1.4 无穷小放飞互联网，究竟为了什么？

    坦率地说，七年前，我们放飞无穷小到国内互联网的目的是：进一步改进国内高校微积分的教学改革，提高数学的创新能力。

   国内数学不能故步自封。请见本文附件。

袁萌  陈启清  10月7日

附件：

1.4 The purpose of nonstandard analysis

After the digression of the preceding section let us now contemplate the purpose of nonstandard analysis. Starting from IN, the sets ZZ,Q and IR (andC, but below complex numbers will be ignored) have been introduced in classical mathematics in order to enrich mathematics with more tools and to rene existing tools. The introduction of negative numbers, of fractions, and of irrational numbers is felt as a strong necessity, and without it mathematics would only be a small portion of what it actually is. The introduction of ∗IN, ∗ZZ, ∗Q, and ∗IR, however, was not meant at all to enrich mathematics (at least not when it all started), but only to simplify doing mathematics. For as soon as notions like limit and continuity are involved, denitions in nonstandard analysis can be given a simpler form, and theorems can be proved in a simpler way. Often the simplications are considerable. In one case the proof of a classical conjecture was found by means of nonstandard analysis, after which a classical proof was found as well. Moreover, both denitions and proofs receive a more natural appearance. This may even enhance the discovery of new facts.