Principia Mathematica
Perhaps the most serious attempt to formalize number theory was by the two mathematicians Bertrand Russell and Alfred North Whitehead. They created a work, Principia Mathematica, which derived number theory by the manipulation of symbols using formal logic. This work was very detailed, and they spent the better part of a decade in writing it. It wasn't until page 379 of the first volume that they were even able to prove that 1+1=2.
Russell's philosophy of mathematics was not formalist, however; it is usually considered a form of logicism. He strongly criticized Hilbert's formalism.
Read more about this topic: Formalism (mathematics)