Caption: An ancient Greek hoplite.

Features:

1. The Aristotelian universe was a finite bounded sphere. But it was also eternal---unbounded in time. Aristotle (384--322 BCE) had arguments for these conclusions---but we won't go into them.

2. Aristotle explicitly concluded that beyond the celestial sphere of the stars (the outermost celestial sphere of Aristotelian cosmology) there was nothing---NOT even empty space---just nothing (e.g., Furley, D. J. 1987, The Greek Cosmologists, p. 137).

   The physical celestial sphere of the stars mutated into the modern imaginary, but very useful, celestial sphere.

3. But what if you were at the celestial sphere of the stars and thrust outward with a spear. What would happen?

   Aristotelian cosmology offers no explanation to the spear paradox.

4. Some (e.g., Archytas (428--347 BCE) and Lucretius (c.95--c.55 BCE) in De Rerum Natura) concluded that there should be space beyond the celestial sphere of the stars---and the spear or whatever would just pass through and there was no boundary to outer space.

5. Even some dyed-in-the-wool Aristotelians were troubled by the spear paradox.

6. If there is space beyond the celestial sphere of the stars, why NOT an infinite universe.

   The atomists---most prominently Leucippus (first half of 5th century BCE, Democritus (c.460--c.370 BCE), Epicurus (341--271 BCE), and Lucretius (c.95--c.55 BCE)---did posit an infinite universe.

   An infinite universe is hard to imagine, but a bounded finite Aristotelian universe seems even less imaginable.

7. Since 1917, general relativity has rescued the finite universe as a possibility.

   If the spatial geometry is curved, then the universe could be the surface 3-sphere, a "sphere" in 4-dimensional mathematical space that has a 3-dimensional surface. A 3-sphere is 3-dimensional n-sphere (or hypersphere) is also called a glome.

   A 3-sphere universe would be finite, but unbounded. The spear paradox would be avoided.

   The 3-sphere universe is possible in theory, but we don't know if the universe is actually like this. At present, it seems NOT.

   If it were, we could roam in the glome as well reave in The Glebe---just a little Ottawa in-joke.

8. A 3-sphere Aristotelian universe was, of course, beyond the conceptions of Greco-Roman antiquity.