  We review recent progress in understanding the meaning of mutual information
in natural language. Let us define words in a text as strings that occur
sufficiently often. In a few previous papers, we have shown that a power-law
distribution for so defined words (a.k.a. Herdan's law) is obeyed if there is a
similar power-law growth of (algorithmic) mutual information between adjacent
portions of texts of increasing length. Moreover, the power-law growth of
information holds if texts describe a complicated infinite (algorithmically)
random object in a highly repetitive way, according to an analogous power-law
distribution. The described object may be immutable (like a mathematical or
physical constant) or may evolve slowly in time (like cultural heritage). Here
we reflect on the respective mathematical results in a less technical way. We
also discuss feasibility of deciding to what extent these results apply to the
actual human communication.
