The Calculus of Identity

I've been thinking of boats and rivers lately.

Particularly of Heraclitus' river and Theseus' boat.

They both deal with identity. Heraclitus claims that you can not step into the same river twice because the waters are constantly flowing. The river holds different water, and is thus, a different river.

On Theseus voyage, the young Athenians he carried with him replaced every decaying board with new and strong timber. By the time they returned to home port from their long journey, they had replaced every board leaving philosophers to ponder--was it the same ship that had sailed out, or was it an entirely different one?

To answer whether they're the same ship or river, let's find out what identity is. From a philosophical perspective, identity, is whatever makes something definable or recognizable. Heraclitus argued that change of components made it impossible for us to step into the same river. Several centuries later, Newton and Leibnitz Calculus. I feel that Calculus is an apt perspective to take on the problem since it is "the mathematics of change" and I think change is the core variable to study when looking at identity.

I agree that, for the most part, we identify entities based on the sum of their components. A forest is its trees, a river is its waters. Or precisely:

x = all parts of x

Heraclitus argues that after time, the identity is broken:

x =/= x'

(Where x' is x after some time has passed.)

I argue however, that the river remains the same if you observe it as it goes through sufficiently small, continuous intervals. Precisely:

1 ~ 1.000000001 ~ 1.000000002

(Where ~ means approximately equal to.)

The closer the change in x gets to 0, the closer we get to being able to say x and x' are the same. Our question really becomes, "When is the change in x insignificant?" 1 meter of water flow? 1 millimeter? 300 kilometers? So the ability to step in the same river twice depends on your perspective of the changing water in the river.

Applied to Theseus' ship: If we stand on the dock as the Athenians leave in the first ship (ship y) and the next time we see the ship sail in, it's made entirely of new timber (ship y'), then it would seem to us that the ship is not the same (that is, y =/= y'). However, if we embark along with Theseus on ship y, we see the ship change very slowly and continuously. It would seem that we've been riding the same ship the whole time.