Recent scholarship warns that computing work has treated problematic features of the status quo as fixed, failing to address and often even exacerbating deep patterns of injustice and inequality. This begs the question: what roles, if any, can computing play to support and advance fundamental social change? Through an analysis informed by critical scholarship, we articulate four such roles -- computing as a diagnostic, formalizer, rebuttal, and synecdoche. We then illustrate how mathematical and computational tools can aid in understanding and tackling poverty and social inequities through the role of computing as a formalizer. We close with a discussion on the Mechanism Design for Social Good (MD4SG) research community, which works to bridge research and practice to ensure that such insights can be leveraged to advance social change.