Le definizioni in matematica is a short foundational essay on the logical character of mathematical definitions. Its central question is how mathematics should introduce new symbols and concepts, and what counts as a legitimate definition. Peano sets out from the ordinary model of a nominal definition: a new sign is equated with an expression composed of symbols already available. Such a definition functions as an explicit abbreviation, so that the newly introduced sign can in principle be eliminated in favor of the older expressions.
The essay argues, however, that not every mathematical idea can be handled in this way. Some concepts are reached by abstraction rather than by direct nominal definition. Peano is especially concerned with the case in which an operation associates with an underlying object \(u\) a new object \(\varphi u\). Because the abstract object is of a different kind from the objects already under consideration, it cannot simply be equated with a known expression. Instead, one defines equality between abstract objects. Under a hypothesis on two underlying objects \(u\) and \(v\), the equality \(\varphi u = \varphi v\) is defined to mean a previously understood relation \(P(u,v)\) between \(u\) and \(v\). This relation must satisfy the formal properties of equality: reflexivity, symmetry, and transitivity. Peano explains that the abstract object \(\varphi u\) is then what results from considering in \(u\) all and only those properties that it shares with every \(v\) such that \(\varphi u = \varphi v\). In other words, the abstract object is not introduced by an explicit abbreviation but by specifying an equivalence relation on prior objects.
The essay illustrates this procedure with Euclid’s treatment of the ratio of two magnitudes in Book V of the Elements. The discussion is thus less a systematic treatise on definitions than a methodological examination of how mathematics can introduce abstract objects without pretending to give them ordinary nominal definitions. Peano treats definition by abstraction cautiously: it defines equality among abstracta by reducing it to a relation on the original domain, rather than defining the abstract object itself in the manner of a nominal abbreviation. The work’s main contribution is this careful separation between explicit definitions and abstraction-based introductions of concepts, together with the formal conditions that make the latter mathematically usable.
Peano’s account makes the legitimacy of a definition depend on two things: the prior availability of the symbols used in the defining expression, and the formal correctness of any relation used to introduce an abstract object. A nominal definition is treated as a linguistic convention that adds no new mathematical content; it is an abbreviation whose definiendum can be replaced by its definiens wherever it occurs. A definition by abstraction is more delicate, because it does not give a synonym for the abstract sign. It fixes instead the conditions under which two abstract signs are equal. The question “what is the abstract object?” is therefore separated from the mathematically operative question “when are two such objects identical?” Peano’s answer is that the latter must be settled by an already understood relation on the underlying objects, and that this relation must have the formal character of equality.
The abstraction procedure also imposes an invariance condition. The abstract object \(\varphi u\) is not allowed to depend on the particular representative \(u\) beyond what the equivalence relation permits: it is determined by the properties shared by all objects equivalent to \(u\). This is what keeps the introduction of the abstract sign from being arbitrary. The underlying relation \(P(u,v)\) must be meaningful for the objects under consideration and must be reflexive, symmetric, and transitive; only then can equality between abstracta inherit the usual logical properties. In this way Peano treats definition by abstraction not as a mysterious mental act but as a controlled mathematical construction whose use is justified by the equivalence relation on the original domain.
The Euclid example functions as a test case for this method. A ratio of magnitudes is not plausibly introduced as a simple nominal abbreviation of a pair of magnitudes, but equality of ratios can be reduced to a relation among pairs of magnitudes. The abstraction is admissible because that relation supplies the identity conditions for ratios and possesses the required formal properties. The example thus shows how mathematics can speak of abstract objects while keeping their equality conditions anchored in prior, more concrete material.
The essay’s significance lies in this sharp methodological separation between explicit definitions and abstraction-based introductions of concepts. It treats definitions as subject to logical criteria—eliminability in the nominal case, and equivalence-structured identity conditions in the abstract case—rather than as merely explanatory prose. Peano thereby clarifies how mathematics may legitimately introduce concepts that are not simple abbreviations, while still requiring that every such introduction be justified by a relation on objects already understood. The result is a precise account of definition as a tool of symbolic economy and logical rigor, with abstraction governed by formal conditions rather than left to informal intuition.
