Department of Philosophy, Carnegie Mellon University, USA.
Stud Hist Philos Sci. 2024 Aug;106:196-207. doi: 10.1016/j.shpsa.2024.06.006. Epub 2024 Jul 25.
The first formal definition of randomness, seen as a property of sequences of events or experimental outcomes, dates back to Richard von Mises' work in the foundations of probability and statistics. The randomness notion introduced by von Mises is nowadays widely regarded as being too weak. This is, to a large extent, due to the work of Jean Ville, which is often described as having dealt the death blow to von Mises' approach, and which was integral to the development of algorithmic randomness-the now-standard theory of randomness for elements of a probability space. The main goal of this article is to trace the history and provide an in-depth appraisal of two lesser-known, yet historically and methodologically notable proposals for how to modify von Mises' definition so as to avoid Ville's objection. The first proposal is due to Abraham Wald, while the second one is due to Claus-Peter Schnorr. We show that, once made precise in a natural way using computability theory, Wald's proposal constitutes a much more radical departure from von Mises' framework than intended. Schnorr's proposal, on the other hand, does provide a partial vindication of von Mises' approach: it demonstrates that it is possible to obtain a satisfactory randomness notion-indeed, a canonical algorithmic randomness notion-by characterizing randomness in terms of the invariance of limiting relative frequencies. More generally, we argue that Schnorr's proposal, together with a number of little-known related results, reveals that there is more continuity than typically acknowledged between von Mises' approach and algorithmic randomness. Even though von Mises' exclusive focus on limiting relative frequencies did not survive the passage to the theory of algorithmic randomness, another crucial aspect of his conception of randomness did endure; namely, the idea that randomness amounts to a certain type of stability or invariance under an appropriate class of transformations.
随机性的第一个正式定义，被视为事件序列或实验结果的属性，可以追溯到理查德·冯·米塞斯（Richard von Mises）在概率论和统计学基础方面的工作。冯·米塞斯（von Mises）引入的随机性概念如今被广泛认为过于薄弱。在很大程度上，这要归因于让·维勒（Jean Ville）的工作，人们常常形容他的工作给冯·米塞斯（von Mises）的方法带来了致命一击，并且对算法随机性的发展起到了至关重要的作用——即现在对概率空间元素的随机性的标准理论。本文的主要目标是追溯历史，并深入评估两个鲜为人知但在历史和方法论上值得注意的建议，以修改冯·米塞斯的定义，从而避免维勒的反对。第一个建议来自亚伯拉罕·沃尔德（Abraham Wald），第二个建议来自克劳斯-彼得·施诺尔（Claus-Peter Schnorr）。我们表明，一旦使用可计算性理论以自然的方式精确化，沃尔德的建议就构成了对冯·米塞斯框架的比预期更为激进的背离。另一方面，施诺尔的建议确实为冯·米塞斯方法提供了部分辩护：它表明，通过将随机性表述为极限相对频率的不变性，可以获得令人满意的随机性概念——实际上，是一种规范的算法随机性概念。更一般地说，我们认为，施诺尔的建议，以及一些鲜为人知的相关结果，表明在冯·米塞斯方法和算法随机性之间存在比通常承认的更多的连续性。尽管冯·米塞斯（von Mises）对极限相对频率的排他性关注在算法随机性理论中并未幸存下来，但他对随机性概念的另一个关键方面却得以保留；即随机性相当于在适当的变换类下的某种类型的稳定性或不变性。
