Please read my answers here where I explain what a physics theory is.
There exists a thermodynamics theory, which has the axiomatic laws given here. Uning these laws a plethora of mathematical relations and definitions for simplicity can be defined, and rigorously proven. Thermodynamics is used extensively and its predictions continually validated. The increase in entropy of isolated systems can be shown using these laws and mathematics.
Therefore the entropy of an isolated system either increases or, in the limit, remains constant
It is shown inductively using the laws and the mathematics of the theory of Thermodynamics.
In summary, the question is, the idea of Entropy is assumed to be true, Scientifically or Statistically?
Entropy is a mathematical definition using the variables of the theory of Thermodynamics, and is directly derivable form its laws. It is true mathematically, since the theory is considered valid, laws and all. Within its region of validity, Thermodynamics has not been falsified.
To answer the title question:
How do we know that the idea of entropy is true?
In physics one uses concepts in models and as long as the models are validated, the concepts are considered valid. Truth is an attribute that can given to mathematical theorems. Assumptions about natural observables can either be validated or falsified. At the moment thermodynamics is validated by an enormous amount of data. Observations not validating thermodynamics are mostly in the small dimensions and in the framework of quantum mechanics in general ( like superfluidity). This reduces the phase space of variables where thermodynamics is a valid description of nature. So the concept of entropy is valid in the realm of the phase space of variables where the laws of thermodynamics are valid.
