"An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity." (Howard Eves, "In Mathematical Circles", 1969)
"God emerges as a complicated master programmer." (Howard W Eves, "In Mathematical Circles: Quadrants III and IV", 1969)
"Every student of college mathematics learns of the remarkable mathematical accomplishment made toward the end of the nineteenth century wherein, by starting from a postulational development of the whole numbers (that is, positive integers) and making no further assumptions, one first obtains the set of all integers, then the set of all rational numbers, and then the set of all real numbers. Since the irrational numbers, like √'2, are among the real numbers, we see that, at least so far as the real number system is concerned, the ancient Pythagorean belief that everything depends upon the whole numbers is today justified." (Howard W Eves, "In Mathematical Circles: Quadrants I and II", 1969)
"In the beginning, man considered only concrete mathematical problems, which presented themselves individually and with no observed interconnections. When human intelligence was able to extract from a concrete mathematical relationship a general abstract relationship containing the former as a particular case, mathematics became a science. This stage of mathematics may be called empirical, or scientific, mathematics, for mathematical findings were discovered by trial and error, experimentation, and other empirical or laboratory-type procedures." (Howard W Eves, "In Mathematical Circles: Quadrants I and II", 1969)
"Mathematicians have found various significant differences between the older and the newer mathematics. Some have found the major difference to lie in the growing inter connectedness of the subject. Others have found the major difference to lie in the changing methods of the subject. And still others have found the major difference to lie in the altering viewpoints of the subject." (Howard W Eves, "In Mathematical Circles: Quadrants III and IV", 1969)
"Mathematics may be likened to a large rock whose interior composition we wish to examine. The older mathematicians appear as persevering stone cutters slowly attempting to demolish the rock from the outside with hammer and chisel. The later mathematicians resemble expert miners who seek vulnerable veins, drill into these strategic places, and then blast the rock apart with well-placed internal charges." (Howard W Eves, "In Mathematical Circles", 1969)
"Older mathematics appears static while the newer appears dynamic, so that the older mathematics compares to the still-picture stage of photography while the newer mathematics compares to the moving-picture stage. Again, the older mathematics is to the newer much as anatomy is to physiology, wherein the former studies the dead body and the latter studies the living body. Once more, the older mathematics concerned itself with the fixed and the finite while the newer mathematics embraces the changing and the infinite." (Howard W Eves, "In Mathematical Circles", 1969)
"The development of mathematics over the ages may be viewed as a continent slowly rising from the sea. At first perhaps a single island appears, and, as it grows in size, other islands emerge at varying distances from one another. As the continent continues to rise, some of the islands become joined to others by isthmuses that widen until pairs of islands become single large islands. At length a point is reached where the shape of the continent is essentially defined, and there remain only a number of lakes and inland seas of various sizes. As the continent further rises, these lakes and seas shrink and vanish one by one. The older mathematics compares to the situation when the general shape of the rising continent is still undefined and the land area consists largely of islands of different sizes. The newer mathematics compares to the situation when the general shape of the rising continent has become essentially clear, with most of the former islands now joined by stretches of land." (Howard W Eves, "In Mathematical Circles", 1969)
"Mathematics may be likened to a large rock whose interior composition we wish to examine. The older mathematicians appear as persevering stone cutters slowly attempting to demolish the rock from the outside with hammer and chisel. The later mathematicians resemble expert miners who seek vulnerable veins, drill into these strategic places, and then blast the rock apart with well-placed internal charges." (Howard W Eves, "In Mathematical Circles", 1969)
"Older mathematics appears static while the newer appears dynamic, so that the older mathematics compares to the still-picture stage of photography while the newer mathematics compares to the moving-picture stage. Again, the older mathematics is to the newer much as anatomy is to physiology, wherein the former studies the dead body and the latter studies the living body. Once more, the older mathematics concerned itself with the fixed and the finite while the newer mathematics embraces the changing and the infinite." (Howard W Eves, "In Mathematical Circles", 1969)
"The development of mathematics over the ages may be viewed as a continent slowly rising from the sea. At first perhaps a single island appears, and, as it grows in size, other islands emerge at varying distances from one another. As the continent continues to rise, some of the islands become joined to others by isthmuses that widen until pairs of islands become single large islands. At length a point is reached where the shape of the continent is essentially defined, and there remain only a number of lakes and inland seas of various sizes. As the continent further rises, these lakes and seas shrink and vanish one by one. The older mathematics compares to the situation when the general shape of the rising continent is still undefined and the land area consists largely of islands of different sizes. The newer mathematics compares to the situation when the general shape of the rising continent has become essentially clear, with most of the former islands now joined by stretches of land." (Howard W Eves, "In Mathematical Circles", 1969)
"The mathematicians, on the other hand, are too often unpleasant to be with; they frequently exude self-importance, are professionally opinionated, tend to bicker and quarrel among themselves and to say unkind things of one another, take an almost gleeful pleasure in unearthing an error in another's work, and are often quite boring to their nonmathematical acquaintances." (Howard W Eves, "In Mathematical Circles: Quadrants III and IV", 1969)
"The two most outstanding mathematicians of the eighteenth century were Euler and Lagrange, and which of the two is to be accorded first place is a matter of debate that often reflects the varying sensitivities of the debaters. Euler certainly published far more than Lagrange, and worked in many more diverse fields of mathematics than Lagrange, but he was largely a formalist or manipulator of formulas. Lagrange, on the other hand, may be considered the first true analyst and, though his collection of publications is a molehill compared with the Vesuvius of Euler's output, his work has a rare perfection, elegance, and exactness about it. Whereas Euler wrote with a profusion of detail and a free employment of intuition, Lagrange wrote concisely and with attempted rigor." (Howard W Eves, "In Mathematical Circles: Quadrants III and IV", 1969)
"There is probably no greater source of confusion in the history of mathematics (indeed, in the history of any subject) than the perpetuation of an error of fact unknowingly made by some authority in the field. Such errors are innocently carried on and multiplied by subsequent writers leaning either on the original authority or on someone who did lean on the original authority. Anyone who works in the history of mathematics soon becomes aware of this situation, for there is hardly a corner of the subject that is not crisscrossed by it. A writer in the history of mathematics just cannot always start from the very beginning; he must constantly borrow from the existing reservoir of written information." (Howard W Eves, "In Mathematical Circles: Quadrants III and IV", 1969)
"A good problem should be more than a mere exercise; it should be challenging and not too easily solved by the student, and it should require some ‘dreaming’ time." (Howard W Eves)
"It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. […] Every new discovery in mathematics, results from an attempt to solve some problem." (Howard W Eves)
"A good problem should be more than a mere exercise; it should be challenging and not too easily solved by the student, and it should require some ‘dreaming’ time." (Howard W Eves)
"It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. […] Every new discovery in mathematics, results from an attempt to solve some problem." (Howard W Eves)
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