I kind of love this Notices article, “Fingerprint Databases for Theorems,” by Sara Billey and Bridget Tenner. They make the excellent point that the OEIS is a really amazing database, not only of integer sequences, but of theorems; if you find some sequence of integers appearing in a computation you’re doing, the OEIS not only helps you guess further terms, but tells you what else is known about that sequence, and where to find it.
Billey and Tenner ask a very natural question I’d never thought of — how much of the rest of mathematical practice can be catalogued in this way? It is a very hard problem, of course, to develop a full machine-readable database of theorems, which would allow you to ask whether whatever you were trying to prove about the Laurent phenomenon for cluster algebras (say) was already somewhere in the literature. But Billey and Tenner argue convincingly that the perfect shouldn’t be the enemy of the good here — rather than fantasizing about ultra-intelligent oracles, why not catalog the things we can catalog? Integer sequences are one such thing. So are permutation avoidance patterns, which are recorded in a database Tenner maintains.
A fingerprint need not determine the object up to isomorphism. Billey and Tenner give the example of finite graphs: you can certainly store a big list of graphs, but checking whether some graph you encounter in nature is isomorphic to something on the list is intractably hard. On the other hand, storing data like a list of vertex degrees might be good enough in practice to determine whether the graph you’re studying is one with theorems already attached to it.
Readers: what are good examples of fingerprintable mathematical objects? Representations of symmetric groups? (Of course one can in some sense list them all, but the same is true for integer sequences — the point would be to catalog those which actually arose somewhere, and say where.) Transitive permutation groups? Hyperbolic 3-manifolds?