Gauss’s theorem of quadratic reciprocity “like none other has left its mark on the development of algebraic number theory,” , writes Jürgen Neukirch, in his celebrated Algebraic Number Theory. Davenport calls it “one of the most famous theorems in the whole of the theory of numbers.”  Gauss himself calls it the fundamental theorem in his Disquitiones Arithmeticae, and privately he referred to it as The Golden Theorem.  Gauss discovered the law at the age of 19.
Gauss offered seven proofs of the theorem during his lifetime. Each relied on very different techniques.  Even after the emergence of various proofs, Davenport writes, “[t]he desire to find what lies behind the law has been an important factor in the work of many mathematicians, and has led to far-reaching discoveries.”  Quadratic reciprocity has undergone several successive sweeping generalizations — see the reciprocity laws of Eisenstein, Kummer, and Hilbert — culminating in Artin’s reciprocity law and even the titanic modern Langlands program.
Why didn’t the work stop after Gauss’s first proof? Gauss, Artin and Langlands weren’t after proof. They were after understanding.
What are mathematicians after when they seek to prove things? The above anecdote demonstrates that we’ve got a puzzle on our hands. We can infer one thing. They seek more than the proof itself. In which capacities are theorems valuable? Why do mathematicians seek them? What role do they play? We’ll find several answers.
It’s a traditional refrain that the factors that determine which mathematical facts are worthy of interest are Truth, Beauty, and Applications. When a fact is interesting for this third reason — application — then a proof of this fact can be valuable for precisely the same reason.
In short, in order to apply a fact, we’d like first to be sure that it’s true. Though examples of this phenomenon abound, the yet-outstanding Hodge Conjecture — which, of course, is valuable for many other reasons — is famous in particular for the quantity and the importance of the applications which hinge upon it. Until the conjecture is proven, these applications will remain off limits.
So dire is this situation — and so formidable the conjecture — that many have devoted energy to discovering mathematical workarounds for the processes which the Hodge Conjecture envisions possible. “In despair,” writes Pierre Deligne, in the official statement of the conjecture, “efforts have been made to find substitutes for the Hodge conjecture.”  In a widely read MathOverflow response, Matthew Emerton writes, “If we had these conjectures available, it would be an incredible enrichment of our understanding of these worlds [of period integrals and of Galois representations]; as it is, people expend a lot of effort to find ways to pass between the two worlds in the way the Hodge and Tate conjectures predict should be possible.”  The Hodge Conjecture is valuable, not in the least, for its applications.
And yet when mathematical results apply to the physical, computing, or life sciences — as they often do — as truth alone often suffices for these applications, and as many facts are known to be true long before proof arrives, proofs tend to become less essential. This hints at a flaw in the Applications explanation. Truth is all we need for application. Why do the mathematicians seeking to apply the Hodge Conjecture so ardently desire its proof? Our characterization of proof’s value must expand.
Proof as the standard for acceptance as mathematical fact
Math is unique among the academic disciplines in that its results are exclusively established by logical proof. Proof, in the mathematical community, is the means by which results are incorporated into the ever-growing body of mathematical knowledge. Mathematicians seek to prove things, perhaps, from a desire to include them in the realm of the known.
The legendary Dirichlet problem for harmonic functions — which belongs properly to complex analysis, and yet touches on the much larger field of partial differential equations — might seem to corroborate this point. Picture a flat sheet of metal with a curving, blob-like outline, and imagine that each point on the boundary of the sheet is heated to a specified temperature, which is then maintained. As the heat spreads through the film, the distribution of heat will eventually achieve a smooth, gradual character. Or so the physicists thought. Mathematicians asked a much more precise question: can the interior of any region be filled in by a harmonic function which achieves specified values on the boundary? The answer: not always. It depends in a very precise way on the smoothness of the boundary and of its prescribed values.
Though the Dirichlet problem for harmonic functions has no physical application — no physical medium could be that precise — it was valuable solely for contribution it made to the mathematical literature.
This explanation would, indeed, explain the frenzy surrounding the Hodge conjecture. But it fails for the Dirichlet problem. Crucially, the result has been proven many times in many different ways! If expansion of the literature alone were the goal, then one proof alone should have sufficed. Instead, we see, well, solutions using: the Perron method, “Sobolev spaces for planar domains”, “reproducing kernels of Szegő and Bergman”, “classical methods of potential theory”… The list goes on . Something else is afoot.
We want to see a puzzle solved
Perhaps we simply want to witness a puzzle’s solution. The Riemann-Roch Theorem — “one of the central results of the theory of algebraic curves,” Shafarevich writes  — might attest to this point.
First developed in the 1850s by Riemann and his student Gustav Roch, the theorem soon attained universal intrigue. “Various mathematicians over the years took the theorem to be central to their researches in complex function theory,” writes Jeremy Gray, a historian of mathematics . Interest in the theorem was acute. Gray, in fact, cites the classical theorem as “one of the most instructive examples in the history of mathematics of how a result stays alive in mathematics by admitting multiple interpretations.”  People wanted to see the puzzle solved.
This proposal, for one, explains the behavior surrounding the Hodge Conjecture and Dirichlet problem. Yet even this theory fails to account for the subtleties of Riemann-Roch. The theorem has undergone several significant generalizations. It was generalized to surfaces by the Italians, and by Hirzebruch to varieties of all dimensions. In a still-further generalization, due to Grothendieck, the theorem began to take on an abstract form entirely unlike its original. Grothendieck introduced his modern language of schemes, in fact, to provide natural environments to prove such facts; today, mathematicians like Jacob Lurie reformulate algebraic geometry in still-higher terms.
The desire to solve a puzzle alone cannot account for mathematical activity which extends mathematical results into territory which did not yet exist at the time of the results’ original formulation. Which puzzles are we trying to solve? The puzzle hasn’t yet been made. We should refine our definition further.
The desire for understanding
The most subtle explanation — and that which, I think, most closely reflects the beliefs of practicing mathematicians — is that mathematicians desire to understand, in some sense. Mathematicians make conjectures on the basis of empirical evidence. They then seek to prove these conjectures simply to attain feelings of understanding regarding them. One proof is good; many proofs are better; some proofs are better than others. Proofs aren’t application-permission-granters, or yes-or-no tickets into the mathematical corpus, or puzzle-solving tricks. They are tools for understanding.
The colossal Langlands Program is a prime example. Tracing its roots through the reciprocity laws of Gauss and Artin, and generalizing massive swaths of number theory ranging from Fermat’s Last Theorem to the Birch and Swinnerton-Dyer Conjecture, the Langlands Program seeks to understand vast arrays of striking mathematical harmony.
This proposal explains this example as well as the previous three.
This is not particularly controversial. Philosophers have studied mathematical explanation for some time. I won’t attempt to define what these others could define better. This article isolates explanation as the critical element in the mathematical endeavor.
The most fascinating part is that to explain might require one to travel arbitrarily deep into the mathematical expanse. There’s no telling how far one will have to go. Somewhere down there, though, awaits the solution, an argument so abstract and universal that it will explain everything we’ve seen so far and then some. That was Gauss’, Artin’s, and Langland’s goal. That’s what proof aims to do.
- Neukirch: Algebraic Number Theory
- Davenport: The Higher Arithmetic
- Wikipedia: Quadratic Reciprocity
- Clay Math’s official statement of the Hodge Conjecture, by Pierre Deligne
- Matthew Emerton on the Hodge Conjecture
- Wikipedia: Dirichlet Problem
- Shafarevich: Basic Algebraic Geometry
- Jeremy J Gray: The Riemann-Roch Theorem and Geometry, 1854-1914