The book has the enticing title Space, Time & Stuff. At first sight it seems to be a contribution to the philosophy of space and time. However, the first impression soon evaporates. By the author's own admission, this book ended up as a text for philosophers with a taste for its rather heady mixture of mathematical physics and fundamental metaphysics and who regard this approach as a valuable exploration of epistemic possibilities. The author invites the reader to see ‘much of this book (…) as an attempt to reduce all of physics to geometry’ – a claim, which will raise eyebrows amongst researchers who either work on statistical mechanics, thermodynamics and cosmology or investigate the foundational problems in these areas. (Readers who are interested in wider philosophical views on space and time or the question of arrows of time will be disappointed since these topics are left aside.) The book claims to be about the structure of space and time, or rather space-time, but many other (loosely) related topics are also discussed (metaphysical issues in relativity theory, quantum mechanics, topology, gauge theories, calculus).

Chapter I assesses Julian Barbour's idea that time has no structure – a view, which is found to be viable but implausible. In this chapter the author also discusses Newtonian and relativistic physics, especially in terms of the distinction between the topology and metric of time. From the point of view of the general reader, and barring some reservations mentioned below, it is one of the more accessible parts of the book. Especially the discussion of the topology and metric of time is useful. Chapter II is devoted to a discussion of the geometric structure of space-time, i.e. the question whether the topology of space-time is separable – ‘what happens in space-time is entirely determined by what is the case at each point in space-time’ (37) – or whether it only gives rise to what the author calls ‘neighbourhood separability’. Arntzenius argues in favour of neighbourhood separability: a ‘topology is separable in the sense that given any division of the total space into open neighbourhoods, the topology of each of the neighbourhoods determines the overall topology’ (47). In simpler terms this view implies that the world, including the classical domain, has ‘irreducibly holistic properties’. Rather strangely, David Lewis is introduced as ‘perhaps the best-known proponent’ of space-time substantivalism (rather than, say, John Earman). Perhaps this attribution reveals the author's metaphysical preferences. Substantivalism, in various forms, is defended throughout the book. Chapter III turns to quantum mechanics and asks what quantities should be taken as ‘fundamental’, in a metaphysical sense. Before the author enters the metaphysical jungle of different kinds of realism and addresses the question of separability and the fundamental space for quantum mechanics, he provides an excellent discussion of the basic mathematics of quantum mechanics. Chapter IV is devoted to the topic of ‘pointless spaces’: the question whether there are instants of time, points in space and space-time or whether space-time exists of finitely extended regions. Although the author regards the latter view as defensible, in the end ‘pointy spaces’ appear preferable to him. It is a general feature of the book that many chapters end in such inconclusive outcomes, which is probably a consequence of the programme of investigating epistemic possibilities. Chapter V returns to the much discussed debate between substantivalism and relationism in space-time theories. Although it defends a version of substantivalism, and produces a number of arguments against relationism (the hand argument, parity violations) its discussion is a bit sloppy: Newton's bucket experiment is not an argument for space but an argument for absolute space and Leibniz's relationism does not amount to a denial of space, since Leibniz characterizes space as the order of coexisting things (and time as the succession of the order of coexisting things). This chapter does not engage with Leibniz's own writings but repeats the usual, rather narrow-minded view of relationism as a view about spatial-temporal relations between material bodies occupying space-time regions. This characterization has become popular in the English-speaking world since the work of M. Friedman and J. Earman and is used by these authors to criticize Leibnizian relationism for its claimed inability to ground a proper theory of motion. Chapter VI discusses the implication of gauge theories for the conception of properties – are velocities properties? – and whether we need to assume the existence of fibre-bundle space over and above space-time. The notion of fibre bundles is well explained. The chapter defends fibre-bundle substantivalism against gauge relationism, which is taken to be the denial of fibre bundles. ‘Gauge relationism (…) is the view that there are no fibre bundles, that the states of gauge fields and connection fields do not correspond to geometric features of fibre bundles, but instead correspond to certain properties of paths in space-time’ (194). Chapter VII is entitled ‘Directions, hands and charges’. It discusses the CPT theorem – which states that ‘any relativistic world is invariant under space-time mirroring’ - and claims that from it valid inferences about space-time can be drawn. But it ends in a now familiar, ambivalent conclusion. Chapter VIII distinguishes various versions of nominalism (hard, soft) and asks whether physics could be done without the help of mathematics. According to the author the ‘incursion of mathematics into physics’ is worrying for two reasons: the relations which mathematical entities bear to physical objects, and the very existence of mathematical entities (214–5). The author admits that scientists are not interested in such nominalistic projects, which attempt to rewrite physics in terms of concrete physical entities only. Indeed, the chapter sits rather uncomfortably with the preceding chapters, which discuss mathematical issues in great technical detail.

The author starts this book with the frank admission that he failed in his original intention to write an accessible book for ‘anybody who is interested in the structure of space and time and knows a bit of calculus.’ In fact from the point of view of the general reader or even a reader with an interest in the philosophy of physics generally, the books fails on various levels.

• • Despite the often excellent discussions of technical issues, the level of explanation is uneven. For instance various notions of space-time (Newtonian, Neo-Newtonian, Minkowski, relativistic space-times) are introduced in the early parts of the book but are not sufficiently clearly distinguished. This will make it hard for the general reader to follow the discussion in this valuable first chapter, whilst the informed reader will find no new material in these passages. Later on, both the Stern-Gerlach experiment and the violation of parity symmetry would have deserved a fuller exposition for the benefit of the less technically informed reader.

• • The book offers detailed discussions of very arcane subjects in metaphysics – for instance, whether time is structureless or whether space-time has a neighbourhood topology, whether velocities are properties, whether pointless spaces can be envisaged, whether space-time and fibre-bundle spaces exist, - which will be of interest to only a small readership of devotees. At the same time topics, which are highly relevant and of great interest to the question of physical time, like thermodynamics, statistical mechanics, quantum cosmology, and the important mechanism of decoherence are completely ignored.

• • Finally the book is based on the naïve assumption that the metaphysical structure of reality is a deductive consequence of the mathematical structure of physical theories. For instance, the author wishes to understand quantum theories ‘as accounts of what there is, fundamentally speaking, and of the pattern of fundamental properties and relations’ (269). He claims repeatedly that ‘this book is mostly about the question as to what the fundamental structure of the world is’ (79). What is meant by ‘fundamental’ structure is not physical structure. Rather ‘what we can know and do know about the metaphysical and physical possibilities derives from our knowledge of what the fundamental objects and predicates are, and what the fundamental laws are in which they figure’ (1). Unsurprisingly, hardly any chapters end in a clear, unambiguous conclusion. The reader is left with a bewildering choice of metaphysical positions – for instance amplitude realism, liberal amplitude realism, density-operator realism, flash realism, various versions of substantivalism (including super-substantivalism) and relationism – demonstrating clearly that the so-called metaphysical structure of reality cannot simply be read off the mathematical structure of physical theories. The author seems to concede this point on almost the last page. ‘Mathematics is a stupendously useful tool. It describes and systematizes a vast array of possible structures, any one of which we might find ourselves having reason to ascribe to part of the real world. We should feel free to avail ourselves of these options, without fearing that once we start doing so, we will somehow end up having to believe that every one of those structures is instantiated somewhere in reality’ (270). The inflation of metaphysical positions stands in stark contrast to the value of simplicity, which the author praises as one of the great virtues of science because ‘simple theories are more likely to be true’.

One chapter of the book is entitled ‘Pointlessness’. Many pragmatically or scientifically minded readers may feel that this term would have been an appropriate title for the whole book. Members of the metaphysical club, however, will embrace the claim in the blurb ‘that when trying to extract the fundamental structure of the world from theories of physics one ignores philosophy at ones’ peril!’ This reviewer, for one, walked away from this book without suffering lasting damage.