对算术命题先天综合性质的系统论证

A Systematic Demonstration of the Synthetic Apriority of Arithmetical Propositions

康德的先天综合命题思想是哲学上的重要创举。然而,囿于时代之限,康德对算术命题先天综合性质的阐述暴露其数学上理解的偏差。基于现代数学尤其是皮亚诺算术公理的视角,借鉴彭加勒对数学归纳法先天综合性质的论述,可以发现,算术命题是先天综合的。对算术命题先天综合性质的系统论证由最小数原理的先天综合性质始,经由数学归纳法的先天综合性质而至皮亚诺算术公理的先天综合性质,最后到达算术命题的先天综合性质。其中对综合性的论证,涉及非概念包含关系、无限性、数学公理的不可化约性、自然数的两种定义、哥德尔不完全性定理、数字和运算的不可化约性,等等。

Kant’s thought of the synthetic a priori is a pioneering work of great philosophical importance.However,due to the limitations of his epoch,Kant’s exposition of the synthetic apriority of arithmetical propositions reveals the bias of his mathematical understanding.Based on the perspective of modern mathematics,especially of Peano axioms,and using Poincaré’s elaboration of the synthetic apriority of mathematical induction,it can be found that arithmetical propositions are indeed synthetic a priori.The systematic demonstration of the synthetic apriority of arithmetical propositions begins with the synthetic apriority of the principle of least number,then,via that of mathematical induction,moves to that of Peano axioms,and finally arrives at that of arithmetical propositions.The arguments for the syntheticity involve the conceptual non-containment relation,the infinity,the irreducibility of mathematical axioms,two definitions of natural numbers,G?del’s incompleteness theorem,the irreducibility of numbers and operations,etc.