In this talk, we will first introduce the fundamental ideas of Constructive Mathematics through basic examples taken from ordinary mathematical practice and focus on its computational aspect. Secondly, we will review how the logical principle of the Excluded Middle, dating back to Aristotle, and, more generally, the concept of negation play a crucial role in distinguishing Constructive Mathematics from Classical Mathematics. Finally, we will give a non-technical overview of Gödel’s double-negation interpretation of arithmetic and present our new result, which generalises it to the “Minimalist Foundation”, a foundation for constructive mathematics designed in Padua by M.E. Maietti and G. Sambin.