It is widely believed that mathematics provides the fundamental basis for physics. On the contrary, it is argued here that pictures of real objects moving in real space provide the proper basis for physics, and that mathematics merely provides quantitative models for calculating the dynamics of these objects. Such models may distort or even hinder the development of new physics, particularly if a consistent physical picture is lacking. This is discussed in connection with quantum mechanics, which discarded realism in favor of mathematical abstraction almost a century ago. A realistic, spin-quantized wave picture of quantum mechanics is presented that avoids the paradoxes and abstractions of the orthodox quantum theory. Quantum indeterminacy stems from an inappropriate application of a statistical point-particle model to extended soliton-like wave packets. Quantum transitions are continuous, rather than the abrupt transitions of the Hilbert space model. Quantum entanglement is an artifact of mathematical constructions incompatible with local realism. These are not merely matters of philosophical interpretation; several experimental implications are presented. It is time to remove the mathematical blinders that have prevented consideration of realistic quantum pictures.