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A scientist is a person who is an expert in at least one area of science and who uses the scientific method to research that area. Upon the request of the poet Samuel Taylor Coleridge in 1833, William Whewell invented the English word "scientist"; before this, the only terms in use were "natural philosopher" and "man of science".
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Traditionally, mathematics has been grouped with the sciences, but in modern times people tend not to regard mathematicians as scientists. Mathematical discoveries generally appear to be arrived at differently than scientific ones, and experiments as they are usually conceived are unable to supply mathematical proof. Yet the class of people called "scientists" includes theorists who never do experiments, and even pure experimentalists often employ mathematics and deduction to arrive at their conclusions. At the same time, a mathematical proof may proceed as a proof by construction, an idea that shares something in common with experiments. Mathematicians usually have hunches about what is "true", and then conduct numerous symbolic or computational experiments in order to establish it.
There is a continuum from the most theoretical to the most empirical sciences, with no clear-cut boundaries. It is hubris to insist on re-categorizing mathematics, which plays such a central role in all quantitative reasoning. In terms of personality, interests, training, and professional activity, there is little difference between mathematicians and theoretical physicists, and even theoretical engineers. (An interest in music is common.) There are many notable examples of people who have moved back and forth among these disciplines. Can you wake up as a mathematician and go to bed as a physicist?
One may argue that there is a clearer distinction between science and engineering. Engineers are concerned with the design of a solution to a practical problem. A scientist may ask "why?" and proceed to research the answer to the question. By contrast, engineers want to know how to solve a problem and how to implement the solution. (On the other hand, this is also what most scientists do routinely.) The common wisdom is that scientists investigate phenomena, whereas engineers invent new solutions to problems or improve upon old ones. But structural and aerospace engineers are usually at the forefront of investigating new phenomena. In the course of their work, scientists may have to complete engineering tasks, such as designing experimental apparatus or building prototypes, while some engineers do first-rate research.
More substantial questions: Is history a science? Is medicine? Is psychoanalysis?