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Peano, Giuseppe

mathematics continuous axioms logic

[pay ah noh] (1858–1932) Italian mathematician: introduced the Peano axioms into mathematical logic.

Peano grew up on a farm and from the age of 12 was taught privately in Turin. On winning a scholarship to Turin University his talent was fully revealed, and by 32 he held a professorship there. The lack of rigour in mathematics provoked him to try to unravel the areas where intuition had concealed the logic of analysis. During the 1880s he looked at the integrability of functions; he proved that first order differential equations y ’= f (x,y) are always solvable if the function f is continuous. In 1890 this was generalized to the first statement of the axiom of choice. In the same year he demonstrated a curve that is continuous but filled space, indicating that graphical methods are limited in the analysis of continuous functions.

Peano worked on the application of logic to mathematics from 1888, producing a new notation. He also wrote down a set of axioms that covered the logical concept of natural numbers (Peano axioms), later acknowledging that had anticipated him in this. Peano’s work in this area was probably more important than that of the other great figures – , F L G Frege (1848–1925) and B Russell (1872–1970).

Peano also initiated geometrical calculus by applying the axiomatic method to geometry. However after 1900 his interests shifted and he did no more creative mathematics, working on a general European language (Interlingua) and on the history of mathematics.

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