  We review different studies of the Periodic Law and the set of chemical
elements from a mathematical point of view. This discussion covers the first
attempts made in the 19th century up to the present day. Mathematics employed
to study the periodic system includes number theory, information theory, order
theory, set theory and topology. Each theory used shows that it is possible to
provide the Periodic Law with a mathematical structure. We also show that it is
possible to study the chemical elements taking advantage of their
phenomenological properties, and that it is not always necessary to reduce the
concept of chemical elements to the quantum atomic concept to be able to find
interpretations for the Periodic Law. Finally, a connection is noted between
the lengths of the periods of the Periodic Law and the philosophical
Pythagorean doctrine.

General Mathematics