Toposes, elephants and gorillas

Peter Johnstone's big book on toposes is called Sketches of an Elephant, the reason being a parallel with the story of the blind men and the elephant: each blind man felt a different part of the elephant and had a different idea of what the elephant was like. With toposes too, each investigator understands only a part of the nature of toposes, so different people will explain them in different ways (here is my own experience of this). In fact, Johnstone's book started off as a collaboration between three "blind men" - Johnstone himself, Ieke Moerdijk and Andy Pitts.

I'd like to explain why I imagine a topos as being like a gorilla.

When I was younger, gorillas had a reputation for being very fierce and hostile. Explorers would proudly boast of the heroic struggle by which they overpowered the creatures and brought them back to a zoo to live in a cage - if they condescended not to shoot them first. When the gorilla died it would be stuffed, mounted in a threatening pose with its teeth bared and displayed in a museum to frighten the children.

Toposes are often viewed in a similar way, scary mathematical objects to be left to those with a natural taste for danger and discomfort a long way from civilization.

In 1979, David Attenborough's series Life on Earth for BBC television included the famous sequence of him playing in the most gentle way with a family of gorillas in Rwanda. As Attenborough himself said, "It seems really very unfair that man should have chosen the gorilla to symbolise everything that is aggressive and violent, when that is the one thing that the gorilla is not - and that we are." If you get to know gorillas in their natural environment, and gain their trust, then you begin to appreciate their gentleness and can play with them.

What makes me say toposes are like that?

The "natural environment" for toposes is a matter of logic. Each topos comes with its own logic, and its own rules of mathematical deduction, and the rules of ordinary "classical" mathematics do not apply in every topos. It is possible to display toposes in the world of classical mathematics, and to believe that is exactly what any right-thinking person would do, but in many ways that is alien to the toposes. If you can confine yourself to the rules of topos-valid logic, or even to those of geometric logic (harder and not always possible, but when it works it rewards your patience), you will see toposes at their most warm and playful.