论自然数列的二重性与双相无限性及其对数学发展的影响

On the Double Character of the Natural Number Sequence and the Double-phased Infinity Concept, and Their Influence on the Development of Mathematics

自然数列的潜无限观与实无限观是众所周知的。正是两种无限观的对立,形成了近代数学史上直觉主义派与公理主义派及逻辑主义派在数学基础问题上的根本分歧及对重建基础的不同主张。本文旨在揭示出自然数列的二重性,即内蕴性和排序性的关系;从而引出“双相无限性”概念,由此也就澄清了诸流派在无限观上形成分歧的根本原因。这对教学和数学史研究都有相当意义。本文主要内容曾于1994年11月上旬在南开数学研究所举行的“首届数学哲学与方法论研讨会”上报告过。

It is pointed out that the natural number sequence (as given by Peano's axioms) possesses a kind of double character that is generated by both the inner implication-property and the ordering property, with which the so-called 'double-phased infinity' concept is naturally connected. Also explained is the reason why most experts seriously doing discrete mathematics support the intuitionists' view regarding the potential infinity concept of natural numbers, and most analysis accept the view about the true infinity of integers. The last section of this article provides a detailed analysis of L. J. Brouwer's counter-example to the trichotomy law of real numbers. Surely the counter-example would not hold from the view-point of Platonism, which asserts that every real irrational number has a completed decimal expansion with truly infinitely many decimal places.