Formalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects. 1 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism (advocated by Heine), game formalism (advocated by Thomae) and instrumental formalism (advocated by Hilbert). After describing these views, I note some basic points of similarity and difference between them. In the remainder of the paper, I turn my attention to Hilbert’s instrumental formalism. My primary aim there will be to develop its formalist elements more fully. These are, in the main, (i) its rejection of the axiom-centric focus of traditional model-construction approaches to consistency problems, (ii) its departure from the traditional understanding of the basic nature of proof and (iii) its distinctively descriptive or observational orientation with regard to the consistency problem for arithmetic. More specifically, I will highlight what I see as the salient points of connection between Hilbert’s formalist attitude and his finitist standard for the consistency proof for arithmetic. I will also note what I see as a significant tension between Hilbert’s observational approach to the consistency problem for arithmetic and his expressed hope that his solution of that problem would dispense with certain epistemological concerns regarding arithmetic once and for all.

数学哲学中的形式主义采取了多种形式，并出于广泛不同的原因而受到提倡。昆虫。1 和 2，我简要介绍了 19 世纪末和 20 世纪初的主要形式主义学说。这些就是我所说的经验语义形式主义（海涅所提倡的）、博弈形式主义（托马所倡导的）和工具形式主义（希尔伯特所倡导的）。在描述了这些观点之后，我注意到它们之间的一些基本相同点和不同点。在本文的其余部分，我将注意力转向希尔伯特的工具形式主义。我的主要目标是更充分地发展其形式主义元素。这些主要是（i）它拒绝以公理为中心的传统模型构建方法来解决一致性问题，(ii) 它背离了对证明基本性质的传统理解，以及 (iii) 它在算术一致性问题方面具有独特的描述性或观察性取向。更具体地说，我将强调我所看到的希尔伯特的形式主义态度与他的算术一致性证明的有限主义标准之间的显着联系点。我还将注意到希尔伯特对算术一致性问题的观察方法与他表达的希望他对该问题的解决方案将一劳永逸地免除某些关于算术的认识论问题之间的显着紧张关系。我将强调我所看到的希尔伯特的形式主义态度与他的算术一致性证明的有限主义标准之间的显着联系点。我还将注意到希尔伯特对算术一致性问题的观察方法与他表达的希望他对该问题的解决方案将一劳永逸地免除某些关于算术的认识论问题之间的显着紧张关系。我将强调我所看到的希尔伯特的形式主义态度与他的算术一致性证明的有限主义标准之间的显着联系点。我还将注意到希尔伯特对算术一致性问题的观察方法与他表达的希望他对该问题的解决方案将一劳永逸地免除某些关于算术的认识论问题之间的显着紧张关系。