Mathematical Physics (MP) simply put is the study of Nature using advanced mathematics. As our knowledge about Nature advances, so our needs of higher mathematics become more severe.

The idea of MP is actually not new. Plato had stated that everything in this world is governed by mathematics by his famous sentence

              God ever geometrizes.
    

In 1623, Galileo published his book Il Saggiatore. Although the book was an unfair polemic against the treatise on the comets by Orazio Grassi since Galileo was actually wrong, the book was a sensation for its passionate linguistic style. It is still a great work for the definition of mathematical physics. In it we read one of the most famous passages ever written for the explanation of the use of mathematics in physics:

   
    Philosophy  is written in this grand book -- I mean the universe -- 
    which stands continually open to our gaze,  but it cannot be 
    understood unless one first learns to comprehend the language 
    and interpret the characters in which it is written. It is written in 
    the language of mathematics, and its characters are triangles, 
    circles, and other geometrical figures, without which it is humanly 
    impossible to understand a single word of it; without these, one 
    is wandering around in a dark labyrinth.
     

More recently, Dirac has expressed the use of mathematics in physics in a very concise and self-explanatory way:

 
        The steady progress of physics requires for its theoretical
        formulation a mathematics that gets continually more advanced.
        This is only natural and to be expected. What, however,
        was not expected by the scientific workers of the last century [19th century]
        was the particular form that the line of advancement of the
        mathematics would take, namely, it was expected that the
        mathematics would get more complicated, but would rest on
        a permanent basis of axioms and definitions, while actually
        the modern physical developments have required a mathematics
        that continually shifts its foundations and gets more abstract.
        Non-euclidean geometry and non-commutative algebra, which
        were at one time considered to be purely fictions of
        the mind and pastimes for logical thinkers, have now been found
        to be very necessary for the description of general facts of the
        physical world. It seems likely that this process of increasing
        abstraction will continue in the future and that advance in
        physics is to be associated with a continual modification
        and generalization of the axioms at the base of mathematics
        rather than with logical development of any one mathematical
        scheme on a fixed foundation.

        There are at present fundamental problems in theoretical
        physics awaiting solution, e.g., the relativistic formulation of quantum 
        mechanics and the nature of atomic nuclei (to be followed by more 
        difficult ones such as the problem of life), the solution of which 
        problems will presumably require a more drastic revision of our 
        fundamental concepts than any that have gone before. Quite likely 
        these changes will be so great that it will be beyond the power of
        human intelligence to get the necessary new ideas by direct
        attempts to formulate the experimental data in mathematical
        terms. The theoretical worker in the future will therefore have
        to proceed in a more indirect way. The most powerful method 
        of advance that can be suggested at present is to employ all the 
        resources of pure mathematics in attempts to perfect and generalise 
        the mathematical formalism that forms the existing basis of theoretical 
        physics, and after each success in this direction, to try to interpret 
        the new mathematical features in terms of physical entities.
      

Dirac's suggestion `to employ all the resources of pure mathematics in attempts to perfect and generalise the mathematical formalism' has been enthusiastically carried out by mathematical physicists. Today, there are proposed Theories of Everything in physics that cannot be fully understood due to the lack of the necessary mathematical formalism. Many physicists and mathematicians are involved in creating new mathematics just to advance these theories and, perhaps, discover the ultimate Theory of Everything.

Why mathematics is so powerful and useful no one really understands. Wigner has put it in the most eloquent way:

    The miracle of the appropriateness of the language of mathematics for
    the formulation of the laws of physics is a wonderful gift which we
    neither understand nor deserve. We should be grateful for it and hope
    that it will remain valid in future research and that it will extend,
    for better or worse, to our pleasure even though perhaps also to our
    bafflement, to wide branches of learning.