We need a critical mathematics. Our ability to enact social change through teaching mathematics and through critical research in mathematics education is severely limited by the canon of mathematical knowledge and the ideology that declares mathematical knowledge as certain. In this dissertation I extend Carspecken's critical action theory to articulate a philosophy of mathematics that hinges on our existential needs for recognition, the experience of error, and the limits of knowledge. Those limits stand contrary to the canon, where what counts as mathematical knowledge is either provable or instrumentally useful. But proof and instrumental utility must come after communicative rationality in the order of explanation, just as the scientific method must come after communicative rationality in that same order (Habermas, 1971). The story of communicative rationality is the story of getting it wrong; rationality is structured by uncertainty and driven by our existential needs for recognition and the experience of error. Uncertainty, existential needs, and error are the shadows that bind mathematical systems together across individuals and ages -- we all do math wrong sometimes. Why does the experience of error drive communicative rationality? What is at stake? The stakes are our being as normative subjects. We live in existential fear as error threatens our social identity, the {me}. The {me} is a complex bundling of commitments, errors, and successful acts of consumption that, for consciousness, are finite forms generally expressed through language. But the {me} is not all that we are - we are also the {I}. The {I}, I shall endeavor to express, is "in"finite - a bursting of finite form. Knowledge of that {I} is emancipatory, or critical (P. Carspecken, 2009). With this in place I will develop a philosophy of number and operation, consider how to incorporate the vast body of research that documents student errors into the system as described, and describe a research method for analyzing student work using projective inferences (Brandom, 2000). Finally, I explore second-person transactional thought, answering the question "Who are you?". [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]