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Archimedes of Syracuse

Time is substantival and continuous — the background against which physical processes (floating, sinking, equilibrium, motion along spirals) unfold. Archimedes's physics is static (statics, hydrostatics) rather than dynamic, so time is present but rarely foregrounded. Deterministic: physical laws hold necessarily. The Sand Reckoner shows him conceiving cosmological time-scales (the Aristarchean heliocentric universe) with equanimity.

Archimedes of Syracuse

Substantival, three-dimensional, flat (Euclidean geometry throughout), local. Archimedes works with finite spatial domains — the surface of a sphere, the volume of a paraboloid, the extent of a fluid — but the mathematical space in which these objects sit is implicitly Euclidean and unlimited. The Sand Reckoner estimates the size of the universe as finite but vast.

Archimedes of Syracuse

Substantival, finite, conserved. On Floating Bodies treats fluids as continuous matter with definite weight; On the Equilibrium of Planes treats solids as having centres of gravity and definite mass. Matter is local: forces act at definite points. Conservation is implicit: the fluid displaced equals the volume submerged; weight is neither created nor destroyed.

Archimedes of Syracuse

The mathematician-physicist who reasons from postulates to theorems and verifies by mechanical experiment. Embodied and active: Archimedes builds machines, tests propositions, and communicates results to correspondents (Dositheus, Eratosthenes). Metaphysical agency is unaddressed — Archimedes does not theologise; his gods, if any, are irrelevant to his physics.

Archimedes of Syracuse

Finite, substantival, conserved. The principle of the lever — "equal weights at equal distances balance" — is an implicit energy-conservation principle (no work is done in equilibrium). Buoyancy is a balance of forces. Reversible: raising and lowering a body in fluid are symmetric operations. Archimedes does not have the concept of energy, but his mechanics is entirely consistent with it.

Archimedes of Syracuse

Mathematical truths are substantival, universal, and conserved — they hold always and everywhere. The Method reveals Archimedes's heuristic process (balancing infinitesimal slices), showing that mathematical information has both a discovery-context and a proof-context. Continuous granularity: Archimedes works with continuous magnitudes, not discrete units.