Theory of Infinite Aristotle and Galileo

Abnormal and paradoxical traits of the idea of ​​infinity were subjected to hardships brilliant minds of the greatest thinkers of the time. True mathematical mysteries and, in general, the world is at the frontier of our thinking. Exceeded, we go beyond what we know is possible and we can begin to explore the wonders of the universe to the extreme. Since the ancient Greeks, scholars have questioned the infinite, but the first man that really caught glimpses of details available about the concept represented by an "eight" horizontally disposed Galielo Galilei was a remarkable visionary.

 

Infinity is. possible

Infinity has always been treated with a mixture of fascination and humility. Some have associated the idea of ​​divinity, while others regard it as a concept with no practical value in the real world. The latter argue their position saying that even math seemingly infinite dependence, can be done by resorting to endless amounts but finite. The ancient Greeks were somewhat uncomfortable concept, given the time which i have assigned "Apeiron", a notion with the same negative connotations modenra civilization understands the word "chaos". "Apeiron" lacked the control, wild and dangerous.

The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set which is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC) only by showing that it follows from the existence of the natural numbers.
A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number.
If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset.
If a set of sets is infinite or contains an infinite element, then its union is infinite. The powerset of an infinite set is infinite. Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped onto an infinite set is infinite. The Cartesian product of an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets each containing at least two elements is either empty or infinite; if the axiom of choice holds, then it is infinite.
If an infinite set is a well-ordered set, then it must have a nonempty subset which has no greatest element.
In ZF, a set is infinite if and only if the powerset of its powerset is a Dedekind-infinite set, having a proper subset equinumerous to itself. If the axiom of choice is also true, infinite sets are precisely the Dedekind-infinite sets.
If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.

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Aristotle (384-322 BC), one of the greatest philosophers of mankind, a disciple of Plato and teacher of Alexander the Great

As such, Aristotle "tuned" so strong that hardly infinity has been considered by anyone until the eighteenth century. His approach has proved extremely pragmatic. Aristotle decided that there must be infinite because time seems to have no beginning and no end. Also, it was possible for someone to claim that a count could have ever completed. If there had been some the greatest number - "maximum" What was wrong with saying "maximum + 1" or "maximum + n"? On the other hand, the infinite can not exist in the real world. If there were, for example, a physical body indefinitely, says Aristotle, it should be without borders - though, by definition, to be an object body must have edges.
Premise, otherwise intelligent, Aristotle was therefore that there are infinite while there. Instead of being a true real property, he argued, there is only infinite possibility. He can exist in principle, but practically never happened. And plastic gave a great example of the great philosopher goes like this: "If a man from another part of the world would come and we would now ask Olympics to show him that we are so proud, I could not do it . At this point, they do not exist in reality, but there as a possibility. Similarly, and infinity is in a state of case ". Everyone was satisfied with this "proof" for nearly 2,000 years from that time until Galileo Galilei entered the scene.

Galileo's imperfect wheel

After his house arrest in 1634 Galielo of due process that was initiated for his heretical heliocentric theory, the scientist has not only become inactive. At the time he wrote the book considered to be the primary work of his scientific activity, called "Discourses and mathematical demonstrations concerning two new sciences", equivalent in value to the "Principia" of Newton. The book was written as a conversation between a number of characters, particularly addressing important issues scientifically. In stringing narrative after they debated and debated about what you knit material, the characters have a deviation apparently more for the sake of art, nature infinitului.Galileo brought in this regard a number of novel approaches, but two of them deserve noted in particular. The first involves the movement of a wheel.
Galileo starts to rotate imperfect, with few sides, as for example in the form of hexagons. The wheels are three-dimensional - to imagine them as if they be made of marble. Hezagonul lower is fixed in the sea and each moves on its own horizontal route. Rotate unit to the side so that they move on the next side. Moving, big wheel pivots on one corner and a long path towards the sides. What happened to the smaller wheel?

Not just the big wheel on the distance moved, but small wheel, must be so, because they are fixed together. I mean both wheels must have traveled exactly the same distance between their position and the previous now. Landed on its side next small wheel executed sixth of a turn and seems to have moved its route as its edge length, but things did not happen because she was moved by remote side large wheels. So to execute the movement in surplus and to respect the laws of physics, small wheel was completely built from the route and laid on its side next. All that, the distance between its initial position and the current must be equal to the size of big wheel side.

Here comes the clever artifice. Galileo imagined an increased number of sides. The more sides, the multiple sets of small movements of the wheel route and small jumps lower own route as it spins. Finally, imagine that the number of sides go to infinity. End up with circular wheels.

My infinity is bigger than your infinity

Spin the wheel again two connectors on those routes. Again, both the same distance, say, a quarter of the circumference of large wheels. Or it should be, because now, something strange happened. Edge big wheel was rotated a quarter of the circumference of her own route. Lower edge of the wheel just spun as a quarter of its own circumference smaller. Yet she must have traveled the same distance as the big wheel, but without the exit route. There are no jumps, or so it seems.
What Galileo imagined it happening here is that as the wheel spins less, an infinite number of infinitesimally small hops is going to cover the difference between small wheel circumference and the distance that it travels. Infinite entered the scene through a physical device capable of doing something seemingly impossible happen. Conclusions drawn characters Galileo's Simplicio and Salviati were there an infinite number of points on a circular wheel and an infinite number of points on the other. But somehow, although each has an infinite number of points, one has traveled a greater distance than the other. An infinity was so like the other, yet higher.

Sounds confusing, because it is a problem to manage the infinite with our finite minds, as Salviati admits in the book. The second model proposed by him is the square, not geometric shape, but the square of a number, meaning any number multiplied by itself. So imagine integers, each with inmultinu it themselves. For everything there is a square integer. We have an infinite number of integers and therefore an infinite number of squares on a one to one correspondence. But here's grasp. There are a lot of numbers that are not perfectly square for nothing. So, although there is a square integer for anything - an infinite set - there are even more individual numbers than perfect squares. Again, different infinities. Galileo discovered something very special about infinity. Normal rules of arithmetic do not apply. There may actually infinities 'smaller' and infinities "older" one replaced the other, which is the same size with him forever. The real implications of Galileo's Thoughts It took over 300 years to come to light, but even so, he planted the seed of everything that was to follow in connection with the infinite.
Fibbonacci the proportion of gold

A painting, a sculpltura an architectural work are all organized by gracefully balanced measures and reports. Infinity itself, in mathematics is hidden, just in the art of proportions. Which rectangle has the most pleasant relation between length and width, for example? Anyone can make an experiment in this direction can try alone or accompanied, to choose the report that finds the best. Is the ratio of width and length close to 2x3, 3x5, 5x8 - standard size notebooks and photos? Or close to another pair of adjacent numbers in the sequence: 1,1,2,3,5,8,13,21,34,55 ...? Don Leonardo Pisa, nicknamed Fibonacci was an Italian mathematician from the beginning of the thirteenth century, which showed how the elegant form sequence of numbers connected to our understanding of what it means pleasing proportions. Secret training is training the famous sequence of numbers each number by adding the two previous.
What connection exists between the array and proprotii suitable? Great Piero della Francesca wrote a book "On the Divine Proportion," and in his paintings and framed the whole parts in boxes with Fibonacci ratios. Leonardo da Vinci noted that branches of trees, climbing spiral stem, the distances between them exactly under the same proprotii. Virtually all the artists working on these principles, whether or not they give out. Cones and snail shells, deer antlers, the lines between rows of sunflower seeds - and again these Fibonacci ratios appear in nature. But only in mathematics, art infinite, these ratios are far past the invisible visible (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, etc.) to define a specific value, called path or Golden ratio (about 1618), the proportion describing the ideal proportions in art and nature finite only approximate.

Desai n-dimensional cube?

But is this ideal of divine or diabolical? Pentacle, pentagram, mark the Black Arts Mephistopheles trap is made ​​of segments that follows the golden ratio. Maybe you are angels of light or the darkness behind mathematician in search of infinity? Art and mathematics are both dependent on balance, and balance is stored in ecuatii.Ecuatiile mathematics are mathematics Cubism. Take for example the five Platonic solids: tetrahedron, cube, octahedron, dodecahedron and icosahedron) that Kepler saw as emblematic of the universe. We can find them everywhere in nature and in art, are what some call "bricks area." How are different from each other, however, a single equation bring them to a common denominator. Enough to count over all forms and denote the result by C. sides and add the number to call the result of L, and the number of veneers, F. What we? For Tetrahedron: C = 4, L = 6, M = 4. Cube: C = 8, L = 12, M = 6. Few things in common. Yet we find, and this also applies where the 3 remaining polyhedra and of all over them, as in each case C-L + F = 2. It is a form of infinity, one that is behind all our arts, and even music, whose harmonies are audible expression of proportionality.
I have just spoken cube has 8 corners, 12 edges and 6 veneers. What can we say but about cubes in four dimensions? What about just four dimensions? Amazing, but enough. We can watch a four-dimensional cube, but we can think about it, researchers say. It has 16 corners, 32 edges and 24 faces. Continue? A cube in seven dimensions are 672 girls. One decadimensional has 5,120 sides. And it can continue because the true nature of mathematics we limit our thinking. Therefore, infinity should not be regarded as a mathematical puzzle to be untied, but as the key to life itself, which, by understanding it, we are free of all constraints of our limited minds and we can have and understand the universe. Because infinity is not only in space whose borders do not perceive, but even the last grain of sand under our feet.