How did Peano’s postulates connect logic with the natural numbers?

Giuseppe Peano’s postulates provided a rigorous foundation for the natural numbers by defining them through a finite set of symbols and rules, effectively allowing an infinite set to be generated logically.1

These five axioms establish the structure of arithmetic by defining zero, the concept of a successor, and the principle of induction. By formalizing these properties, the postulates became a crucial link between logic and mathematics.12

The connection was significant because many scholars believed that all of mathematics could be derived from the theory of natural numbers. Consequently, if the Peano postulates could be successfully derived from logic—or from logic combined with set theory—it would demonstrate that mathematics itself rests upon a logical foundation. While Peano himself was not focused on the logicist program, his work, alongside similar efforts by Dedekind and Peirce, provided the essential framework that later influenced foundational research by figures such as Bertrand Russell.2