What Have You Learned about Math Today?

Covers of grade 1 and 2 math textbooks from the former GDR which include pictures of a carousel and of a roll call.
Covers of grade 1 and 2 mathematics textbooks from the former G.D.R.

George Orwell was not a mathematician.

And neither am I.

James Lindsay, however, was trained as one. What has he been teaching us about mathematics lately?

Over the past weeks, a corner of academic Twitter has been in deep debate about the possibility of 2+2 not always being equal to 4. It doesn’t matter much where we say the debate started (if it can be said to start anywhere in particular), but the sides in the debate seem quite clear.

On the one side, we have those who sternly remind us that 2+2 is always equal to 4. Claiming anything else is portrayed as needless confusion, abhorrent wokeism, and the undoing of civilization. That is the side where James Lindsay — 2010 PhD in combinatorics — responds to alternative examples by Kareem Carr — MSc in mathematics and MA in statistics — with, “You can’t be this stupid, which means you’re completely dishonest,” or “Wrong. Lie. Stupid.” It is the side where Lindsay writes a long essay that absurdly frames the other side’s discussion as part of a potentially violent revolution, but does not deign to provide evidence for such claims. In this essay he also, albeit reluctantly, explains relevant mathematical notation and bemoans how “tedious” and “stupid” it is that he has to do so.

On the other side, we have those who delve into explanation of how axioms and definitions determine how mathematical operations are carried out, who give examples of how variation of axioms and definitions can lead to different results, who illustrate how mathematical operations can be tied to the cultural contexts in which they might be applied, and who remind us of the interesting history of debate behind 2+2. If the recent surge of this debate has a Twitter starting point, then perhaps in the concern expressed by Laurie Rubel — professor of secondary education — about how language of mathematics, including stern repetition of 2+2=4 always, can align with other expressions of dominance. This concern is tied to the desire of mathematics educators to understand, and alleviate, how the examples and language chosen in mathematics teaching can close the field off for students for whom those examples and that language pose barriers to learning. That is not a controversial position.

As a non-mathematician, I have learned from the latter side about possibilities in translating applied issues into mathematical operations and about how numbers and letters are used in different base systems of counting. Whenever someone more of a mathematician than I posted an equation in which 2+2 did not equal 4, I asked myself: what would be the conditions under which this is the case? And I have learned by following links to find an answer to that question. In the process, I have become aware of the deep history of discussion behind 2+2=4 and 2+2=5. It has been fun.

I can’t say I have learned much about mathematics from James Lindsay’s side in the debate — or, not until weeks into the debate, when he discussed in a long blog post, and with much protestation, how mathematics handles the notation issues posed by the 2+2=5 examples that were brought forward. I have noted, however, the outrage provoked by the suggestion that a natural science discipline as pure as mathematics is part of and shaped by the cultural contexts that produce and use it. I have also noted the — perhaps we should call it vestigial? — racism that can underlie outraged reaction to the idea that success in mathematics teaching and research depends on cultural and political contexts, and that in order to resist the effects of colonization, white supremacy, and discrimination, one might need to question how research and teaching in mathematics is currently carried out.

It is not always obvious to everyone that James Lindsay’s protestations can provide harbour to white nationalist or white supremacist ways of thinking, but there is evidence that sometimes they do. In a rare, non-fraudulent academic publication titled “Decolonizing the Curriculum,” Lindsay clearly implied that the contemporary writing of Ta-Nehisi Coates was somehow less part of Western culture than the historical writing of John Stuart Mill. He claimed, “Changing specifics about the curriculum, say by replacing John Stuart Mill with Ta-Nehisi Coates, is merely a proximal tool in a far more radical project that could threaten not only Western culture but the tremendous strides into modernity some aspects of that culture have enabled much of the world to take.” Race also seems a factor in how dismissively Lindsay has turned against every of Kareem Carr’s tweets in the 2+2 discussion.

In a recent podcast episode, Aaron Rabinowitz carefully examined the implications of Lindsay’s collaboration with a Christian nationalist organization called Sovereign Nations. In his long essay on “2+2 Never Equals 5,” Lindsay makes the unsupported claim that the recent Twitter discussion seeks a “radical rewriting of the entire rational project, and any reason that doesn’t forward [the activists’] favored actors as the sole arbiters of what is true and correct needs to be deconstructed by rhetorical tricks and marginalized by moral, and, perhaps, physical force and intimidation.” He says that critical approaches in mathematics education are an “existential risk to advanced modern civilization” and lead to a “breakdown of the fundamental logic of civilization.” With such over-dramatic and fearful thinking about the immanent destruction of Western civilization, white nationalist thought is often not far behind.

George Orwell was not a mathematician. When he wrote about 2+2=5 in his novel 1984, he used this equation to satirize the potential of authoritarian power to force people to repeat and thereby to believe something that goes against established knowledge. In this fictional process, authoritarian power does not tolerate any questions or resistance, and tries to coerce its subjects into repeating and believing something that it itself knows very well to be untrue. Through imagining this particular act of coercion, Orwell considered how sheer force can distort and curtail public knowledge.

When mathematicians write on Twitter about possible conditions under which 2+2 could be said to equal 5, there is not any such coercion nor any withholding of what these writers think is true. Rather, they use the phrase 2+2=5 to playfully demonstrate that mathematical operations can look differently depending on their axioms and cultural contexts. They do not ask anyone to believe that 2+2 does not equal 4 in base 10. Instead, they show how public knowledge can be expanded by thinking in different bases and different definitions. They reassure their audience that even outlandish questions can be asked and entertained in that process. That kind of flexibility and playfulness in thinking — while still asserting what is established knowledge — is a mark of being a good educator.

I see two general approaches. One takes a mocking and righteous attitude, the other an attitude of educational playfulness. Which one is more effective in engaging a wide audience in questions on how thinking in mathematics works? I suggest James Lindsay’s mocking approach sounds like it is trying to make others accept his view without also encouraging the asking of curious, perhaps even outlandish, questions. In contrast, the playfulness of taking 2+2=5 seriously for a moment encourages expression of a diversity of ideas while also marking the boundaries of how and whether these ideas can be considered true.

Thanks to Nicolas Bray for feedback on this piece.

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