In a series of lectures given in 2003, soon after receiving the Fields Medal for his results in the Algebraic Geometry, Vladimir Voevodsky (1966–2017) identifies two strategic goals for mathematics, which he plans to pursue in his further research. The first goal is to develop a ‘‘computerised library of mathematical knowledge,’’ which supports an automated proof-verification. The second goal is to ‘‘bridge pure and applied mathematics.’’ Voevodsky's research towards the first goal brought about the new Univalent foundations of mathematics. In view of the second goal Voevodsky in 2004 started to develop a mathematical theory of Population Dynamics, which involved the Categorical Probability theory. This latter project did not bring published results and was abandoned by Voevodsky in 2009 when he decided to focus his efforts on the Univalent foundations and closely related topics. In the present paper, which is based on Voevodsky's archival sources, I present Voevodsky's views of mathematics and its relationships with natural sciences, critically discuss these views, and suggest how Voevodsky's ideas and approaches in the applied mathematics can be further developed and pursued. A special attention is given to Voevodsky's original strategy to bridge the persisting gap between pure and applied mathematics where computers and the computer-assisted mathematics play a major role.