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Peano was the founder of symbolic logic and his interests centred on the foundations of mathematics and on the development of a formal logical language. Peano studied mathematics at the University of Turin and joined the staff there in 1880, being appointed to a chair in 1890. In 1889 Peano published his famous axioms, called Peano axioms, which defined the natural numbers in terms of sets. In 1891 he founded Rivista di matematica , a journal devoted mainly to logic and the foundations of mathematics.In 1886 Peano proved that if f(x,y) is continuous then the first order differential equation dy/dx = f(x,y) has a solution. The existence of solutions with stronger hupothesis on f had been given earlier by Cauchy and then Lipschitz. Four years later Peano showed that the solutions were not unique, giving an example.
Peano introduced the basic elements of geometric calculus and gave new definitions for the length of an arc and for the area of a curved surface. He invented 'space-filling' curves in 1890, these are surjective mappings from [0,1] onto the unit square. Hilbert, in 1891, described similar space-filling curves.
He produced an axiomatic definition of the natural number system and showed how the real number system can be derived from these postulates.
Peano was also interested in universal, or international, languages and created the artificial language Interlingua in 1903. He compiled the vocabulary by taking words from English, French, German and Latin. It has been further developed by Alexander Gode. Peano, however, considered his work in mathematical analysis to be of greater significance.
Although Peano is a founder of mathematical logic, the German mathematical philosopher Gottlob Frege (1848-1925) is considered the father of mathematical logic.
