Reverse Mathematics

研究者认为，数学逻辑需要回到它在数学基础中的根源，在弗雷格、罗素、希尔伯特、图灵和德尔的伟大传统中。《数学基础》中的一个基本问题是：需要哪些集合存在公理来证明核心数学的特定定理？这里的“核心数学”包括分析、代数、拓扑/几何等标准主题。研究者和他的同事们从二阶算术子系统的角度来研究这个问题，正如研究者最近发表的同名研究专著中所阐述的那样。一系列广泛的案例研究表明：(i)许多核心数学定理在逻辑上等同于证明它们所需的集合存在公理，（ii）只有少数集合存在公理以这种方式出现，（iii）相应的子系统通过逻辑蕴涵线性排序。这将数百个核心数学定理进行了广泛的分类，将其分成少数几个类，这些类是根据弱基系统上的逻辑等价来定义的。这个正在进行的分类项目被称为逆向数学。它对哲学动机的基础程序有许多影响，如建构主义（Bishop）、可计算数学（paul - el /Richards）、有限还原论（Hilbert）、预测论（Weyl/Feferman）和预测还原论。逆向数学分类项目也是具有挑战性的技术问题的丰富来源。研究者和他的同事们正在研究其中的几个，重点是分析、几何和可数组合学。未来的一个方向是削弱基础系统，以便大大拓宽分类项目的范围，提供与数学的其他部分（如数论和计算复杂性）的重要接触点。数学基础是研究数学最基本的概念和逻辑结构，着眼于人类知识的统一。这条研究路线处理了一些基本问题，如：数学证明的本质是什么？数学的逻辑结构是什么？什么是适合数学的公理？《数学基础》是一门有着悠久历史的丰富学科，可以追溯到亚里士多德和欧几里得，并在弗雷格、罗素、希尔伯特、图灵和德尔等杰出的现代人物手中继续发展。研究者和他的同事们继续这一基础传统，通过追求一个深远的分类项目被称为逆向数学。具体的数学定理是根据证明它们所需的公理分类的。这揭示了数学中一个非常简单的逻辑结构。这种结构的存在具有许多深刻的含义。正在进行的研究有助于澄清无限在数学中的作用、数学结构的本质、数学中不可预知定义的作用以及相关问题。

The investigator believes that mathematical logic needs to return to its roots in Foundations of Mathematics, in the great tradition of Frege, Russell, Hilbert, Turing, and G del. A basic question in Foundations of Mathematics is: Which set-existence axioms are needed to prove specific theorems of core mathematics? Here ``core mathematics'' comprises standard topics in analysis, algebra, topology/geometry, etc. The investigator and his colleagues study this question in terms of Subsystems of Second Order Arithmetic, as exposited in the investigator's recently published research monograph of that title. An extensive series of case studies reveals that (i) many core mathematical theorems are logically equivalent to the set-existence axioms needed to prove them, (ii) only a handful of set-existence axioms arise in this way, (iii) the corresponding subsystems are linearly ordered by logical implication. This gives a far-reaching classification of hundreds of core mathematical theorems into a small number of classes, the classes being defined in terms of logical equivalence over a weak base system. This ongoing classification project is known as Reverse Mathematics. It has many implications for philosophically motivated foundational programs such as constructivism (Bishop), computable mathematics (Pour-El/Richards), finitistic reductionism (Hilbert), predicativism (Weyl/Feferman), and predicative reductionism. The Reverse Mathematics classification project is also a rich source of challenging technical problems. The investigator and his colleagues are pursuing several of these, with emphasis on analysis, geometry, and countable combinatorics. A direction for the future is to weaken the base system, in order to greatly broaden the scope of the classification project, providing significant points of contact with other parts of mathematics such as number theory and computational complexity.Foundations of Mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. This line of research deals fruitfully with fundamental questions such as: What is the nature of mathematical proof? What is the logical structure of mathematics? What are the appropriate axioms for mathematics? Foundations of Mathematics is a rich subject with a long history, going back to Aristotle and Euclid and continuing in the hands of outstanding modern figures such Frege, Russell, Hilbert, Turing, and G del. The investigator and his colleagues continue in this foundational tradition by pursuing a far-reaching classification project known as Reverse Mathematics. Specific mathematical theorems are classified according to the axioms needed to prove them. This reveals a remarkably simple logical structure within mathematics. The existence of such a structure has many profound implications. The ongoing research contributes to clarification of the role of the infinite in mathematics, the nature of mathematical constructions, the role of impredicative definitions in mathematics, and related issues.