Mathematics is the language used for compact description of the order in nature, especially the laws of physics. This was noted and advocated by Pythagoras, Plato, Galileo, and Newton.
Physics theories use mathematics to obtain order and provide precise formulas, precise or estimated solutions, quantitative results and predictions. Experiment results in physics are numerical measurements. Technologies based on mathematics, like computation have made computational physics an active area of research. The distinction between mathematics and physics is clear-cut, but not always obvious, especially in mathematical physics.
Ontology is a prerequisite for physics, but not for mathematics. It means physics is ultimately concerned with descriptions of the real world, while mathematics is concerned with abstract patterns, even beyond the real world. Thus physics statements are synthetic, while math statements are analytic. Mathematics contains hypotheses, while physics contains theories. Mathematics statements have to be only logically true, while predictions of physics statements must match observed and experimental data.
The distinction is clear-cut, but not always obvious. For example, mathematical physics is the application of mathematics in physics. Its methods are mathematical, but its subject is physical. The problems in this field start with a "math model of a physical situation" and a "math description of a physical law". Every math statement used for solution has a hard-to-find physical meaning. The final mathematical solution has an easier-to-find meaning, because it is what the solver is looking for.
Physics is a branch of fundamental science, not practical science.Physics is also called "the fundamental science" because the subject of study of all branches of natural science like chemistry, astronomy, geology and biology are constrained by laws of physics,similar to how chemistry is often called the central science because of its role in linking the physical sciences. For example, chemistry studies properties, structures, and reactions of matter (chemistry's focus on the atomic scale distinguishes it from physics).
Physics is applied in industries like engineering and medicine.
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