发明与发现之间:论维特根斯坦的数学观

Invention or Discovery:Wittgenstein’s View on Mathematics

众所周知,数学家的工作是发现数学真理,但中后期维特根斯坦却认为:不论是数学实体还是数学真理都是数学家的发明而非发现。他还呼吁数学家应当限制这种数学发明活动,以免发明出不可判定的拟数学命题。这些观点颇受学界争议,但也颇具启发性。虽然数学概念的定义是一种发明过程,但一旦数学实体被发明出来,它们的性质和关系就随之而被确定,从而数学真理是已然在那数学实在之中等候数学家去发现的。因此,数学并不完全是发现也不纯粹是发明,而是居于发明和发现之间的学科。

As we know,mathematicians’ job is to discover mathematical truth. But middle and later Wittgenstein claimed that all mathematical entities and truths are mathematicians’ inventions rather than discoveries. And he proposed that mathematicians should restrain their mathematical invention to avoid undecidable pseudo-propositions. Although these views are controversial in the literature,they are quite enlightening. Once those mathematical entities are invented through definitions,their properties and relations are also determined. So mathematical truths are already there in the mathematical realm waiting for mathematicians’ discovery. Hence,mathematics is neither complete discovery nor pure invention,but a discipline between invention and discovery.