Alexander A. Lokshin
Moscow, MAKS-Press, 2012 , 40 pages (abstract)
 The general idea of the booklet is as follows. All main mathematical
notions ( such as infinity, variable, integer number) implicitly
depend on the notion of free will. Therefore a scientist employing
mathematics when modeling nature cannot deny the existence of the free
will. (Unfortunately , Stephen Hawking made this incorrect conclusion
in “The Grand Design”).
To make the general idea more clear , suppose we are proving the well-
known formula  S = ah/2 for the area of a triangle. Our proof will
necessarily begin as follows:
“Let us consider AN ARBITRARY triangle…” Here we obviously apply the
operator of the free will choice which cannot be replaced by the
random choice. In fact, let us imagine that our proof begins in such a
way : “Let us consider A RANDOMLY SELECTED triangle…”  Surely, such a
beginning will not lead us to the desired proof. The formula obtained
for a randomly selected triangle is not necessarily valid for all
On the other hand when proving the formula S=ab/2, obviously, it is
impossible to consider all the triangles simultaneously. Thus the
operator of the free will choice must be used inevitably.
More widely, let us consider a variable x which is running about a
sphere of radius 1. Let us pose a question: what does x  denote?
a) x does not denote an object,
b) x does not denote a multitude,
c) x does not denote a physical process.
In my opinion, x denotes the free will choice which the reader of the
mathematical text must do. So, the notion of a variable inevitably is
based on the notion of the free will.
Moreover, considerations adduced in the book show that the notion of
an integer number is based on the notion of the free will as well.
In fact, when constructing integers by means of one-to-one
correspondence we implicitly assume that the mentioned correspondence
is realized by means of continuous lines connecting pairs of objects.
But to ensure continuity of lines above-mentioned one must consider an
arbitrary point on such a line!
A new approach to the Alan Turing problem (how to distinguish a person
from an android) is also proposed ; this approach is based on the idea
that an android cannot generate the notion of an arbitrary object.

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