This is a marvelous coffee table book that brings a cultural history of Western civilization with in particular the intertwining of mathematics, philosophy, religion and the work of artists that were inspired by mathematics. It is argued that philosophical and religious views have great influence on how mathematics is conceived. Following the braided threads of mathematics, philosophy, art and to some extent also religion, the reader is guided through history from page to page and from illustration to illustration. To bring the story from the prehistory till the present day is an enormous enterprise, and thus the 560 pages of large format (the book's measures are 24.1 x 30.7 x 4.3 cm and weighs 3 kg) are barely sufficient. Certainly the visual arts like paintings, sculpture, and architecture, are easily represented by wonderful pictures, but also music, poetry, and even dance, and other less obvious connections are represented. But clearly every example is just picking a drop from an ocean. It is the presentation of the idea, the doctrine, the philosophy, that prevails throughout the text. It might be expected that ancient history is illustrated with ancient art, which holds true to some extent, but it can also be modern art that is inspired by an ancient historical event. There is some chronological order in the chapters, but many chapters are organized around some concept and sweeps though its own time line.

A brief introduction to Babylonian calculus and Egyptian pyramids is followed by the Greek philosophers. Platonism assumes that abstract objects like mathematical concepts do actually exist in a reality outside of our sensible world. Democritus and Lucretius on the other side had a mechanical atomistic view that was picked up later by Descartes after Kepler and Galileo found the mechanical laws that governed the interaction between the Sun and the planets, the 'particles' of our solar system. Euclid's *Elements* of geometry was dominant and geometry was applied in astronomy, the oldest scientific discipline where mathematics played an important role. Islam and the Arabs brought algebra, equations, and the decimal system into Western culture, and of course they reintroduced the forgotten Greek philosophy. What the *Elements* were for the Western civilization was the *Jiuzhang suanshu* for the East. During the Middle Ages mainly architecture produced the marvelous cathedrals, but mathematics basically got stuck with the Greek approach until the Enlightenment, when Western science was boosted in a Scientific Revolution. The early cosmological models of Kepler were still based on Greek geometrical ideas but soon evolved to proper experiment-based mechanical laws. For a thorough study of the Scientific Revolution, one might want to read *The Invention of Science. A New History of the Scientific Revolution* by D. Wootton. This period is well known to be influenced by the Catholic Church as illustrated by the fate of Kepler living in a protestant Germany and Galileo living in a Roman Catholic Italy. All of these elements were essential to shape modern Western science, art and culture.

The second chapter is taking a closer look at the concept of **proportion**. This can be linked to the discovery of perspective. For paintings, this was of course a major step, and so it is well represented in this kind of book. On the other hand it also relates to the famous Golden Ratio that presumably was dominant in classic architecture and was taken up again during the Renaissance with for example Da Vinci's Vitruvius Man. Mathematically, the Golden Ratio results from ratios of Fibonacci numbers. Gamwell finds it a misconception that the Golden Ratio was considered during antiquity as an ideal of beauty and perfection in art. It was suggested to be God's fingerprint that the ideal human body had this Divine Proportion all over. This is not true, says Gamwell, since Darwin's *Origin of Species* illustrates that the ratios in the human body do change with time.

Chapters 3-10 are each circling some central concept. The first one deals with the concept of **infinity**. There are Newton and Leibniz inventing calculus which requires the infinitely small to arrive at continuous changes, there are the irrational numbers with an infinity of non-repeating digits and there is of course Cantor's transfinite arithmetic. This concept of infinity and the absolute brings us again into the realm of religion, free will, statistics (the law of large numbers), and even psychology.

**Formalism** is the keyword for chapter 4. This applies to mathematics with Hilbert and Bertand Russell as the protagonists, but also to art where it represents the emphasis on abstract 'structure' rather than 'imitation' or 'meaning'. In mathematics this entailed a detachment from 'reality' leading to, for example, non-Euclidean geometries. Formalism was so influential that one started analyzing and formalizing music and linguistics. The reaction was constructivism as preached by Brouwer. Gamwell illustrates these elements with several tendencies in the Russian and Polish art scene of the early twentieth century.

The next issue is **logic**. Boolean algebra, Lewis Carroll's paradoxes, and Frege's predicate calculus, and of course the *Principia Mathematica* by Whitehead and Russell are considered. The artistic counterpart is that for example the human body is reduced to its essence (Henri Moore), and paintings are seen as a collection of elementary parts (Cézanne), and flat, emotionless characters populate the literature (Eliot, Joyce, Woolf).

The **intuitionism** of Brouwer and his topology is framed in the psychology debate of his time and the writings of Frederik Van Eeden. This chapter explores the Dutch contributions. As Freudian psychoanalysis swept The Netherlands and abstract art was emerging. Steiner and Blavatsky were promoting theosophy. The key figure here is Piet Mondrian whose art and essays, like Brouwer's philosophy, were rooted in German Romanticism of Goethe.

The microscope allowed to discover crystals, whose **symmetries** were studied with group theory. Klein had his Erlangen program where he wanted to classify all transformations that left a geometric structure invariant. This idea of invariance is also underlying Einstein's relativity theory: describing the world from any four-dimensional space-time reference framework. Later this evolved in an ongoing search for a theory of everything. Symmetry of course plays an important role in art as well, in particular the decorative art. Escher and the Alhambra immediately pop to your mind, but it is also present in Bach's compositions. Symmetry was observed in nature and biology and the human sensitivity to the perception of symmetry was studied in *Gestalt Psychology*. Gamwell links this to the Concrete Art in Switzerland of the 1930s and 40s.

Chapter 8 returns to Germany in the post World War I period. Fatalism was the general teneur of the *Neue Sachlichkeit* when Gropius founded the Bauhaus school. In art the opposing groups were formed by Enlightenment rationality and Romantic expression. Two groups of scientists attempted to develop a unifying language for science and mathematics: there was Planck, Einstein and others developing quantum physics and Mach, Carnap and others, known as the Vienna Circle, whose credo was logical positivism.

Self reflection on mathematics made Wittgenstein and Gödel throw over Hilbert's tower of mathematical certainties. Self reflection was also a tendency in art. Magritte's *Ceci n'est pas une pipe* or Escher's self-drawing hands are examples. But also other surrealists painted impossible objects, and Language Art included letters, numbers and sentences in paintings.

Turing and the construction of the first computers are definitely associated with Gödel's incompleteness theorems. Music was formalized (Schoenberg) and later computerized. Computers were used to generate art and to analyze and understand art. Gamwell at this point makes the link with Japanese traditions and how this influenced the New Age movement.

So far, the discussion has been mainly concentrating on 'Germanic' culture, somewhat neglecting the Latin (French, Italian, Spanish) and the Anglo-Saxon (UK, USA) evolution. This is corrected in the remaining chapters. After the disaster of World War II, the French group Bourbaki embraced Hilbert's formalism and the literary equivalent Oulipo produced constraint writing, and everywhere artists expressed their believe in order and rationalism by creating abstract and optical art. Britain meets Process Art where 'the making of' is the main focus, much more than the finished object. Constructivism and Concrete Art was exported to South America, and in North America artists imported abstract and algorithmic processes to produce their *objet d'art*. The idea of reducing the essence in mathematics to a set of axioms was also reflected in American Minimal Art.

Since 1950 computers were taking over. They were used to generate proofs in mathematics (the four color problem), but fractal geometry was also useful to create realistic graphics, and via digital photography and computer animation they enter the artist's studio. Knots, graph theory, and network analysis appeared in mathematics and know many applications, and they all were used by artists in their work. Origami is just one example.

In a last chapter, Gamwell reflects on Platonism in a Postmodern era. After the 1950s, the mathematical center shifted from Europe to the United States. Current research is still asking the old questions: How do we know the world? What is everything made of? Mathematicians and theoretical physicists struggle with the standard model and supersymmetry, different interpretations of quantum physics exist, fusion experiments show that we are made of star dust, the belief in a general truth is crumbling. In *The Mathematical Experience* (1981) Davis and Hersch state that mathematics is an ideology, a religion, or an art form, dealing with human meanings, intelligible only within a context of culture.

This is an overwhelming book, very well researched and documented. It touches upon so many fundamental questions that philosophers, scientists, mathematicians and artists have asked since antiquity. But yet it guides the reader smoothly through all these competing visions and theories without becoming dull or getting lost in abstraction. This is the history of Western civilization with particular interest in art and mathematics, illuminating and instructive, and all wrapped up in a rich, colorful, and fancy book.