Legend has it that a long time ago, in a village, there was only one barber, and he stipulated that only those who didn't cut their own hair should be cut. This leads to a question: should he cut his hair? Or ask: Who should have a haircut?
If he cuts his own hair, then he violates his own regulations, because according to the regulations, he should not cut his own hair; If he doesn't cut his own hair, he also violates his own regulations, because according to the regulations, he must cut his own hair, so he has to cut his own hair. The barber is puzzled: no matter what he does, he "slaps himself".
In logic, if you admit that a proposition is true, but it is false; If you admit that it is false, but it is true. Such a proposition is called paradox or paradox. The above story is called "barber paradox".
190 1 In June, the British mathematician and philosopher Russell (1872 ~ 1970) discovered the "Russell paradox" named after him, which is a paradox in set theory, so it is also called "set paradox". Its basic content is: if all sets are divided into two categories: A and B, A can take itself as its own element, and B cannot take itself as its own element; So, is the set of all B sets Class A or Class B? If all sets of class B sets belong to class A, since class A can take itself as its own element, then the set of class B sets should belong to class B. If all sets of class B sets belong to class B, then it can obviously be included in all sets of class B sets, making it meet the requirements of class A and belong to class A. From this point of view, all sets of class B are both class A and class B, which creates insurmountable logical contradictions. In 19 18, Russell popularly interpreted the paradox in this relatively advanced set theory as the aforementioned "Barber Paradox", so many literatures compared these two paradoxes, the essence of which was to make logic fall into an unavoidable "strange circle".
So, how did the barber paradox cause the crisis? It did lead to the "crisis"-"the third mathematical crisis". There are insurmountable logical contradictions in set theory, which fundamentally endanger the certainty and rigor of the whole mathematical system. How can this not be a "crisis"?
However, there is a very important historical background here, that is, why did this crisis happen at the beginning of the 20th century, that is, when Russell Paradox was put forward?
It seems that it can come earlier, because mathematical paradoxes in history have already been discovered and countless. For example, Aubrey or Cicero in ancient Greece (BC 106 ~ 43), the "Baldness Paradox" of German philosopher Hegel, the "Paradox of Natural Number Equal to Complete Square Number" of Galileo in Italy, and the German mathematician Schwartz (1843 ~ 19265438). The reason why these paradoxes failed to cause "crisis" is that mathematicians are not confident enough about themselves, because such problems as "paradoxes" abound in mathematics and are not worth mentioning. The second reason why there is no crisis is that some paradoxes have been "overcome". Once overcome, there will be no "crisis". For example, the ancient Greek mathematician Zhi Nuo (about 496-429 BC) put forward four paradoxes: one is the well-known paradox that Achilles, who runs well in ancient Greek mythology, can never catch up with the tortoise, which was solved in the19th century; Others failed to attract enough attention. Therefore, before the 20th century, this "crisis" did not come.
1874, Cantor published a paper on a property of all real algebraic number sets in Klier magazine, which marked the birth of set theory. The establishment of set theory reversed many predecessors' ideas and conflicted with traditional mathematical concepts, so it was criticized by opponents from the beginning. However, when the first international congress of mathematicians was held in Zurich, Switzerland in 1897, German mathematician leonid hurwicz (1859 ~19) and French mathematician Adama (1865 ~ 1963) were full. In addition, with the appearance of "Piano Arithmetic Axiomatic System", the theory of natural numbers is simplified into a set of undefined concepts and several related axioms, and the arithmetic theory is axiomatized. In this way, the basis of mathematics is set theory.
In this way, in the second half of the19th century, mathematicians began to be intoxicated: the foundation of mathematics has been extremely solid, and the rigor of mathematics has also reached. However, almost at the same time, there are some things that make mathematicians less intoxicated: 1897, the Italian mathematician Blary Forti (1861~1931) put forward a paradox named after him; 1899, Cantor also put forward the paradox of maximum cardinal number and maximum ordinal number. These paradoxes in set theory have not been solved, and some people are confused.
However, these have not stopped people's self-confidence. 1900 At the Second International Congress of Mathematicians held in Paris, Poincare, a famous French mathematician and physicist (1854 ~ 19 12) declared: "Now, we can say that complete rigor has arrived." Then came Russell Paradox and the third mathematical crisis.
It can be seen that when people mistakenly think that the mathematical foundation is solid, so they are blindly optimistic, the "third mathematical crisis" is inevitable, but at this time they encounter an insurmountable "paradox".
However, although the emergence of the "third mathematical crisis" shocked the western mathematics, philosophy and logic circles, it did not make them lose their minds. Because people already have the historical experience of the first two crises. Therefore, they worked hard to eliminate this crisis, and this effort has continued to this day. However, in the first 30 years of the 20th century, they invested the most and had the fiercest debate, so many great achievements were produced one after another. One of the achievements is the birth of three schools of mathematics-logicism, intuitionism and formalism.
193 1 year, the Austrian mathematician godel (1906 ~ 1972) published the paper "On the Formal Indefinite Proposition of Mathematical Principles and Related Systems" and gave two "Incompleteness Theorems", which is "an epoch-making greatness for the foundation of mathematics and logic. Godel's first theorem overturns the strong belief that all mathematical fields can be completely axiomatized; The second theorem destroyed all hopes of proving the internal compatibility of mathematics along the route envisaged by Hilbert and others. Since then, all the efforts of the three major schools of mathematics to overcome the "crisis" and find a reliable mathematical foundation have gone up in smoke! As a result, mathematicians once again fell into confusion, and people entered a new round of thinking and exploration along the guiding light of incomplete theorem in confusion.
Incomplete theorem shows that any so-called strict formal system is not perfect, no important department can guarantee that it has no internal contradictions, and the source of human wisdom cannot be fully axiomatized; New proof principles are waiting for us to discover or invent. Some accepted mathematical philosophies should be re-evaluated, and some will be updated or abandoned. This leap in epistemology has opened a broad field of vision for us.
It took nearly a century to explore and overcome the "crisis" in the field of mathematics, which led to the "paradox", perfected the three schools of mathematics, destroyed the illusions of these schools, produced Godel's incomplete theorem, and led to the unfinished exploration so far. So, does this "strange circle" only exist in the field of mathematics?
No, this "strange circle" is universal, it exists in the fields of art, music and so on.
From 65438 to 0979, Douglas Hofstadter, an American mathematician, wrote a book called GEB-Eternal Golden Belt. The title of this book is as curious as the content. This caused a sensation in the United States and won the Pulitzer Prize. The Pulitzer Prize was founded by Pulitzer, a Hungarian who went to the United States (1847 ~1911). Although the prize named after this newspaper publisher is only 1000 dollars each, it is the highest prize in the press. "G" refers to the indexer Godel, "E" refers to the painter Meurice Goroni Reeves escher, and "B" refers to Bach, the father of music.