  The purpose of this paper is to emphasize the role of language in the process
of teaching and learning mathematics. We will begin with the definition of
mathematics given by Cassiodorus (in its essential features repeated in
Kolmogorov's famous article "Mathematics" from the "Great Soviet Encyclopedia"
and explored in the author's book "Humanitarian oriented mathematics teaching
in general school" (Ryazan, Ryazan Institute of Education Development, 2000,
112 p.). This definition illuminates the exceptional role of language in the
building, discovering, understanding, teaching, and learning of mathematics. We
show that this role is predetermined by the abstract character of mathematics.
The abstract object cannot be shown; it can only be described with the help of
language. This gives rise to the problem of identifying abstract entities on
the grounds of their descriptions. We accentuate the way the language of set
theory helps finding the solution to this problem. We illustrate this through
the examples from elementary geometry and elementary algebra. And we especially
underline the importance of this solution when working with events in the
probability theory. After that, we are going to show how proof and language
help to form and develop imagery thinking and, thereby, illuminate the
possibility of the left brain hemisphere to participate in the formation of
imagery thinking.
  Keywords: mathematics, teaching, and learning, language, imagery thinking,
functional brain asymmetry
