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…and in turn “geometric algebra” is the physicist’s name for a subset of the even more general mathematicians Clifford algebra (which of course is a subspace of

by Jenz 3y ago

…and in turn “geometric algebra” is the physicist’s name for a subset of the even more general mathematicians Clifford algebra (which of course is a subspace of the tensor algebra over the underlying vector space…)

jiggawatts 3y ago
Clifford Algebra is basically just Geometric Algebra, except with complex numbers shoved into where there ought to be real numbers... because no Mathematician could ever resist doing so. As a programmer / physicist, the analogy I use is that if I had to represent an 8-dimensional parameter as an array, I would write: double foo[8]; but a mathematician would be unable to resist writing: complex foo[4]; They're the same, but the latter is more complex (hah!) for no real (haha!) benefit.
- aleph_minus_one 3y ago
  > They're the same, but the latter is more complex (hah!) for no real (haha!) benefit. When you consider problems in real geometry, this is plausible. But on the other hand, not all mathematics is real geometry. For some applications in mathematics, the complex-geometric perspective is more natural. P.S. Of course p-adic (and in particular 2-adic) geometry is even cooler than anything real or complex ... ;-)
  - jiggawatts 3y ago
    It's type confusion caused by duck-typing. Just because can you can add, subtract, multiply, and divide complex numbers doesn't mean they're a natural substitute for real numbers in all scenarios. Sometimes you have '2n' real numbers, not 'n' complex numbers. If mathematicians designed computer software, their data structures would look like: struct Person { char name; complex name_height; }
    - eigenspace 3y ago
      > It's type confusion caused by duck-typing No it's not. This is such a typical and braindead HN take.
    - aleph_minus_one 3y ago
      > doesn't mean they're a natural substitute for real numbers in all scenarios. I claimed the exact opposite: whether real or complex numbers as basic objects are more natural depends on the problem that you consider.

by Jenz 3y ago

jiggawatts 3y ago

Clifford Algebra is basically just Geometric Algebra, except with complex numbers shoved into where there ought to be real numbers... because no Mathematician could ever resist doing so. As a programmer / physicist, the analogy I use is that if I had to represent an 8-dimensional parameter as an array, I would write: double foo[8]; but a mathematician would be unable to resist writing: complex foo[4]; They're the same, but the latter is more complex (hah!) for no real (haha!) benefit.

aleph_minus_one 3y ago
> They're the same, but the latter is more complex (hah!) for no real (haha!) benefit. When you consider problems in real geometry, this is plausible. But on the other hand, not all mathematics is real geometry. For some applications in mathematics, the complex-geometric perspective is more natural. P.S. Of course p-adic (and in particular 2-adic) geometry is even cooler than anything real or complex ... ;-)
- jiggawatts 3y ago
  It's type confusion caused by duck-typing. Just because can you can add, subtract, multiply, and divide complex numbers doesn't mean they're a natural substitute for real numbers in all scenarios. Sometimes you have '2n' real numbers, not 'n' complex numbers. If mathematicians designed computer software, their data structures would look like: struct Person { char name; complex name_height; }
  - eigenspace 3y ago
    > It's type confusion caused by duck-typing No it's not. This is such a typical and braindead HN take.
  - aleph_minus_one 3y ago
    > doesn't mean they're a natural substitute for real numbers in all scenarios. I claimed the exact opposite: whether real or complex numbers as basic objects are more natural depends on the problem that you consider.