This is a historical review of the evolution of mathematics in physics and philosophy. The author and publishers have used a catchy title for the book to enhance its marketability. I was looking for a philosophical analysis of the basic laws (and equations) of physics and how it influenced the thought on physical reality. There is no discussion of how consciousness fit in within all this. If God used mathematics to create the laws of physics, then how did he create consciousness? Did he use mathematics to link consciousness with the physical reality? What are the roles of dimensionless physical constants such as structural constant and the value of Pi that God created? These questions are not fully explored.
There are nine chapters in the book, and a significant part of the book gives a historical account of the work of early Greek philosophers leading up to the work of modern philosophers, mathematicians and physicists. There is fair amount of discussion on the theory of curves, analytical geometry, Cartesian coordinate system, Pythagoras theorem, the evolution of calculus and differential equations.
The author proposes that mathematical theories have two aspects; active and passive. In active theories, laws of nature are formulated in applicable mathematical terms. The terms include mathematical entities, relations, and equations that were developed with an application of mind for the topic under consideration. The researchers tend to perceive the similarities between the properties of the mathematical concepts and the observed phenomenon. One could conclude the theories were tailored to the observations (E.g., Newtonian Physics). The passive effectiveness refers to cases in which abstract mathematical theories were developed with non-intended applications for possible use in future models, such as knot theory, and Riemannian geometry. Invention are; calculus by Newton, and topological (geometrical) ideas in the context of string theory; or the application of Riemannian geometry in general relativity, and group theory in particle physicists are examples of mathematical discoveries. The accuracy and predictive power of mathematics are equally important. There are numerous examples for predictive power such as; prediction of antiparticles, Maxwell's prediction of waves associated with electrical and magnetic fields, prediction Bosons, and W particles by electroweak theory, and the quantum electrodynamics (QED) predicted the magnetic moment of an electron with a great accuracy.
Does mathematics has an independent existence from human mind or they have application beyond the context they were originally developed? Platonists view mathematics as discovery because it dwells in the abstract eternal world of mathematical forms. Some Platonists believe that mathematical structures are in fact a real part of the natural world. Max Tegmark of MIT states that the nature is mathematics, period. The answer to mathematics being "invented or discovered" question can therefore be gleaned from a careful examination of clues from a variety of domains. Since this physical world is entirely independent of humans, Tegmark maintains, its description must be free of any human conceptions. In other words, the final theory cannot include any concepts such as "subatomic particles," "vibrating strings," "warped spacetime," or other humanly conceived constructs. He concludes that the cosmos involves only abstract concepts and the relations among them. The author believes that math¬ematics is a combination of inventions and discoveries; the axioms of Euclidean geometry as a concept is an invention, just as the rules of chess are an invention. The axioms are supplemented by a variety of invented concepts, such as triangles, parallelograms, ellipses, the golden ratio, and so on. The theorems of Euclidean geometry, on the other hand, were by and large discoveries; they were the paths linking the different concepts. In some cases, the proofs generated the theorems mathematicians examined what they could prove and from that they deduced the theorems. Humans invent mathematical concepts and discover the relations among these concepts. Some empirical discoveries surely preceded the formation of concepts but concepts lead to theorems.
The limited explanatory power of mathematics in biology or medicine is a problem for mathematics to have universal role in physical reality. Because evolutionary biologists argue that the human evolution naturally selected them for survival since they had the best models of reality in their minds. Hence human logic was forced on us by the physical world through the process of natural selection.
The mathematics is effective in explaining the physical world because the natural world is not random; it has structure, organization and patterns, mathematics is a logically relevant. Atoms behave in precise mathematical ways when they emit and absorb energy. String theory (if proved correct) will prove that the universe is a geometrical structure and physical reality is mathematical. On a hypothetical note, if were contacted by aliens, communication could be a problem, but aliens will have the same laws of physics, and the common language would be mathematics.
1. Converging Realities: Toward a Common Philosophy of Physics and Mathematics
2. A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality
3. Reality's Mirror: Exploring the Mathematics of Symmetry (Wiley Science Editions)