# [FOM] Number theory proof mentioned by Frege

Vaughan Pratt pratt at cs.stanford.edu
Thu Apr 17 16:24:45 EDT 2008

```John Baldwin wrote:
> the quaternions lose commutativity, and with more losses you get degree 8
> and I think 16.  Then no more.

What one gives up here is by no means uniquely determined.  You're
presumably referring to the octonions, which retain the existence of
multiplicative inverses of nonzero elements in exchange for giving up
associativity of multiplication, and the sedenions, which give up
inverses as well.

For those for whom associativity is important and who are willing to
accept a few zero divisors (as anyone accepting sedenions must), there
are Clifford algebras, very popular with physicists.  Those over the
reals are the main ones---Clifford algebras over the complex numbers
amount to a more simply structured subfamily of the real ones since both
are representable as square real matrices.

Finite-dimensional Clifford algebras are generated by p generators with
square 1 and q generators with square -1.  The case (p,q) = (0,1) gives
the complex numbers while (0,2) gives the quaternions, (2,0) gives the
2x2 real matrices, and (1,1) is isomorphic to (2,0).

The only commutative Clifford algebras are (0,0) (the reals), (0,1) (the
complex numbers), and direct sums thereof (so-called by analogy with
vector spaces, they're really direct products of associative algebras)
which includes (1,0) as the direct sum of (0,0) with itself, aka the
hyperbolic numbers.

Vaughan Pratt
```