Four dimensions, same problem
A paper on Sider's four-dimensionalist treatment of the Ship of Theseus. Written as a state and a transition function, the four-dimensional view turns out to need exactly what the three-dimensional causal unity account needs, so it inherits that problem rather than escaping it.
Written for a PHL233 paper assignment, submitted April 17, 2024. The assignment set a passage and asked two questions in 500 words: what is the author’s argument, and what is a reasonable objection to it. Rewritten here from the original, with the citations turned into inline links and a looser register; the argument is unchanged, and the paper as submitted is available as a PDF. The passage is Sider, “The Four-Dimensional Picture”, from “Let us return to the question of whether The Ship of Theseus” to “The only remaining question is the merely conceptual one of which of these spacetime worms counts as a ship.” The copy I worked from is here.
On the four-dimensional view an object is a sequence of temporal stages, each made of spatial parts. The Ship of Theseus becomes a series of stages, each one the planks standing in the harbour at some moment.
Several sequences of stages, several spacetime worms, can be drawn through the same planks, and they end in different places. Sider’s move is that choosing among them is a matter of what we mean by “ship”. He works through two candidate meanings, the Replacement Worm and the Same Plank worm, each picking out its own final stage, and allows a third answer where the question is Indeterminate because more than one worm is equally good.
The metaphysical question, which one is the Ship of Theseus, has become a conceptual question about what a ship is. Sider grants that this does not dissolve the paradox, and claims the conversion itself is the achievement: the metaphysical difficulty is gone.
Restating the view
There is not much here that is deductively valid, so the passage is best read as an inference to the best explanation. That is the claim I want to test.
Take an object to be an initial state , a set whose elements are the temporal parts that constitute it, together with a transition function , which takes a stage and returns the next one. That is equivalent to what Sider describes, and it puts the weight somewhere useful. The transition function is not a trivial object, and it is doing all the work. Whether it can be defined at all for the Ship of Theseus is exactly the open question.
Causal unity, written the other way
Now a three-dimensional view: an object is a causally unified body of matter. That account works only if “causal unity” can be spelled out, which is the standard complaint against it.
Suppose a predicate returning true when the parts in are causally unified. From it a transition function follows immediately: let return the causally unified stage sharing the most elements with its input. Now suppose instead we have . A predicate follows just as fast: let return true when matches the current stage, meaning the output of the last call to .
The two are codependent. Assume either is achievable and the other is constructed in a line.
What the trade actually buys
So a three-dimensional account built on causal unity and a four-dimensional account built on a transition function stand or fall together. If the causal unity problem is bad enough to reject the first, the same problem is sitting inside the second under a different name.
Sider’s conversion of the metaphysics into a question about the word “ship” looks like progress because the difficulty has been moved rather than answered. Deciding which spacetime worm the planks trace is deciding which successive stages belong to one object, and that is the causal unity question in different notation. The four-dimensional picture does not resolve the puzzle. It relabels it.
How formalizing gave it away
I started by taking Sider at his word: the ship is a spacetime worm, which worm it is depends on what we mean by “ship”, and the metaphysics has turned into a question about a word.
To test whether that is really the best explanation I wrote his view out as a state and a transition function, and the difficulty concentrated there at once. Nothing in the four-dimensional picture says why one stage is followed by this next stage and not another.
Then the thing the paper is built on. The three-dimensional view has something of the same shape in causal unity, and the two might be one problem in two notations. The interdefinability of the predicate and the function follows from that, and so does the answer to the complaint that one of them is stable while the other moves with time.
Four days before the paper was due I put it to a friend in a group chat. The reply was that defining the transition function is the question of how to understand the ship, asked in different words, so nothing has been solved.