paradox.....

[ashkanOo]

they are interesting

[ashkanOo]

Mathematics transcends individual civilizations and specific languages. It is one large system of logic - a kind of universal language. As such, certain paradoxes and contradictions have arisen that have troubled mathematicians from ancient times to the present. Some are false paradoxes: they do not present actual contradictions, and are merely slick logic tricks. Others have shaken the very foundations of mathematics - requiring brilliant, creative mathematical thinking to resolve. Others remain unresolved to this day.

[ashkanOo]

Paradox 1: The Motionless Runner

A runner wants to run a certain distance - let us say 100 meters - in a finite time. But to reach the 100-meter mark, the runner must first reach the 50-meter mark, and to reach that, the runner must first run 25 meters. But to do that, he or she must first run 12.5 meters.

Since space is infinitely divisible, we can repeat these 'requirements' forever. Thus the runner has to reach an infinite number of 'midpoints' in a finite time. This is impossible, so the runner can never reach his goal. In general, anyone who wants to move from one point to another must meet these requirements, and so motion is impossible, and what we perceive as motion is merely an illusion.

[ashkanOo]

paradox :http://www.haverford.edu/physics-astro/ ... adoxes.pdf

[jhvh]

فارسی میذارم
مشکل اینجاست که حرکت متحرک نقطه ای در نظر گرفته میشه
متحرک اگه حرکت کنه مثل مسافت : اگه از ا به ب بره بی نهایت نقطه رو طی میکنه
تا اینجا خوب بحث کردی بی نهایت نقطه وجود داره ولی چه ایرادی داره طی کنه؟توضیح میدی؟
در ضمن فاصله در فیزیک نوین حد اقل مقدارش حد پلانکه

[ashkanOo]

فارسی میذارم
مشکل اینجاست که حرکت متحرک نقطه ای در نظر گرفته میشه
متحرک اگه حرکت کنه مثل مسافت : اگه از ا به ب بره بی نهایت نقطه رو طی میکنه
تا اینجا خوب بحث کردی بی نهایت نقطه وجود داره ولی چه ایرادی داره طی کنه؟توضیح میدی؟
در ضمن فاصله در فیزیک نوین حد اقل مقدارش حد پلانکه

The great Greek philosopher Zeno of Elea (born sometime between 495 and 480 B.C.) proposed four paradoxes in an effort to challenge the accepted notions of space and time that he encountered in various philosophical circles. His paradoxes confounded mathematicians for centuries, and it wasn't until Cantor's development (in the 1860's and 1870's) of the theory of infinite sets that the paradoxes could be fully resolved.
Zeno's paradoxes focus on the relation of the discrete to the continuous, an issue that is at the very heart of mathematics. Here we will present the first of his famous four paradoxes.
Zeno's first paradox attacks the notion held by many philosophers of his day that space was infinitely divisible, and that motion was therefore continuous.

plz discuss in english

[ashkanOo]

Are there more integers or more even integers?
Seems like a simple question, right? After all, every even integer is an integer but what about all the even integers? So there are more integers than there are even integers, right? But wait a second. How many even integers are there? An infinite number. And how many integers are there? An infinite number. Hmmmm....

"Infinity," says math student A, "is just a term... there's no way you can actually show me that there is the same number of each."

"Okay, let's play..." says math student B. "Give me an integer, and I'll give you an even integer that corresponds to it. And if two of your integers are different, I guarantee that my two even integers will be different."

Math Student A: Okay... 1

Math Student B: 2

A: 2

B: 4

A: 18

B: 36

A: -100

B: -200

A: n

B: 2n

A: I'm beginning to see what you mean. But let's consider some of the set theory we learned in math class. The set of even integers is contained in the set of integers, but is not equal to that set. So the two sets can't be the same size.

(Who's right? What kind of sets did the teacher put on the board in class? How do these sets differ from those?)

The paradox characterized by the above problem puzzled mathematicians for centuries. At its core lay that troubling concept that haunts all of mathematics: infinity. In 1874 Georg Cantor worked out a system of degrees of infinitythat solved the problem once and for all and greatly increased mathematicians' understanding of infinity and set theory.

[ashkanOo]

Source:

math forum

[ashkanOo]

Archaeologists have found evidence of games of chance on prehistoric digs, showing that gaming and gambling have been a major pastime for different peoples since the dawn of civilization. Given the Greek, Egyptian, Chinese, and Indian dynasties' other great mathematical discoveries (many of which predated the more often quoted European works) and the propensity of people to gamble, one would expect the mathematics of chance to have been one of the earliest developed. Surprisingly, it wasn't until the 17th century that a rigorous mathematics of probability was developed by French mathematicians Pierre de Fermat and Blaise Pascal.

The Problem of Points

The problem that inspired the development of mathematical probability in Renaissance Europe was the problem of points. It can be stated this way:
Two equally skilled players are interrupted while playing a game of chance for a certain amount of money. Given the score of the game at that point, how should the stakes be divided?

In this case 'equally skilled' indicates that each player started the game with an equal chance of winning, for whatever reason. For the sake of illustration, imagine the following scenario.

Pascal and Fermat are sitting in a cafe in Paris and decide, after many arduous hours discussing more difficult scenarios, to play the simplest of all games, flipping a coin. If the coin comes up heads, Fermat gets a point. If it comes up tails, Pascal gets a point. The first to get 10 points wins. Knowing that they'll just end up taking each other out to dinner anyway, they each ante up a generous 50 Francs, making the total pot worth 100. They are, of course, playing 'winner takes all'. But then a strange thing happens. Fermat is winning, 8 points to 7, when he receives an urgent message that a friend is sick, and he must rush to his home town of Toulouse. The carriage man who has delivered the message offers to take him, but only if they leave immediately. Of course Pascal understands, but later, in correspondence, the problem arises: how should the 100 Francs be divided?


In a letter to Pascal, Fermat proposes this solution:

Dearest Blaise,

As to the problem of how to divide the 100 Francs, I think I have found a solution that you will find to be fair. Seeing as I needed only two points to win the game, and you needed 3, I think we can establish that after four more tosses of the coin, the game would have been over. For, in those four tosses, if you did not get the necessary 3 points for your victory, this would imply that I had in fact gained the necessary 2 points for my victory. In a similar manner, if I had not achieved the necessary 2 points for my victory, this would imply that you had in fact achieved at least 3 points and had therefore won the game. Thus, I believe the following list of possible endings to the game is exhaustive. I have denoted 'heads' by an 'h', and tails by a 't.' I have starred the outcomes that indicate a win for myself.

h h h h * h h h t * h h t h * h h t t *
h t h h * h t h t * h t t h * h t t t
t h h h * t h h t * t h t h * t h t t
t t h h * t t h t t t t h t t t t
I think you will agree that all of these outcomes are equally likely. Thus I believe that we should divide the stakes by the ration 11:5 in my favor, that is, I should receive (11/16)*100 = 68.75 Francs, while you should receive 31.25 Francs.