**Robin Wilson and Amirouche Moktefi, eds., The Mathematical World of Charles L. Dodgson (Lewis Carroll), (Oxford: Oxford University Press, 2019), xiv+266pp.**

Many people's first introduction to the oeuvre of Lewis Carroll is through the *Alice* books. I tried them as a child and never succeeded (nor as an adult, either); my first proper introduction to him was through "What the Tortoise Said to Achilles", perhaps the most influential piece of scholarship I've ever read – it's what made me go, as a 14 year old, "THIS. This is what I want to do with my life; I want to learn how to think like that." And thus began the first step on a journey to receiving a doctorate in logic (with an emphasis on the history of logic) and a career teaching logic to undergraduates – and reading them this story. So Carroll as a logician holds a special place in my heart, and I was delighted to be offered the opportunity to read and review this wonderful book on the mathematical context he worked in and contributed to. Luckily for Wilson and Moktefi, norms of book reviews have changed from the 19^{th} C, when Carroll reviewed a geometrical treatise by James Maurice Wilson by saying "My ultimate conclusion on your Manual is that it has *no claim whatever* to be adopted as *the* Manual for purposes of teaching and examination" (p. 27). Luckily for me, even if they had remained the same, I was given no cause for such a harsh accounting.

In this book, the editors bring together "the best authorities on Dodgson's mathematics in their areas of expertise" (p. viii). Each of the eight chapters is devoted to a different aspect of Carroll's mathematical enterprises. (Though the editors used both of his names for the title, throughout I'm simply going to refer to Carroll – not only because we know him better this way, but because this was the name he used for his public-facing books, such as "his books on logic" (p. 16).) Some touch on topics familiar to anyone who has studied higher mathematics, such as logic and algebra. Others cover topics some might be less familiar with, such as voting.

The book is beautifully typeset with many photographs and prints – it has all the charm of a coffeetable book, and yet it also manages to be learned and scholarly in its approach, never sacrificing accuracy in the name of accessibility. It is full of delightful anecdotes (even when the anecdotes are denied by Carroll himself), such as the one involving Queen Victoria having enjoyed *Alice* and wanting to read the next book by him, being "somewhat surprised to receive a copy of *An Elementary Treatise on Determinants*" (p. 57), as well was lucid discussions of complex mathematical theories. Occasionally, when I looked up a note in the back of the book, I was directed to a secondary source (often something by Wilson or Moktefi) rather than to the primary source where the quote ostensibly originally came. From the coffee-table point of view, this is not real detriment; from the point of view of an academic researcher, I was mildly peeved.

Though all the chapters are written by different people, the style throughout is remarkably uniform, and gives the impression of being written as a coherent whole – a very difficult feat to pull off, and both the authors and the editors should all be commended.

The early chapters (Wilson and Moktefi's biography of Carroll and Wilson's on geometry) paint a picture of a scholar with wide-ranging interests and a lack of desire to follow a traditional path. Carroll was not a typical mathematician for his period, in that he never joined any of the scholarly mathematical societies, nor generally participated in the activities that research-active mathematicians of his time did (e.g., writing articles for journals, presenting at the learned societies). Much of his work was driven by intensely internal motivations: Specific concrete teaching needs, a desire to solve puzzles, his closely held principles as to the proper approach to proofs, both logical and geometrical. This latter point is illustrated in the geometry chapter, where Wilson frankly accounts for Carroll's lack of legacy in the field of geometry (in his immediate time) by his insistence on strict methods of proof and his introduction of nonstandard terminology.

I was particularly impressed with Rice's chapter on algebra, which endeavors to introduce the casual, non-mathematician, reader to the theory of determinants – with high success, in my opinion. Rice manages to convey the magic that is Carroll's "most significant research-level contribution" (p. 57), a 'condensation' method for linear equations which "never requires the computation of anything more sophisticated than 2 x 2 determinants" (p. 71). One need not have understanding of matrices or their applications to be able to appreciate the value of Carroll's discovery.

Being a logician myself, Moktefi's chapter on Carroll's logic naturally excited me the most. It was perhaps also the chapter that disappointed me the most, in that its history of logic leading up to Carroll's developments suffers from the same defect that pretty much every history of logic suffers: The Middle Ages are not even mentioned. Now, I'll be the first to admit that the medieval developments in logic are unlikely to have influenced Carroll. But when one is purporting to give a history of the discipline of logic, one cannot just skip centuries of the most important developments in the field before modern times. And without a clear picture of the history of logic, it becomes harder to see just how genuinely novel Carroll's game-based approached to solving multiple premise syllogisms was in fact.

Despite having read extensively on voting theory, including Carroll's contributions, I found McLean's chapter more confusing than illuminating; for someone who does not already have solid footing in the material, I suspect parts of it will be truly opaque. The presentation of the material was repetitive, as the chapter both tried to trace the historical developments of the field and then repeated them in the context of Carroll's own contributions. There were also too many concepts that McLean assumed knowledge with (social choice, game theory); and one does not come away with a clear idea of what Carroll's brilliant contributions were. I was also a bit uncomfortable by the fact that McLean *first* highlighted Carroll's being "strikingly eccentric" (p. 123) before even discussing his scholarly contributions; such an analysis is both irrelevant and problematic reinforcing of the "brilliant genius" stereotype of (usually masculine) scientists.

The chapter on "recreational mathematics" was the one that I look forward to the most.[1] The most brilliant facet of Carroll was his ability to make things into games, whether they are the nonsense games of *Alice in Wonderland*, the logic games of his textbooks, or the tournament insights that underpinned his voting theory. Games are often considered pejorative, because childish; but in fact game-based developments in mathematics and logic (e.g., game theory, game-theoretic semantics, etc.) have provided some of the most important advances in the mathematical disciplines in the 20^{th} and 21^{st} century. In that respect, Carroll in the late 19^{th} century was ahead of the game (yes, yes, pun intended). What sets Carroll's recreational mathematics apart from his more erudite contributions is not that they are childish but that they are "aimed at the non-mathematician, thereby making them accessible to all" (p. 142). On the one hand, the individual games outlined in the chapter are relatively trivial, in the sense that they are easily and enjoyably played with children. On the other hand, the number theory that underpins the success of these games is by no means trivial, and the games themselves provide a way of opening up a conversation that can end up in number theory. It's one thing to say that a particular solution to a game works; it's entirely another to say *why* it works.

Unfortunately, many of the puzzles and their solutions displayed in this chapter are done so via photos of the manuscripts, without any transcription provided. While Carroll's hand is not as bad as some, it is difficult enough to puzzle out sometimes, and it would have been an improvement if the author had taken the time to do so for his readers.

From geometry to logic, algebra to arithmetic, from voting theory to number games, Carroll's mathematical output is extensive. But what of his legacy? One thread that ran throughout the entire book was an emphasis on the idiosyncracies of his approaches – whether it involved introducing new terminology or new notation, or insisting upon using old methods, such as his insistence on the rigorous Euclidean proofs – and how these directly obstructed many of his contributions from being taken up by his successors. The final chapter, on Carroll's "Mathematical Legacy" addresses this issue head on: "What mathematics did [Carroll] do? How good a mathematician was he? How influential was his work?" (p. 178). Abeles' approach is systematic, looking at the state of the art at the time of Carroll's contributions in each of seven different sub-fields of mathematics (geometry, trigonometry, algebra, logic, voting, probability, and cryptology; all of these except the last receiving fuller treatment in a separate chapter of the book) and identifying how the advances he made in each field were taken up by successors, both in the 19^{th} century and into the 20^{th} and 21^{st}.

What I found most interesting in this discussion of Carroll's legacy was the importance that he placed on what Abeles calls the "psychological element" involved in accepting an axiom or a postulate. It is not enough for axioms to express unshakable truths; they must also be the sort of thing that we, limited finite beings can accept and recognise as true; this is why Carroll rejected a version of the parallel postulate that depended on the existence of infinitesimals and infinities, which he believed the human mind could not grasp. What is not emphasised in this chapter is the importance given to the question of what the relationship between logic and psychology was in the 19^{th} century: This is seen in Boole's *Laws of Thought*, wherein logic is firmly rooted in human psychology, and in Mill's philosophy of logic as witnessed in his *System of Logic*. But it is also seen in Frege's anti-psychologism, and his arguments that logic and mathematics, being the most exact of the sciences, *cannot* be founded in psychology, one of the least exact of the sciences. And it is firmly in this context that Abeles points to a concrete influence of Carroll's views upon those who came after, namely, on Hugh McColl's conception of what it meant to be an axiom.

The book wraps up with a detailed descriptive bibliography of Carroll's mathematical works, ordered by date and including published works (both books, journals, and pamphlets), as well as important manuscripts and galley proofs (pp. 217-237). Minor pieces occurring only his diaries or letters are omitted, as well as some fragmentary work. The bibliography also contains a short section of secondary work relevant to the study of Carroll, his life, and his mathematical works. Following the bibliography there are further "further reading" sources (pp. 239-240), as well as endnotes (alas!) for the chapters and references (pp. 240-260), biographical notes for the contributors (pp. 261-262), and an index (pp. 263-266).

**Sara L. Uckelman**, Department of Philosophy, Durham University, s.l.uckelman@durham.ac.uk

[1] Unfortunately, the "puzzles and games [that] are concerned with language and logic...are not within the remit of this chapter" (p. 142), and indeed, no chapter covers them, which is a shame; while the language ones might be excluded on the basis that this book is concerning his *mathematical* world, it's hard to exclude the logic puzzles on that basis.