Peano on Symbolization, Design Principles for Notations, and the Dot Notation

Abstract

Peano a été l'une des forces motrices dans le développement du formalisme mathématique actuel. Dans cet article, nous étudions son approche particulière de la conception notationnelle et présentons quelques caractéristiques originales de ses notations. Pour motiver l'approche de Peano, nous présentons d'abord sa vision de la logique comme méthode d'analyse et son désir d'un symbolisme rigoureux et concis pour représenter les idées mathématiques. Sur la base à la fois de sa pratique et de ses réflexions explicites sur les notations, nous discutons des principes qui ont guidé Peano dans l'introduction de nouveaux symboles, le choix des caractères et la mise en forme des formules. Enfin, nous examinons de plus près, d'un point de vue systématique et historique, l'une des innovations les plus marquantes de Peano, à savoir l'usage de points pour regrouper des sous-formules.

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.…”

“…The intended interpretation, on the other hand, may vary in those limited cases such that the formal laws—even the theorems and their proofs—do not differ. Indeed, Peano was generally very thoughtful in his selection and use of symbols (see Schlimm, 2021, for an insightful discussion). The “design principles” Peano employed explain, for instance, why he chose pairs of inverted symbols for dual operations like ‘ ’ and ‘ ’ 22…”

“…For example, in his “Logic of Relations” (Russell, 1901a) Russell uses dots for conjunction and for scope, reserving concatenation for relational products. In contrast, Peano never used dots for conjunction: he only used dots for scope (Schlimm, 2021, pp. 118–119).…”

“… 1 For an extended discussion of these two design principles, see Schlimm (2018) on Frege, and Schlimm (2021) on Peano. We take the phrase “design principles” from Schlimm.…”

“… 4 It should be noted that neither Peano nor Russell followed this principle quite perfectly. Peano used ‘ ’ for both implication and deductive consequence (Schlimm, 2021, sec. 3.2.1).…”