The title of the book may suggest that this is about numbers, but there are nu numbers in the book, at all. So the subtitle: "How modern mathematics reveals nature's secrets", is a better description of the content. Because the book is about "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" as Eugene Wigner formulated it back in 1960. Especially with physics, there has been a close and successful interaction with mathematics. But Farmelo explains that this is not so unreasonable and in fact it goes also the other way around since there is also "A Reasonable Influence from Theoretical Physics on Mathematics". Perhaps, there can even be a superstructure that has mathematics and nature as two of its realizations that we humans, with our limited intellectual capabilities, are able to experience, without yet understanding the superstructure, an idea that has been proposed before by Max Tegmark in Our Mathematical Universe.
Clearly both communities, physicists and mathematicians, have their own culture. Mathematics expands in the minds of mathematicians that are driven by abstraction and a mathematical result is true and remains true beyond discussion after it has been proved once and for all. The (traditional) physicists are driven in their urge to explain nature. They care somewhat less about rigour and their models are inspired by observation, and perhaps less by strict abstract deduction. Their models are accepted if they are confirmed by nature itself after appropriate experiments, but acceptance of the model is only guaranteed until it is contradicted by new or better experiments. The latter situation seems to have changed for current theoretical physics that has moved to the mathematical approach.
Since antiquity mathematics and physics have developed in parallel. Even mathematics developed and progressed often driven by practical physical problems. Yet, Greek philosophers already discussed whether mathematics was created by humans or provided by nature and left for humans to be discovered. Farmelo sketched in the first half of the book this on-off relationship between mathematics and physics up till about the 1970's. He tells the history by staging the people who have contributed to the major steps in the evolution of both mathematics as well as physics.
Of course Newton and the scientists of the Enlightenment who envisioned a mechanical clockwork world. It was promoted by Laplace that if all the details of the current state age given, then this would allow to predict the future perfectly. Much of this resided on an atomic idea about the world consisting of particles subject to forces.
The electromagnetic theory of Maxwell introduced the concept of a field. His theory is condensed in the Maxwell equations named after him, but written down by Heaviside. The "beauty" and symmetry in the equations were a source of inspiration for later developments.
Gravity, the source of inspiration for Newton, coupled to a geometric vision, and thought experiments, were the instruments used by Einstein to develop his special relativity theory, only to be confirmed afterwards by practical experiments. At first he was not convinced that physics required advanced mathematics but he had to abandon this idea when he got in trouble for the development of his general relativity theory. Early twentieth century were turbulent times, shaking the foundations of both physics and mathematics. This was promptly followed by a steep increase in our understanding of the world.
Heisenberg and Schrödinger developed quantum mechanics, but it was Dirac who could link this new model, which inherently involved uncertainty, with classical world view of Laplace. He endorsed Einstein's revised view that mathematics is unavoidable for the development of physics. But then, during the war and in the post war period came, what Farmelo calls, "The Long Divorce" where mathematics and theoretical physics each went their own way for about two decades. However, physicists in search for their unifying theory got stuck and were confronted with a zoo of subatomic particles. Feynman made some progress, and Yang and Mills tried to generalize the symmetry of the Maxwell equations, but Freeman Dyson in his 1972 talk to the AMS pointed to the missed opportunities because the physicists were not aware of the most recent developments in mathematics and mathematicians were not interested in physics. Until in November 1974 (hence called the November Revolution) the psion (J/ψ meson) was discovered, a particle that lived much longer than other elementary particles, and gauge theory became the common interest of physicists and mathematicians, also because of the Atiyah-Singer theorem which had shown the power of differential geometry in explaining subatomic quantum mechanics. The Standard Model was realized in the 1970's.
This is where the first part of the book ends, surveying about 3 centuries from Newton till the Standard Model. The second half deals with the 4 decades that follow. Veneziano had written down the formula forming the model for string theory on a napkin already in 1968. It was however abandoned because quantum chromodynamics and quantum gravity (the quantum mechanical approach to study gravity near black holes) had stolen the hearts and minds of physicists. But strings were later reinstalled as the road to take for a Theory of Everything. String theory introduced extra dimensions because supersymmetry is the only way to extend the symmetry between space and time as in Einstein's special relativity theory. Farmelo continues to illustrate the intense interaction between theoretical physicists and mathematics. They mutually helped each other to make progress, with Witten, Deligne, Seilberg, Penrose, Arkani-Hamed and many others as main contributors. However the Large Hydron Collider (LHC) of CERN didn't provide the many particles that were predicted by the theory, the detection of the Higgs boson in 2012 being the last success.
However the state of affairs have brought mathematicians and theoretical physicists closer together than ever before. They are collaborating in the new emerging field of mathematical physics. In the last chapter Farmelo concludes his arguments defending what he has illustrated in this book: it is predestined and the fate of mathematics and physics to work together. He even makes some predictions about ideas that will stand the test of time like space and time are not fundamental but are aspects of a more fundamental concept. He also believes that supersymmetry will be verified experimentally thereby affirming the beauty of mathematics to be basic. And he has a few more like those. So with this book he contradicts Sabine Hossenfelder who in her book Lost in Math complains about the state of affairs that physics is at a dead end caught up in theoretical imaginations remote from reality and just because this vision happens to be mainstream, it absorbs all the research money. Who is right? Time should bring the answer, but in my opinion it doesn't look like it will come in the very near future, despite what mathematicians and physicists may think or hope for.
It is remarkable that this book, that is from the first till the last page about mathematics and mathematical physics, has no formulas (well almost none, I counted five very simple ones and that includes E = mc²). Farmelo does not go into technical details in the sense that he avoids confusing the reader with technicalities. If that reader does not know the exact meaning of the terms (gauge theory, quark, gluon,...) then it does not really harm the basic story that he wants to tell and it does not hinder reading on. He keeps the reader hooked. This alone is a tour de force. Farmelo's style is very entertaining, describing the moments and the circumstances when it was realized that some breakthrough had been found. This is only possible because he interviewed the people involved or he himself was a witness of the events in the second half of his book. He also frames the time and the setting by referring for example to the fact that some physics event took place "in the year that the Beatles produced their first LP" or "when nearby a large group of music lovers flocked together" (referring to Woodstock), or "a few weeks after Obama was inaugurated". He discusses throughout the book the, sometimes subtle, interplay between the mathematics and the physics, and he is as generous about mathematics as he is about physics. I also appreciated how he analyses the important lectures of Dyson, Witten, and others where they made some important statements about the state of affairs. He claims that it is the book he has been writing since his childhood, and I can believe that. A recommended read, and Hossenfelder can be a comparable complementary read to keep the balance.