uname reports on the OS you're running, not the hardware. So you'll need to look at a hardware information program to learn the exact nature of your processor. The major desktops have information tools, but for cli, check out lshw.
Anyway, if you have a 64bit cpu (assuming the common amd64-compatible type), then it should have a set of 32bit emulation extensions, allowing you to run a 32 bit OS instead if you want. But the reverse is not true. You can't run a 64bit OS or program on a 32 bit processor.