Mathematics: Queen and Servant of Science by E.T.Bell

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Mathematics is queen of the sciences and arithmetic the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but under all circumstances the first place is her due." 

So wrote the master mathematician, astronomer, and physicist, Gauss (1777-1855) over a century ago. Whether as history or prophecy, Gauss' declaration is far from an overstatement. Time after time in the nineteenth and twentieth centuries, major scientific theories have come into being only because the very ideas in terms of which the theories have meaning were created by mathematicians years, or decades, or even centuries before anyone foresaw possible applications to science. 


Without the geometry of Riemann, published in 1854 X or without the theory of in variance developed by the mathematicians Cayley (1821-1895), Sylvester (1814-1897), and a host of their followers, the general theory of relativity and gravitation of Einstein in 1916 could not have been stated. Without the whole mathematical theory of boundary value problems to use a technical term which need not be explained now originating with Sturm (1803-1855) and Liouville (1809-1882), the far-reaching wave mechanics of the atom of the past five years would have been impossible. 

The revolution in modern physics which began with the work of W. Heisenberg and P. A. M. Dirac in 1926 could never have started without the necessary mathematics of matrices invented by Cayley in 1858, and elaborated by a small army of mathematicians from then to the present time. 

The concept of invariance, of that which remains unchanged in the ceaseless flux of nature, permeates modern physics, and it originated in 1801 in the purely arithmetical work of Gauss. 

These are but a few of many similar instances. In none of the scores of anticipations of fruitful applications to science was there any thought of what might come out of the pure mathematics. Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now as in the past are inspired by the art of mathematics rather than by any prospect of ultimate usefulness. However it may be in  engineering and the sciences, in mathematics the deliberate attempt to create something of immediate utility leads as a rule to shoddy work of only passing value. The important practical and scientific applications of mathematics are unsought byproducts of the main purposes of professional mathematicians.






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