"That mathematics is a handmaiden of science is a commonplace; but it is less well understood that experiments stimulate mathematical imagination, aid in the formulation of concepts and shape the direction and emphasis of mathematical studies. One of the most remarkable features of the relationship is the successful use of physical models and experiments to solve problems arising in mathematics. In some cases a physical experiment is the only means of determining whether a solution to a specific problem exists; once the existence of a solution has been demonstrated, it may then be possible to complete the mathematical analysis, even to move beyond the conclusions furnished by the model-a sort of boot-strap procedure. It is interesting to point out that what counts in this action and reaction is as much the 'physical way of thinking', the turning over in imagination of physical events, as the actual doing of the experiment." (James R Newman, "The World of Mathematics" Vol. II, 1956)
"The ultimate origin of the difficulty lies in the fact" (or philosophical principle) that we are compelled to use the words of common language when we wish to describe a phenomenon, not by logical or mathematical analysis, but by a picture appealing to the imagination. Common language has grown by everyday experience and can never surpass these limits. Classical physics has restricted itself to the use of concepts of this kind; by analysing visible motions it has developed two ways of representing them by elementary processes; moving particles and waves. There is no other way of giving a pictorial description of motions - we have to apply it even in the region of atomic processes, where classical physics breaks down." (Max Born, "Atomic Physics", 1957)
"Computers do not decrease the need for mathematical analysis, but rather greatly increase this need. They actually extend the use of analysis into the fields of computers and computation, the former area being almost unknown until recently, the latter never having been as intensively investigated as its importance warrants. Finally, it is up to the user of computational equipment to define his needs in terms of his problems, In any case, computers can never eliminate the need for problem-solving through human ingenuity and intelligence." (Richard E Bellman & Paul Brock, "On the Concepts of a Problem and Problem-Solving", American Mathematical Monthly 67, 1960)
"I discovered that a whole range of problems of the most diverse character relating to the scientific organization of production" (questions of the optimum distribution of the work of machines and mechanisms, the minimization of scrap, the best utilization of raw materials and local materials, fuel, transportation, and so on) lead to the formulation of a single group of mathematical problems" (extremal problems). These problems are not directly comparable to problems considered in mathematical analysis. It is more correct to say that they are formally similar, and even turn out to be formally very simple, but the process of solving them with which one is faced [i. e., by mathematical analysis] is practically completely unusable, since it requires the solution of tens of thousands or even millions of systems of equations for completion." (Leonid V Kantorovich, "Mathematical Methods of Organizing and Planning Production", Management Science 6(4), 1960)
"In bringing techniques of logical and mathematical analysis gives men an opportunity to bring conflicts up from the level of fights, where the intellect is beclouded by passions, to the level of games, where the intellect has a chance to operate." (Anatol Rapoport, "The Use and Misuse of Game Theory", Scientific American 207, 1962)
"The probability concept used in probability theory has exactly the same structure as have the fundamental concepts in any field in which mathematical analysis is applied to describe and represent reality." (Richard von Mises,"Mathematical Theory of Probability and Statistics", 1964)
"Mathematicians, on the other hand, often regard all of physics as a kind of divine revelation or trickery, where mathematical morals are irrelevant, so that if they enter this red-light district at all, it is only to get what they want as cheaply as possible before returning to the respectability of problems purely mathematical in the older sense: analysis, probability, differential geometry, etc." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)
"For a long time the aim of combinatorial analysis was to count the different ways of arranging objects under given circumstances. Hence, many of the traditional problems of analysis or geometry which are concerned at a certain moment with finite structures, have a combinatorial character. Today, combinatorial analysis is also relevant to problems of existence, estimation and structuration, like all other parts of mathematics, but exclusively for finite sets." (Louis Comtet, "Advanced Combinatorics", 1974)
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