---
tag: G1M5ZZ70WB
kind: statement
role: proposition
name: Peano's The Principles of Arithmetic 1889
sources:
  - wikipedia: en:36595472@r1359939615
    locator: History of the function concept, section Logicians' function 1850–1950 > Peano's The Principles of Arithmetic 1889
    note: adopted
    mention: wikipedia://en/36595472#YOUR
    sections: [Statement]
inventory: [C87MAKV9EV]
---

## Statement

Peano defined the notion of "function" in a manner somewhat similar to Frege, but without the precision. First Peano defines the sign "K means *class*, or aggregate of objects", the objects of which satisfy three simple equality-conditions, *a* = *a*, (*a* = *b*) = (*b* = *a*), IF ((*a* = *b*) AND (*b* = *c*)) THEN (*a* = *c*). He then introduces φ, "a sign or an aggregate of signs such that if *x* is an object of the class *s*, the expression φ*x* denotes a new object". Peano adds two conditions on these new objects: First, that the three equality-conditions hold for the objects φ*x*; secondly, that "if *x* and *y* are objects of class *s* and if *x* = *y*, we assume it is possible to deduce φ*x* = φ*y*". Given all these conditions are met, φ is a "function presign". Likewise he identifies a "function postsign". For example if *φ* is the function presign *a*+, then φ*x* yields *a*+*x*, or if φ is the function postsign +*a* then *x*φ yields *x*+*a*.