Why were Peano’s postulates important to attempts to ground mathematics in logic?

Giuseppe Peano’s postulates were pivotal to the logicist program because they provided a rigorous, axiomatic foundation for the natural numbers, which many logicians believed served as the bedrock for all of mathematics.12

By defining the structure of natural numbers through a finite set of symbols and rules, the Peano axioms offered a clear target for foundational research. It was widely held that if these postulates could be successfully derived from logic—or from logic augmented by set theory—it would demonstrate the feasibility of grounding all mathematical truth in logical principles.12

While Peano himself was not primarily focused on this logicist agenda, his work—alongside similar contributions from Dedekind and Peirce—became the essential link between mathematics and logic. His notation, particularly the use of quantifiers and the symbol for set membership, was subsequently adopted by Bertrand Russell and Alfred North Whitehead, further cementing the role of his axioms in the development of modern symbolic logic.2