Applications: Mathematical Proofs and Automated Reasoning 253 Giuseppe Peano (27.8.1858–20.4.1932) was an Italian mathematician, one of the founders of mathematical logic, best known for the axiom system, now called Peano Arithmetic, that he proposed Peano was born and grew up in a farm in north Italy He graduated from the University of Turin in 1880 and then took up a position and worked there for most of his career as Professor in Mathematics Peano was strongly influenced by the logical system developed by Frege and, along with Russell, did a lot to popularize and develop further his ideas In 1889 Peano published his famous axioms, which defined the natural numbers in terms of sets Peano’s axioms, where “number” means “natural number,” are as follows Zero is a number The successor of any number is another number There are no two numbers with the same successor Zero is not the successor of a number (Axiom of Induction) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S Much later in 1930 Kurt Gödel proved his First Incompleteness Theorem, showing not only that Peano’s system of axioms is incomplete but that it cannot be extended to a complete system in any “reasonable” way, so that the axioms are effectively (algorithmically) recognizable Another mathematical result for which Peano is known is his construction of a space-filling curve – a continuous curve filling the entire unit square, an early example of a fractal – as a counterexample to the claim that a continuous curve can be enclosed in an arbitrarily small region Since 1892 Peano embarked on an extremely ambitious project called Formulario Mathematico, intended as an encyclopedia of all mathematical formulae and theorems expressed in a symbolic language Many symbols that Peano used in the Formulario are still in use today His monumental work, completed in 1908, had a strong influence due to its rigorous exposition and modern style, but Peano’s use of it for teaching was not liked by his students and colleagues In 1903 Peano started developing an international language called Latino sine flexione, later also called Interlingua, using Latin vocabulary but with simplified grammar to make it easier to learn He published the final edition of Formulario Mathematico in Latino sine flexione, which was an added reason for its lack of popularity Eventually, Peano received wide recognition for his crucial contributions to modernizing mathematics and making mathematical notation and reasoning more rigorous, based on mathematical logic