Peano, Giuseppe (1858–1932)

Peano curve

Peano curve.

Giuseppe Peano (pronounced "piano") was an Italian pioneer of mathematical logic and the axiomatization (see axiom) of mathematics. He studied at the University of Turin, later becoming a professor there. In 1889, he published his first version of a system of mathematical logic in Arithmetices principia, which included his famous axioms of natural numbers, now known as Peano's axioms. Two years later, he established a journal, the Rivista, in which he proposed the symbolizing of all mathematical propositions into his system. The project, which became known as the Formulario, was his chief focus for the next fifteen years. Peano's ideas profoundly influenced those of Alfred North Whitehead and Bertrand Russell. He also devised the international language Interlingua, still occasionally used.


Giuseepe Peano


Peano curve

The Peano curve is the first known example of a space-filling curve. Discovered by Peano in 1890, its effect was like that of an earthquake on the traditional structure of mathematics. Commenting in 1965 on the impact of the curve in Peano's day, N. Ya Vilenkin said: "Everything has come unstrung! It's difficult to put into words the effect that Peano's result had on the mathematical world. It seemed that everything was in ruins, that all the basic mathematical concepts had lost their meaning."


Today, the Peano curve is recognized as just one of an infinite class of familiar objects known as fractals. But at the end of the nineteenthth century it was an extravagant, completely counterintuitive thing; indeed, it was something that had been believed impossible. Writing of Peano's result in Grundzüge der Mengenlehre in 1914, Felix Hausdorff said: "This is one of the most remarkable facts of set theory."


Originally, the Peano curve was derived purely analytically, without any kind of drawing or attempt at visualization. But the first few steps in drawing it, as shown in the diagrams, are easy enough, even though the finished product is unattainable in this way – and totally unimaginable. To fill the unit square, as the Peano curve does, without leaving any holes, the curve has to be both continuous and self-intersecting.