"How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play." (Tobias Danzig, "Number: The Language of Science", 1930)
"Pure mathematics and
physics are becoming ever more closely connected, though their methods remain
different. One may describe the situation by saying that the mathematician plays
a game in which he himself invents the rules while the while the physicist
plays a game in which the rules are provided by Nature, but as time goes on it
becomes increasingly evident that the rules which the mathematician finds
interesting are the same as those which Nature has chosen." (Paul A M Dirac, "The Relation Between Mathematics and
Physics", Proceedings of the Royal Society of Edinburgh, 1938-1939)
"God is a child; and
when he began to play, he cultivated mathematics. It is the most godly of man’s
games." (Vinzenz Erath, "Das Blinde Spiel" ["The Blind Game" ] , 1954)
"When we propose to apply mathematics we are stepping outside our own realm, and such a venture is not without dangers. For having stepped out, we must be prepared to be judged by standards not of our own making and to play games whose rules have been laid down with little or no consultation with us. Of course, we do not have to play, but if we do we have to abide by the rules and above all not try to change them merely because we find them uncomfortable or restrictive." (Mark Kac, "On Applying Mathematics: Reflections and Examples", Quarterly of Applied Mathematics, 1972)
"Some people think that mathematics is a serious business that must always be cold and dry; but we think mathematics is fun, and we aren’t ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive." (Donald E Knuth et al, "Concrete Mathematics: A Foundation for Computer Science", 1989)
"And you should not think that the mathematical game is arbitrary and gratuitous. The diverse mathematical theories have many relations with each other: the objects of one theory may find an interpretation in another theory, and this will lead to new and fruitful viewpoints. Mathematics has deep unity. More than a collection of separate theories such as set theory, topology, and algebra, each with its own basic assumptions, mathematics is a unified whole." (David Ruelle, "Chance and Chaos", 1991)
"Mental imagery is often useful in problem solving. Verbal descriptions of problems can become confusing, and a mental image can clear away excessive detail to bring out important aspects of the problem. Imagery is most useful with problems that hinge on some spatial relationship. However, if the problem requires an unusual solution, mental imagery alone can be misleading, since it is difficult to change one’s understanding of a mental image. In many cases, it helps to draw a concrete picture since a picture can be turned around, played with, and reinterpreted, yielding new solutions in a way that a mental image cannot." (James Schindler, "Followership", 2014)
"Mathematicians start by playing around with ideas to get a feel for what might be possible, good and bad."
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