The formulation of Hippasus' Paradox is closely related to the discovery of the Pythagorean Theorem. Therefore, we start with the hook theorem. The Hook Theorem is one of the most famous theorems in Euclidean geometry. The astronomer Kepler once called it one of the two bright pearls of Euclidean geometry. It has an extremely wide range of applications in mathematics and human practical activities, and is also one of the first plane geometry theorems recognized by mankind. In China, the earliest astronomical and mathematical work "Zhou Thigh Calculation Classic" has already had a preliminary understanding of this theorem. However, the proof of the collinear theorem in China was a later event. It was not until Zhao Shuang in the Three Kingdoms period that the first proof of it was supplemented by area cuts.
In foreign countries, the earliest proof of this theorem is the ancient Greek Pythagoras. Therefore, it is generally called "Pythagorean Theorem". And it is said that Pythagoras was ecstatic after completing the proof of this theorem, and killed a hundred cows to celebrate. Therefore, this theorem has also gained a mysterious title: "The Hundred Cows Theorem".
Pythagoras
Pythagoras was a famous mathematician and philosopher in ancient Greece in the fifth century BC. He founded the Pythagorean school of mysticism, a trinity of politics, scholarship and religion. The famous proposition put forward by Pythagoras that "everything is number" is the philosophical cornerstone of this school. "All numbers can be expressed as integers or ratios of integers" was the mathematical belief of this school. However, the Pythagorean Theorem, which was established by Pythagoras, dramatically became the "gravedigger" of the mathematical beliefs of the Pythagorean school. After the Pythagorean Theorem was formulated, one of the members of the Pythagorean school, Hipparchus, considered the question: what is the length of the diagonal of a square with side length 1? He discovered that this length could be expressed neither as an integer nor as a fraction, but only as a new number. Hippasos' discovery led to the creation of the first irrational number √2 in the history of mathematics. The appearance of the tiny √2 created a huge storm in the mathematical world of the time. It directly shook the mathematical beliefs of the Pythagorean school, which was greatly frightened by it. In fact, this great discovery was not only a fatal blow to the Pythagoreans. It was a great shock to all the ancient Greek conceptions of the time. The paradoxical nature of this conclusion is shown by its conflict with common sense: any quantity can be expressed as a rational number to any degree of accuracy. Not only was this a generally accepted belief in Greece at the time, but today, with the highly developed technology of measurement, this assertion is true without exception! But the assertion which is so well established by our experience, and which is entirely in accordance with common sense, is actually disproved by the existence of the tiny √2! How contrary to common sense, how absurd this should be! It simply overturns what was previously known at all. Worse still, there was nothing one could do in the face of this absurdity. This led directly to a crisis in people's understanding at the time, which led to a big storm in the history of Western mathematics, known as the "First Mathematical Crisis".
Eudoxus
Two hundred years later, around 370 BC, the brilliant Eudoxus established a complete theory of proportions. His own writings have been lost, and his results are preserved in the fifth book of Euclid's Principia Geometrica. Eudoxus's ingenious method solved the mathematical crisis caused by the appearance of irrational numbers by avoiding the "logical scandal" of irrational numbers and retaining some of the conclusions associated with them. Eudoxus' solution, however, was achieved by avoiding the direct appearance of irrational numbers by means of a geometrical approach. This rigidly mutilates numbers and quantities. Under this solution, the use of irrational numbers is permissible and legal only in geometry, illegal and illogical in algebra. Or irrational numbers were treated only as mere symbols attached to geometric quantities, not as real numbers. It wasn't until the 18th century, when mathematicians proved that fundamental constants such as pi were irrational, that there were more advocates for the existence of irrational numbers. To the second half of the nineteenth century, the present sense of the theory of real numbers was established, irrational number nature was completely clarified, irrational numbers in the garden of mathematics really took root. The establishment of the legitimate position of irrational numbers in mathematics, on the one hand, so that human understanding of numbers from the rational numbers to the expansion of the real numbers, on the other hand, but also the first mathematical crisis really completely and satisfactorily resolved.
Becquerel's Paradox and the Second Mathematical Crisis
The Second Mathematical Crisis resulted from the use of the tools of calculus. Almost at the same time in the seventeenth century, along with an increase in theoretical and practical scientific understanding, the sharp mathematical tool of calculus was discovered independently by Newton and Leibniz. As soon as this tool was introduced, it showed its extraordinary power. Many difficult problems became easy to solve with this tool. But neither Newton nor Leibniz created a rigorous theory of calculus. Both theories were based on the analysis of infinitesimals, but their understanding and use of infinitesimals as a fundamental concept was confused. As a result, calculus was opposed and attacked by a number of people from its inception. The most violent of these attacks was by the English Archbishop Berkeley.
Bishop Beckley
In 1734, Beckley published a book with a long title, "The Analyst; or an Essay to a Disbelieving Mathematician, in which he examines whether the objects, principles, and assertions of modern analysis are more clearly expressed, or the essentials of faith, than those of the mysteries of religion. Points have been more clearly expressed, or more obviously reasoned. In this book, Berkeley attacked Newton's theories. For example, he accused Newton of calculating the derivative of, say, x2 by taking x in non-zero increments Δx, obtaining 2xΔx + (Δx2) from (x + Δx)2 - x2, dividing by Δx to obtain 2x + Δx, and then suddenly making Δx = 0 to obtain the derivative as 2x. This is "relying on a double error to get an unscientific but correct result". Because the infinitesimal quantity is said to be zero in Newton's theory, and then it is said not to be zero. Therefore, Beckley ridiculed infinitesimal quantities as "the ghosts of dead quantities". Although Beckley's attack came from the purpose of defending theology, it really captured the flaws in Newton's theory and was to the point.
The history of mathematics calls Beckley's problem "Beckley's paradox". Generally speaking, Beckley's paradox can be expressed as "infinitesimal quantity is 0" problem: on the infinitesimal quantity in the practical application of the time, it must be both 0, and is not 0. But from the formal logic, this is undoubtedly a contradiction. This question caused some confusion in the mathematical world at that time, which led to the second mathematical crisis.
Newton and Leibniz
In response to Berkeley's attack, both Newton and Leibniz attempted to solve it by refining their own theories, but neither was entirely successful. This left the mathematicians in an awkward position. On the one hand calculus was a great success in its applications, but on the other hand it had a logical contradiction of its own, the Berkeley paradox. Where does one go from here in terms of the trade-offs for calculus in this situation?
"Go forth, go forth, and you will gain faith!" D'Alembert blew the horn of courage, inspired by this horn, eighteenth-century mathematicians began to disregard the foundation of the rigors of the argument, but more reliance on intuition to open up new mathematical territory. New methods, new conclusions, and new branches of mathematics emerged. After more than a century's long journey, the efforts of several generations of mathematicians, including D'Alembert, Lagrange, the Bayesian family, Laplace, and Euler, who was the greatest of all, a staggering amount of unprecedented virgin land was reclaimed, and the theory of the calculus was enriched in an unprecedented way. the 18th century has sometimes even been called the "century of analysis". The 18th century is sometimes even called "the century of analysis". However, at the same time, the 18th century rough, not rigorous work also led to more and more fallacies in the situation, the dissonance of the ear began to vibrate the nerves of mathematicians. Here is just one example of an infinite series.
Infinite series S = 1-1 + 1-1 + 1 ......... exactly equal to what?
At the time it was thought that on the one hand S = (1-1) + (1-1) + ......... = 0; on the other hand, S = 1 + (1-1) + (1 On the other hand, S = 1 + (1-1) + (1-1) + ......... = 1, so is it not 0 = 1. This contradiction has puzzled mathematicians like Fourier, and even Euler, who is called the hero of mathematicians by the later generations, has made an unforgivable mistake here. He got
1 + x + x2 + x3 + ..... = 1/(1- x)
and then made x = -1, resulting in
S = 1-1 + 1-1 + 1......... = 1/2! From this example, it is easy to see the confusion that arose in the math at that time. The seriousness of the problem lies in the fact that any of the more delicate problems of analysis, such as gradations, convergence of integrals, commutation of differential integrals, the use of higher-order differentiation, and the existence of solutions to differential equations ...... were virtually unanswered. Especially by the beginning of the nineteenth century, Fourier theory led directly to a complete exposure of the problems underlying mathematical logic. It thus became an urgent task for mathematicians to remove the dissonance and put analysis back on a logical basis. By the nineteenth century, the necessary period of critique, systematization, and rigorous argumentation had descended.
Corsi
The first major step toward making the foundations of analysis rigorous was taken by the famous French mathematician Corsi. Cauchy began publishing several landmark books and papers in 1821. In them, he gave rigorous definitions of a series of fundamental concepts of analysis. For example, he began to characterize limits in terms of inequalities, reducing the operation of infinity to the derivation of a series of inequalities. This is the so-called "arithmeticization" of the concept of limit. Later, the German mathematician Weierstrass gave a more refined version of the "ε-δ" method that we currently use. In addition, thanks to the efforts of Cauchy, the concepts of continuity, derivatives, differentiation, integration, and sums of infinite series were also put on a firmer footing. However, under the circumstances, it was not possible to perfect Cauchy's theory of limits because the rigorous theory of the real numbers had not been established.
After Cauchy, Weierstrass, Dedekind, and Kantor, each after their own independent and in-depth study, all attributed the basis of analysis to the theory of real numbers, and in the seventies they each established their own complete system of real numbers. Weierstrass's theory can be summarized as the principle of existence of the limit of an increasing bounded series; Dedekind established the famous Dedekind partition; Cantor proposed to define irrational numbers in terms of rational "fundamental sequences"; in 1892, another mathematician created the "principle of the interval set" to establish the theory of real numbers. In 1892, another mathematician created the "interval set principle" to establish the theory of real numbers. Thus, along the path opened by Cauchy, the rigorous theory of limits and the theory of real numbers were established, completing the logical foundation work of analysis. The problem of the non-contradiction of mathematical analysis was reduced to the non-contradiction of the theory of real numbers, so that the calculus, an unprecedentedly majestic edifice in the history of human mathematics, was built on a firm and reliable foundation. The important but difficult task of rebuilding the foundations of calculus was thus triumphantly accomplished through the efforts of many outstanding scholars. The establishment of a firm and solid foundation for calculus put an end to the temporary chaos in mathematics, and at the same time announced the complete resolution of the second mathematical crisis.
Russell's Paradox and the Third Mathematical Crisis
In the second half of the nineteenth century, Cantor created the famous set theory, which had been violently attacked by many when it was first produced. But soon this groundbreaking result was accepted by a wide range of mathematicians and was widely and highly acclaimed. Mathematicians found that the whole edifice of mathematics could be built up from the natural numbers and Cantor's set theory. Thus set theory became the cornerstone of modern mathematics. The discovery that "all mathematical results can be built on the basis of set theory" mesmerized mathematicians, and in 1900, at the International Congress of Mathematicians, the famous French mathematician Poincaré declared enthusiastically: "...... ...with the help of the concept of set theory we can build the whole edifice of mathematics ...... Today we can say that absolute rigor has been attained ......"
Cantor
But it didn't last long.In 1903, news broke that shocked the mathematical world: set theory was flawed! This was the famous Russell's Paradox, proposed by the British mathematician Russell.
Russell constructed a set S: S consists of the set of everything that is not an element of itself. Then Russell asked: does S belong to S? According to the law of the middle of the row, an element either belongs to a set or does not belong to a set. Thus, for a given set, it makes sense to ask whether it belongs to itself. However, the answer to this seemingly reasonable question leads to a dilemma. If S belongs to S, by definition S does not belong to S. Conversely, if S does not belong to S, again by definition S belongs to S. Either way it is a contradiction.
Russell
In fact, paradoxes were found in set theory before Russell. For example, in 1897, Braley and Forti proposed the maximal ordinal paradox. in 1899, Cantor himself discovered the maximal base paradox. However, because these two paradoxes are involved in the set of many complex theories, so only in the mathematical world uncovered a small ripple, failed to attract great attention. Russell's paradox is different. It is very simple and easy to understand, and involves only the most basic things in set theory. Therefore, Russell's paradox caused a great shock in the mathematical and logical circles at that time. For example, G. Frege said sadly after receiving Russell's letter introducing this paradox: "There is nothing more disagreeable to a scientist than the collapse of his foundations towards the end of his work. A letter from Mr. Russell has placed me in just that position." Dedekind has thus postponed the reprinting of his essay "What is the Nature and Role of Numbers". The paradox was, so to speak, like a boulder thrown on the calm waters of mathematics, and the enormous repercussions it caused led to the Third Mathematical Crisis.
After the crisis arose, mathematicians came up with their own solutions. It was hoped that Cantor's set theory could be revamped to rule out paradoxes by placing restrictions on the definition of sets, which would require the establishment of new principles. "These principles must be sufficiently narrow to ensure the exclusion of all paradoxes; on the other hand, they must be sufficiently broad so that everything of value in Cantor's set theory is preserved." In 1908, Zemero based this principle of his on the first system of axiomatized set theory, which was later improved by other mathematicians and became known as the ZF system. This axiomatized set system largely compensated for the shortcomings of Cantor's plain set theory. In addition to the ZF system, there are various other axiomatic systems of set theory, such as the NBG system proposed by Neumann and others. The establishment of the axiomatic set system successfully ruled out the paradoxes appearing in set theory, thus solving the third mathematical crisis more satisfactorily. But on the other hand, Russell's paradox had an even more profound impact as far as mathematics is concerned. It brought the problem of the foundations of mathematics before mathematicians for the first time in the guise of a most urgent need, leading to the study of the foundations of mathematics by mathematicians. Further developments in this area, in turn, have had an extremely profound impact on mathematics as a whole. For example, around the mathematical foundations of the dispute, the formation of modern mathematical history of the famous three major schools of mathematics, and the work of each school in turn have contributed to the great development of mathematics, and so on.
The above briefly introduces the history of mathematics due to the mathematical paradox of the three mathematical crisis and spend, from which we can easily see the mathematical paradox in the promotion of the development of mathematics in the huge role. It has been said that: "to raise the problem is half of the solution", and mathematical paradoxes put forward precisely so that mathematicians can not avoid the problem. It says to the mathematician, "Solve me or I will swallow your system!" As Hilbert pointed out in his essay "On Infinity," "It must be admitted that the situation in which we now find ourselves cannot long be endured in the face of these paradoxes. One imagines that the conceptual structures and methods of reasoning that everyone learns, teaches, and applies in mathematics, the model of reliability and truthfulness that it claims to be, could lead to irrational results. If even mathematical thinking fails, where should one look for reliability and truthfulness?" The emergence of a paradox forces mathematicians to devote their greatest enthusiasm to solving it. In the process of solving the paradox, various theories came into being: the first mathematical crisis led to the birth of axiomatic geometry and logic; the second mathematical crisis led to the improvement of the basic theory of analysis and the creation of set theory; the third mathematical crisis led to the development of mathematical logic and a number of modern mathematics. Mathematics has thus flourished, which is perhaps where the importance of mathematical paradoxes lies.
Paradoxes at a glance
1. The barber's paradox (Russell's paradox): only one person in a village barber, and the village people need a haircut, the barber rules, to and only to the village does not give their own haircuts. Ask: does the barber give a haircut to himself or not?
If the barber gives himself a haircut, he is breaking his promise; if the barber does not give himself a haircut, then according to his rule, he should give himself a haircut. Thus, the barber is in a dilemma.
2. Zeno's Paradox - Achilles and the Tortoise: In the 5th century BC, Zeno used his knowledge of infinity, continuity, and partial sums to come up with the following famous paradox: He proposed to have a race between Achilles and the tortoise, and to have the tortoise start 1,000 meters in front of Achilles. It is assumed that Achilles can run 10 times faster than the tortoise. The race starts, and when Achilles has run 1000 meters, the tortoise is still 100 meters ahead of him; when Achilles runs the next 100 meters, the tortoise is still 10 meters ahead of him ...... So, Achilles can never catch up with the tortoise.
3. The Liar's Paradox: In the 6th century B.C., the ancient Greek philosopher of Crete, Epimenides, had this to say: "Every word spoken by all Cretans is a lie."
If this statement is true, then that means that Ibimenides the Cretan has spoken a true statement, but in contradiction to his true statement that every word spoken by all Cretans is a lie; and if this statement is not true. That is to say, Ibimenides the Cretan told a lie, then the truth should be that every word spoken by all Cretans is true, and the two again contradict each other.
So it is difficult to justify it in any way, which is the famous liar's paradox.
In the 4th century B.C., the Greek philosophers came up with another paradox: "This statement I am now making is false." Ditto, which is again difficult to justify!
The Liar's Paradox continues to haunt mathematicians and logicians to this day. The Liar's Paradox takes many forms. E.g., I predict, "What you are going to say below is 'no', right? Answer with 'yes' or 'no'."
Another example is, "My next statement is wrong (right) and my previous statement was right (wrong)."
4. Paradoxes related to infinity:
{1, 2, 3, 4, 5, ...} is the set of natural numbers:
{1, 4, 9, 16, 25, ...} is the set of numbers squared by natural numbers.
These two sets of numbers can easily form a one-to-one correspondence, so are there as many elements in each set?
5. Galileo's paradox: We all know that the whole is greater than the parts. By connecting lines from points on line BC to vertex A, each line intersects line DE (point D on AB, point E on AC), so DE is as long as BC, which is contradictory to the diagram. Why?
6. Paradox of the Unexpected Exam: A teacher announces that an exam will be held on one of the five days (Monday through Friday) of the following week, but then tells the class, "You cannot know which day it is, and you will only be informed at 8:00 a.m. on the day of the exam that it will be held at 1:00 p.m.. "
Can you tell why this exam could not take place?
7. The Elevator Paradox: In a skyscraper, there is an elevator that is run by a computer that stops at every floor and stays for the same amount of time. However, Mr. Wang, whose office is near the top floor, said, "Whenever I have to go downstairs, I have to wait for a long time. The elevators that stop always go up and rarely go down. How strange!" Ms. Li is also unhappy with the elevator; she works in an office near the ground floor and has to go to the restaurant on the top floor for lunch every day. She said, "No matter when I have to go upstairs, the elevator that stops always goes down and rarely goes up. It's so annoying!"
What's going on here? The elevator obviously stops at each floor for the same amount of time, but why does it make people near the top and bottom floors wait?
8. The Coin Paradox: Two coins are placed flat together, and the top coin makes a half-circle around the bottom coin, resulting in the pattern in the coin being in the same position as it was at the beginning; however, common sense would dictate that the pattern of the coin that has made a half-circle around the circumference should be facing down! Can you explain why?
9. The Heap Paradox: Obviously, 1 grain is not a heap;
If 1 grain is not a heap, then 2 grains are not a heap;
If 2 grains are not a heap, then 3 grains are not a heap;
......
If 99999 grains are not a heap, then 100000 grains are not a heap;
If 99999 grains are not a heap, then 100000 grains are not a heap. grains are not a heap, then 100000 grains are not a heap;
......
10. The pagoda paradox: If you take one brick from a tower of bricks, it doesn't collapse; if you take two bricks, it doesn't collapse; ...... draw the Nth brick when the tower collapses. Now start drawing bricks in a different place, and unlike the first time, the tower collapses when the Mth brick is drawn. When the tower collapses, there are L fewer bricks. And so on, with each change of location, the tower collapses with a different number of bricks. So just how many bricks are drawn before the tower collapses?
Tired of pulling!!!!