 As Detlefsen points out, Poincaré’s most interesting argument does not concern the status of the basic laws of the logical systems, but rather the “place of logical inference in mathematical proof” (Detlefsen 1992: 349). According to Poincaré, today’s admitted validity of a logical, i.e., topic-neutral, inference from a proposition \(p\) to a proposition \(q\) (in Tarski’s conception: every model of \(p\) is a model of \(q)\) is not enough to sustain epistemic growth of mathematical knowledge, which requires rather that “\(p\) be seen mathematically to imply \(q\) (i.e., that \(p\) and \(q\) be seen as united by a common mathematical structure)” (Detlefsen 1993: 272). Poincaré insists on the non-invariance of mathematical reasoning with respect to its content and advances, so to speak, a local conception of reasoning (for example, the mathematical induction principle for arithmetical reasoning). According to this view, a gap in a proof is no longer a logical gap, but a gap in the mathematical understanding (Detlefsen 1992: 360, 366). If “formalization” and a universal calculus are, then, no longer the criterion of mathematical rigor, how shall rigor be maintained? Poincaré’s response mentions in particular the condensing of the content of the premises to the content of the conclusion by means of intuition. Rigor is obtained by combining Ernst Mach’s principle of the economy of thought with the concept of harmony, that is, thanks to “the fortunate inventions of language” that introduce an order-structure, the complexity of a domain of objects is made more harmonious by the introduction of an invariance (see Poincaré 1908: 23–30; 1913b: 371–375). Thus, in mathematical reasoning, rigor is placed in the context of a “mathematical architecture” whose simplest expression is found in mathematical induction.