The practice of first developing a clear and precise definition of a process without regard for efficiency, and then using it as a guide and a test in exploring equivalent processes possessing other characteristics, such as greater efficiency, is very common in mathematics. It is a very fruitful practice which should not be blighted by premature emphasis on efficiency in computer execution.
General Systems Theory is a name which has come into use to describe a level of theoretical model-building which lies somewhere between the highly generalized constructions of pure Mathematics and the specific theories of the specialized disciplines. Mathematics attempts to organize highly general relationships into a coherent system, a system however which does not have any necessary connections with the "real" world around us. It studies all thinkable relationships abstracted from any concrete situation or body of empirical knowledge.Kenneth Boulding (1956) "General systems theory - the skeleton of science" in: Management Science, Vol.2. No. 3. (April 1956) p.197: Opening sentences
Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.Richard Courant in: The Australian Mathematics Teacher, Volumes 39-40, Australian Association of Mathematics Teachers, 1983, p.3
At present, when the prevailing forms of society have become hindrances to the free expression of human powers, it is precisely the abstract branches of science, Mathematics and theoretical physics, which ... offer a less distorted form of knowledge than other branches of science which are interwoven with the pattern of daily life, and the practicality of which seemingly testifies to their realistic character.max horkheimer
Even in Mathematics his whimsical fancy was sometimes suffered to peep out, and little girls who learnt the rudiments of calculation at his knee found the path they had imagined so thorny set about with roses by reason of the delightful fun with which he would turn a task into a joy. But when the fun was over the little girl would find that she had learnt the lesson (all unknowingly) just the same. Happy little girls who had such a master.Page 5 (The Story of Lewis Carroll (1899))
The land of easy Mathematics where he who works adds up and he who retires subtracts.Núria Añó, in the short story 2066. Beginning the age of correction
The subject of management science has evolved for more than 60 years and is now a mature field within the broad category of applied Mathematics. This book will emphasize both the applied and mathematical aspects of management science.Wayne Winston, ?S. Albright (2011) Practical Management Science. p. 3
It is noteworthy that modern Platonists , almost without exception, are ignorant of Mathematics, in spite of the immense importance that Plato attached to arithmetic and geometry, and the immense influence that they [these studies] had on his philosophy. This is an example of the evils of specialization: a man must not write on Plato unless he has spent so much of his youth on Greek as to have no time for the things that Plato thought important.Book One, Part II, Chapter XV, The Theory of Ideas, p. 132
Hungary, severed from half of the population and most of the natural resources that it had once claimed, had now to practice a sort of economic acupuncture, striving to know the magic nodes in the global energy flow where a pinprick could alter the workings of a major organ. Mathematics was one of the few disciplines where it was possible to exert that degree of leverage, and so the Hungarians had become phenomenally good at teaching it to their children.neal stephenson
Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.W. S. Anglin, in Mathematics and History, elucidating the symmetry between the creative and logical aspects of mathematics.
[There is] a delusion that macro-economics is both viableanduseful (a delusionencouraged by its extensive use of Mathematics, which must always impress politicians lacking any mathematical education, and which is really the nearest thing to the practice of magic that occurs among professional economists).1989 The Fatal Conceit: The Errors of Socialism, ch.6.
Three principles — the conformability of nature to herself, the applicability of the criterion of simplicity, and the "unreasonable effectiveness" of certain parts of Mathematics in describing physical reality — are thus consequences of the underlying law of the elementary particles and their interactions. Those three principles need not be assumed as separate metaphysical postulates. Instead, they are emergent properties of the fundamental laws of physics.murray gell-mann
For all the sublimity of art, physics, music, Mathematics, and other manifestations of human genius, everything depends on the mundane, frustrating, often debased vocation known as politics (and its most exacting subspecialty – statecraft). Because if we don't get politics right, everything else risks extinction.charles krauthammer
The temptation to use Mathematics is irresistible for economists. It appears to convey the appropriate air of scientific authority and precision to economists' musings.Part I, Chapter 3, The Roots of Economic Orthodoxy, p. 43
One of the most frequently mentioned equations was Euler's equation, Respondents called it "the most profound mathematical statement ever written"; "uncanny and sublime"; "filled with cosmic beauty "; and "mind-blowing". Another asked: "What could be more mystical than an imaginary number interacting with real numbers to produce nothing ?" The equation contains nine basic concepts of Mathematics once and only once in a single expression. These are: e (the base of natural logarithms); the exponent operation; ?; plus (or minus, depending on how you write it); multiplication; imaginary numbers; equals; one; and zero.Robert P. Crease, in "The greatest equations ever" at PhysicsWeb (October 2004)
The terms synthesis and analysis are used in Mathematics in a more special sense than in logic. In ancient Mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid , XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it."p. 30 (The Greeks)
He filled his writings with mathematical discoveries, and exhibited on every occasion the remarkable connection between Mathematics and philosophy.Eudemus of Rhodes (ca. 340 BC), as quoted in Proclus' commentaries on Euclid, referred to as the Eudemian Summary by Florian Cajori in A History of Mathematics (1893) p.30
It is a well-known experience that the only truly enjoyable and profitable way of studying Mathematics is the method of "filling in details" by one's own efforts.Cornelius Lanczos, Applied Analysis, Prentice-Hall, 1956
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