philosophical thought on the problem of how to mathematically representat space:
We are all thought that space is like a grid, taken to the limit where the voxel size goes to zero. At first this seems correct. But upon closer look, there is some doubt:
- Computers have shown us all, via pixelated 2D graphics and minecraft-style 3D graphics, that shapes actually composed on a grid seem unnatural (to the point of being comical)
- Cantor's diagonalization proof has shown us that the number line of real analysis is uncountable. It involves not one, but two infinities, with all of the paradoxes that that brings. Further investigation into these paradoxes in foundational studies of set theory and logic have not been able to resolve these paradoxes to general satisfaction yet, and this failure can be taken as evidence that perhaps the paradoxes are a fundamental and inescapable consequence of the real number line. The consensus viewpoint is that this implies that physical reality is deeply paradoxical, but another possiblity is that perhaps the number line does not, in fact, represent physical reality.
- Perhaps higher math does offer alternative conceptualization of the nature of space, for example, in the subject of topology. I need to learn more about this. And what other alternatives are there?
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