Poincaré's conventionalism establishes that geometric properties like finiteness are underdetermined by experience, so unboundedness is a structural feature independent of metric size.
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Underdetermined by experience(explains why we have choices in geometry)
When our observations and experiments can't force us to believe one explanation over another—multiple different theories could all fit the same data equally well.
conventionalism(Philosophy of language debate in Plato's Cratylus)
The view that the correctness of names is determined by social consent and agreement rather than by natural resemblance or description
finiteness(Kleene 1967)
The property that an algorithm consists of a finite set of instructions and terminates in a finite number of steps.
unboundedness(Riemann distinguishes unboundedness (an empirical certainty) from infinitude (not entailed by unboundedness).)
The property of space having no edges or boundaries, such that one can travel indefinitely without reaching an end — distinct from infinite spatial extent.