The influence of Platonism on mathematics research and theological beliefs

An age-old debate in the philosophy of mathematics is whether mathematics is discovered or invented. There are four popular viewpoints in this debate, namely Platonism, formalism, intuitionism, and logicism. A natural question that arises is whether belief in one of these viewpoints affects the mathematician’s research? In particular, does subscribing to a Platonist or a formalist viewpoint influence how a mathematician conducts research? Does the area of research influence a mathematician’s beliefs on the nature of mathematics? How are the beliefs regarding the nature of mathematics connected to theological beliefs? In order to investigate these questions, five professional research mathematicians were interviewed. The mathematicians worked in diverse areas within analysis, algebra, and within applied mathematics, and had a combined 160 years of research experience. Although none of the mathematicians wanted to be pigeonholed into any one category of beliefs, the study revealed that four of the mathematicians leaned towards Platonism, which runs contrary to the popular notion that Platonism is an exception today. This study revealed that beliefs regarding the nature of mathematics influenced how mathematicians’ conducted research and were deeply connected to their theological beliefs. The findings are presented in the form of vignettes that give an insight into the mathematical and theological belief structures of the mathematicians.     

理念主义与形式主义的巴别塔

圣经里讲了一个关于语言的故事,早期的先民们使用相同的语言,尝试建立通天的巴别塔,这个计划因为语言变乱而失败。这个故事或许隐喻着:人类是不可能通过语言来达到真理的。然而,人类自从吃了知识果,走出了伊甸园,就担当起自己的运命。从理念主义(Rationalism)和形式系统(Formalism)这两条线索,人类尝试着重建新的巴别塔。我们从理念主义的线索,沿着柏拉图的理念世界,到康德的先天准则,再到胡塞尔的本质直观,评论了理念主义的不自明性,提出了使用处于构造之中的"共相"来代替"理念"或者"本质"。维特根斯坦提出"家族相似",反对理念主义和本质主义,但"家族相似"可能过于宽泛。另一方面,从形式主义的线索,沿着莱布尼兹的通用计算,到弗雷格和罗素的逻辑主义,到希尔伯特的形式主义纲领,再到哥德尔不完备性定理,评论了逻辑主义和形式主义的失败。形式主义失败之后,人们转向形式主义的能力范围研究,从而提出了图灵机,为计算机科学的出现奠定了理论基础。基于以上的思考,文中最后提出一些有待研究的问题,比如基于关系的数学,基于谓词的逻辑,矛盾但完备的形式系统。