  The theories of quantum mechanics and relativity dramatically altered our
understanding of the universe ushering in the era of modern physics. Quantum
theory deals with objects probabilistically at small scales, whereas relativity
deals classically with motion in space and time.
  We show here that the mathematical structures of quantum theory and of
relativity follow together from pure thought, defined and uniquely constrained
by the same elementary "combining and sequencing" symmetries that underlie
standard arithmetic and probability. The key is uncertainty, which inevitably
accompanies observation of quantity and imposes the use of pairs of numbers.
  The symmetries then lead directly to the use of complex "$\surd\mathord-1$"
arithmetic, the standard calculus of quantum mechanics, and the Lorentz
transformations of relativistic spacetime. One dimension of time and three
dimensions of space are thus derived as the profound and inevitable framework
of physics.
