The finite-infinite interplay is central in the human thinking, from ancient philosophers and mathematicians (Zenon, Pythagoras), to modern mathematics (Cantor, Hilbert) and computer science (Turing, Goedel). Recent developments in mathematics and computer science suggest radically new answers to classical questions such as: Does infinity exist? Where does infinity come from? How can we reconcile the finiteness of the human brain with the infinity of ideas it produces? Well-known authors from around the world, many of them architects of the mathematics and computer science for the new century, contribute to the volume. While mathematical in spirit, contributions have many connections with computer science, cognitive science, linguistics, philosophy, physics, biology and semiotics.