We begin with the idea that lines of reasoning are continuous mental processes and develop a notion of continuity in proof. This requires abstracting the notion of a proof as a set of sentences ordered by provability. We can then distinguish between discrete steps of a proof and possibly continuous stages , defining indexing functions to pick these out. Proof stages can be associated with the application of continuously variable rules, connecting continuity in lines of reasoning with continuously variable reasons. Some examples of continuous proofs are provided. We conclude by presenting some fundamental facts about continuous proofs, analogous to continuous structural rules and composition. We take this to be a development on its own, as well as lending support to non-finitistic constructionism.

中文翻译：

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连续推理：连续证明的正式构造

我们从推理线是连续的心理过程的想法开始，并发展出证明连续性的概念。这需要将证明的概念抽象为一组按可证明性排序的句子。然后我们可以区分证明的离散步骤和可能的连续阶段，定义索引函数来挑选这些。证明阶段可以与连续可变规则的应用相关联，将推理线的连续性与连续可变的原因联系起来。提供了一些连续证明的例子。我们最后介绍了一些关于连续证明的基本事实，类似于连续结构规则和组合。我们认为这本身就是一种发展，并为非有限建构主义提供支持。