Explicit versus tacit knowledge in history of mathematics: the case of Girolamo Cardano

The solution formula of the cubic equation confronted Cardano with the problem of studying expressions of the form a±b√(-1), an obstacle that prevented him from giving a really general solution to equations of third and fourth degree. Before investigating foundational questions on the nature of “radices sophisticae” and their geometric representation, Cardano provides the rules for operating arithmetically with these strange expressions, but he immediately encountered the problem of establishing which sign they had, and he was not able to solve this problem. Bombelli reconsidered the problem of the sign of these expressions introduced the signs "più di meno'' and "meno di meno'', for which he established appropriate rules of multiplication. On this basis, Bombelli founded an arithmetic of Cardano's sophistical quantities allowing him to make sense of the irreducible case of cubic equations. Thus, in the late Renaissance, both in the case of Cardano and Bombelli "complex numbers'' were solutions to problems, making sense of the solution formula for cubic equations, even if their formal properties were still unclear.