  Understanding and creating mathematics using natural mathematical language -
the mixture of symbolic and natural language used by humans - is a challenging
and important problem for driving progress in machine learning. As a step in
this direction, we develop NaturalProofs, a multi-domain corpus of mathematical
statements and their proofs, written in natural mathematical language.
NaturalProofs unifies broad coverage, deep coverage, and low-resource
mathematical sources, allowing for evaluating both in-distribution and
zero-shot generalization. Using NaturalProofs, we benchmark strong neural
methods on mathematical reference retrieval and generation tasks which test a
system's ability to determine key results that appear in a proof. Large-scale
sequence models show promise compared to classical information retrieval
methods, yet their performance and out-of-domain generalization leave
substantial room for improvement. NaturalProofs opens many avenues for research
on challenging mathematical tasks.
