Notations de logique mathématique is a compact methodological exposition of Giuseppe Peano’s symbolic language for mathematics. Its subject is not mathematics as a whole but the notation in which mathematical propositions, definitions, and proofs can be written. The work sets out Peano’s signs for logical operations and for the basic objects of mathematical discourse—propositions, classes, relations, variables, and functions—and explains the conventions by which formulas are formed and read. It is intended as an introduction to the symbolic apparatus that underlies the larger Formulaire de mathématiques project rather than as a complete logical treatise or a systematic development of a mathematical discipline.
The book presents Peano’s notation as an “ideography” for mathematics: a written language designed to make logical structure visible. It explains signs for negation, conjunction and disjunction, implication and equivalence, and for universal and existential quantification. It shows how class membership, inclusion, intersection, union, and complement are expressed, and how relations and functions are represented. Peano also explains the use of dots and other grouping conventions in place of ordinary parentheses, and he stresses the importance of fixing variables and of introducing new symbols by explicit definition. The aim is economy and precision: primitive ideas and signs should be reduced to the smallest possible number, and everything else should be built up by definition from them.
A central theme of the work is that logical notation is not mere shorthand. Peano treats symbolism as an instrument of research as well as of expression. Translation into symbols, he argues, condenses mathematical writing, exposes the logical form of arguments, and reveals that many propositions which appear distinct in ordinary language are in fact the same proposition. It can also expose redundancy in verbal theories and help isolate the assumptions on which a proof depends. The book therefore combines a technical presentation of notation with a methodological defense of its use: definitions are to be stated exactly, proofs are to proceed by substitution and logical transformation, and mathematical statements are to be analyzed into their simplest constituent ideas.
The work includes specimen translations of mathematical propositions into symbolic form and illustrates how the notation can be applied beyond elementary logic—for example in arithmetic, analysis, and geometry. These examples are not intended to exhaust the mathematical field; they demonstrate how ordinary mathematical language can be reduced to a small set of logical signs and how the resulting formulas can be manipulated. Peano presents lists of logical propositions and shows how class calculus and the logic of relations can express mathematical statements that would otherwise require lengthy verbal paraphrase. The treatment is deliberately preliminary and programmatic: the book establishes the notation, explains its grammar, and displays its usefulness, while leaving the full codification of mathematics to the collaborative Formulaire.
The book’s scope is thus limited and technical. It is an introduction to Peano’s formal language, its conventions, and its rationale, together with illustrative applications. It does not contain a narrative, a biography, or a general philosophical treatise; nor does it aim to develop all of mathematics symbolically within its own pages. Its contents are the notation itself, the rules for its use, the argument for its superiority over ordinary language in rigorous exposition, and examples showing how mathematical propositions and proofs are to be translated into that notation.
Within this program, definition is treated as a rule of abbreviation: a defined sign is to be eliminable in favor of primitive signs, so definitions add no new content and serve only to shorten formulas. Peano accordingly keeps primitive ideas and primitive propositions to a minimum and presents the rest as derived by substitution and logical transformation. This gives the work its characteristic austerity: the notation is not an ornament on ordinary mathematics but a device for displaying exactly which signs and assumptions are doing the work.
Propositional logic and the calculus of classes are developed in parallel. The same patterns of negation, conjunction, disjunction, implication, and equivalence serve for propositions and for classes; inclusion corresponds to implication, intersection to conjunction, union to disjunction, and complement to negation. Peano uses this correspondence to make class reasoning algebraic and to reduce verbal arguments to formulas that can be checked by rules. The logic of relations extends the same approach: relations are treated as classes of ordered pairs, and the notation is meant to express converse, relative product, and functional dependence in a uniform way, so that order, correspondence, and functions fall within the same calculus.
The treatment of quantification is tied to the careful separation of real from apparent variables. Quantifiers bind variables and determine scope; dots and grouping conventions make that scope visible, while substitution governs how formulas may be transformed. Peano stresses that ordinary language often leaves scope and hypotheses implicit, whereas the ideographic formula states them explicitly—often by writing a theorem as a conditional whose antecedent lists the hypotheses. The book’s significance lies in this practical synthesis: it turns logical notation into a working calculus for classes and relations, makes formal derivation a matter of manipulating visible formulas, and provides the linguistic foundation for the larger Formulaire without claiming to be a complete formalization of mathematics.
