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Objects change their properties continuously over time, yet we treat them as the same object. What makes an object at one time identical to an object at another time despite such qualitative changes? This puzzle about diachronic identity concerns temporal continuity and the criteria for object identity over time.
Examine the three-dimensionalist and four-dimensionalist approaches. Consider cases of gradual change (growing, aging) versus sudden change. Work through the role of spatiotemporal continuity.
That persistence requires unchanging properties. That three- and four-dimensionalism necessarily conflict. That memory or psychological continuity is the only basis for persistence.
Your prerequisite work on persistence and change established the basic puzzle: things change over time, yet we treat them as the same thing. You may also have encountered it through personal identity—what makes you the same person you were as a child, when nearly every atom in your body has been replaced and your beliefs, memories, and desires have substantially changed? Persistence through change generalizes this puzzle from persons to all objects, and the solutions on offer organize around two fundamentally different pictures of what it means for an object to exist in time.
The first picture is three-dimensionalism (or endurantism): an object is wholly present at each moment of its existence. When you look at a tree in spring and again in autumn, you are seeing the very same tree—not a different temporal stage of it, but the complete tree existing at different times with different properties. The challenge for three-dimensionalism is explaining how one and the same object can have contradictory properties: the leaf is green in spring and brown in autumn, but the same leaf cannot be both green and brown. Endurantists resolve this by indexing properties to times—the leaf is green-at-t1 and brown-at-t2—which are compatible because they are relativized to different moments.
The second picture is four-dimensionalism (or perdurantism): objects are extended through time just as they are extended through space, and they persist by having distinct temporal parts at each time. The leaf at t1 and the leaf at t2 are different temporal stages of one four-dimensional spacetime worm. There is no contradiction because the green stage and the brown stage are literally different objects (temporal parts) of the same extended whole. This resolves the problem of change elegantly, but it multiplies entities and raises the question of what the four-dimensional whole really is if it is never encountered all at once.
The Ship of Theseus thought experiment cuts to the heart of the debate. If you replace planks of the ship one by one until every original plank is gone, most people say it is still the same ship—spatiotemporal and functional continuity seems sufficient for persistence. But then someone reassembles all the original planks into a ship: which is the true Ship of Theseus? Your answer reveals what you think the persistence criterion is. Spatiotemporal continuity favors the repaired ship; material composition favors the reconstructed one. The thought experiment shows that persistence conditions depend on what kind of thing we are individuating—which connects back to your earlier work on personal identity, where psychological continuity, not physical composition, was the dominant criterion for persons.
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