## Friday, May 31, 2019

### Leibniz on infinite downward complexity

Leibniz famously thinks that ordinary material objects like trees and cats have parts, and these parts have parts, and so on ad infinitum. But he also thinks this is all made up of monads. Here is a tempting mental picture to have of this:

• Monads, …, submicroscopic parts, microscopic parts, macroscopic parts, ordinary objects.

with the “…” indicating infinitely many steps.

This is not Leibniz’s picture. The quickest way to see that it’s not is that organic objects at each level immediately have primary governing monads. There isn’t an infinite sequence of steps between the cat and the cat’s primary monad. The cat’s primary monad is just that, the cat’s primary monad. The cat is made up of, say, cells. Each cell has a primary monad. Again, there isn’t an infinite sequence of steps between the cat and the primary monads of the cells: there might turn out to be just two steps.

In fact, although I haven’t come across texts of Leibniz that speak to this question, I suspect that the best way to take his view is to say that for each monad and each object partly constituted by that monad, the “compositional distance” between the monad and the object is finite. And there is a good mathematical reason for this: There are no infinite chains with two ends.

If this is right, then the right way to express Leibniz’s infinite depth of complexity idea is not that there is infinite compositional distance between an ordinary object and its monads, but rather than there is no upper bound on the compositional distance between an ordinary object and its monads. For each ordinary object o and each natural number N, there is a monad m which is more than N compositional steps away from o.