*February 2, 2012 21:11*

Currently, the concept of "free will" does not have the most reliable and well defined. To date, the question of whether there will in living creatures are concerned not only philosophers, but also experimental.

Experiments conducted on flies and people did not give a precise answer to this question. One reason for this may be called the very concept of "free will" has some problems.

Stephen Hawking — famous scientist, wrote that **free will** is an illusion similar.

Of his claims that the existing comprehensive physical theory, the control of all things, determines our actions. But the corollary of this theory can not be predicted for a person, as it is a complex organism, and in addition, this same theory has a certain element of chance, which corresponds to the quantum-mechanical effects. That is what gives the right to say that the declaration of free will stem from the inability to predict a person's actions.

And Academician BB Kadomtsev believes that free will — the freedom of action and freedom to choose. He also explained that man has free will and is free in the commission of the acts. Even if a man's actions affect any circumstances, the choice is still with him.

Perhaps this statement is not all prefer the truth. Moreover, according to Schopenhauer, it can be argued that analyzes only the will of man and the decision may not be verifiable.

However, we will focus on the fact that man is free in his actions, and is responsible for them, he also independently. But what we did not do action, do not forget to hurt animals.

The claim that a person has free will, and must take responsibility for their own actions rather difficult amenable to evidence if we take the argument of Hawking.

Kadomtsev, bringing the claim that free will — is the freedom of choice does not explain the definition of freedom of choice. The same can be said about this definition: "Free will — the ability to choose their own actions." In this case, it is not clear what actually does the word "self." Such verbal definition of freedom of speech reminded, we can say, a vicious circle. The most reliable would make the definition stable and repetitive context.

This idea can be explained as an example of "a duel with pistols." Dantes and Pushkin shoot. Lensky and Onegin shoots. Martynov and Lermontov shoot. This is — a duel with pistols.

The paradox lies in the fact that there are no stable and repetition contexts to define "free will." All because of the free will of each person individually, and no reasons are verifiable.

However, in an area such as math, this context can still be found, as in mathematics, thought-used tools based on the existence of free thought.

The name of the tool — The operator of Free Choice. The action of this operator is saying "let's — an element of X, and is arbitrary." (*) Consider the use of the operator (*), proving the theorem school.

Theorem: The area of each triangle is? the product of its height to the base. Proof: Consider an arbitrary triangle and call it ABC. Carry out the construction and calculation, it is necessary to prove the theorem to the triangle ABC. It follows, as an arbitrary **triangle** ABC, it means that the area of any triangle will be determined by the same formula. Theorem.

Note: The "arbitrary choice" in the proof can not be replaced by a "random selection." Try to prove the theorem: let ABC — triangle, selected randomly. Carry out the construction and calculation, it is necessary to prove that the required formula is suitable for the triangle. Since the triangle ABC is chosen randomly, then … the proof can not be completed. If a formula is true for a random triangle, not the fact that it will be true for all triangles.

This example shows how to "randomly selected element" focuses on a single object, so that the result of analysis can be applied to all objects.

Consider another example of the application of the operator of free choice, but only in mathematical physics. Take the argument of the book by Richard Courant.

And so, we have to define what a function U (Q), defined on the interior of the ball when the point Q tends to the boundary of the ball, will be closer to the given boundary values. Proof Folding way: Let P — point on the boundary, chosen arbitrarily, and Q — point inside the ball, also randomly selected. Evidence is the reliance on geometrical considerations that U (Q) has a fairly small differences on the boundary value at the point P, if Q is sufficiently close to P. Courant concluded: "This proves," without taking into account, as too obvious, the argument "because P — proizvrolnaya point on the boundary of the ball."

Note: The words "because the element's — arbitrary, the above argument is true test for all x», coincide with the second part of (*). They should be the completion of the evidence, starting with the application of (*). Formalized mathematical texts in this consideration is omitted. In the logic, called the theory of predicate logic manipulation mentions — the rule of generalization.

Mathematical text, means there is the presence of (*), convinces the reader into equity of certain conclusions, and this means the free will of the reader. In other words, mathematics, writing texts using the operator (*) to readers who have assumed the existence of **free will**.

The concept of "free will" — a significant tool not only in mathematics but also in mathematical physics. Thanks to this concept has been much results that allow a physical check and kept it. It gives the right to say that free will — not an illusion, as in mathematical papers present the predictive power.

"Variable"

Back in the III century Diophantus meant unknown letters in the solution of equations, but note that the system of letter designations in algebra was formed later. This was after the works of Vieta, who used to refer to the letter of the known quantities.

Further Mathematics zaimela concept of variable, denoted by letters. Now math is unthinkable, if you imagine it without the concept of variable.

In mathematical logic of our modern concept of variable virtually replaced the concept of the unknown.

For example: X +5 = 5 (1) the equation when considering similar predicates, that is, an expression that depends on x and is transformed into a statement (true or false), which is substituted for x any numeric variable. And the solution of equation (1), it is seen as an element of truth, which corresponds to the predicate (that is, the set of all x for which there is a transformation to a true statement of the predicate (1)). In our case, this is a lot of truth in the singular 2.

This view eliminates the "extra" unknown concept, and unify the presentation of material that relates to the theme of "predicates". But the math does not work in the sphere of pure logic to solve the equations give preference to the case of unknown x than x.

When viewed from the point of view of mathematics, these two approaches equivalent, but from an educational **point** of view, the difference between them is significant. The thing is that the "unknown" — a concept more simple, if "variable." Order solution examples, this explains the arrival of the mathematical **concept** of "the variable" only after more than a thousand years after the "unknown."

If x — unknown, which has one solution, then x — the name of the individual object. If the equation has multiple solutions, it is possible for a while to focus on a single solution, assuming x — name of the individual object.

If x — variable, x will not be the name of an individual object, the name of a set of objects, and not the name of a physical process.

If we assume that the variable x — point, running through the field and, if you ask about the location of this point at a certain point and the direction of its movement in the two mathematicians, it is likely that the answers will be different.

But what actually is a variable?

x — an element that is selected from the set X.

Of this concept can be seen that the "variable" is not "primary" and is determined by completing the procedure of choice.

Even showed up here a free choice, even if hidden. This means that free will — not an illusion.