This is a smart book for two reasons, content and form. The former stems from the nature of the subject matter – the concept of number – reputed as one of the most difficult philosophical entanglements in Late Antiquity. The latter reflects the erudition and command with which K. transforms the at times impenetrable fabric of the topic into a clearly articulated systematisation of divergent and opaque views, with far-reaching intellectual impact. K. succeeds in his goal to offer a narrative that is ‘comprehensible, intellectually engaging, and a pleasure to read’, as intimated in the acknowledgements. Why comprehensibility and readability matter in this case more than in any other book devoted to this period will become clear as we proceed.

The book presents K.'s 2006 doctoral dissertation, ‘Formation of the Early Christian Theology of Arithmetic: Number Symbolism in the Late Second and Early Third Century’. The printed result does not suffer from the shortcomings expected of a work of this kind.

The thesis of the book is two-pronged: (1) ‘the principal impetus for this [early Christian] theology of arithmetic was the Platonic and Pythagorean literary tradition, in which numbers were a driving force of metaphysics, symbolism, and interpretation’ (p. 1); and (2) ‘the second-century Christian debate over numbers eventually influenced the very tradition that inspired it’ (p. 2). The first claim is indisputable, the second not so. I will air my scepticism on the latter below.

There are eight chapters (including an introduction), three supplementary essays and one appendix of excerpts from Irenaeus' Revelation to Marcus and Proclus' Commentary on the Timaeus. The introduction sets the stage by exposing the gaps in the current understanding of the subject in the Christian literature of the first four centuries after Christ and the lack of terminological precision. As explained (p. 3), K. restrains from using the term ‘gnostic’ as an umbrella term covering the wide and disparate range of early Christian texts, with strong arithmological motifs. He also institutes, in his words, ‘provisionally’ the term ‘orthodox’ to distinguish the above group of texts, exhibiting novel cosmogonical ideas about number, from the mainstream attitude towards it in Irenaeus, Clement, Tertullian and Hippolytus (pp. 3–4). Lastly, he attempts to delineate the terminological tableau of the pertinent disciplines: arithmetic, mathematics, numerology, arithmology, psephy and number symbolism, the last among which he favours (pp. 4–5). He sticks with his choice throughout, although his preferred term inherently carries the possible misunderstanding that numbers are nothing more than substitutes for hidden entities. If these texts are indeed built upon the Platonic and Pythagorean views on number, as K. rightfully posits, then this streamlining of terminology is oversimplified and more importantly fails to convey the quintessential tenet of the above traditions that number is not a quantitative but relational expression of higher realities.

The latter is the ‘Schicksalsfrage der Philosophie’ about the relation between One and Many which determines the post-Platonic and post-Aristotelian fate of the subject. K. outlines the history of this development in Chapter 2 by focusing on two less standard representatives, as opposed to Speusippus and Xenocrates, of the early interpreters of the Platonic status quaestionis on number: Philo of Alexandria and Plutarch of Chaeronea. There follows a brief section on Pythagoreanism in the first and second centuries in the spotlight of which are the anonymous Pythagorean Memoirs, Eudorus and Moderatus. Numenius of Apamea, whose idea of the three Gods ordering the universe decisively influences the later (Neo)Platonic unfolding of the concept (fr. 21 [des Places]), and Nicomachus of Gerasa, whose Introduction to Arithmetic is one of the few complete extant sources we have about the Pythagorean revival in this period, do not appear in this background chapter. The former is fleetingly mentioned as a prelude to the Neoplatonic history of the concept in Chapter 8 (pp. 154–5), but his presence in this preliminary stage of the investigation would be more beneficial because Numenius' commitment to the ontological importance of number underlines precisely the counter-positions the texts under investigation here reckon with.

The next five chapters present the evidence in support of the first part of the thesis, i.e. early Christian texts comprising works attributed to the Valentinians (Chapter 3), Marcus ‘Magus’ (Chapter 4), Monoïmus and the anonymous Paraphrase of the ‘Apophasis Megale’ (Chapter 5), and their refutations by Irenaeus of Lyons (Chapter 6) and Clement of Alexandria (Chapter 7), derive their foundation from the (Middle) Platonic and (Neo)Pythagorean views on number. In these core chapters K. shows his forte in outlining as clearly as possible a picture of interlaced and obscure material. His mastery enlivens it, for example the discussions of Epiphanes' protology (pp. 31–3), the Valentinian Ogdoad (pp. 33–7), Irenaeus' refutation of the Valentinian Pleroma (pp. 37–48), by making it accessible to subsequent analysis. In cases in which the primary sources are less known, there is a fine line between presenting what the originals say and construing an argument, based on them. K. is aware of this fine line and at times veers towards exposition more than towards analysis because, as he notes (p. 5), this is the first presentation of this material and he is building the ground for future studies. The bottom line of his exegesis is that the plethora of early Christian texts with strong ‘number symbolism’ overtones initially ‘resembles lowbrow, pop mysticism’, but upon a closer inspection, it becomes clear that these texts ‘used subtle terms and concepts to signal to the cultural elite that they wanted to engage with them in science and philosophy’ (p. 48). This proposition is appealing and holds a significant degree of truth within the early Christian intellectualising community.

The last chapter deals with the Nachleben of the above ideas in the Neoplatonic thinkers of the third and fourth centuries, particularly Plotinus and Iamblichus, and in the later Christian and Byzantine traditions, broadly defined, without specific authors. The first half of the chapter presents the second part of K.'s thesis, i.e. that the early Christian engagement with number directly influenced the Neoplatonists' growing interest in the subject, especially as found in Iamblichus who, in K.'s eyes, belongs to Irenaeus' and Clement's highbrow ‘purity’ camp against the vulgarisation of the concept. The question of the Neoplatonic understanding of number and its origin is far more complex than the characterisation of it as being reactionary to contemporary Gnostic trends. It is true that Plotinus' focal treatment of number in Enn. 6.6[34] follows chronologically his openly anti-Gnostic treatise, Enn. 2.9[33], and that the latter alludes to the aforementioned treatise but, considering the deep ontological roots of Plotinus' understanding of number and his painstaking elaboration of its intelligible nature, the talk of any Gnostic ‘influence’ on its formulation is not plausible. But the spark which ignites Plotinus' determination to pose the question exclusively on Aristotelian and Platonic terms, i.e. arithmetical vs ontological number, perhaps?

The book is well executed, with few editorial mishaps, such as the mismatching in fig. 1 of the entries in the left column with the triangles representing the male gender (p. 34) or the conceptual infelicity of conceiving Plotinus' second hypostasis as four metaphysical levels (pp. 155–6). The latter stems from the fact, as K. declares from the beginning (p. 2), that his interest in the concept of number is from the perspective of intellectual history, not philosophy.