By Carl B. Boyer, Uta C. Merzbach, Isaac Asimov
Boyer and Merzbach distill hundreds of thousands of years of arithmetic into this interesting chronicle. From the Greeks to Godel, the math is incredible; the solid of characters is uncommon; the ebb and circulation of principles is all over obtrusive. And, whereas tracing the advance of eu arithmetic, the authors don't forget the contributions of chinese language, Indian, and Arabic civilizations. absolutely, this is—and will lengthy remain—a vintage one-volume background of arithmetic and mathematicians who create it.
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There are various bits and items of folklore in arithmetic which are handed down from consultant to pupil, or from collaborator to collaborator, yet that are too fuzzy and non-rigorous to be mentioned within the formal literature. often, it was once an issue of good fortune and placement as to who discovered such folklore arithmetic.
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On Proof and Progress in Mathematics 21 One of Thurston’s most incisive ideas is that human thinking and understanding do not follow a single track. There are several different thinking faculties at play. Among these are: (1) Human Language: “Our linguistic facility is an important tool for thinking, not just for communication” [THU, p. 164]. The language that we use reflects both the level and the profundity of our thinking. We learn the quadratic formula, as a simple example, almost as a chant.
But Kepler’s mathematical maturity had the flaw that he could not understand John Napier’s theory of logarithms. In this sense Napier was ahead of Kepler. If collaboration were more the norm in the seventeenth century, as it is today, then perhaps Napier and Kepler could have worked together, and could have produced much more scientific work more efficiently. It is fun to rewrite history, and to speculate on what might have happened. ✐ ✐ ✐ ✐ ✐ ✐ “master” — 2011/11/9 — 15:21 — page 26 — #44 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ “master” — 2011/11/9 — 15:21 — page 27 — #45 ✐ CHAPTER ✐ 2 Math Concepts If we desire to form individuals capable of inventive thought and of helping the society of tomorrow to achieve progress, then it is clear that an education which is an active discovery of reality is superior to one that consists merely in providing the young with ready-made wills to will with and ready-made truths to know with .
And to do so with panache and flair. Someone who is struggling to follow the reasoning step-by-step is not going to be able to do it. Instead the situation calls for someone who is similar to a chess master. That is, it requires someone who can see ahead five or six moves, who knows where the line of reasoning is heading, who can see the forest for the trees. The student sees only the next fallen limb in the path, and likely as not trips over it. The professional sees the long-term goal, and nimbly walks around—or jumps over—that pesky limb.