This article is about deterministic models, what they are, why they are useful, and what their limitations are. First, the article emphasizes that determinism is a property of models, not of physical systems. Whether a model is deterministic or not depends on how one defines the inputs and behavior of the model. To define behavior, one has to define an observer. The article compares and contrasts two classes of ways to define an observer, one based on the notion of “state” and another that more flexibly defines the observables. The notion of “state” is shown to be problematic and lead to nondeterminism that is avoided when the observables are defined differently. The article examines determinism in models of the physical world. In what may surprise many readers, it shows that Newtonian physics admits nondeterminism and that quantum physics may be interpreted as a deterministic model. Moreover, it shows that both relativity and quantum physics undermine the notion of “state” and therefore require more flexible ways of defining observables. Finally, the article reviews results showing that sufficiently rich sets of deterministic models are incomplete. Specifically, nondeterminism is inescapable in any system of models rich enough to encompass Newton’s laws.

中文翻译：

---

决定论

这篇文章是关于确定性模型，它们是什么，为什么有用，以及它们的局限性是什么。首先，文章强调确定性是模型的属性，而不是物理系统的属性。模型是否具有确定性取决于如何定义模型的输入和行为。要定义行为，必须定义观察者。这篇文章比较和对比了两类定义观察者的方法，一类基于“状态”的概念，另一类更灵活地定义可观察对象。“状态”的概念被证明是有问题的，并导致不确定性，当可观察量的定义不同时，这种不确定性就可以避免。这篇文章探讨了物理世界模型中的决定论。令许多读者惊讶的是，它表明牛顿物理学承认非确定性，而量子物理学可以被解释为确定性模型。此外，它表明相对论和量子物理学都破坏了“状态”的概念，因此需要更灵活的方式来定义可观测量。最后，文章回顾了结果，表明足够丰富的确定性模型集是不完整的。具体来说，在任何足以包含牛顿定律的模型系统中，非确定性都是不可避免的。