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This paper challenges some of the common assumptions underlying themathematics used to describe the physical world. We start by reviewing many ofthe assumptions underlying the concepts of real, physical, rigid bodies and thetranslational and rotational properties of such rigid bodies. Nearly allelementary and advanced texts make physical assumptions that are subtlydifferent from ours, and as a result we develop a mathematical description thatis subtly different from the standard mathematical structure. Using thehomogeneity and isotropy of space, we investigate the translational androtational features of rigid bodies in two and three dimensions. We find thatthe concept of rigid bodies and the concept of the homogeneity of space areintrinsically linked. The geometric study of rotations of rigid objects leadsto a geometric product relationship for lines and vectors. By requiring thisproduct to be both associative and to satisfy Pythagoras' theorem, we obtain achoice of Clifford algebras. We extend our arguments from space to includetime. By assuming that ct=l and rewriting this in Lorentz invariant form asc^2t^2-x^2-y^2-z^2=0 we obtain a generalization of Pythagoras to spacetime.This leads us directly to establishing that the Clifford algebra CL(1,3) is anappropriate mathematical structure to describe spacetime. Clifford algebras arenot division algebras. We show that the existence of non-invertible elements inthe algebra is not a limitation of the usefulness to physics of the algebra butrather that it reflects accurately the spacetime properties of physicalsystems.

Assumptions and Axioms: Mathematical Structures to Describe the Physics of Rigid Bodies