Permanence as a principle of practice

Abstract

Dedicated to the memory of Mic Detlefsen The paper discusses Peano's argument for preserving familiar notations. The argument reinforces the principle of permanence, articulated in the early 19th century by Peacock, then adjusted by Hankel and adopted by many others. Typically regarded as a principle of theoretical rationality, permanence was understood by Peano, following Mach, and against Schubert, as a principle of practical rationality. The paper considers how permanence, thus understood, was used in justifying Burali-Forti and Marcolongo's notation for vectorial calculus, and in rejecting Frege's logical notation, and closes by considering Hahn's revival of Peano's argument against Pringsheim' reading of permanence as a logically necessary principle. Der Beitrag diskutiert Peanos Argument für die Bewahrung vertrauter Notationen. Es geht um das Prinzip der Permanenz, das im frühen 19. Jahrhundert von Peacock artikuliert, dann von Hankel angepasst und von vielen anderenübernommen wurde. Während es im allgemeinen als ein Prinzip der theoretischen Rationalität betrachtet wurde, interpretierte Peano es anknüpfend an Mach und gerichtet gegen Schubert als ein Prinzip der praktischen Rationalität. Der Beitrag untersucht, wie Permanenz, so verstanden, von Burali-Forti und Marcolongo zur Rechtfertigung von Bezeichnungen in der Vektorrechnung verwendet wurde und zur Zurückweisung der logischen Notation Freges führte. Die Arbeit schließt mit einer Erörterung der Wiederbelebung des Peanoschen Arguments in Hahns Kritik von Pringsheims Interpretation der Permanenz als ein logisch notwendiges Prinzip.

The Genealogy of ‘’

“…The story has two parts: The first part concerns the introduction of ‘ ’ by Whitehead and Russell, who were working within the notational and formal logical tradition of Peano. We suggest here that the initial introduction of ‘ ’ was motivated by a shared commitment to two design principles for logical symbolisms (Schlimm, 2018, 2021; Toader, 2021). 1 The first principle is that symbols should be unambiguous —one shouldn’t use the same symbol in different meanings.…”

“…Since the laws governing class union and intersection, and the laws for propositional disjunction and conjunction are the same, this justifies the use of the same symbols for both. By contrast, the arithmetical operations and obey different laws, e.g., idempotence fails (see Toader, 2021, for discussion). Moreover, context disambiguates between the different meanings of ‘ ’ and ‘ ’.…”

“…We take the phrase “design principles” from Schlimm. Toader (2021) highlighted the second principle.…”

“… 2 Peano puts this in terms of a principle of permanence : “ when establishing a new system of notations, or a new calculus, it is convenient to do it such that the new calculus be similar as much as possible with old calculi, so that the student does not have to learn a whole new calculus, but only the differences from the theory known to him” (quoted from Toader, 2021, p. 85). …”