From Fermat to Gauss: Indefinite Descent and Methods of Reduction in Number Theory

The sizable (vii + 574 pages) opus under review, From Fermat to Gauss: Indefinite Descent and Methods of Reduction in Number Theory , by Paolo Bussotti, is a serious and interesting contribution to the history of mathematics, specifically the history of number theory, spanning the era from Fermat (1601–1665) through Gauss (1777–1855). These two and a half centuries are famously noted for magnificent growth in not only number theory, whose birth is often identified with the work of Fermat, but in mathematics in general: this was a time when giants walked the earth.

Bussotti properly identifies Euler (1707–1783) and Lagrange (1736–1813), in addition to Fermat and Gauss, as rounding out a quartette of titans for whom arithmetic was a particular favorite (one is reminded of Gauss’s famous aphorism about “the queen of mathematics”), and we are all familiar with a number of their contributions in this area.

A major methodological innovation, the method of in(de)finite descent, originating with Fermat, saw a number of beautiful and fascinating applications at the hands of all four of these major figures and Bussotti devotes a chapter — more aptly described as a section, I think — to the contributions of each of them. The problems and theorems Bussotti considers are beautiful and, given their historical status, accessible, and his book can therefore be used to great advantage for a number of purposes, ranging from arithmetical browsing par excellence to historical reading (or research).

Beyond this, Descent and Methods of Reduction in Number Theory can certainly be used as a supplementary source in seminars of various flavors. As far as number theory goes, I’d locate the book at a junior or senior undergraduate level, for students and auditors who already know a good deal of elementary number theory; as a source for work in the history of number theory, given that this is a work of specific historical scholarship, I’d suggest a more mature audience (for what it’s worth: I’m no one’s idea of a historian).

First, a bit of logic is perhaps in order, for the sake of completeness and continuity. The basic method of descent proceeds as follows: Prove ∀n∈N , P(n). Proof method: Suppose (*) [∃n such that ~P(n) ]. Prove (†) that (*) implies ∃m < n such that ~P(m). Iterate (i.e., get descent). But every nonempty subset of N has a least element, so the descent has to stop. But this contradicts (†). So (*) is false and we get ∀n∈N, P(n). There’s an obvious partner to this line of argument starting with the indicated negative statement.

To get back to the arithmetical content of the book, however, it is worth noting that several of the theorems covered are amenable to many different approaches and comparing some of these in a seminar or directed research setting would be most instructive. For example, I recently had occasion to direct a master’s thesis on the geometry of numbers which included coverage of a (to me at least) somewhat unexpected geometrical proof of the famous and supremely elegant fact that a prime of the form 4n+1 is always the sum of two squares: my student found a proof in the literature (I believe in a book by J. W. S. Cassels, now out of print) deriving this result from the Minkowski theorem on lattice points in convex bodies. My own first exposure to this theorem occurred in my introductory undergraduate number theory course and, if memory serves (over thirty years having gone by), the proof given was by the method of descent. (I can still hear my one of my favorite professors, the late Ernst Straus, say that “there is no infinite descent in the natural numbers…,” which is of course part and parcel of the logical interlude presented in the previous paragraph.) In Bussotti’s book this theorem is dealt with in § 2.4.2 and on pp. 482–495, in the Appendix by Sergio Paolini, whose recent work figures prominently in Bussotti’s exposition. (Paolini’s role in the story is very interesting: see the footnote on p. 9.)

Other theorems dealt with include the impossibility for the area of a right triangle with integer sides to have a square area (§ 2.2.3), Pell’s equation (§ 2.4.3), polygonal numbers (§ 2.4.4, § 4.3), Euler (and Weil!) on Fermat’s Last Theorem for n = 3 (§ 4.4), and Gauss on the same assertion and the case n = 5 (§ 6.3). By the way, I picked up the following bit of interesting historical minutiae from Bussotti as regards Fermat’s Last Theorem: whereas Fermat only mentioned the full assertion in the notorious margin of Bachet’s copy of the Arithmetika by Diophantus of Alexandria, he addressed the particular cases of n = 3,4 very often in his correspondence, for instance in letters to Mersenne. Cool. (More about Fermat himself and the cubic case of his last theorem is available in e.g. § 2.4.5 of Descent and Methods of Reduction in Number Theory.)

Bussotti caps off his ambitious undertaking with a section labeled “Conclusion” in which he begins by recapitulating his objectives in writing the book. Briefly, they are threefold: to reassess the status of the method of descent in the 17^th century, with Fermat as the focus; to analyze the spread of the method in the 18^th century; and to “clarify the logic” of the method, always (and naturally) centering on problems and on the styles of the main players in the game, Fermat, Euler, Lagrange, and Gauss.

The author then goes on to appraise his findings and deductions, supplementing these remarks by a commentary on how the moves instigated by Fermat fit with Kronecker’s Mathematikanschauung: “Kronecker’s opinion — who anyway recovered many elements of Gauss’ thought — is quite more extreme than Gauss’: Kronecker explicitly addressed the idea that not only arithmetic must use elementary methods, but also that… analysis must be based on arithmetical procedures (not only arithmetical objects), otherwise analysis has no foundation and its results are not legitimate. Kronecker’s position can be understood if we think of his polemic against the ‘arithmetizators’ [sic?] of … analysis (Weierstrass, Cantor, Dedekind).” The third chapter of Descent and Methods of Reduction in Number Theory presents Bussotti’s dissection of this aspect of a famously bellicose era in the history of modern analysis and engenders a particularly apt departure from arithmetic per se. The point to be taken is that Descent and Methods of Reduction in Number Theory is dense with historical scholarship and sound mathematical development and evaluation of marvelous arithmetical results from a heroic era populated by figures all of us hold very dear. Recommended.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles.