps! Mathematically one can represent this limit of an infinite sequence of actions as the sum of the series (1/2 + 1/4 + 1/8 + ...). One modern tendency is to attempt to resolve Zeno's paradox by insisting that this sum of infinite series 1/2 + 1/4 + 1/8 is equal to 1. If each step is done in half as much time, the actual time to complete the infinite series is no different than the real time required to leave the room.
However, this approach may not provide a satisfying resolution because it does not explain how one is able to finish going through an infinite number of points, one after another. Today, mathematicians make use of infinitesimal (unimaginably tiny quantities that are almost but not quite zero) to provide a microscopic analysis of the paradox. Coupled with a branch of mathematics called nonstandard analysis and, in particular, internal set theory, we may have resolved the paradox, but debates continues. Some have also argued that if space and time are discrete, the total number of jumps in going from one point to another must be finite.

Pythagoras Founds Mathematical Brotherhood

Around the year 530 B.C., the Greek mathematician Pythagoras moved to Croton, Italy, to teach mathematics, music, and reincarnation. Although many of Pythagoras's accomplishments may actually being due to his disciples, the ideas of his brotherhood  influenced both numerology and mathematics for centuries. Pythagoras is usually credited with discovering mathematical relationships relevant to music harmonies. For example, he observed that vibrating strings produce harmonious sounds when the ratios of the length of the strings are whole numbers. He also studied triangular numbers ( based on patterns of dots in a triangular shape) and perfect numbers (integers that are the sum of their proper positive divisors). Although the famous theorem that bears his name, a^2 + b^2 + c^2 for a right  triangle with legs a, b and hypothenuse c, may have been known to the Indians and Babylonians much earlier, some scholars have suggested that Pythagoras or his students were among the first Greeks to prove it.
To Pythagoras and his followers, numbers were like gods, pure and free from material change. Worship of the numbers 1 though 10 was kind of polytheism for the Pythagoreans. They believed that numbers were alive, with a telepathic form of consciousness. Humans could relinquish their three-dimensional lives and telepatize with those numbers by using various forms of meditation.
Some of these seemingly odd ideas are not foreign to modern mathematicians who often debate whether mathematics is a creation of the human mind or if it's simply a part of the universe, independent of human thought. To the Pythagoreans, mathematics was an ecstatic revelation. Mathematical and Theological blending flourished under the Pythagoreans and eventually affected much of the religious philosophy in Greece, played a role in religion of the Middle Ages, and extended to philosopher Immanuel Kant in modern times.  Bertrand Russell mused that if it were not for Pythagoras, theologians wouldn't not have so frequently sought logical proofs of God and immortality.

Go

Go is a two-player board game that originated in China around 2000 B.C. The earliest written references to the game are from the earliest Chinese work narrative history, Zuo Zhuan ( Chronicles of Zuo), which describes a man in 548 B.C. who played the game. The game spread to Japan, where it became popular in the thirteenth century. Two players alternately place black and white stones on intersections of a 19x19 playing board. A stone or a group of stones is captured and removed if it's tightly surrounded by stones of the opposing color. The objective is to control a larger territory than one's opponent.
Go is complex for many reasons, including its large game board, multifaceted strategies, and huge numbers variations in possible games. After taking symmetry into account here, there are 32,940 opening moves, of which 992 are considered to be strong ones. The number of possible board configurations is usually estimated to be one of the order 10^172, with about 10^768 possible games. Typical games between talented players consist of about 150 moves, with an average of about 250 choices per move. While powerful chess software is capable of defeating top chess players, the best Go program often lose to skillful children.
Go-playing computers find it difficult to "look ahead" in the game to judge outcomes because many more reasonable moves must be considered in Go than in chess. The process of evaluating favorability of a proposition is also quite difficult because a difference of a single unoccupied grid point can affect large group of stones.
In 2006, two Hungarians researchers reported that an algorithm called UCT (for Upper Confidence bounds applied to Trees) could compete with professional Go players, but only on 9x9 boards. UCT helps the computer focus its search on the most promising move.