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One of the things that's become very clear to me from the big science project that I've been doing for the past nine years is that such a view of mathematics is really not correct. Mathematics, as it's practiced, isn't about arbitrary abstract systems. It's about the particular abstract system that happens to have been historically studied in mathematics. And if one traces things back, there seem to be three basic traditions from which essentially all of mathematics as we know it emerged: arithmetic, geometry, and logic.

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Free College Essay Compare and Contrast Past and Present

Things got a little more definite in the 1950s, when Chomsky and Backus, essentially independently, invented the idea of context-free languages. The idea came out of work on production systems in mathematical logic, particularly by Emil Post in the 1920s. But, curiously, both Chomsky and Backus came up with the same basic idea in the 1950s.

AQA AS French past essay questions - TES Resources

The real question for the future, however, is the degree to which Soviet elites have assimilated the consciousness of the universal homogenous state that is post-Hitler Europe. From their writings and from my own personal contacts with them, there is no question in my mind that the liberal Soviet intelligentsia rallying around Gorbachev have arrived at the end-of-history view in a remarkably short time, due in no small measure to the contacts they have had since the Brezhnev era with the larger European civilization around them. "New political thinking," the general rubric for their views, describes a world dominated by economic concerns, in which there are no ideological grounds for major conflict between nations, and in which, consequently, the use of military force becomes less legitimate. As Foreign Minister Shevardnadze put it in mid-1988:

Past essay questions for AS French collated into one document, sorted by topic area.

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Most mathematical notation now in use is between one and five hundred years old. I will review how it developed, with precursors in antiquity and the Middle Ages, through its definition at the hands of Leibniz, Euler, Peano and others, to its widespread use in the nineteenth and twentieth centuries. I will discuss the extent to which mathematical notation is like ordinary human language—albeit international in scope. I will show that some general principles that have been discovered for ordinary human language and its history apply to mathematical notation, while others do not.

Given its historical basis, it might have been that mathematical notation—like natural language—would be extremely difficult for computers to understand. But over the past five years we have developed in capabilities for understanding something very close to standard mathematical notation. I will discuss some of the key ideas that made this possible, as well as some features of mathematical notation that we discovered in doing it.

Large mathematical expressions—unlike pieces of ordinary text—are often generated automatically as results of computations. I will discuss issues involved in handling such expressions and making them easier for humans to understand.

Traditional mathematical notation represents mathematical objects but not mathematical processes. I will discuss attempts to develop notation for algorithms, and experiences with these in APL, , theorem-proving programs and other systems.

Ordinary language involves strings of text; mathematical notation often also involves two-dimensional structures. I will discuss how mathematical notation might make use of more general structures, and whether human cognitive abilities would be up to such things.

The scope of a particular human language is often claimed to limit the scope of thinking done by those who use it. I will discuss the extent to which traditional mathematical notation may have limited the scope of mathematics, and some of what I have discovered about what generalizations of mathematics might be like.

AP Biology Essay Questions The following is a comprehensive list of essay questions that have been asked on past AP exams. The questions are organized according to …

Past Exam Questions and Sample Answers - …

For the past six years, he has taught literacy and creative writing in correctional facilities. As Assistant Professor of English and Director of the Writing Program at Regis College, he teaches courses in creative nonfiction, writing as community service, and copyediting as well as freshman composition.

Lord of the Flies: Past Essay

AP essay questions - Biology Junction

There is no making up for past mistakes, because these are perceived as symptoms and not mistakes.
The acid test, however, comes when one tries to defend a person under attack, especially when she's not there, If such a defense is taken seriously, and some concern expressed for hearing all sides and gathering all evidence, trashing is probably not occurring.

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IN WATCHING the flow of events over the past decade or so, it is hard to avoid the feeling that something very fundamental has happened in world history. The past year has seen a flood of articles commemorating the end of the Cold War, and the fact that "peace" seems to be breaking out in many regions of the world. Most of these analyses lack any larger conceptual framework for distinguishing between what is essential and what is contingent or accidental in world history, and are predictably superficial. If Mr. Gorbachev were ousted from the Kremlin or a new Ayatollah proclaimed the millennium from a desolate Middle Eastern capital, these same commentators would scramble to announce the rebirth of a new era of conflict.