Thus, abstract mathematical entities we think we are familiar with can nevertheless surprise or disappoint us. They can pop up unexpectedly in new guises, or disguises. They can be inexplicable, and then later conform to a new explanation. So they are complex and autonomous, and therefore by Dr Johnson’s criterion we must conclude that they are real. Since we cannot understand them either as being part of ourselves or as being part of something else that we already understand, but we can understand them as independent entities, we must conclude that they are real, independent entities.
