Who is the mathematician?

                                                   

A mathematician is a person with an extensive knowledge of mathematics who use this knowledge in their
work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, collection, quantity, structure, space, and change.

Mathematicians involved with solving problems outside of pure mathematics are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of the imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, and localized constructs, applied mathematicians work regularly in the study and formulation of mathematical models.

The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry; thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on problems, often concrete but sometimes abstract. As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science, engineering, business, and other areas of mathematical practice.

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Quotations about mathematicians

                                      

Some  quotations about mathematicians, or by mathematicians.

1-    Each generation has its few great mathematicians...and [the others'] research harms no one.

        —Alfred W. Adler (1930- ), "Mathematics and Creativity"

2-    In short, I never yet encountered the mere mathematician who could be trusted out of equal roots, or one who did not clandestinely hold it as a point of his faith that x squared + px was absolutely and unconditionally equal to q. Say to one of these gentlemen, by way of experiment, if you please, that you believe occasions may occur where x squared + px is not altogether equal to q, and, having made him understand what you mean, get out of his reach as speedily as convenient, for, beyond doubt, he will endeavor to knock you down.

        —Edgar Allan Poe, The purloined letter

3-    A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

        —G. H. Hardy, A Mathematician's Apology

4-    Some of you may have met mathematicians and wondered how they got that way.

        —Tom Lehrer

5-    It is impossible to be a mathematician without being a poet in soul.

        —Sofia Kovalevskaya

6-    There are two ways to do great mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else—but persistent.

        —Raoul Bott

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Differences with scientists

                                          

Mathematics differs from natural science in that scientists subject truth claims to tests by experiments, while mathematical propositions are conclusions of mathematical proofs.

If a definite statement is believed plausible by some mathematicians but has been neither proved nor disproved, it is called a conjecture, as opposed to an ultimate goal: a theorem that has been proved.

Scientific theories change when new information about the world is discovered. Mathematics changes in a similar way: New ideas do not falsify old ones, but new concepts refine old concepts and old theories, attaining a fuller understanding of the truth. One method of refinement is generalization, for example widening the scope of a concept. For instance, calculus (in one variable) generalizes to multivariable calculus, which generalizes to analysis on manifolds. The development of algebraic geometry from its classical to modern forms is a particularly striking example of the way an area of mathematics can change radically in its viewpoint, without making what was correctly proved before in any way incorrect; of course mathematical progress clarifies gaps in previous proofs, often by exposing hidden assumptions, which progress has revealed worth conceptualizing.

A theorem is true, and was true before we knew it and will be true after humans are extinct. Of course, our understanding of what the theorem really means gains in profundity as the mathematics around the theorem grows. A mathematician feels that a theorem is better understood when it can be extended to apply in a broader setting than previously known. For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.

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