Composition of Binary Quadratic Forms and the Foundations of Mathematics

3. Conclusion

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 obviously an associative binary operation, and this property easily translates into the property Gauss uses.¹⁰

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.

Since multiplication of modules can be used to establish the theory of composition of forms, Gauss’s proof of quadratic reciprocity using composition of forms can be deduced in this way. However, quadratic reciprocity can be proved working directly with multiplication of modules. Therefore, if the goal is quadratic reciprocity, one can dispense with quadratic forms altogether. Other aspects of Gauss’s theory can be revisited in a similar way. See [Edwards 2005].