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Mathematics has been defined as "the logic of drawing conclusions from arbitrary assumptions." And those conclusions remain valid forevermore! But, mathematics does not supply the assumptions. The engineer does that, but then the conclusions are unambiguous. These mathematical models of the physical world have been remarkably useful. Witness the plethora of equations of all sorts, such as the Navier-Stokes equations, the harmonic oscillator equation, the Korteweg de Vries equation, and so on. All have proven effective in predicting real-world phenomena. But we should not forget that they are not the phenomena, but only models of it. Further physical information often leads to improved mathematical models.
A definition of mathematics by the American philospher Charles Peirce (1839 - 1914):
Mathematics: The conceptual or hypothetical science, studies what is and what is not logically possible, without making itself responsible for any actual existence.
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