  Thermodynamics as limiting behaviors of statistics is generalized to
arbitrary system with probability {\it a priori} where thermodynamic
infinite-size limit is replaced by multiple-measurement limit. A duality
symmetry between Massieu's and Gibbs' entropy arises in the limit of infinitely
repeated observations, yielding the Gibbs equation and Hill-Gibbs-Duhem
equation (HGDE) as dual pair. If a system has thermodynamic limit satisfying
Callen's postulate, entropy being an Eulerian function, the symmetry is lost:
the HGDE reduces to the Gibbs-Duhem equation. This theory provides a
de-mechanized foundation for classical and nanothermodynamics and offers a
framework for distilling emergence from large data, free from underlying
details.
