"But there is another kind of change, too, change that is less suited to mathematical analysis: the abrupt bursting of a bubble, the discontinuous transition from ice at its melting point to water at its freezing point, the qualitative shift in our minds when we 'get' a pun or a play on words. Catastrophe theory is a mathematical language created to describe and classify this second type of change. It challenges scientists to change the way they think about processes and events in many fields." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)
"Physics defines itself as the science devoted to discovering, developing and refining those aspects of reality that are amenable to mathematical analysis." (John M Ziman, "Reliable Knowledge: An Exploration of the Grounds for Belief in Science", 1978)
"Having vegetated on the fringes of mathematical science for centuries, combinatorics has now burgeoned into one of the fastest growing branches of mathematics – undoubtedly so if we consider the number of publications in this field, its applications in other branches of mathematics and in other sciences, and also the interest of scientists, economists and engineers in combinatorial structures. The mathematical world was attracted by the successes of algebra and analysis and only in recent years has it become clear, due largely to problems arising from economics, statistics, electrical engineering and other applied sciences, that combinatorics, the study of finite sets and finite structures, has its own problems and principles. These are independent of those in algebra and analysis but match them in difficulty, practical and theoretical interest and beauty." (László Lovász, "Combinatorial Problems and Exercises", 1979)
"Another direction of research is fuzzy systems. This will greatly increase the use of mathematics from the inanimate to the animate. In the past, mathematics has been used for the analysis of physical systems. With fuzzy systems and computer simulation we can study many processes in the social sciences." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)
"To me, that the distribution of prime numbers can be so accurately represented in a harmonic analysis is absolutely amazing and incredibly beautiful. It tells of an arcane music and a secret harmony composed by the prime numbers." (Enrico Bombieri, "PrimeTerritory", The Sciences, 1992)
"The acceptance of complex numbers into the realm of algebra had an impact on analysis as well. The great success of the differential and integral calculus raised the possibility of extending it to functions of complex variables. Formally, we can extend Euler's definition of a function to complex variables without changing a single word; we merely allow the constants and variables to assume complex values. But from a geometric point of view, such a function cannot be plotted as a graph in a two-dimensional coordinate system because each of the variables now requires for its representation a two-dimensional coordinate system, that is, a plane. To interpret such a function geometrically, we must think of it as a mapping, or transformation, from one plane to another." (Eli Maor, "e: The Story of a Number", 1994)
"By studying analytic functions using power series, the algebra of the Middle Ages was connected to infinite operations" (various algebraic operations with infinite series). The relation of algebra with infinite operations was later merged with the newly developed differential and integral calculus. These developments gave impetus to early stages of the development of analysis. In a way, we can say that analyticity is the notion that first crossed the boundary from finite to infinite by passing from polynomials to infinite series. However, algebraic properties of polynomial functions still are strongly present in analytic functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"I see some parallels between the shifts of fashion in mathematics and in music. In music, the popular new styles of jazz and rock became fashionable a little earlier than the new mathematical styles of chaos and complexity theory. Jazz and rock were long despised by classical musicians, but have emerged as art-forms more accessible than classical music to a wide section of the public. Jazz and rock are no longer to be despised as passing fads. Neither are chaos and complexity theory. But still, classical music and classical mathematics are not dead. Mozart lives, and so does Euler. When the wheel of fashion turns once more, quantum mechanics and hard analysis will once again be in style." (Freeman J Dyson, "Book Review of ‘Nature’s Numbers’", The American Mathematical Monthly, Vol. 103" (7), 1996)
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