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文件名称:The.Princeton.Companion.to.Mathematics.2008

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The.Princeton.Companion.to.Mathematics.2008 普林斯顿数学指南.
It is notoriously hard to give a satisfactory answer to the question, “What is mathematics?” The approach of this book is not to try. Rather than giving a definition of mathematics, the intention is to give a good idea of what mathematics is by describing many of its most impor- tant concepts, theorems, and applications. Nevertheless, to make sense of all this information it is useful to be able to classify it somehow.
The most obvious way of classifying mathematics is by its subject matter, and that will be the approach of this brief introductory section and the longer section enti- tled some fundamental mathematical definitions [I.3]. However, it is not the only way, and not even obvi- ously the best way. Another approach is to try to clas- sify the kinds of questions that mathematicians like to think about. This gives a usefully different view of the subject: it often happens that two areas of mathematics that appear very different if you pay attention to their subject matter are much more similar if you look at the kinds of questions that are being asked. The last sec- tion of part I, entitled the general goals of mathe- matical research [I.4], looks at the subject from this point of view. At the end of that article there is a brief discussion of what one might regard as a third classi- fication, not so much of mathematics itself but of the content of a typical article in a mathematics journal. As well as theorems and proofs, such an article will contain definitions, examples, lemmas, formulas, conjectures, and so on. The point of that discussion will be to say what these words mean and why the different kinds of mathematical output are important.
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