I will talk about the mathematical
aspects of Shannon's theory of
information. In 1954, at the
International Congress of Mathematicians,
A. N. Kolmogorov talked about
Shannon theory, addressing to probability
theorists who con- sidered Shannon's
work mostly as engineering. In the
preface to the Russian translation
of Shannon's works on information
theory he remarked that "Shannon's
mathematical intuition was amaz-
ingly precise" and that "he can
ranked as both one of the lead- ing
mathematicians and one of the leading
en- gineers of his time." In
this connection, we can now say
that Shannon theory presently appears
almost in all works on information
theory and has itself developed with
them. For example, owing to Shannon
theory, ergodic theory got a
new impulse. Shannon theory is
also closely connected with mathematical
statistics, particularly, with nonparametric
statistics. We can find a bound
on risk by estimating the capacity
of channels in the epsilon-entropy
of messages. It is used by mathematical
statisticians. Such methods helped
to solve many problems in mathematical
statistics. More- over, constructive
methods in Shannon theory work for
many problems in networks.
Some people think it is information
theory only if it has something
to do with applications. It is
very important to think about applications,
but it is also very important
to understand that information-
theoretic research is a very im-
portant mathematical area.