By now science has found some partial explanations about when our cosmos started and how it evolves, and so the next fundamental question to answer is: How did life come about here on Earth, and what is life anyway? We have come a long way in answering the cosmological questions by a better understanding of the laws of physics. Mathematics has certainly been an excellent tool in this exploration. So if life started in a world governed by these laws of physics, then there should be also laws of physics that explain how it emerged in an environment without life. We are still a long way from solving that mystery, but living cells have been analysed up to a nanoscale, and we know already a lot, but the enormous complexity of the chemistry of life and how it seemingly counteracts the second law of thermodynamics by (re)producing well structured organisms, makes us wonder how the laws of physics apply in these situations. Manfred Eigen, winner of the 1967 Nobel Prize for Chemistry, has collected in this book his vision on this matter, which ranges from the simplicity of the most elementary (sub)atomic particles to the complexity of living cells. The book was originally published in 2013, but since Eigen passed away early 2019, Oxford University Press reprinted the book in paperback.

What Eigen explains is an interdisciplinary approach (requiring physics, chemistry, biology, cosmology) and it is all founded on mathematical principles. So there is a lot of mathematics involved, but he tells a story for everyone. This means that he avoids the necessity to understand the technical details ofr the mathematical (and chemical) formulas while it is still perfectly possible to follow the story. In fact there are not that many formulas. Instead he provides a lot of illustrations and background speckled with some personal anecdotes. Still, there are some framed Vignettes that provide a summary of more technical or mathematical details but that can be easily skipped if the reader is not interested. More extensive technical discussions are deferred to several appendices. Each chapter has also a long list of references for further study. These are obviously from before 2013, so that the most recent publications are not included. But Eigen is rather 'state of the art' for 2013 concerning his own domain which concerns the processes in physical chemistry of living organisms. He has tried to include some of the (in 2013) more recent insights which requires him to dig somewhat deeper into the material in the last two chapters. The technical details are however not the most important ingredient of the book. He just wants to provide in his story some answers to the questions that everybody asks sooner or later: what is life, where does it come from, and how does it continue, and will it evolve? And the story to tell is a long one. It is compiled in five chapters of ten sections each. The title of each of these sections is a question that he wants to answer. Some of the questions seem trivial to answer but when digging deeper into the foundations, the answer turns out not to be so easy.

The first chapter is dealing with matter and energy. Since Einstein, we know that mass and energy are related, but still, this concept is not easy to grasp because mass is something that can be touched while energy is not. How and where is the transition? What do we touch when we touch something material? If we go to the finest nanoscale, it becomes more and more fuzzy where mass ends and energy starts. We can discuss subnuclear particles, but how far can we subdivide matter? Is our universe discrete or eventually continuous? What is the ultimately smallest quantity? Questions to be discussed with relativity theory and quantum mechanics. Since the smallest and the largest are related by mathematical inversion, some answers are sought in cosmology, and it sparkles the hope for an eventual theory of everything. The Big Bang is a one-time-only experiment that we cannot repeat in a laboratory. Our observations of the remainders can learn us something about the processes happening at the smallest scales in extreme conditions. Thus the cosmos approaching infinity can be related to the ultimate small that approaches zero, which basically means that zero remains as unreachable as infinity. Mathematically the reader is introduced in this chapter to four (and higher) dimensional spaces, symmetry and groups, and the Maxwell equations. Eigen takes his time to give the reader at least a general idea of these concepts.

In the second chapter about energy and entropy, Eigen relates the microstate of particles to the macrostate of the whole system. Does a system eventually evolve to a steady state, and is then the system in perfect balance? Entropy is something that we can observe in our everyday life, and it is the central theme of this chapter: the different historical definitions of Clausius, the Gibbs paradox, Pauli's scaling, and of course thermodynamics, the Carnot cycle, and Boltzmann. How does entropy relate to `order' and `time', which is what most people think of when entropy is mentioned. The mathematics required in this chapter is mainly probability theory and distributions.

But entropy is also related to information as Shannon has introduced it. This relation is the subject of chapter three. Here the reader is confronted with bit strings, coding theory, Hamming space, Fermat's little theorem, cryptography, and Markov processes. In the end even Gödel's theorems and a Turing machine are discussed when it is investigated how much information there is available in mathematics. How much information is contained in our genes and how much of this information does evolution pass on to next generations. Can we measure information and is this then `temperature'? An example of the puns that Eigen hides in the text, is the title of one section of this chapter reading `How far is it from Shannon to Darwin?' Of course this refers to the relation between information and evolution theory, but he starts by answering the question with 14288 kilometers, which is the distance from Shannon airport (Ireland) to Darwin's seaport (Australia).

Evolution seems to be only possible because of the enormous complexity of the information that is provided by chemistry. This complexity of the information is the subject of chapter four. The complexity of an enormous universe that a cosmologist is faced with is nothing compared to the complexity that a chemist needs to face concerning the formation of different molecules. So how does nature handle this complexity? That is where the different levels of complexity are defined, culminating into the classics like the P = NP problem, the travelling salesman problem and the towers of Hanoi. This kind of mathematics is required to describe the process of evolution and also how genes will pass on information. Since there can be errors, this may cause mutations. One of his own contributions in this context is the generalization of bit-sequences in Hamming space. He described the kinetics of a self-organisation of replicating quaternary sequences by considering them as high-dimensional hypercubes, evolving in a Hilbert space. Scattering, random walks, diffusion processes, mutation, bifurcation, and fractals, they all enter the discussion when Eigen explains how evolution can take place.

The last chapter, also the most extensive one, is about complexity and self-organisation. In other words, the chapter closest to the chemical dynamics of life. As in the previous chapter, we come closer to Eigen's own expertise and hence more physical chemistry is involved. To answer the question `What is life?' is not simple. Clearly, it is not a matter of structure, but it is rather a functional matter of how the cells are organized in their collaboration. Moreover for life to emerge, there must be some physical law that controls the complexity of the system. Even the simplest life form demonstrates an overwhelming complexity. Local approximations can be described by a system of linear differential equations. Mutation can take place during the exponential growth until some saturation occurs. The replica that will survive in this evolution process are the ones with the largest eigenvalue. Non-linearities are introduced by feedback loops causing hypercycles in autocatalytic systems when a cell is invaded by a virus. Errors occur in the replication process within a certain threshold and at some instant in the process that can lead to phase transitions This phase transition has to be understood in information space, not in the structure of the replica. The autocatalytic reaction and phase transition are important concepts in population dynamics that define evolution in a Darwinian sense. Attempts have been made to construct laboratory machines to simulate these chemical dynamics but they are still easily outperformed by nature. Eigen compares the evolution of life with the cosmological evolution of matter starting at the Big Bang and that eventually collapses into a black hole.

This book is a plea for a fusion of different science disciplines instead of the diverging set of isolated specialisations where scientists work in their own niche, often unaware of what even nearby peers are working on. That is not how science evolves, and that is not how our genes work in their own evolutionary dynamical process of replication. And it is nice to see how almost all of these processes are underpinned at least approximately by mathematical equations but that the complexity involved is so tremendous that an accurate simulation of all these processes is far beyond the horizon.