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Abstra t Is there an alternative mathemati s? In parti ular, does intuitionism yield an essentially new approa h that annot be spe i ed within lassi al mathemati s? The intended informal meaning of intuitionisti logi Int was given in the 1930s by the Brouwer-HeytingKolmogorov semanti s whi h understands intuitionisti truth as provability. Moreover, Kolmogorov (and later Godel) suggested interpreting Int via lassi al provability and thus providing a meaningful semanti s for Int independent of intuitionisti assumptions. Natural attempts to formalize this semanti s met serious diÆ ulties related to Godel's in ompleteness phenomenon. In this le ture we will talk about re ent advan es in this area that have bridged the in ompleteness gap and provided an adequate formalization of the propositional Brouwer-Heyting-Kolmogorov semanti s based on lassi al provability.