A bronze lump exactly constitutes a statue at a time with all the same parts and location, yet they differ in identity conditions—the statue ceases to exist if melted down but the lump continues. This pressures theories of identity and composition, forcing choice between denying that constitution and identity differ, or explaining why they come apart.
Your prerequisites have given you two key tools. From the study of constitution versus identity, you know that two things can be constitutionally related — made of exactly the same matter, occupying the same space — without being identical. From mereology and composition principles, you know the questions around what makes parts compose a whole, and under what conditions two collections of parts constitute one object versus two. The material constitution problem is where these threads collide most dramatically.
Imagine a sculptor who shapes a lump of bronze into a statue of Hermes. At time T, the bronze lump and the statue share all their matter, every particle, and occupy exactly the same region of space. And yet — here is the puzzle — they appear to be two distinct objects with different persistence conditions. Melt the statue down, and the statue ceases to exist; the bronze lump continues. Chip a fragment off the statue, and the statue may survive but the original lump is gone — replaced by a smaller lump. Their fates can come apart even when they share all the same matter, which means they cannot be strictly identical. If the lump and the statue were the same object, they would have to share all their properties — including their persistence conditions — and they do not.
This creates what philosophers call the problem of coincident objects: two numerically distinct objects occupying the same region at the same time. This violates a widespread intuition — that space does not allow two material objects to fully occupy the same location simultaneously. We need to choose between abandoning that intuition or finding some way to explain why the lump and the statue do not really coincide. The major responses divide into three families. Temporal parts theories hold that objects are four-dimensional entities extended through time; the lump and the statue share spatial parts but differ in their temporal parts, so they do not fully coincide. Sortal-relative identity theories hold that identity is always identity relative to a sortal concept (statue, lump, organism), so asking whether the lump is the statue simpliciter is a category error. Eliminativist views deny that there is a statue over and above the lump — ordinary object talk about statues is a useful fiction grounded in lump-facts.
The identity-of-indiscernibles principle you encountered in prerequisites is directly relevant here. Leibniz's principle states that if A and B share all their properties, then A = B. If the lump and statue share all properties, they must be identical. But they seem to differ — at minimum, the statue has the property "would cease to exist if melted down" while the lump lacks it. If these are genuine properties, the indiscernibility of identicals (the reverse of Leibniz's law) entails they are not identical. Defenders of strict identity must argue that modal and sortal properties like "would survive melting" are not genuine first-class properties that individuate objects — which requires a substantive and contested metaphysical commitment.
The puzzle generalizes far beyond bronze statues. Organisms and the masses of cells that compose them, rivers and the water molecules that constitute them, persons and the bodies that constitute them — all face analogous questions. The statue case is philosophically vivid because we can stipulate exactly when the statue comes into existence and goes out of existence, making the identity conditions maximally concrete. But the underlying issue — the relationship between material composition and object identity — is one of the deepest and most unresolved problems in contemporary metaphysics.
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