威廉·阿克曼Wilhelm Ackermann

威廉·阿克曼是一位数学逻辑学家，曾在哥廷根与大卫·希尔伯特合作，但职业生涯中一直担任高中教师。

阿克曼于1914年进入哥廷根大学学习数学、物理学和哲学。然而，他开始学习后不久，第一次世界大战爆发。阿克曼于1915年被征召入伍，并继续服役直到1919年，他才得以回到哥廷根继续学业。他于1925年获得博士学位，学位论文为Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit Ⓣ（使用大卫·希尔伯特的一致性理论为‘排中律’奠定基础），在大卫·希尔伯特的指导下完成。该论文提供了无需归纳法的算术一致性证明。它旨在成为初等分析的一致性证明，尽管该证明包含重大错误。Richard Zach在[6]中解释了该论文的背景：-

阿克曼1924年的学位论文特别令人感兴趣，因为它是大卫·希尔伯特所认为的有限主义一致性证明的第一个非平凡例子。冯·诺伊曼1927年的论文是20世纪20年代对证明论的另一项主要贡献，但它并不完全符合大卫·希尔伯特学派的传统，而且我们没有证据表明大卫·希尔伯特在其写作中的参与程度。后来的一致性证明，特别是格哈德·根岑和拉斯洛·卡尔马的那些，是在库尔特·弗雷德里希·哥德尔的不完备性结果已经广为人知、其含义已被证明论者理解之后写成的。另一方面，阿克曼的工作完全源于大卫·希尔伯特的研究项目，并且有充分证据表明大卫·希尔伯特了解该证明的范围和细节。

提交学位论文后，阿克曼前往英国剑桥，在那里度过了1925年上半年。他获得了洛克菲勒奖学金以支持这次旅行，大卫·希尔伯特写信支持他的申请，表明了他对Ackerman工作的高度评价[6]：-

在其学位论文“使用大卫·希尔伯特的一致性理论为‘排中律’奠定基础”中，阿克曼在最一般的情况下表明，使用“所有”和“存在”这些词以及“排中律”是无矛盾的。该证明仅使用原始和有限的推理方法。可以说，一切都是在数学形式系统上直接展示的。阿克曼在这里克服了相当大的数学困难，解决了一个对现代为数学提供新基础的努力至关重要的问题。

阿克曼也是被称为epsilon演算的逻辑系统发展的主要贡献者，该系统最初由大卫·希尔伯特提出。这种形式系统构成了尼古拉·布尔巴基的逻辑和集合论的基础。

从1929年到1948年，他在Burgsteinfurt和Luedenscheid的Arnoldinum 文理中学（Gymnasium）任教。他是哥廷根科学院通讯院士，并且是明斯特大学名誉教授。

1928年，阿克曼注意到$A(x, y, z)$，即$x$以$y$进行的$z$次迭代幂，是一个非原始递归的递归函数的例子。$A(x, y, z)$被罗莎·培特简化为一个二元函数$P(x, y)$，其初始条件又被拉斐尔·罗宾逊简化。今天教科书中所出现的阿克曼函数正是后者。同样在1928年，大卫·希尔伯特和阿克曼所著、屡次重印的Grundzüge der Theoretischen Logik Ⓣ（理论逻辑原理）问世。

阿克曼后来的工作包括集合论（1937年）、完整算术（1940年）和无类型逻辑（1952年）的一致性证明。此外还有集合论的一种新公理化（1956年），以及一本书Solvable cases of the decision problem（North Holland，1954年。第二版，1962年）。

集合论的新公理化由阿克曼在Zur Axiomatik der MengenlehreⓉ（《论公理化集合论》）（1956）中提出。Dana Scott写道[5]：-

提出了一个关于集合论系统的极其简单的公理化，评论者认为它值得认真考虑。该系统在一个带等词的应用一阶演算中形式化，使用一个二元谓词$\in$（属于关系）和一个一元谓词M（是一个集合）。对于该系统的相容性来说，M不能用$\in$来定义是相当关键的。外延公理被假定，使得所有个体都可以被视为个体的汇集，但从公理很容易证明存在不是集合、甚至包含非集合的汇集。该理论中这种非正常汇集存在的必要性，使得与标准集合论系统的比较有些困难。

Rudolf Grewe于1966年以阿克曼的集合论为题写了博士学位论文，他在[2]中给出了该理论的模型。其他几位作者也研究过这个系统，参考文献见[2]。

1957年，阿克曼发表了Philosophische Bemerkungen zur mathematischen Logik und zur mathematischen Grundlagenforschung Ⓣ（关于数理逻辑和基础数学研究的哲学评论）及其英译本Philosophical observations on mathematical logic and on investigations into the foundations of mathematics。这篇为非专业人士撰写的论文出色地概述了阿克曼如何看待数理逻辑。弗瑞兹·约翰 Van Heijenoort写道[3]：-

对数理逻辑的一个反对意见是，它不同于构成我们思想基础、并且对思维来说唯一必需的哲学逻辑。阿克曼指出，传统的推理模式被包含在数理逻辑中，此外还有许多其他模式，如命题逻辑或关系逻辑。只要数学推理没有得到充分分析，人们就会有一种在数学中用“亚里士多德逻辑”就能应付的错觉。

对数学逻辑的另一个反对意见是它是不完备的，意思是，根据库尔特·弗雷德里希·哥德尔1931年的结果（原文说1932年），直觉上正确的定理不能在给定系统内证明。但直觉思维并不一致；悖论出现在从朴素直觉观点看必须被认为是正确的推理模式的基础上。如果我们限制直觉推理以消除悖论，我们就得到一个形式系统，而不完备性是我们为一致性必须付出的代价。

数学逻辑在澄清“必然性”和“可能性”的概念、“分析”和“综合”之间的区别方面有用。讨论逻辑的“平凡性”，作者提出了判定问题。

数学今天被视为对结构的研究；但与自然数概念相关的明显性独立于任何公理化引入的结构。阿克曼提出了直觉主义，它构建了一个具有最少逻辑的数学，以及数概念的戈特洛布·弗雷格-伯特兰·罗素分析。他为这一分析辩护，反对某些反对意见（循环性、无穷公理的必要性）。他指出，勒伊岑·布劳威尔将数直觉归结为两个概念：统一性和反复区分一个统一性与另一个的可能性。在这些概念中“没有与逻辑无关的东西”。他得出结论：“在自然数理论中，我们有一个领域，它能够以[勒伊岑·布劳威尔]意义上的最少逻辑进行直觉基础，并且如果我们预设一个广泛的逻辑，也能纯粹通过逻辑定义达到。”
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论文以关于几何学的评论结束，作为传达外部世界知识的科学。超越任何几何学的公理化处理，有直觉的几何印象以强制力量强加于我们。为了获得一个系统而完整的整体，我们可以添加其中出现自由构造的原则。但是，尽管界限可能难以追踪，仍然存在一个强制性直觉元素的核心。因此，作者勾勒出一个立场，一方面不同于康德意义上的全面先验论，另一方面不同于彻底的约定论。

Wilhelm Ackermann was a mathematical logician who worked with David Hilbert in Göttingen but spent his career as a high school teacher.

Ackermann entered the University of Göttingen in 1914 to study mathematics, physics and philosophy. However soon after he began his studies Word War I started. Ackermann was drafted into the army in 1915 and continued to serve until 1919 when he was able to return to his studies in Göttingen. He received his doctoral degree in 1925 with a thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit Ⓣ written under David Hilbert's supervision. It provided a proof of the consistency of arithmetic without induction. It was intended to be a consistency proof for elementary analysis although this proof contained significant errors. Richard Zach explains the background to the thesis in [6]:-

Ackermann's 1924 dissertation is of particular interest since it is the first non-trivial example of what Hilbert considered to be a finitistic consistency proof. Von Neumann's paper of 1927, the only other major contribution to proof theory in the 1920s, does not entirely fit into the tradition of the Hilbert school, and we have no evidence of the extent of Hilbert's involvement in its writing. Later consistency proofs, in particular those by Gentzen and Kalmár, were written after Gödel's incomplete ness results were already well-known and their implications understood by proof theorists. Ackermann's work, on the other hand, arose entirely out of Hilbert's research project, and there is ample evidence that Hilbert was aware of the range and details of the proof.

After submitting his dissertation, Ackermann went to Cambridge, England, where he spent the first half of 1925. He was awarded a Rockefeller Scholarship to support this trip and Hilbert wrote in support of his application showing his high opinion of Ackerman's work [6]:-

In his thesis "Foundation of the 'tertium non datur' using Hilbert's theory of consistency," Ackermann has shown in the most general case that the use of the words "all" and "there is," of the "tertium non datur," is free from contradiction. The proof uses exclusively primitive and finite inference methods. Everything is demonstrated, as it were, directly on the mathematical formalism. Ackermann has here surmounted considerable mathematical difficulties and solved a problem which is of first importance to the modern efforts directed at providing a new foundation for mathematics.

Ackermann was also the main contributor to the development of the logical system known as the epsilon calculus, originally due to Hilbert. This formalism formed the basis of Bourbaki's logic and set theory.

From 1929 until 1948 he taught as a teacher at the Arnoldinum Gymnasium in Burgsteinfurt and in Luedenscheid. He was corresponding member of the Akademie der Wissenschaften in Göttingen, and was honorary professor at the Universität Münster.

In 1928, Ackermann observed that $A(x, y, z)$, the $z$-fold iterated exponentiation of $x$ with $y$, is an example of a recursive function which is not primitive recursive. $A(x, y, z)$ was simplified to a function $P(x, y)$ of two variables by Rozsa Peter whose initial condition was simplified by Raphael Robinson. It is the latter which occurs as Ackermann's function in today's textbooks. Also in 1928 the often reprinted book Grundzüge der Theoretischen Logik Ⓣ by Hilbert and Ackermann appeared.

Among Ackermann's later work are consistency proofs for set theory (1937), full arithmetic (1940) and type free logic (1952). Further there was a new axiomatization of set theory (1956), and a book Solvable cases of the decision problem (North Holland, 1954. Second edition, 1962).

The new axiomatization of set theory was presented by Ackermann in Zur Axiomatik der Mengenlehre Ⓣ (1956). Dana Scott writes [5]:-

A remarkably simple axiomatization of a system of set theory is presented which the reviewer feels deserves serious consideration. The system is formalized in an applied first-order calculus with identity using a binary predicate $\in$ (membership) and a singulary predicate M (being a set). It is quite essential for the consistency of the system that M is not definable in terms of $\in$. The axiom of extensionality is assumed so that all the individuals can be considered as collections of individuals, but it is easily proved from the axioms that there are collections that are not sets and even contain non-sets. The necessity for the existence of such improper collections in the theory makes comparison with the standard systems of set theory somewhat difficult.

Rudolf Grewe, who wrote his doctoral dissertation on Ackermann's set theory in 1966, gives models for the theory in [2]. Several other authors have studied this system and references are given in [2].

In 1957 Ackermann published Philosophische Bemerkungen zur mathematischen Logik und zur mathematischen Grundlagenforschung Ⓣ and its English translation Philosophical observations on mathematical logic and on investigations into the foundations of mathematics. This paper, written for non-experts in the subject, gives an excellent overview of how Ackermann viewed mathematical logic. John Van Heijenoort writes [3]:-

An objection to mathematical logic is that it is not the same as the philosophical logic which forms the foundation of our thought and which alone is necessary for thinking. Ackermann remarks that the traditional modes of inference are included in mathematical logic, besides many others, like the statement logic or the logic of relations. One has the illusion of getting by with "Aristotelian logic" in mathematics just as long as mathematical reasonings are insufficiently analyzed.

A further objection to mathematical logic is that it is incomplete, in the sense that, by Gödel's result of 1931 (the text says 1932), intuitively correct theorems cannot be proved within a given system. But intuitive thinking is not consistent; paradoxes arise on the basis of modes of inference which from the naive intuitive point of view have to be considered correct. If we restrict intuitive inference so that paradoxes are eliminated, we come to a formal system, and incompleteness is the price we have to pay for consistency.

Mathematical logic is of use in clarifying the concepts of "necessity" and "possibility," the distinction between "analytic" and "synthetic." Discussing the "triviality" of logic, the author presents the decision problem.

Mathematics is today viewed as an investigation of structures; but the obviousness connected with the concept of natural number is independent of any axiomatically introduced structure. Ackermann presents intuitionism, which constructs a mathematics with a minimum of logic, and the Frege-Russell analysis of the number concept. He defends this analysis against certain objections (circularity, necessity of an axiom of infinity). He remarks that Brouwer reduced the number intuition to two concepts: unity and possibility of repeatedly distinguishing a unity from another. In these concepts "there is nothing foreign to logic." He concludes that "in the theory of natural numbers we have a domain which is capable of an intuitive foundation with a minimum of logic in the sense of [Brouwer] and also of attainment purely by logical definitions if we presuppose an extensive logic."
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The paper ends with remarks on geometry, as a science conveying knowledge of the external world. Beyond any axiomatic treatment of geometry there are intuitive geometrical impressions which force themselves upon us with compelling power. In order to obtain a systematic and complete whole we may add principles in which free constructions occur. But, although limits may be difficult to trace, there remains a nucleus of obligatory intuitive elements. Thus the author sketches a position distinct, on the one hand, from a comprehensive apriorism in the sense of Kant and, on the other hand, from a thoroughgoing conventionalism.