集合论多宇宙观与形式主义

Set-theoretic Multiverse View and Formalism

数学对象的实在性问题一直是数学哲学中争论的焦点。作为二十世纪数学基础三大流派之一的形式主义常常被认为是反实在论的;而新近颇受瞩目的多宇宙观则似乎应持有实在论立场。本文试图论证,形式主义的要义或许在于形式系统以及元数学,因此经过某种重构后的形式主义在本体论上可以是中立的;另一面,多宇宙观中的核心概念则可视为理想元,因而多宇宙观可以纳入到这一新的形式主义框架中;进而,这两者的结合可以支持、推动当前的数学实践甚至创造新的数学实践形式。

The reality of mathematical objects has always been the focus of debate in the philosophy of mathematics.Formalism,one of the three major schools of foundations of mathematics in twentieth-century,is often regarded as anti-realism.The recently popular multi universe view seems to take a realist position.This paper tries to argue that the essence of formalism lies in the formal system and meta-mathematics,so the formalism after some reconstruction can be ontologically neutral.On the other hand,the core concepts in the multi-universe view can be regarded as ideal elements,so the multi-universe view can be incorporated into this new formalism framework.Furthermore,the combination of these two can support and promote current mathematical practice and even create new mathematical practice forms.