Ian Mueller’s Philosophy of Mathematics and Deductive Structure in Euclid’s Elements is a Dover reprint of the 1981 classic. In a nutshell, this work is an exegetical commentary on the Elements. Its focus is on what an analysis of the text and structure tells us about Euclid’s philosophy of mathematics, revealing both questions and answers as to what Euclid was doing and why. Throughout the book, Mueller provides diagrams showing the deductive structure of each book of the Elements. He offers historical commentary and discusses philosophical issues. Importantly, he provides symbolic logical analyses of the definitions and propositions when this helps distinguish among possible meanings of Euclid’s text. Finally, Mueller uncovers and makes explicit additional assumptions made by Euclid. In the description of each chapter below, I will mention only some of the specific topics Mueller considers — please do understand that in each case, the chapters contain the content described above as well.
Mueller takes us on a tour of Euclid’s propositions. Rather than following the order of Euclid’s thirteen books, he has chosen to organize his discussion of them with the goal of highlighting their deductive structure. In Chapter 1, Mueller explores the first book of the Elements. He suggests that the motivation for the structure and results of Book I is to prove Proposition I, 45: “To construct in a given rectilineal angle a parallelogram equal to a given rectilineal (figure or area).” (322) He also discusses the contrasts between the program of David Hilbert’s axiomatization of geometry, stemming from the modern interest in structure in mathematics, and Euclid’s descriptive intensions. Further topics of discussion include the stylized form of Euclid’s proofs, the concepts of existence and constructibility, and Euclid’s use of superposition. Mueller concludes the chapter with a first look at the idea of geometric algebra, especially in Euclid II, 1–7. He ultimately rejects the hypothesis of any algebraic intent in favor of geometric results.
In Chapter 2, Mueller examines Euclid’s arithmetic as developed in Books VII, VIII and IX. The philosophical discussions include Euclid’s implicit use of induction on the number of terms in an assertion rather than on the integers themselves. Mueller’s logical and philosophical analysis of Book VII is done very carefully. He points out the puzzling deductive structure of Books VIII and IX. He analyzes the notion of “continuously proportional” in three different ways. Finally, he points out the distinction between ratios, which are relations rather than objects, and fractions, which are objects.
Mueller then considers Euclid’s Book V, having ordered his own chapters to allow him to explore the analogies between Euclid’s treatment of magnitudes in Book V and of numbers in Book VII. He discusses Dedekind’s definition of a real number, explaining why a comparison with Euclid’s/Eudoxus’s definition of proportional ratios can be made while also discussing the limits of its appropriateness. Three special topics appear at the end of the chapter: (1) a development of operations with ratios: this wasn’t done by Euclid, nor apparently did he intend to do it, but Mueller shows how it could be done; (2) a discussion of what magnitudes are: they are geometric objects, not numbers; and (3) an analysis showing in what ways Euclid did and did not formulate the Archimedean condition on magnitudes.
Mueller looks at Book VI, which is based on Book I and the concept of proportion from Book V, in his Chapter 4. He admits that it is not sensible to give a complete deductive diagram for Book VI. Mueller provides a discussion of the procedures used by the Babylonians to carry out quadratic computations and compares these with Euclid VI, 28 and 29. He also argues here that Euclid is not concerned with the calculation of areas.
Books III and IV, concerning circles, are Mueller’s subject in Chapter 5. Mueller (and Euclid) give special attention to the construction of the inscribed regular pentagon, which is carried out without reference to the theory of proportion. In fact, Euclid’s interest in avoiding and delaying the use of proportion is an important factor in the structure of Books I-IV. Mueller also discusses the topic of circles in the geometry of Hippocrates.
Mueller’s Chapter 6, concerning Books XI and XII on solid geometry and the method of exhaustion, includes an analysis of the results on areas in Books I, III, VI, and XII and compares them with their analogues for volumes in Book XI. A crucial point is that Euclid did not make full use of compounding proportions, and he was not as systematic in the arrangement of his arguments as one might have expected. Mueller finds that a constructive view of existence is not in play in Book XII. He provides a very nice discussion of the method of exhaustion versus the modern version of limits. Finally, he sees Book XII as a culminating point of the Elements — it is a deductive terminus, in that the results of Book XII are not used in Book XIII, and it is an “outer limit” of Euclid’s geometric, rather than computational, style of mathematics.
In Chapter 7, Mueller notes that it is clear that Euclid is not thinking algebraically in his treatment of Book XIII, which concerns the Platonic solids, constructing them and comparing their edge values if they were inscribed in the same sphere. In the context of the side length of an inscribed icosahedron, Mueller also discusses here most of Book X, stating that “Book X appears to be an expedient for dealing with a particular problem and at the same time a mathematical blind alley.” (271) This discussion also leads back to results in Book II. The complicated structure of Book X can be seen in the fact that Mueller provides no deductive diagram for all of Book X. Instead, he lists the foundations for Book X in six different categories, then makes comments on the deductions for propositions within nine different sets. According to Mueller, Euclid’s interest in the relations of Book X is basically qualitative and classificatory, rather than quantitative or computational.
According to Proclus, the construction of the Platonic solids in Book XIII is Euclid’s ultimate goal in the Elements. Mueller sees this as an exaggeration while discussing why Proclus might say this. For Mueller, “the significance of the Elements lies less in its final destination than in the regions traveled through to reach it. To a greater extent than perhaps any other major work in the history of mathematics, the Elements are a mathematical world.” (303)
Appendices to Mueller’s book provide a list of the definitions, postulates, common notions, and propositions of the Elements, making this book largely self-contained. The appendices also provide a handy reference to the various abbreviations and additional assumptions that Mueller introduces.
There has certainly been a great deal of further scholarship concerning Euclid and the Elements in the 35 years since Mueller’s book first appeared. Nevertheless, it remains a useful guide to the Elements. It would be an important supplementary text for a course on Euclid and an important reference for courses in the history of mathematics in general.
Joel Haack is Professor of Mathematics at the University of Northern Iowa.