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The conquest of several mathematical areas by arithmetic, which we mentioned in the introduction, was thus achieved through two rival ways, both stemming from Gauss’s work. The first one, followed by Weierstrass, Dedekind and Cantor, borrowed the conception of number from Gauss. This conception entailed a development of mathematics through the of this concept of number and the introduction of new numbers. At the close of the conquest, which was to lead, in particular, to the arithmetization of analysis, “the captive analysis captured its savage victor,” and arithmetic was reduced to a simple province of analysis. The second way, followed by Kronecker who rejected this annexation of arithmetic by analysis, borrowed other concepts and methods, also from Gauss. These concepts, more related to operations than to objects, had to avoid the extension of the concept of number and the proliferation of new numbers. Thus Kronecker proposed this deployment of operations as an alternative to the development of the mathematical subject matter through conceptual extension. These two approaches — conceptual extension (and the consequent widening of the domain of objects) or operative deployment — appear more generally as distinct paths to mathematical progress. These two ways, usually coming one after the other, are combined in Gauss’s work. Thus this work stops the pendular swing from the concept (or the object) to the operation and vice versa.

There are a few respects which Cantor’s, Dedekind’s, and Kronecker’s arithmetization programmes share, in spite of all their manifest incompatibility with respect to finitist or constructivist requirements. First, all three authors considered mathematics as a science with a clearly defined domain of objects: as mentioned before, Kronecker viewed mathematics as a natural science; Dedekind considered his analysis of continuity via cuts as expressing the essence of this concept; Cantor seems to have considered even his transfinite numbers as something that he discovered, rather than invented. For all three authors arithmetization reduced the irrational numbers to the rational — or all the way to natural — numbers whose existence was taken to be evident. Second, they all executed this reduction to elementary given objects in away that they considered naturally adequate for the problem at hand: for Kronecker, this meant indeterminates and congruences , for Dedekind grouping together sets of primary objects was just as naturally adequate a procedure as the consideration of series of rational numbers was to Cantor. The overall image that this suggests of the movement of arithmetization in the 1870s and 1880s is therefore that of a novel theory of objects that had formerly been understood in terms of extrinsic notions (magnitudes), this novel theory being founded on an independently accepted basis (the natural numbers), and proceeding with ingredients or methods deemed to be acceptable. From this point of view, arithmetization, in spite of all its novelty, appears not as an expression of modernity — indeed, as far as new objects were created, they were not purely formal, nor were they objectivized tools, but regularly formed from existing integers — but as a new type of solid building, erected on a traditional base in a controlled and supposedly innocuous and stable construction.

Our periodization has allowed us to isolate a transitional phase of arithmetization where Gaussian influence is detectible at least in two of the major authors. This influence operated via diverging fundamental positions (Kronecker’s constructivism vs. Dedekind’s completed infinites), but always in the direction of a novel but object-oriented rewriting of analysis. Gauss’s after-effect ended with the onset of purely set-theoretic, axiomatic or logicist approaches, i.e., at the same time as the Göttingen nostrified image of arithmetization took shape.

I hope that the use of module multiplication in some measure simplifies Gauss’s theory of composition of forms. For example, it clarifies the difficult associativity property described and proved by Gauss in art. 240. Multiplication of modules is an associative binary operation, and this property easily translates into the property Gauss uses.

But, more importantly, I hope that by focussing attention on Gauss’s composition of actual forms — as opposed to equivalence classes of forms as in the modern theory — I have highlighted Gauss’s great achievement in giving a rigorous treatment of the composition of binary quadratic forms in the greatest possible generality.

His theory is “rigorous,” not only in the usual sense that it is mathematically convincing, but also in the literal sense that it makes great demands on the reader. The second kind of rigor has caused succeeding generation of mathematicians to devote some of their best efforts to avoiding it. But it is the first kind of rigor that makes Gauss the great master. It is based on his mastery of the computational structure of his subject and his ability to explain that structure in the most general circumstances. While they may seek to avoid the difficulties of Gauss’s theory, succeeding generations should never cease to admire it.

We have tried to showthe beginnings of the intervention of the theory of elliptic functions in arithmetic: complex multiplication, -calculus, and the functional equation for the theta functions, Kronecker’s limit formula, addition theorem and Diophantine analysis. The germs of most of these ideas were already present in Gauss’s work. Gauss clearly saw the necessity to use methods not restricted to the consideration of natural numbers, in order to prove properties of natural numbers.

After Abel and Jacobi, the mathematicians who developed these theories in the first two thirds of the XIX century were Eisenstein, Kronecker, and Hermite. We saw the particular importance of Kronecker’s work which vindicates this author’s insistence on formulae as the essence of mathematics.

At the end of the XIX century, and throughout the XX century, the theories mentioned have continually developed and this growth continues to this day. This is a testimony to the continuing presence of Gauss’s work.

The conquest of several mathematical areas by arithmetic, which we mentioned in the introduction, was thus achieved through two rival ways, both stemming from Gauss’s work. The first one, followed by Weierstrass, Dedekind and Cantor, borrowed the conception of number from Gauss. This conception entailed a development of mathematics through the of this concept of number and the introduction of new numbers. At the close of the conquest, which was to lead, in particular, to the arithmetization of analysis, “the captive analysis captured its savage victor,” and arithmetic was reduced to a simple province of analysis. The second way, followed by Kronecker who rejected this annexation of arithmetic by analysis, borrowed other concepts and methods, also from Gauss. These concepts, more related to operations than to objects, had to avoid the extension of the concept of number and the proliferation of new numbers. Thus Kronecker proposed this deployment of operations as an alternative to the development of the mathematical subject matter through conceptual extension. These two approaches — conceptual extension (and the consequent widening of the domain of objects) or operative deployment — appear more generally as distinct paths to mathematical progress. These two ways, usually coming one after the other, are combined in Gauss’s work. Thus this work stops the pendular swing from the concept (or the object) to the operation and vice versa.

I hope that the use of module multiplication in some measure simplifies Gauss’s theory of composition of forms. For example, it clarifies the difficult associativity property described and proved by Gauss in art. 240. Multiplication of modules is an associative binary operation, and this property easily translates into the property Gauss uses.

But, more importantly, I hope that by focussing attention on Gauss’s composition of actual forms — as opposed to equivalence classes of forms as in the modern theory — I have highlighted Gauss’s great achievement in giving a rigorous treatment of the composition of binary quadratic forms in the greatest possible generality.

His theory is “rigorous,” not only in the usual sense that it is mathematically convincing, but also in the literal sense that it makes great demands on the reader. The second kind of rigor has caused succeeding generation of mathematicians to devote some of their best efforts to avoiding it. But it is the first kind of rigor that makes Gauss the great master. It is based on his mastery of the computational structure of his subject and his ability to explain that structure in the most general circumstances. While they may seek to avoid the difficulties of Gauss’s theory, succeeding generations should never cease to admire it.

I hope that the use of module multiplication in some measure simplifies Gauss’s theory of composition of forms. For example, it clarifies the difficult associativity property described and proved by Gauss in art. 240. Multiplication of modules is an associative binary operation, and this property easily translates into the property Gauss uses.

But, more importantly, I hope that by focussing attention on Gauss’s composition of actual forms — as opposed to equivalence classes of forms as in the modern theory — I have highlighted Gauss’s great achievement in giving a rigorous treatment of the composition of binary quadratic forms in the greatest possible generality.

His theory is “rigorous,” not only in the usual sense that it is mathematically convincing, but also in the literal sense that it makes great demands on the reader. The second kind of rigor has caused succeeding generation of mathematicians to devote some of their best efforts to avoiding it. But it is the first kind of rigor that makes Gauss the great master. It is based on his mastery of the computational structure of his subject and his ability to explain that structure in the most general circumstances. While they may seek to avoid the difficulties of Gauss’s theory, succeeding generations should never cease to admire it.

We have tried to showthe beginnings of the intervention of the theory of elliptic functions in arithmetic: complex multiplication, -calculus, and the functional equation for the theta functions, Kronecker’s limit formula, addition theorem and Diophantine analysis. The germs of most of these ideas were already present in Gauss’s work. Gauss clearly saw the necessity to use methods not restricted to the consideration of natural numbers, in order to prove properties of natural numbers.

After Abel and Jacobi, the mathematicians who developed these theories in the first two thirds of the XIX century were Eisenstein, Kronecker, and Hermite. We saw the particular importance of Kronecker’s work which vindicates this author’s insistence on formulae as the essence of mathematics.

At the end of the XIX century, and throughout the XX century, the theories mentioned have continually developed and this growth continues to this day. This is a testimony to the continuing presence of Gauss’s work.

There are many ways to measure the reception — and success — of a text, or, more precisely, the ideas of a text. We have elaborated here, in part, on Martin Kneser’s personal remarks and contributions at Oberwolfach, in June 2001. That Gauss’s question on the composition of forms could lead to the construction of a complex, yet stunningly elegant, algebraic theory of composition for binary quadratic modules over an arbitrary commutative ring with unity certainly testifies — again — to the breadth and depth of Gauss’s original ideas. Hurwitz’s private thoughts on Gauss’s composition of forms at the close of the nineteenth century, Bhargava’s more contemporary results on the arithmetic of number fields and Kneser’s extension of the theory via Clifford algebras provide further evidence of the lasting impact of the .

We have tried to showthe beginnings of the intervention of the theory of elliptic functions in arithmetic: complex multiplication, -calculus, and the functional equation for the theta functions, Kronecker’s limit formula, addition theorem and Diophantine analysis. The germs of most of these ideas were already present in Gauss’s work. Gauss clearly saw the necessity to use methods not restricted to the consideration of natural numbers, in order to prove properties of natural numbers.

After Abel and Jacobi, the mathematicians who developed these theories in the first two thirds of the XIX century were Eisenstein, Kronecker, and Hermite. We saw the particular importance of Kronecker’s work which vindicates this author’s insistence on formulae as the essence of mathematics.

At the end of the XIX century, and throughout the XX century, the theories mentioned have continually developed and this growth continues to this day. This is a testimony to the continuing presence of Gauss’s work.