The Pythagorean World

In this book the theoretical physicist and philosopher Jane McDonnell offers a metaphysical framework to explain why mathematics is unreasonably effective in physics. She starts with an analysis of Eugene Wigner’s famous essay on the subject and argues that the core issues are: the existence of the laws of nature and our minds’ ability to understand them; the use of formal mathematics to discover things about the quantum world; the need for deep mathematics to describe fundamental physics; and the asymptotic nature of our quest for knowledge. She goes on to describe a philosophy that is an extension of the idea from the Pythagorean school that everything is number.

She says that she has developed “a metaphysical framework which is a version of idealism. It is a view of the universe as an abstract mind: a mind constituted by monads and their states … The structure contains all of mathematics, all possible forms, and physical reality condenses out by a process of self-actualization in thought.”

McDonnell argues that the unreasonable effectiveness of mathematics in theoretical physics has become evident through the growth of our understanding of symmetry principles in nature and the expanding collection of mathematical concepts and tools developed to extend that understanding. She believes that this gives credence to the final dream of a theory of everything in a form with mathematical beauty, simplicity and logical inevitability. This is reinforced in her mind by the notion that physical theories applicable at different scales are grounded in an ultimate theory that is applicable at the highest energy levels. Unification at the highest energy levels and symmetry breaking at lower and lower levels would lead to the physical phenomena we observe.

If mathematical principles underlie physical reality, a Pythagorean view may demand that we accept the existence of a unique mathematical truth (a “one true mathematics” as she calls it.) Gödel’s incompleteness theorem raises a potential obstacle here, and independence results further suggest that a pluralistic view of mathematics may be more realistic. McDonnell devotes a long, very technical chapter to this question. She believes that a universalist view — a one true mathematics — is correct, and she bases her belief on Hugh Woodin’s Ω-logic, and on his Ultimate L version of set theory.

From this speculative version of set theory we move on to an even more speculative metaphysics. According to McDonnell there is the One (the empty set in mathematics and the Big Bang in physics) and the Many (the universe of sets in mathematics and the physics of the cooled universe). In between is Being (the mind) and Becoming. The One and the Many are not reachable, and thus not real, and everything happens in between these two. Mathematics is the structure of Being and physical reality is its mental interpretation.

Next she discusses what she calls quantum monadology, a fusion of Leibnitz’s monad theory and a more contemporary quantum theory of consistent histories. She proposed that a monad is a consistent history that exists in the Being. It begins with “I exist” and builds up by projecting quantum collapses onto Being, thus learning concepts (a monad has a mind) while building up its history toward a universal theory that will explain the whole world.

Not everyone is convinced of the unreasonable effectiveness of mathematics in physics, let alone that mathematics is the foundation of reality. Richard Hamming argues in a 1980 Monthly article that, while we may be impressed at the wide applicability of mathematics, we might be deluded. He notes that we see what we look for, we select the kind of mathematics that we use, and that science and mathematics are capable of answering comparatively few questions. One might naturally ask, for example, why mathematics is not unreasonably effective in biology.

While this is a thought provoking book, its speculations and arguments often strain credibility. The metaphysical interpretation of set theory is a serious obstacle, and the introduction of Leibnitz’s monadology in the context of quantum mechanics only makes it worse.

One practical issue with the book is that page references in the index are mostly incorrect. Somehow the index must have been created for an earlier version of the text. In any case, it is exceptionally difficult to track back to find specific items. 

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.