Petr Vopenka's Reflections on Bolzano

For Petr Vopenka, Bolzano was not only an important figure in the history of mathematics, but also a source of inspiration and a locus of confrontation. Almost in every one of his works, Bolzano takes an essential place, especially when Vopenka talks about set theory. He dedicated a whole book Podivuhodny kvet ceskeho baroka (The Wonderful Flower of the Czech Baroque, Praha, Karolinum 1998) to Bolzano's post humous Paradoxy nekonecna (Paradoxes of the Infinite, 1851). There he returns to the native grounds of modern science, namely the discussions and quarrels between different Christian sects in the first four centuries, followed after more than a thousand years by Protestantism. His pilgrimage leads us to Cervantes, the Spanish mystics and the Spanish baroque. All this goes together with a discussion of the infinite according to Augustine, Thomas Aquinas, Giordano Bruno, Galilel, Rodrigo de Arriaga and oth ers. While Medieval scholasticism showed reticence towards actual infinity, modern science (Gauss, Cauchy, etc.) has refused it without hesitation and has returned to Aristotelian potential infinity. Bolzano, educated in the 18th century neo-scholastic tradition in Bohemia, was the first important mathematician to have introduced actual infinite collections and sets into mathematics. In section 20 of his Paradoxes, he stated a characteristic property of infinite sets: their reflexivity (the possibility to put into 1-1 correspondence a set with some of its infinite subsets). Vopenka interprets this text as a general collapse, namely the possibility to reduce all Cantorian cardinalities of infinity to one: that of the natural numbers. He also shows, why Bolzano sought his infinite numbers not among Cantor's cardinalities, but among infinite sums of real numbers.