Our digression from Fermat's Last "Theorem" about the global geometry of the universe got me thinking. Euclidean space has zero curvature, hyperbolic space has negative curvature and elliptical space (the sphere) has positive curvature. Treating curvature as a parameter that can vary, as it increases in the positive direction, the sphere comprising space gets smaller, as it increases in the negative direction, space "undulates" more, and at zero it's flat. It seems remarkable to me that of all the real number values the curvature of our universe could be, it's either zero, or has an absolute value very close to zero.
There's the theory of the Fine-tuned Universe, which states,
The fine-tuned Universe is the proposition that the conditions that allow life in the Universe can only occur when certain universal fundamental physical constants lie within a very narrow range, so that if any of several fundamental constants were only slightly different, the Universe would be unlikely to be conducive to the establishment and development of matter, astronomical structures, elemental diversity, or life as it is presently understood. The existence and extent of fine-tuning in the Universe is a matter of dispute in the scientific community. The proposition is also discussed among philosophers, theologians, creationists, and intelligent design proponents.
Martin Rees formulates the fine-tuning of the Universe in terms of the following six dimensionless constants,
N = ratio of the strengths of gravity to that of electromagnetism;
Epsilon (ε) = strength of the force binding nucleons into nuclei;
Omega (ω) = relative importance of gravity and expansion energy in the Universe;
Lambda (λ) = cosmological constant;
Q = ratio of the gravitational energy required to pull a large galaxy apart to the energy equivalent of its mass;
D = number of spatial dimensions in spacetime.
Perhaps the curvature of the universe is another one of these dimensionless constants, or perhaps those constants are somehow dependent on the curvature.
If an uncountable infinite number of universes exist that have curvatures that range along the real numbers, then it is statistically impossible that our universe should happen to have zero, or nearly zero curvature. Proponents of "Intelligent Design" would cite that as evidence that our universe was "designed" by an outside agency. However, I think that the "Intelligent Design" argument is intellectually lazy - just because we currently lack understanding of a phenomena, we shouldn't just "throw up our hands" and give up.
A more interesting line of thought would be that the geometry of the universe, manifesting in its curvature, permits some or all of those other dimensionless constants to attain those fine-tuned values that permit the conditions of life to exist and evolve to the point where conscious beings can ask and answer these questions, which is an example of the anthropic principle.
In astrophysics and cosmology, the anthropic principle is the philosophical consideration that observations of the physical Universe must be compatible with the conscious life that observes it. Some proponents of the anthropic principle reason that it explains why the Universe has the age and the fundamental physical constants necessary to accommodate conscious life. As a result, they believe it is unremarkable that the universe's fundamental constants happen to fall within the narrow range thought to be compatible with life.