At the end of the 19th century there was a crisis in the epistemology of mathematics. Previously, it had always been assumed that mathematics was just obviously and intuitively correct. There was criticism of some mathematical concepts, such as Leibniz's infinitesimal numbers, but for the most part what mathematicians did was taken to be both necessarily true and indubitably knowable.
Then, in the 19th century, along came some problems for this view. Several mathematicians developed non-Euclidean geometries. Gauss appears to have been the first, but he was deterred from publishing his work, apparently because he thought it would damage his repuation if he was interpreted as saying that Euclid was wrong. Later in the 19th century, non-Euclidean geometry became well established and turned out to have useful applications.
Another problem arrived when Russell demonstrated his eponymous paradox in set theory. The crisis was now established. If mathematics was supposed to be intuitively obvious, why was Euclid seemingly wrong and why was there a contradiction in set theory?
The upshot was an attempt by several mathematicians to set mathematics on a sound footing. Whitehead and Russell's Principia Mathematica, building on the logic of Frege, proposed to resolve the issue by attempting to reduce all of mathematics to logic. Brouwer developed intuitionism as an attempt to show that mathematics is founded on what is provable by human cognition. Hilbert challenged mathematicians to undertake a program to formalise all of mathematics into axiomatic theories using finitary methods of proof.
The logicist program is mostly regarded as a bust today. Some of the axioms Russell relied upon, particularly the axiom of reducibility, were widely criticised and Russell himself acknowledged that he could not see a reason to accept it as necessary. Gödel's incompleteness theorems punched a hole in the project. However, there are some neo-logicists who continue to fly the flag for a weaker version of logicism. Intuitionism in mathematics is not highly popular, though it remains an important minority view, and logicians such as Dag Prawitz and Per Martin-Löf have worked on it.
It is debatable whether reducing mathematics to logic really solves the epistemological issue. Arguably, it just pushes the question back a step to how we explain and justify logic itself.