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"One of the best ways to sharpen your brain, and to develop intelligence, is to study mathematics. It challenges and strengthens your mind in a way that very few other things do. It’s like going to the gym -- but for your brain!"
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Mathematics"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 ones own efforts."
Mathematics is a field of study that discovers and organizes methods, theories, and theorems that are developed and proved either in response to the needs of empirical sciences or the needs of mathematics itself. There are many areas of mathematics, including number theory, algebra, geometry, analysis, and set theory.
"One of the best ways to sharpen your brain, and to develop intelligence, is to study mathematics. It challenges and strengthens your mind in a way that very few other things do. It’s like going to the gym -- but for your brain!"
"Euclid alone has looked on Beauty bare."
"The thorough analysis of even simple problems in arithmetic may require the application of advanced mathematics."
"You cannot read mathematics superficially; the inescapable abstraction always has an element of self-torture in it, and the one to whom this self-torture is joy is the mathematician."
"No part of mathematics came to birth in the form that it now appears in a modern textbook: mathematical creativity can be slow, sometimes messy, often frustrating."
"Mathematics in general is fundamentally the science of self-evident things."
"The absolute requisites for the study of this work... are a knowledge of algebra to the binomial at least, plane and solid geometry, plane trigonometry, and the most simple part of the usual applications of algebra to geometry. ...A. De Morgan. London July 1, 1836"
"A free people will always refuse to put up with preventable poverty. If freedom is to be saved and enlarged, poverty must be ended. There is no other solution. The problem of how to prevent these three forces from coming into head-on collision is the principal study of the more politically conscious Conservative leaders. How can wealth persuade poverty to use its political freedom to keep wealth in power? Here lies the whole art of Conservative politics in the twentieth century."
"A clear prediction in an area undergoing vigorous study permits doctrines to be subject to disproof. The last posture a bureaucratic religion wishes to find itself in is vulnerability to disproof, where an experiment can be performed on which the religion stands or falls."
"Spencer argued in a very articulate way for the communality of these processes of self-organization, and used his ideas to make a theory of sociology. However, he was not able to put these ideas into mathematical form or argue them from first principles. And no one else has, either — doing so is perhaps the central problem in the study of complex systems."
"The commercial exploitation of the web has become a growing facet of the world economy, particularly in the last several years. In June 1999 NUA Internet Surveys estimated that 179 million people are connected to the Internet worldwide. A recent study by the sponsored by estimated that the "Internet Economy" generated $300 billion in revenue in the United States alone."
"In biology the Cartesian view of living organisms as machines, constructed from separate parts, still provides the dominant conceptual framework. Although Descartes simple mechanistic biology could not be carried very far and had to be modified considerably during the subsequent three hundred years, the belief that all aspects of living organisms can be understood by reducing them to their smallest constituents, and by studying the mechanisms through which these interact, lies at the very basis of most contemporary biological thinking. This passage from a current textbook on modern biology is a clear expression of the reductionist credo: One of the acid tests of understanding an object is the ability to put it together from its component parts. Ultimately, molecular biologists will attempt to subject their understanding of cell structure and function to this sort of test by trying to synthesize a cell."