巴拿赫—塔斯基佯谬的接受历程与哲学意义探析

Banach-Tarski’s Paradox:A Discussion on Its History of Acceptance and Significance in Philosophy

数学的视野扩大到纯数学的发展历程暗藏了认识或价值倾向变化。巴拿赫-塔斯基佯谬断言,我们在数学上能将一个球体拆成有穷片,再重组为两个原球完全一样的球体。该命题及其相关命题起初因“反直观性”而被视为悖论;后又被接受为定理;至今又已启发了有穷可加测度和顺从群方面的成果。这些成果巩固了巴-塔佯谬的合法地位,也可授信于推演它所需的公理。这一事实不仅可影响到数学哲学中“公理辩护”的探讨,也为思考价值如何影响认知,这一更具一般性的主题,贡献了一则似乎更先天的例子。

A change of value disposition or way of perception happened implicitly when mathematics expanded into the field of pure mathematics.Banach-Tarski paradox,which promises we may mathematically decompose a sphere into finite pieces and then recombine them forming two original spheres,serves a good witness.This proposition and similar ones were initially judged as paradoxes for their‘anti-intuitiveness’,yet well received as theorems later,and till now have inspired various results concerning,for example,finite additive measure and amenable groups.The results fortify the legitimacy of BT paradox,as well as add credits to the axioms needed to deduct it.This fact not only proves pertinent to the discussion about‘axiom justification’in philosophy of mathematics,but also contribute a seemingly a priori case,of more general interest,on how value could affect perception.