"It appears that in everything the interest of ideas is in inverse proportion to the usefulness they have in practice. This is not surprising when we consider that the human intellect, when working for its own satisfaction, should encounter the greatest intellectual beauties rather than when guided by an external motive [...]" (Sophie Germain, [letter to Gauss] 1809)
"Every mathematical method has its inverse, as truly, and for the same reason, as it is impossible to make a road from one town to another, without at the same time making one from the second to the first. The combinatorial analysis is analysis by means of combinations; the calculus of generating functions is combination by means of analysis." (Augustus de Morgan, "The Differential and Integral Calculus", 1836)
"It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the Disquisitiones Arithmeticae; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory." (George B Mathews, "Theory of Numbers", 1892)
"If we study the history of science we see happen two inverse phenomena […] Sometimes simplicity hides under complex appearances; sometimes it is the simplicity which is apparent, and which disguises extremely complicated realities. […] No doubt, if our means of investigation should become more and more penetrating, we should discover the simple under the complex, then the complex under the simple, then again the simple under the complex, and so on, without our being able to foresee what will be the last term. We must stop somewhere, and that science may be possible, we must stop when we have found simplicity. This is the only ground on which we can rear the edifice of our generalizations." (Henri Poincaré, "Science and Hypothesis", 1901)
"In general, complexity and precision bear an inverse relation to one another in the sense that, as the complexity of a problem increases, the possibility of analysing it in precise terms diminishes. Thus 'fuzzy thinking' may not be deplorable, after all, if it makes possible the solution of problems which are much too complex for precise analysis." (Lotfi A Zadeh, "Fuzzy languages and their relation to human intelligence", 1972)
"A complex system that works is invariably found to have evolved from a simple system that worked. The inverse proposition also appears to be true: A complex system designed from scratch never works and cannot be made to work. You have to start over, beginning with a working simple system." (John Gall, "General Systemantics: How systems work, and especially how they fail", 1975)
"The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them." (George Pólya, "Mathematical Discovery", 1981)
"The ballast of factual information, so far from being just about to sink us, is growing daily less. The factual burden of a science varies inversely with its degree of maturity. As a science advances, particular facts are comprehended within, and therefore in a sense annihilated by, general statements of steadily increasing explanatory power and compass - whereupon the facts need no longer be known explicitly, that is, spelled out and kept in mind. In all sciences we are being progressively relieved of the burden of singular instances, the tyranny of the particular. We need no longer record the fall of every apple."
"The interpreter of the wonders of nature is experience. It never misleads us, only our grasp can do it with us. Until we can establish a general rule, we must accept the help of experience. Although nature begins with the cause, and with the experiment, we must do it inversely, we must discover the cause with experiments." (Leonardo da Vinci)
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