Murad's Chamcha Rayhan was Fucked by all Max Boys Brian Davis (Brian Davies), a professor of the Royal College of London. The paper, entitled "Whither Mathematics?" ( "Where is mathematics?"), Argued that the XX century, the most accurate of the exact sciences suffered fractures, which fundamentally changes the nature of the results obtained therein. In the future, according to Professor Davis, mathematics will be very much different from the one science that has been known for the past two thousand years.
For thousands of years it was believed that mathematics offers irrefutable eternal truth. Many remarkable mathematical allegations such as Euclidean geometry theorems were correct in our day, in exactly the same as two thousand years ago. Yet in the XX century mathematics experienced three deep crisis, which substantially altered the status of mathematical research.
The first of these crises linked to Gödel's incompleteness theorem, which argues that any sufficiently rich axiomatic system is a proposal that its can neither prove nor disprove. Although Gödel theorem has had relatively little impact on the practical work of mathematicians, it is directly linked to the ontological status of the problem of mathematical objects.
Most mathematicians, says Brian Davis, adheres to the concept intuitively known as Platonism. Under this concept, and construction of mathematical essence, like platonovym ideas have some kind of objective existence, such as the logical possibilities. But the objective essence of all properties must be very clearly defined, that hardly stick to Gödel theorem.
Just four colors to color a map of Great Britain so that no two neighboring counties not painted in a single color. So you can color any map on the plane. The theorem was formulated in 1852 and proved in 1976 using the computer (Fig. Simon Singh from the book "The Great Fermat theorem", Moscow, 2000)
The second crisis Brian Davis connects to the invasion of computers in math. Considering the example of the color map theorem four colors, he recalled that the full redundancy of all branches of the proof had accomplished only on the computer. But many mathematicians serious question arises how such evidence can be trusted, which have never been fully tested "by hand".
Criticism here has several aspects. First, the computer can fail to make calculations. Even if the result is checked several times, it only increases the probability of correctness of the evidence, but did not make it absolutely safe. Secondly, the processor and support programmes (compiler, libraries, etc.) may contain (and even probably contains) errors, and can not be completely excluded their impact on the accuracy of the evidence. And finally, most importantly: the program itself, which was written for search or test evidence, too, may contain errors. Strictly mathematically verify that it is fully consistent with the specifications, is as hard as to check manually executed with the help of the evidence (and perhaps more difficult). Suffice it to say that the description of the languages in which the programs are written, contain hundreds of pages is not always perfectly clear text. The inclusion of such descriptions in the wording of the theorem denies any prospect for evidence.
All these considerations led to the fact that a number of extremely pure mathematicians sceptical about the evidence obtained through the use of computers. Yet, in recent decades, an increasing number of theorems, which is neobozrimy evidence for the human mind, if not enhance his computer.
As an example, Davis leads the so-called decision task Kepler most dense packing of spheres. In 1998, Thomas Hales (Thomas Hales) presented in the journal Annals of Mathematics proof approval, which took more than 250 pages and included a geometric reasoning, along with the results of extensive computer calculations. A group of twenty experts began to analyze the evidence, finally disintegrated in 2004, has not come to a final conclusion about the accuracy of the evidence.
Yet as the culmination of a true nightmare "complexity" Brian Davis gives another example - a problem known as the classification of finite simple groups. For the issue is not so important, what is the problem itself. Importantly, the group theory is the basis of many areas of research in physics and mathematics, and therefore the classification of groups considered to be very important.
For his decision in 1970 - was recovered's a kind of international consortium of mathematicians. Some theorists hundreds divided the work among themselves and started to address the problem. It is probably the only example in history of such "industrial" approach to solving mathematical problems. Gradually, has been allocated infinite family of three teams and 26 special cases of finite groups (the existence of the largest of them found only through computers).
Since then, the issue of exhaustive proof of this classification. When different groups began to unite in one common proof, have detected many gaps. The greater part of them was able to gradually close. Yet for the moment - 25 years after the first announcement that the theorem is proved - published only 5 of 12 volumes of full evidence.
According to experts, evidence can be considered relatively stable. But that only means that now known gaps in the evidence does not look principled and, apparently, can be closed at the cost of moderate effort, and without changing the overall strategy evidence. Nevertheless, the very existence of these gaps, said that it was not guarantee the reliability of the evidence giant as a whole. But even worse than that, in time, even if all the gaps in the evidence could be close, it is unlikely the whole earth there at least a dozen mathematicians sufficiently understand the logic monstruoznogo evidence.
So, faced with the problem of mathematics almost insurmountable difficulty of evidence. Decision important task, which is formulated in a few sentences, can take tens of thousands of pages that actually makes it impossible to complete his record and understanding.
Concluding his article Brian Davis describes the nature of the changes taking place in math. "In 1875, every person who is capable in math, for a few months to fully understand the evidence most famous theorems. By 1975, ... Mathematics yet to fully understand any evidence proven theorems. Q in 2075, many areas of pure mathematics will depend on the theorems, which are not understood none of mathematicians - either individually or collectively. ... The usual affair will be a formal verification of complex evidence, but there will be many results, the recognition of which will be based on a social consensus in not less than on strict proof. "
Like engineers, mathematicians will say no knowledge of the firm, and the extent of confidence in the reliability of their results. That could bring together mathematics and other disciplines could lead to the lifting of the philosophical question of the special ontological status of mathematical objects.
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