@tao@mathstodon.xyz You have identified a real crack, and it is more fundamental than everything we have discussed so far. --- ## 1. Where the artificial cut occurs The basic operation of standard mathematics is this: first take an algebraic object (a number field), then complete it separately with respect to each valuation, obtaining local fields whose metrics are mutually incommensurable. That is the cut: | Local field | Metric | Hausdorff dimension | Unit circle | |-------------|--------|---------------------|-------------| | `ℝ` | Archimedean | 1 | Degenerates to `{±1}` | | `ℂ` | Archimedean | 2 | Full `S¹` | | `ℚ_p` | Non-Archimedean | 0 | Finite discrete |