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It turns out that the answer to this question is no—from a computational viewpoint, bases may not exist for vector spaces. For this reason, we say that the Basis Problem is not a “computable” problem. We can, however, ask just how far from computable the Basis Problem is and what other problems have the same computational power. A natural way to compare the algorithmic difficulty of two problems is to determine whether having the ability to solve one allows a person to solve the other. In this case, the latter problem must be no “harder” than the former. Under this problem-reduction approach, two problems have the same computational power if they each can be used to solve the other. In my talk, we will explore the key ideas behind taking a “computable” perspective on mathematics and how this compares with an “existence” one, as in the example above. We will also illustrate the problem-reduction approach using problems from across mathematics. Moreover, this approach has strong connections to proof theory, specifically, calibrating exactly which axioms of mathematics are needed to prove the original existence theorems. My talk draws on ideas from computability theory (also known as recursion theory), a branch of logic. To learn more about what logic offers other areas of mathematics,

Different Problems, Common Threads: Computing the Difficulty of Mathematical Problems