Jane McDonnell has a PhD in theoretical physics (U. of Cambridge, 1987) and a PhD in Philosophy (Monash U., 2015). In this book she places the philosophy of science and mathematics in a broader metaphysical framework. She has three sources that triggered her ideas: Wigner's well known quote asking about the `unreasonable effectiveness of mathematics in physics' (1960), Plato's Timaeus (360 BC), in which the nature of the physical world is explained, and Leibniz's Monadology (1714) where he explains his philosophy. First she provides a speculative framework that is inspired by set theory, then combines this with elements from the monadology and from quantum theory to arrive at her synthesis that she calls quantum monadology. I am sure this requires some explanation.
Pythagoras' view was that everything is number, which resounds in the contemporary view of Max Tegmark who claims that our world is just a mathematical structure. In McDonnell's view, the Pythagorean vision should be combined with 'mind' to explain our world. In the first chapter she examines Wigner's paper, and the quest of physicists for a theory of everything (TOE). It is the intention to arrive at a consistent Pythagorean metaphysical framework which will be the proper explanation to Wigner's problem. This leads to claims such as 'there is nothing special about the applicability of mathematics in general, but there is certainly something curious about the applicability of mathematics to cutting-edge fundamental physics'.
This is explained by the growing role of symmetry in theoretical physics. A short survey follows how symmetry was introduced and how it gave rise to quantum chromodynamics (QCD). It is clear that here mathematics took the leading role and that particles are proposed that are not observable anymore. QCD is almost a Pythagorean theory, although string theory or M-theory and multiverses pose new challenges. There is a layered hierarchy as more symmetry is required in the theories to explain physics at growing energy levels and this symmetry leads to simplification of the theory. Either there is a 'fundamental layer' which will be the TOE, or there is a nesting: the theory explaining next energy level contains the theory at a previous level as a special case. Thus also in the latter case there is some 'limit theory' that should be fully symmetric and hence extremely simple to explain everything. This is an explanation of Wigner's mystery. Mathematics and fundamental physics both rely on the same principles that can be experienced in the physical world and in our mind.
If mathematical principles underly a physical reality, should we then accept a Pythagorean view of mathematics, i.e., accept the existence of a unique mathematical truth. Here pluralists and universalists have different views. Clearly McDonnell is in favour of the latter. According to Gödel's theorems there are statements that cannot be proven to be true or false. Second order logic may resolve this, but it will have its own unresolved statements, etc. So, each time one can accept one of two possibilities, true or false, and this may lead to the pluralist view of (infinitely) many different mathematics that may exist independently. All this is explained in a very technical chapter discussing set theory, the Zemelo-Fraenkel theory completed with the axiom of choice (ZFC) and the axiom of infinity. Besides other other independence results that are, or are not, compatible with the axiom of choice, the hope for those adhering the Pythagorean idea is to extend the ZF(C) set theory with additional axioms that will eventually lead to a unique kind of mathematics. Several possibilities are described. McDonnell seems to hope that the the winner in this pluralist-universalist controversy in set theory will come from Hugh Woodin and his Ω-logic.
In the next chapter we learn about McDonnell's own speculative metaphysics that lean towards a realistic structuralism. There is the One (in math the empty set, in physics the Big Bang physics) and there is the Many (in math the universe of sets, in physics the physics of the cooled universe) and in between is Being (the mind) and Becoming. The One and the Many are not reachable, and thus not real, but everything happens in between these. So, if mathematics is the structure of being, there must be one, true mathematics, and the physical reality is its mental interpretation. This universal mathematics is not the mathematics that we are used to. There is indeed some human mathematics that can be fictional, but there may exist rational beings that have a deeper insight in this true universal mathematics where no absolute undecidable propositions exist.
A more concrete proposal if this idea is given in the penultimate chapter. Two paradigms are discussed: Leibniz' monad theory (rooted in classical physics and following the structure of set theory) and the consistent histories quantum theory (CHQT) of Griffiths, Gell-Mann, Hartle and Omnès. This CHQT is a generalization of the classical Copenhagen interpretation which allows an almost natural interpretation of quantum cosmology because probabilities of the histories are assume additive. Eventually the two paradigms are melted into what McDonnell calls quantum monadology. In her proposal a monad is described by a consistent history that exists in the Being. It starts from One: 'I exist' and builds up a history by projecting quantum collapses onto Being, thus learning concepts (a monad has a mind) while building up its history towards a universal theory that will explain its whole world. Depending on stochastic outcomes, monads may develop different histories with different theories. This raises a question whether an optimal evolution exists and whether or not this is chosen by some transcendental entity. Are we living in the best of all possible worlds? Her answer: yes we do.
The concluding chapter gives and overview of the theory with some additional remarks and gives links with other philosophers.
This Pythagorean view is certainly not mainstream and therefore, this book is filling a gap in the literature. Of course, on this subject, it is difficult to propose something that is absolutely true and everything is necessarily speculative. Much will depend on what the next advances in the foundations of mathematics as well as physics will bring. One might expect that, given the background of McDonnell as a theoretical physicist, the quantum theoretical part is the hardest part of the book, but it turns out that the most difficult technical part is the one on set theory. Some of the technical stuff has been moved to an appendix, together with another one about problems with plenitude. When you are an ordinary mathematician, not specializing in the philosophical foundations, I do not think this book will influence in any way what you are doing every day. You will probably loose solid ground after set theory gets its metaphysical interpretation. If you are indeed involved in the philosophy of mathematics or more general philosophy, it is certainly a source that will raise controversy. I can imagine you feel the urge to put your heels in the sand or even to dig in and take a stand preparing for an intense polemic. But that is not exceptional. Most of these philosophical explanations provoke controversy.