  The Turing machine is one of the simple abstract computational devices that
can be used to investigate the limits of computability. In this paper, they are
considered from several points of view that emphasize the importance and the
relativity of mathematical languages used to describe the Turing machines. A
deep investigation is performed on the interrelations between mechanical
computations and their mathematical descriptions emerging when a human (the
researcher) starts to describe a Turing machine (the object of the study) by
different mathematical languages (the instruments of investigation). Together
with traditional mathematical languages using such concepts as 'enumerable
sets' and 'continuum' a new computational methodology allowing one to measure
the number of elements of different infinite sets is used in this paper. It is
shown how mathematical languages used to describe the machines limit our
possibilities to observe them. In particular, notions of observable
deterministic and non-deterministic Turing machines are introduced and
conditions ensuring that the latter can be simulated by the former are
established.
