The Shaping of Arithmetic After C. F. Gauss's Disquisitiones Arithmeticae

Here's a simple story. After some work by Fermat, and more importantly rather more work by Euler and Lagrange, Gauss established number theory as a central subject in mathematics by the sheer profundity of his Disquisitiones Arithmeticae of 1801. This book, and Gauss's many later contributions to the subject, won more and more followers as the 19th Century progressed until the century's end is marked by the equally magnificent Zahlbericht of David Hilbert, published in 1897. The subject then continues triumphantly to the present day. The point of the book under review is not to refute this story, but to show how and in what ways it is an over-simplification amounting to a distortion, and to put in place a number of important insights and additions.

The book opens with a fresh look over the content of the seven sections of the Disquisitiones Arithmeticae. The authors, Goldstein and Schappacher, observe that in fact this great work is incomplete. It was to have contained an eighth section which would have brought the book full circle from the elementary treatment of congruences with which it opens to higher congruences, via quadratic forms, cyclotomy and periods (also taken mod p) and offering four proofs of quadratic reciprocity. They give their reasons for indicating why Gauss might have contemplated such a structure for his book, and in a later chapter Frei discusses in more detail what the eighth section might have contained. Goldstein and Schappacher then describe what happened as a book in search of a discipline. Cyclotomy was taken up first, because it fitted well into contemporary algebra around 1800. Whereas Gauss had explicitly stated that his book was about the advanced part of the study of the integers, the initial responses were largely to subordinate it to an existing domain (algebra), which is what one might well expect if indeed a new discipline is being formed. In the 1820s the new University of Berlin proved receptive to number theory, and progress in mathematical analysis, much of it due to Dirichlet, brought about a union of analysis and arithmetic for a time, forming what Goldstein and Schappacher call the research field of arithmetic algebraic analysis by the 1850s, and which by then embraced a growing literature connecting Gauss's questions with the theory of elliptic functions (many pages of this book are devoted to that topic alone).

Goldstein and Schappacher break off their account here, and resume in the next chapter with an account of 'Several disciplines and a book'. Their first point is that there was a hiatus in research in the late 1850s. Many leading figures died, and only after a quiet period of several years did another generation of mathematicians come along, many of them also drawn to the Göttingen-based project to produce an edition of all of Gauss's works, many of them unpublished. This is the period of Dedekind and Kronecker, but also of a number of figures commonly omitted from the simple story. In fact here it is the diversity of responses to Gauss's Disquisitiones Arithmeticae that is the chief novelty of this exposition, highlighting as it does the sheer richness of Gauss's book and the many responses it brought forth. That said, here as in other chapters the various authors make good substantial sense even of the well-known connections between Gauss's work and that of Dedekind and Kronecker themselves.

An important feature Shaping is the high level of mathematical coverage. Fascinating though this material is, it is not an easy read throughout (no reason why it should be) and how much of it one reads is likely to be determined by how advanced is one's acquaintance with modern number theory. Part II of the book discusses key mathematical techniques in the Disquisitiones Arithmeticae. Olaf Neumann looks at algebraic techniques, roughly those of Galois theory up to the time of Galois. Harold Edwards performs the valuable service of explaining what Gauss's composition of forms actually involves, a task he rightly says has been ducked by most writers from the time of Dirichlet. Della Fenster and Joachim Schwermer then revisit this topic in a way that connects to the modern theory. Finally, as noted, Frei looks at the posthumously published Section Eight of the Disquisitiones Arithmeticae, which he treats as a stage on the way to the study of function fields over a finite field.

There are 14 more essays in the book, which is that rare thing a readable, indeed essential, set of conference proceedings, in this case an Oberwolfach workshop in 1999 and a conference there in 2001 on the occasion of the 200th anniversary of the first publication of the Disquisitiones Arithmeticae. The editors, Goldstein, Schappacher, and Schwermer, are to be congratulated on this achievement. Nothing much would be served by taking each of these 14 essays in turn, and it seems better to group them in some way.

Those by Bölling, Houzel, Piazza, Patterson, Lemmermeyer, and a further essay by Schwermer are more technical. They take us, in this order, from reciprocity laws to ideal numbers, through elliptic functions and arithmetic, through Zolotarev's work, Gauss sums, the principal genus theorem, and to Minkowski's reduction theory for quadratic forms. Higher reciprocity laws figure in various places, too.

The essays by Pieper (Humboldt's relation with number theorists), Décaillot (number theory in the French Association for the Advancement of Science), Brigaglia (arithmetic in Italy), and Della Fenster again (the American reception) are more social-historical. Those by Ferreirós, Boniface, and Petri and Schappacher together are more discursive and philosophical. Ferreirós writes on Gauss's influence on the rise of pure mathematics as arithmetic, Boniface on the concept of number from Gauss to Kronecker, and Petri and Schappacher on arithmetization. Finally, Catherine Goldstein recovers one of the largest single omissions in the simple story, the novel contributions of Charles Hermite and his lifelong identification with the theory of elliptic functions and the Disquisitiones Arithmeticae.

A number of aspects of this book are worth singling out. Many of the more technical essays, without ceasing to be historical, take us quite some way into the 20th Century. This is by no means a book only on the story from Gauss to Hilbert. Indeed, it raises the interesting questions of just how technical the history of substantial branches of 'recent' mathematics will have to be, who will write them and who will read them. Once we leave the confines of a good undergraduate education, can history of mathematics continue to be one source of unity in the spreading tree of mathematics? As noted already, readers with different backgrounds will deal differently with much of the material here. That said, an immense amount of mathematics is presented carefully, and it is hard to imagine anyone wanting to be a number theorist not benefiting from this book.

The sophistication of the historical approach is also significant. There are in any case very few accounts in the history of mathematics of a specific work and its influence, but it is more valuable to observe that there are very few works in the history of mathematics that proceed, as literary critics, music and art historians do, to deal in detail with a work. There are many occasions where a nodding acquaintance with a work suffices, but not many that provide this level of detailed attention. When, as here, the result is readable we can hope that this book produces more of this sort.

But the historical sophistication reaches further, into the formulation of the questions asked, the breadth of the information presented, and above all its variety. What several essays here demonstrate in full mathematical detail is the way Gauss's visions in the Disquisitiones Arithmeticae opened up several inter-related lines of enquiry, and were frequently enriched by later contributions. That there were many hidden connections between seemingly different topics was, of course, one of Gauss's own arguments for finding number theory interesting at all, and he turned out to be more right about this than he can have suspected. Here for the first time this multiplicity and inter-connectedness is reflected in the range of authors treated, and the range of possible readings of the Disquisitiones Arithmeticae that were considered and found to exist or not. Here the discursive or philosophical essays mentioned above are indicative of how important general lessons can be drawn from careful readings of suitable texts.

Should we then dump the simple story, and does this book admit a new equally memorable simplification? As to the first question, that is a matter of taste, but The Shaping of Arithmetic does at least suggest several key additions. One concerns the story or stories of elliptic functions and the (higher) reciprocity laws. Another concerns the reduction theory of quadratic forms, notably in the hands of Hermite and Minkowski. And we should remember tangle and complexity, too, the variety of consequences and influences a major work can have. For reasons that may have more to do with undergraduate education than anything else, the simple story tends to direct attention to the rise of modern algebra. How tempting it is to continue from Dedekind to Hilbert and then to Emmy Noether, who famously said 'Everything is already in Dedekind'. Now we have an account that reminds us forcefully of the deep results in analysis that are also part of the story. Gauss would never have doubted it, and as this book eloquently reminds us, he is still well worth listening to.

Jeremy Gray is Professor of the History of Mathematics at the Centre for the History of the Mathematical Sciences of the Open University, in Milton Keynes, UK. He is the author of many books on the history of mathematics.