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25.01.2008 Kurt Goedel
Category: Mathematics
(b. 1906)
Mathematician
INTUITION AND LOGIC
The scientist who was destined to carry out a revolution in the exact science of mathematics was born in 1906 in the Austro Hungarian Empire, which was close to its collapse.
For the German Kurt Goedel, the Czech city of Brno became his native.
In 1928, Kurt Godel graduated from the Faculty of Physics and Mathematics of the University of Vienna.
In 1940, he went to the United States, to the Princeton Institute for Advanced Study.
And in the interval between these dates —in 1931 he discovered something that made him famous.
His contribution to the very foundations of mathematics is considered revolutionary, pushing the boundaries of this discipline and having a significant impact on the general worldview and culture of the XX century.
The course of geometry, studied in secondary schools around the world, is based on the" Principles " of Euclid.
The ancient Greek, who lived in the third century BC, formulated several axioms regarding the properties of points and straight lines in the plane, from which follows the validity of many useful and important geometric theorems.
Euclid's axioms are simple and unprovable.
One of them, for example, states that only one single straight line can be drawn through two points, the other that parallel lines intersect at infinity.
These statements are accepted as something obvious and do not require proof.
Euclid, in fact, was able to represent all geometry with the help of a small number of true and fundamental statements, expressed very clearly and concisely.
Mathematicians decided, using the" method " of Euclid, to try to represent other branches of mathematics in a similar way.
Let's say the science of numbers.
The Italian mathematician Giuseppe Peano first formulated arithmetic in the language of axioms.
These statements seemed ridiculously obvious — there is a zero, each number is followed by another number— but in fact they were surprisingly exhaustive.
Peano's work was continued by the outstanding German mathematician David Hilbert and his students.
Hilbert tried to reduce all mathematics to a system of axioms.
He believed that in this form it would be possible to enter it into a computer programmed in such a way that, at the order of the operator, it would issue any statements following from these axioms.
Thus, all possible theorems would be issued by a machine, which would make the work of mathematics in general meaningless, reducing it to the role of an operator of a computing center.
If a mathematical robot were created, people would reach the heights of logic and get an electronic oracle capable of answering any question.
But Gilbert's hopes were not destined to come true.
In 1932, Godel's theorem (or the so called incompleteness theorem) appeared.
"Completeness" is an indication that any real theorem can be derived from axioms.
And "consistency" implies the absence of paradoxes, when both some statements and statements that are opposite to them can be derived.
So, it follows from Godel's theorem that there is no complete formal theory where all the true theorems of arithmetic are provable.
Godel's work had the effect of an exploding bomb.
She forced von Neumann to interrupt the course of lectures in Göttingen, and Hilbert to stop working on his program, which seemed to him so promising.
According to Godel, the consistency and completeness of any logical system cannot be proved with the help of auxiliary means of this system itself.
You can, of course, use the methods of a more powerful system to prove it, but this more powerful system itself also cannot prove its consistency with its own methods, which means that the next more powerful system is required.
As G. I. Ruzavin, a Soviet philosopher and mathematician, pointed out, " it turns out a whole hierarchy of formal systems, each of which will surpass the previous one in strength
means of formalization.
Based on this, in our opinion, it can be argued that a complete formalization cannot be completed at a certain historical stage of the development of mathematics."
A machine whose work is based on the axioms of Peano, as Godel argued, will be unable to answer a well defined sequence of questions.
What these questions are, Goedel did not say.
But we can assume that in the Godelian sense, the puzzle below will be unsolvable.
Let's construct a sequence of integers starting with any integer, and each subsequent number must be equal to half of the previous one if it is even, or the previous one multiplied by three and then added with one if it is odd.
Repeating the procedure for calculating subsequent numbers, we will eventually build the entire sequence: 5, 16, 8, 4, 2, 1.
So, we have come to one.
It turns out that, regardless of the number with which the sequence begins, we always come to one, although there is no proof of this fact.
Perhaps this is due to our inability to find it, but maybe it also points to the shortcomings inherent in the very fundamentals of arithmetic.
The result obtained by Godel goes beyond the narrow framework of arithmetic, also influencing cybernetics.
Some time after Godel's discovery, the mathematician Turing noticed that all computing machines can be replaced with just one simple and even very slow calculator, since, if you do not limit the memory used, such a calculator perceives programs of arbitrary length and complexity.
In principle, you can create countless such programs, but, fortunately, they can be combined and stored together, making a complete list of them.
Not all programs will be useful, and because of some, the machine may even enter the mode of continuously and non stop repetitive calculations.
If everything works fine, then in accordance with the orders in the program, the machine prints another number in response to the number entered into it, i.e. it performs calculations: for example, it can print the square of some number, double it, or output a number following the number entered initially.
In general, this machine can calculate incredibly complex functions of the initial number entered into it.
Functions calculated by a " Turing machine "are by definition" computable", so instructions for calculating them can be passed to different machines without fear that errors or ambiguities will arise.
At the same time, there are functions that cannot be calculated, moreover, they make up the vast majority, although it is difficult to define such a function.
Oddly enough, the example of an uncomputable function follows directly from the theory of the "Turing machine".
If you assign the value " one "to an integer corresponding to a normally running machine, then" zero", on the contrary, will correspond to a machine that has entered the mode of non stop repeated calculations.
Thus, we have given an uncomputable function, and this proof repeats the proof given by Kurt Godel for logical systems.
Knowing this function, we can tell in advance, without resorting to running the program itself, whether the corresponding machine will stop or will idle.
This is not an abstract question: it would be very convenient to know in advance whether the program is running or not before running it into the machine.
The Turing result confirmed what machine users already felt intuitively: namely, that there is no way to determine with certainty how a program works, except to test it in practice.
Is there always an unknown function that cannot be calculated?
Godel's answer was simple: even if the first hundred or a thousand values of this function are calculated, we still do not know anything about how to calculate the subsequent value, so it takes human intelligence and creative efforts to get out of the rigid programming framework for a computer.
Again and again, a person is convinced that a computer is surprisingly diligent and at the same time just as stupid: it performs calculations without thinking, only according to a previously compiled detailed instruction.
Godel had to listen to many reproaches for destroying the integrity of the foundation of mathematics.
He invariably replied that, in fact, the foundations remained as unshakable as before, and his theorem led to a reassessment of the role of intuition and personal initiative in one of the fields of science, in that which is governed by the iron laws of logic, which seemingly leave little room for these advantages.
Despite the assurances of idealists, mathematics turned out to be a real art, where there is a place for improvisation, and Godel himself gave a worthy example of creative service to the museum of Mathematics in his works written in a dry language and dealing, at first glance, only with the essence of the problem.
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