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关于非标准数学的感想

(2019-08-06 15:15:25)

关于非标准数学的感想

   一般认为,希尔伯特计划中的数学只有一种,包括非欧几何在内。

   如今,出现了“非标准”数学,真是有点儿匪夷所思也。什么是非标准数学?它是什么东西?从何而来?

   希尔伯特想把全部数学形式化、公理化,哥德尔不完全性定理表明希尔伯特的想法是行不通的。 

   实际情况是,在数学的形式化世界里面潜伏着数学非非标准模型。这一事实等待着人们去发现。谁是幸运儿呢?鲁宾逊也!

    请见本文附件,其中有50多篇“非标准数学”的珍贵历史文献。

袁萌  陈启清  86

附件:

On the Foundations of Nonstandard Mathematics

Mauro Di Nasso Dipartimento di Matematica Applicata, Universit`a di Pisa, Italy E-mail: dinasso@dma.unipi.it

Abstract In this paper we survey various set-theoretic approaches that have been proposed over the last thirty years as foundational frameworks for the use of nonstandard methods in mathematics.

Introduction. Since the early developments of calculus, innitely small and innitely large numbers have been the object of constant interest and great controversy in the history of mathematics. In fact, while on the one hand fundamental results in the dierential and integral calculus were rst obtained by reasoning informally withinnitesimalquantities, itwaseasilyseenthattheirusewithoutrestrictions led to contradictions. For instance, Leibnitz constantly used innitesimals in his studies (the dierential notation dx is due to him), and also formulated the so-called transfer principle, stating that those laws that hold about the real numbers also hold about the extended number system including innitesimals. Unfortunately, neitherhenorhisfollowerswereabletogiveaformaljustication of the transfer principle. Eventually, in order to provide a rigorous logical frameworkforthetreatmentoftherealline, innitesimal numbers were banished from calculus and replaced by the εδ-method during the second half of the nineteenth century. 1 A correct treatment of the innitesimals had to wait for developments of a new eld of mathematics, namely mathematical logic and, in particular, of its branch called model theory. A basic fact in model theory is that every innite mathematical structure has nonstandard models, i.e. non-isomorphic structures which satisfy the same elementary properties. In other words, there are dierent but equivalent structures, in the sense that they cannot be distinguished by means of the elementary properties they satisfy. In a slogan, one could say that in mathematics “words are not enough to describe reality”.

1An interesting review of the history of calculus can be found in Robinson’s book [R2], chapter X.


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