I read book "A primer of infinitesimal analysis" John Bell. I was confused when I saw example with area under curve. In that example author mentioned about nondegenerate triangle of zero area. But triangle of zero area is degenerate triangle.
How is it possible? How to imagine what author meant in that example?
Thanks.
What makes this possible is intuitionistic logic. The law of excluded middle is not accepted. In particular, it cannot be proved that every number is either zero or nonzero. There is a gray area where neither can be proved. Nilsquare infinitesimals are in this gray area. Thus, if $\epsilon$ is a nilsquare infinitesimal of Smooth Differential Geometry (SDG), then one can prove that $\epsilon^2=0$ but one cannot prove that $\epsilon$ itself is zero (nor can one prove that it is nonzero, as mentioned above). Thus, by definition a square of side $\epsilon$ will have area $\epsilon^2=0$. If you divide such a square by a diagonal, you get a pair of right-angle infinitesimal triangles that look perfectly ordinary except that their area is zero :-)
It used to be thought that it is difficult to make infinitesimals rigorous, but there are now axiomatic approaches conservative over ZF where one has effective infinitesimals (the latter approach has nothing to do with SDG).