康德哲学中数学知识的性质与基础——先天综合、纯直观与概念构造的再阐释

The Nature and Ground of Mathematical Knowledge in Kant’s Philosophy—Reinterpreting the Synthetic a Priori, Pure Intuition, and the Construction of Concepts

摘要: 康德在《纯粹理性批判》中以“纯粹数学何以可能”为线索，将算术与几何的基本命题界定为先天综合判断，并据此提出一种以“纯直观–构造”为核心的数学知识论。本文在沿用这一框架的基础上作出三方面推进：其一，澄清“分析/综合”与“先天/后天”的双重区分在康德体系中的功能，说明数学之所以“综合”，并非仅是语义外延的扩展，而是依赖于在纯直观中对概念的构造；其二，区分空间与时间作为感性形式、图式与想象力在证明中的作用，解释为何康德既反对经验主义的归纳根据，也拒绝将数学化约为纯逻辑演算；其三，回应非欧几何与形式主义/逻辑主义对康德论题的挑战，论证应将康德的主张理解为关于人类认知条件的“构成性”论断，并可借助“相对化先验”(relativized a priori)与符号构造的研究对其加以当代化重述。本文旨在表明：即便欧氏几何的唯一性已不可维持，康德关于“构造性–客观有效性”的核心洞见仍为理解数学知识的必然性与可应用性提供了富有解释力的路径。

Abstract: In the “Critique of Pure Reason” Kant famously asks how pure mathematics is possible and argues that basic arithmetical and geometrical propositions are synthetic a priori, grounded in the pure forms of intuition (space and time) and secured through the construction of concepts. Building on this Kantian framework, this paper offers three revisions. First, it clarifies the roles of the analytic/synthetic and a priori/a posteriori distinctions, showing that the “synthetic” character of mathematics is not merely semantic extension but a constructive procedure carried out in pure intuition. Second, it explicates how space, time, schematism, and imagination contribute to mathematical proof, thereby motivating Kant’s rejection of both empiricist foundations and a reduction of mathematics to purely logical calculation. Third, it addresses challenges from non-Euclidean geometry and from logicist/formalist programs by interpreting Kant’s thesis as a constitutive claim about conditions of human cognition, and by drawing on contemporary proposals such as the relativized a priori and accounts of symbolic construction. The upshot is that, even if the uniqueness of Euclidean geometry is no longer defensible, Kant’s core insight about the link between constructability and objective validity remains philosophically fruitful for understanding the necessity and applicability of mathematics.