Both are unavoidable consequences of the geometry of spacetime. I remember Carl Sagan, in his book Contact, suggested that "God's signature" was hidden deep inside of numbers like Pi, in sequences of numbers that would form intelligible images to creatures using base-10 math.
"Pure" math still needs a framework. If there is nothing, the very concept of numbers cannot exist. No numbers, no maths.
That's actually a very great point of debate within mathematical philosophy. But it has nothing to do with the geometry of spacetime.
Hmmmm. A circle is defined as a planar set of points a given distance from a point on that plane. If out in space we built a station to shoot out 360 laser beams in a 360 degree pattern (all in a single plane from the reckoning of the station), and we considered the location of the leading pulses after, say, one week of travel, would that pattern form a circle, a geometric pattern wherein the length of the shape formed by all the endpoints smoothly joined would by equal to 2*pi*one light-week? Would the geometry of spacetime make any difference? Yes, it certainly could. If there was some non-uniform distribution of mass/energy within that one light-week radius, those light rays would've followed very different paths, some longer, some shorter, some straight, some curved. The resulting figure wouldn't necessarily be a nice, perfect circle. The circumference divided by the radius need not equal pi. But a mathematical assumption in defining the circle is that it is created in conditions that mirror flat spacetime perfectly. If spacetime isn't flat, the ratio of a circle's circumference to its diameter may not be pi. So, yes, pi is a product of pure math. But it is a product of pure math in a world where spacetime is flat.
Seems like you are assuming there is some objectively correct reference frame. From the perspective of the person firing our those photons they would seem to have traveled out in a perfect circle.
This is kinda what I'm envisioning... The red circle in the center is the station. The orange paths are the lasers. The green circle is a large gravitational mass. If the orange lasers pass near the mass, their paths curve toward it. Since they must propagate at the speed of light, that means they cannot reach the same extent that the lasers travelling through flat space can (the maximum extent is represented by the yellow circle). The endpoints of the lasers don't form a perfect circle because the lasers curved by gravitation traveled in ways other than directly outward and so can't reach the same extent the others did. It would even be possible to create a situation where the orange lasers crossed. Since, in an Einsteinian sense, the light rays map out the topography of space, this is a "circle" in that curved space. All of the endpoints are a fixed light distance from the station. In a flat space, all the orange lines would be straight and coincide with the yellow circle. So, for a circle to be drawn in this way, and to have its circumference be equal to its diameter times pi, it would have to be in flat space.
