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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.
kenneth e. iversonGeneral Systems Theory is a name which has come into use to describe a level of theoretical modelbuilding 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 sentencesMathematics 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 3940, Australian Association of Mathematics Teachers, 1983, p.3At 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 horkheimerThe 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. 3It 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. 132Mathematics is the gate and key of the sciences... Neglect of Mathematics works injury to all knowledge , since he who is ignorant of it cannot know the other sciences or the things of this world. And what is worse, men who are thus Ignorant are unable to perceive their own ignorance and so do not seek a remedy.
roger baconPerhaps the least inadequate description of the general scope of modern Pure Mathematics I will not call it a definition would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations.
E. W. Hobson "Presidential Address British Association for the Advancement of Science, Section A, (1910}" in: Nature, Vol. 84, p. 287We had principles in Mathematics that were granted to be absolute in Mathematics for over 800 years, but new science has gotten rid of those absolutism, gotten forward other different logics of looking at Mathematics, and sort of turned the way we look at it as a science altogether after 800 years.
mahmoud ahmadinejadAs soon as a thought or word becomes a tool, one can dispense with actually ‘thinking’ it, that is, with going through the logical acts involved in verbal formulation of it. As has been pointed out, often and correctly, the advantage of Mathematics the model of all neopositivistic thinking lies in just this ‘intellectual economy.’ Complicated logical operations are carried out without actual performance of the intellectual acts upon which the mathematical and logical symbols are based. … Reason … becomes a fetish, a magic entity that is accepted rather than intellectually experienced.
max horkheimerIt 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.164In Mathematics the art of asking questions is more valuable than solving problems.
Georg Cantor (1867) Doctoral thesisMystical doctrines as to the relation of time to eternity are also reinforced by pure Mathematics, for the mathematical objects, such as numbers, if real at all, are eternal and not in time. Such eternal objects can be conceived as God's thoughts. Hence Plato's doctrine that God is a geometer, and Sir James Jeans' belief that He is addicted to arithmetic.
Book One, Part I, Chapter III, Pythagoras, p. 37Bacon not only despised the syllogism, but undervalued Mathematics, presumably as insufficiently experimental. He was virulently hostile to Aristotle , but he thought very highly of Democritus , Although he did not deny that the course of nature exemplifies a Divine purpose, he objected to any admixture of teleological explanation in the actual investigation of phenomena; everything, he held, should be explained as following necessarily from efficient causes .
bertrand russellComputer science is neither Mathematics nor electrical engineering
Alan Perlis (1968) title of article "Computer Science is neither Mathematics nor Electrical Engineering" in: A. Finerman (Hg.), University Education in Computing Science, New York, London, pp. 6977Ever since Plato most philosophers have considered it part of their business to produce ‘proofs’ of immortality and the existence of God . They have found fault with the proofs of their predecessors Saint Thomas rejected Saint Anselm's proofs, and Kant rejected Descartes ' but they have supplied new ones of their own. In order to make their proofs seem valid, they have had to falsify logic , to make Mathematics mystical , and to pretend that deep seated prejudices were heavensent intuitions .
Bertrand Russell, in Ie – Evaluating “Proofs” of God’s Existence, Zenofzero.net, 30 October 2012Create and save customized word lists. Sign up today and start improving your vocabulary!