  Introduction to the special issue of Phil. Trans. R. Soc. A 376, 2018,
`Hilbert's Sixth Problem'. The essence of the Sixth Problem is discussed and
the content of this issue is introduced.
  In 1900, David Hilbert presented 23 problems for the advancement of
mathematical science. Hilbert's Sixth Problem proposed the expansion of the
axiomatic method outside of mathematics, in physics and beyond. Its title was
shocking: "Mathematical Treatment of the Axioms of Physics." Axioms of physics
did not exist and were not expected. During further explanation, Hilbert
specified this problem with special focus on probability and "the limiting
processes, ... which lead from the atomistic view to the laws of motion of
continua". The programmatic call was formulated "to treat, by means of axioms,
those physical sciences in which already today mathematics plays an important
part." This issue presents a modern slice of the work on the Sixth Problem,
from quantum probability to fluid dynamics and machine learning, and from
review of solid mathematical and physical results to opinion pieces with new
ambitious ideas. Some expectations were broken: The continuum limit of
atomistic kinetics may differ from the classical fluid dynamics. The "curse of
dimensionality" in machine learning turns into the "blessing of dimensionality"
that is closely related to statistical physics. Quantum probability facilitates
the modelling of geological uncertainty and hydrocarbon reservoirs. And many
other findings are presented.
