Here’s a “quote garden”. As time goes on, some quotes will be added, moved around, removed.
Another motivation for preschemes comes from the possibility of constructing via schemes an explicit and meaningful theory of infinitesimal objects. This is based on the idea of introducing nilpotent functions into the structure sheaf, whose values are everywhere zero, but which are still non-zero sections. (Mumford, The Red Book of Varieties and Schemes, 1999, p.65. Added on May 8th, 2025)
Taniyama was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction and so eventually he got right answers. I tried to imitate him, but I found out that it is very difficult to make good mistakes. (Goro Shimura. Added on May 15th, 2025)