Martin Krieger, USC – Mathematical Foundations

Martin Krieger

Mathematics is a universally important topic.

Martin Krieger, frequent AM contributor, provides a fascinating analysis of mathematics as the important field of research it is.

Martin H. Krieger is professor of planning at the Sol Price School of Public Policy at the University of Southern California. He is trained as a physicist, and has taught in urban planning and policy at Berkeley, Minnesota, MIT, Michigan, and USC. His nine books are about mathematical modeling, environmental policy, and about theories of planning and design. He has been a fellow at the Center for Advanced Study in the Behavioral Sciences and at the National Humanities Center. He is a Fellow of the American Physical Society.

Mathematical Foundations

Mathematics provides natural science with a language that casts a very wide net.

Mathematics demands rigor and clarity, and that is not merely for show. Such precision often reveals vital features about the actual world that we would otherwise not have appreciated.

We know from experience that ordinary matter is stable. Classical mechanics could not prove this, but with quantum mechanics scientists have been trying to prove that ordinary everyday matter composed of electrons and nuclei, attracted by their opposite charges, is stable, and won’t contract or explode.

The mathematical proof by Dyson and Lenard, in 1967, not only shows what we know already. It goes much deeper. In the process of rigorously proving the stability of matter, it turns out that what matters is that electrical forces are said to shield each other, that is, the force of nuclei’s positive charges and the electrons’ negative charges cancel out at a distance.

Gravity has no such negative charges, so it will cause collapse in heavy enough stars, and that is why there are black holes and supernovae.

The Pauli Exclusion Principle says that no two particles can be in the same state. Such particles are called fermions, and electrons are fermions. The Pauli principle accounts for the Periodic Table of the Elements, the build up of electron orbitals as we have heavier and heavier nuclei.

And furthermore, the Dyson-Lenard proof discovered that if electrons were not fermions, ordinary matter would not be stable.  

Thus, the mathematical analysis shows us what is most crucial about our everyday world—electrical forces shield each other, and electrons are fermions.

Physicists might be said to use the recipe books of mathematicians to create delicious dishes that account for our world.