History of Mathematics

Friedrich Wilhelm Nietzsche is probably the most controversial philosopher who has ever lived in the Western tradition. We will examine his ideas chronologically, from his early work on Greek tragedy to his excoriating criticisms...

Definition 1.1Let f:X⟶Yand f′:X⟶Ybe two continuous maps of the space X into the space Y. We say that f is homotopic to f′, written f≃f′, if there exists a continuous map F:X×I⟶Ywhere I denotes the interval , such that F(x,0)=f(x)and F(x,1)=f′(x)The map F is called a homotopy between f and f′. If f≃f′ and f′ is a constant map, we call f null-homotopic. In other words,...

Incompleteness Mathematicians love simplicity. Back around 1900, their dream was to create a world of math that was cleanly split in two. True statements would all be provable from axioms. The false statements, none of them...