Daniel Wieser

The effectiveness of mathematics in the natural sciences is often regarded as philosophically puzzling. If mathematics is an abstract, autonomous domain, why should its structures apply so successfully to physical reality? This question has motivated ontological interpretations ranging from mathematical realism to claims that reality itself is fundamentally mathematical. This paper argues that the appearance of a mystery rests on a mistaken assumption. Mathematics is not an external language that happens to match nature; it functions as a condition of scientific intelligibility. Scientific description, explanation, and law-formation presuppose mathematical articulation in advance. What mathematics provides is not a representation of an independently given reality, but a grammatical framework that fixes what counts as a measurable quantity, a lawful relation, and an objective scientific object at all. On this basis, the so-called “unreasonable effectiveness” of mathematics is dissolved rather than explained. Its effectiveness is not coincidental but constitutive: only phenomena that can be articulated, compared, and transformed within a mathematical framework can enter scientific inquiry as science. As mathematical frameworks expand, the space of scientifically intelligible questions expands with them, not because mathematics reveals new aspects of reality, but because it enables new forms of articulation. The result is a non-ontological, transcendental account of mathematics that preserves scientific objectivity while relocating philosophical explanation from correspondence to normativity.

This paper examines the possibility and limits of abstraction in thinking through a close reading of Ludwig Wittgenstein’s On Certainty. While abstraction is commonly regarded as a precondition ofthought, the paper argues that, for Wittgenstein, abstraction becomesproblematic when detached from practical use. It is shown that think-ing, understood as a rule-governed activity within language-games, de-pends on certainties that function as conditions of sense rather thanas epistemic achievements. By analysing Wittgenstein’s treatment ofdoubt, knowledge, and rule-following, the paper develops the thesisthat complete abstraction does not represent a higher form of think-ing, but leads instead to the self-dissolution of thought. The paper concludes by situating Wittgenstein’s critique of abstraction as a methodological tool rather than a rejection of abstraction as such, and briefly considers its implications for highly abstract forms of thinking.