数学证明的对应问题
A correspondence problem for mathematical proof
AI总结：
本文探讨数学证明中对应性的本质，指出仅存在形式化不足，需通过充分表示和步骤跟踪来确立对应性，揭示现有形式化系统多样性和未来数学发展的新挑战。
摘要：
数学证明常被说通过指示存在对应的正式推导来证明结论。本文认为这种普遍观点依赖于对对应性的未充分考察。仅仅存在形式化不够，必须通过充分表示（原定理）和跟踪（原证明步骤）来确立对应性。对现有形式化系统的考察显示，我们通过各种近经验方式确立这些标准，并指向未来数学自身可能面临的新负担。
英文摘要：
Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for a particular derivation to ''correspond'' to a particular proof. Mere existence of a formalization is not enough, and a substantive account of the required correspondence resolves into two criteria -- adequate representation (of the original theorem) and tracking (of the steps in the original proof). An examination of the actually-existing formalization systems we have today shows the variety of quasi-empirical ways we establish these criteria, and points towards new burdens that may be placed on the future evolution of mathematics itself.
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