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You need formal axioms to do precise and rigorous mathematics, but you don't need any axioms to see and manipulate geometric patterns in our heads. We seem to n
by 4ad 12y ago
You need formal axioms to do precise and rigorous mathematics, but you don't need any axioms to see and manipulate geometric patterns in our heads. We seem to naturally posses this ability (probably because our brains use a lot of pattern-matching).
We already have the geometry manipulation program installed in our head, and maybe this program is even based on Euclid's (or Hilbert's) axioms. But our brains just run this program, they can use this abstraction without having been thought geometric axioms beforehand.
- 3rd3 12y agoNo. Geometric intuition is widely believed to be only a very rudimentary native skill. There appear to be structures in our brain that allow us to think spatially [1], but infants are generally unable to perform even very simple geometric reasoning. Our environments are full of geometric theorems however, that our brains become acustomed to within the first few years of our lifes. [1] http://en.wikipedia.org/wiki/Grid_cell http://en.wikipedia.org/wiki/Grid_cell