On Problem Solving X: Mathematicians II

“More than any other science, mathematics develops through a sequence of consecutive abstractions. A desire to avoid mistakes forces mathematicians to find and isolate the essence of problems and the entities considered. Carried to an extreme, this procedure justifies the well-known joke that a mathematician is a scientist who knows neither what he is talking about nor whether whatever he is talking about exists or not.” (Élie Cartan) 

"Mathematics consists of content and know-how. What is know-how in mathematics? The ability to solve problems." (George Polya) 

"The science of physics does not only give us [mathematicians] an oportunity to solve problems, but helps us also to discover the means of solving them, and it does this in two ways: it leads us to anticipate the solution and suggests suitable lines of argument." (Henri Poincaré)
 
“Solving problems can be regarded as the most characteristically human activity.” (George Polya, 1981)

 “Solving problems is the specific achievement of intelligence.” (George Polya, 1957)

"Mystery is found as much in mathematics as in detective stories. Indeed, the mathematician could well be described as a detective, brilliantly exploiting a few initial clues to solve the problem and reveal its innermost secrets. An especially mathematical mystery is that you can often search for some mathematical object, and actually know a lot about it, if it exists, only to discover that in fact it does not exist at all - you knew a lot about something which cannot be." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.” (Paul Lockhart, “A Mathematician's Lament”, 2009)
 
“To learn mathematics is to learn mathematical problem solving.” (Patrik W Thompson, 1985)

„[...] mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It’s the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.“ (Ivars Peterson, „Islands of Truth: A Mathematical Mystery Cruise“, 1990)

"The real raison d'etre for the mathematician's existence is simply to solve problems. So what mathematics really consists of is problems and solutions. And it is the 'good' problems, the ones that challenge the greatest minds for decades, if not centuries, that eventually become enshrined as mathematical mountaintops." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"For solving problems in the real world, not only a knowledge of mathematics is useful, but also the mathematician is meaningful: the mathematician uses specific ways of thinking such as abstraction, generalization, and extraction of truth from various realities to crystallize to a simple statement as a theorem. Moreover, in various applications, mathematicians do not merely apply knowledge in an existing specific area of mathematics, but can build up a new theoretical system of mathematics for solving concrete problems." (Masahiro Yamamoto, "Mathematics for Industry: Principle, Reality and Practice, from the Point of View of a Mathematician", [in "What Mathematics Can Do for You"] 2013)

"What I really am is a mathematician. Rather than being remembered as the first woman this or that, I would prefer to be remembered, as a mathematician should, simply for the theorems I have proved and the problems I have solved.“ (Julia Robinson)

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