In my thesis, titled “Towards a Critical Mathematics,” I seek to describe the shadow side of mathematical knowledge. This shadow is the desire for recognition, often experienced as existential fear related to the possible loss of recognition or shame that often accompanies error. The ‘growth’ mindset is ubiquitous in mathematics education, but I seek to make explicit a system of mathematical knowledge that acknowledges the ways in which such growth hurts. Poetically, the shadow is the blue in the growing green of mathematical becoming.

To do this, I take self-certainty and the experience of error to be fundamental - they are like meta-axioms which bind together all levels and philosophies of mathematics. Most of us are trained to erase our errors, but this means that we lose the substance of mathematical concepts. My thesis develops a logic based on the exclusion of error, not so they can be erased and forgotten but rather so that they can be elevated and preserved as essential for understanding.

Without knowing what specific errors are to be excluded, the resulting mathematics would be as conceptually empty as the canon mathematics which critical mathematics critiques. So I also demonstrate how the literature of mathematics education, much of which categorizes various kinds of errors one might wish to exclude, can be incorporated into a mathematical system.

Mathematicizing literature brings two types of knowledge into conversation with each other, which is in itself enough to earn the name “critical” but my work is critical in other ways too. Instead of limiting our interest to just the fruit of mathematical thought, we can also express interest in the roots and the soil. It opens mathematical inquiry to why we do mathematics. What existential needs for recognition does mathematical inquiry meet? What prevalent attitudes in mathematics classrooms suppress those existential needs? In what ways might those suppressions be racialized, gendered, ableist or in other ways distorting?

We who say “we” all have an existential need to be recognized as good people, we have a need - at times - to be seen as conforming to some socially acceptable rational norms. Using language means at least partially conforming to normative expectations. But we do not just conform. We also create, destroy, dissolve, undifferentiate, and deconstruct. In short, our existential needs for recognition are boundless. To be human is to exist at once socially as a conformist “me” and singularly as an infinite “I” and so the form of the human subject is both its formlessness and conformity. We are both finite and infinite.

 In explicating this shadow while simultaneously producing a workable pragmatist philosophy of mathematics, my thesis pushes mathematics and mathematics education forward in unexpected and beautiful ways.