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
When I came back from Munich, it was September, and I was Professor of Mathematics at the Eindhoven University of Technology. Later I learned that I had been the Department's third choice, after two numerical analysts had turned the invitation down; the decision to invite me had not been an easy one, on the one hand because I had not really studied Mathematics, and on the other hand because of my sandals, my beard and my "arrogance" (whatever that may be).edsger w. dijkstra
Mathematics would certainly have not come into existence if one had known from the beginning that there was in nature no exactly straight line, no actual circle, no absolute magnitude.Friedrich Nietzsche, as quoted in The Puzzle Instinct : The Meaning of Puzzles in Human Life? (2004) by Marcel Danesi, p. 71.
Mathematics is good for the soul, getting things right enlivens a sense of truth, efforts to understand automatically purify desires.Iris Murdoch, Metaphysics As A Guide To Morals, p. 178
There is no branch of Mathematics, however abstract, which may not some day be applied to phenomena of the real world.As quoted in George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer (1998 ), p. 225; also in Stanley Gudder, A Mathematical Journey, McGraw-Hill (1976), p. 36
It was, however, from Spain, and not from Arabia, that a knowledge of eastern Mathematics first came into western Europe. The Moors had established their rules in Spain in 747, and by the tenth or eleven century had attained a high degree of civilisation.W. W. Rouse Ball, A Short Account of the History of Mathematics (1888), Courier Dover, 1960, p.164
India was the motherland of our race, and Sanskrit the mother of Europe's languages: she was the mother of our philosophy; mother, through the Arabs, of much of our Mathematics; mother, through the Buddha, of the ideals embodied in Christianity; mother, through the village community, of self-government and democracy. Mother India is in many ways the mother of us all.Will Durant,prolific American writer, historian, and philosopher in The Case for India (1931)
Addition and multiplication In our study of oscillating systems we shall have occasion to use one of the most remarkable, almost astounding, formulas in all of Mathematics. From the physicist's point of view we could bring forth this formula in two minutes or so, and be done with it. But science is as much for intellectual enjoyment as for practical utility, so instead of just spending a few minutes, we shall surround the jewel by its proper setting in the grand design of that branch of Mathematics called elementary algebra.Richard Feynman, Robert B. Leighton, ?Matthew Sands (1963), The Feynman Lectures on Physics, Vol. I, p. 22 : on Euler's formula, of which Euler's identity is a special case, in
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)
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."Florian Cajori, A History of Mathematics (1893). p. 30
The apparently ancient reports of the importance of Pythagoras and his pupils in laying the foundations of Mathematics crumble on touch, and what we can get hold of is not authentic testimony by the efforts latecomers to paper over a crack, which they obviously found surprising, by the use of various kinds of reconstruction and reinterpretation. On the other hand, there are ancient and unassailable indications of a Greek Mathematics antedating Pythagoras and quite outside his sphere of influence.Walter Burkert, Lore and Science in Ancient Pythagoreanism (1972)
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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