Some men would be called mathematicians in any man's language; other physicists or engineers. These typical men are differentiated in certain essential respects:Next, we have a list of categories of usefulness for mathematics in service for industry, taken from 'Applications of Undergraduate Mathematics in Engineering' by Ben Noble, pp. 6-8.
The typical mathematician feels great confidence in a conclusion reached by careful reasoning. He is not convinced to the same degree by experimental evidence. For the typical engineer these statements may be reversed. Confronted by a carefully thought-out theory which predicts a certain result, and a carefully performed experiment which fails to produce it, the typical mathematician asks first `What is wrong with the experiment?' and the engineer `What is wrong with the argument?' Because of this confidence in thought processes the mathematician turns naturally to paper and pencil in many situations in which the engineer or physicist would resort to the laboratory. For the same reason the mathematician in his `pure' form delights in building logical structures, such as topology or abstract algebra, which have no apparent connection with the world of physical reality and which would not interest the typical engineer; while conversely the engineer or physicist in his `pure' form takes great interest in such useful information as a table of hardness data which may, so far as he is aware, be totally unrelated to any theory, and which the typical mathematician would find quite boring.
A second characteristic of the typical mathematician is his highly critical attitude towards the details of a demonstration. For almost any other class of men an argument may be good enough, even though some minor question remains open. For the mathematician an argument is either perfect in every detail, in form as well as in substance, or else it is wrong. There are no intermediate classes. He calls this `rigorous thinking,' and says it is necessary if his conclusions are to have permanent value. The typical engineer calls it `hair splitting,' and says that if he indulged in it he would never get anything done.
The mathematician also tends to idealize any situation with which he is confronted. His gases are `ideal,' his conductors `perfect,' his surfaces `smooth.' He admires this process and calls it `getting down to essentials'; the engineer or physicist is likely to dub it somewhat contemptuously as `ignoring the facts.'
A fourth and closely related characteristic is the desire for generality. Confronted with the problem of solving the simple equation x^3-1=0, he solves x^n-1=0 instead. Or asked about the torsional vibration of a galvanometer suspension, he studies a fiber loaded with any number of mirrors at arbitrary points along its length. He calls this `conserving his energy'; he is solving a whole class of problems at once, instead of dealing with them piecemeal. The engineer calls it `wasting his time'; of what use is a galvanometer with more than one mirror?
In the vase army of scientific workers who cannot be tagged so easily with the badge of some one profession, those may properly be called "mathematicians" whose work is dominated by these four characteristics of greater confidence in logical than experimental proof, severe criticism of details, idealization, and generalization. The boundaries of the profession are perhaps not made sharper by this definition, but is has the merit of being based upon type of mind, which is an attribute of the man himself, and not upon such superficial and frequently accidental matters as the courses he took in college or the sort of job he holds.
It is, moreover, a more fundamental distinction than can be drawn between, say, physicist, chemist, and astronomer.
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