物理学理论的Topos基金会：I。物理学的形式语言
A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
摘要：
本文是系列文章中的第一篇，其目标是发展一种构建物理学理论的全新方法。其动机来自于一种渴望，即解决在思考空间和时间的量子理论时出现的某些深层问题。我们的基本论点是，构建一个物理学理论等价于在一个附着于该系统的某种形式语言的主题中找到一个表示。当topos是集合的范畴时，经典物理学就产生了。其他类型的理论采用不同的主题。在本文中，我们讨论了两种不同类型的语言，可以附加到一个系统，S。第一种是命题语言PL（S）;第二种是高阶类型语言L（S）。这两种语言都为演绎系统提供了直觉逻辑。引入PL（S）的原因是，如系列论文II所示，它是理解和扩展拓扑理论和量子物理学早期工作的最简单方法。然而，我们的计划的主旨利用更强大的语言L（S）和它的代表性在一个适当的主题。
英文摘要：
This paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper we discuss two different types of language that can be attached to a system, S. The first is a propositional language, PL(S); the second is a higher-order, typed language L(S). Both languages provide deductive systems with an intuitionistic logic. The reason for introducing PL(S) is that, as shown in paper II of the series, it is the easiest way of understanding, and expanding on, the earlier work on topos theory and quantum physics. However, the main thrust of our programme utilises the more powerful language L(S) and its representation in an appropriate topos.
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