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 8月6日
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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