The War on 2 + 2 = 4

If you can lie about mathematical truths, what can't you lie about?

The Party told you to reject the evidence of your eyes and ears. It was their final, most essential command. [Winston’s] heart sank as he thought of the enormous power arrayed against him… And yet he was in the right! They were wrong and he was right. The obvious, the silly, and the true had got to be defended. Truisms are true, hold on to that! The solid world exists, its laws do not change. Stones are hard, water is wet, objects unsupported fall towards the earth’s center. With the feeling that he was setting forth an important axiom, he wrote: Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.
—George Orwell, 1984

Simple Truths?

Back in the early 1990s, I flirted briefly with working for one of the big U.S. intelligence organizations. No job in the end materialized, but I did get to the point of taking a polygraph (or lie detector) test. The examiner explained to me the difference between true and false: "You see that door over there. It's blue. If you say it's blue, you're telling the truth. Otherwise, you're not."

That seemed straightforward enough at the time. More recently, I came across a former elementary school teacher who used to explain to his younger students the difference between telling the truth and lying: he would tell boys that if they said they were boys, they were telling the truth, but not otherwise. Likewise, for girls, if they said they were girls, they were telling the truth, but not otherwise. It seemed uncontroversial at the time. Aristotle would have approved. He defined truth as to say of what is that it is, and of what is not that it is not.

The color of a door. The sex of a human person. What could be more straightforward? Where could truth be less controversial? Sure, there can be borderline cases. With sex, in rare cases, a blending of sexual characteristics can occur (as with intersex individuals). With colors, ambiguous colors exist where one color seems perceptually to blend into either of two distinct colors, as at the borderline between green and blue. There are also color illusions, where the same swath is perceived as having distinct colors depending on how in context the swath is arranged with other colors. Here’s an example:

But even in all such cases, any lack of clarity underscores that in most cases clarity is evident. The ambiguity in the anomalies puts in sharp relief the clarity of the norm. Moreover, it seems nonsensical to give up on clarity, which applies to the vast majority of cases, because of ambiguity in a tiny minority of anomalous cases.

The War on Biology

The war on door colors is probably not one that the left will fight any time soon, if only because not much is riding on door colors. But the war on biology is a different matter. We are biological creatures. Much of human identity resides in our biology. Biology constrains who we are and what we can do. Such constraints go beyond sex differences. Human intelligence, emotional well-being, social adeptness, athleticism, etc. all depend to some degree on biology. Consequently, differences in these capacities depend to some degree (though not completely) on differences in our biology.

Unfortunately for biological constraints, they stand in the way of bringing about the radical social and cultural change that the left desires. For the left, society needs to be changed because, as it stands now, it gives oppressors too much power to oppress. Changing society thus becomes a matter of justice, delivering the oppressed, unseating the oppressors.

The left therefore has job to do. Yet it can only do that job if polarities of oppression and victimhood are there to be exploited. Without these polarities, the left is out of a job. It's also a reason that redemption is so hard to find on the left. Redemption would mean a reconciliation of oppressed and oppressors, of victims and victimizers. But reconciliation is never the aim of revolution, with its valiant crusade against injustice, its demonization of those deemed reactionary, and its eradication of anything that taints paradise.

Victimizers and oppressors are, for the left, evil. In fact, they are so evil that they cannot be rehabilitated. Rather, they must be neutralized and humiliated. And if that doesn’t work, they must be destroyed. Repentance for the left is never redemptive. Even those who own their guilt in abject confession are not to be reinstated into the cultural and social mainstream. Rather, like corpses left to rot on a gibbet, their punishment becomes a graphic reminder to dissuade others from transgressing.

Invalidation, cancellation, eradication, and indeed annihilation is the proper end for the reactionaries who get in the left's way, reactionaries being those deemed oppressors as well as those deemed complicit with oppressors for not attacking the oppressors with sufficient vigor. The left seeks purity. Indeed, it is implacably committed to purity. And anything that even gestures at an impure thought or motive cannot be tolerated. This readily leads to cannibalism, where members previously in good standing are suddenly no longer pure enough and must face the penalty of their transgressions, with the threshold for what counts as a transgression constantly being lowered.

Happily for the left, victims to be championed are never in short supply because they can always be manufactured. Biology provides one such source of victims. Biological constraints, varying as they do, can be interpreted as consigning certain people to disadvantaged classes, and such classes can then be identified as victims. Now there is a measured approach to dealing with such biological constraints, namely, to admit that they are real but at the same time to try to advance the best interests of those so constrained, and without harming those facing other constraints (we all face constraints).

The other approach is to invalidate biology, simply denying the biological constraints and removing them by fiat. This latter option, now fully embraced by the left, means that those who continue to assert that such biological constraints remain in force must in turn be invalidated — no matter that those who deny the reality of biological constraints often do themselves irreparable harm.

But how to accomplish this repudiation of biological truth? Most people, I submit, start out where my retired elementary school teacher started out: boys are boys and girls are girls, end of story. Clearly, that's no longer the end of the story. But the question is why it's no longer the end of the story. We didn't get to the place where a biological male could assert "I'm a girl" or a biological female could assert "I'm a boy" — and be taken seriously — without some sort of rationale.

Self-Creation and the Social Construction of Knowledge

That rationale for reinventing biology begins with a philosophical doctrine of self-creation. Friedrich Nietzsche resisted values being imposed on people and urged that truly enlightened individuals hammer out their own aesthetic and ethical values. Michel Foucault added to such self-creation the role of power relations, both societal and individual. Toward the end of his life, Foucault focused on “technologies of the self,” which he described as “techniques that allow individuals to effect by their own means or with the help of others a certain number of operations on their own bodies and souls, thoughts, conduct, and way of being, so as to transform themselves in order to attain a certain state of happiness, purity, wisdom, perfection, or immortality.”

Most ordinary people reading these remarks would ask themselves whether any techniques are powerful enough to induce such thorough-going changes in the self. Reality, it seems, could be an obstacle to self-creation at this scale. Enter the social construction of knowledge to remove this pesky obstacle. I first encountered knowledge as a social construction in pursuing a doctorate in philosophy, where I focused on the history and philosophy of science.

During my graduate student days in philosophy, David Bloor and Bruno Latour were key figures in the social construction of knowledge as applied to science. When I first encountered the social construction of knowledge, I thought it insane. Since then I've come to regard it more as an elaborate fraud, as I now appreciate much more the cultural heavy-lifting that this approach to knowledge accomplishes for the left.

According to the social construction of knowledge, knowledge is socially constructed, full stop. In other words, knowledge is what our social group deems to be knowledge—again, full stop. The full stop here is important. It’s not that there is an external reality to hold us to account. It’s not that there are canons of logic to which we must adhere. It’s not that there’s an objective realm to which we must bow the knee. It’s not that there are any areas of knowledge that can command universal intersubjective agreement. Knowledge is what the residents of the asylum take it to be.

This approach to knowledge does come with a backhanded justification in the form of a developed skepticism about any overarching metanarrative. All comprehensive and universal (“totalizing”) approaches to knowledge are thus deconstructed, shown to endlessly defer meaning, lack any compelling foundation, and so allow anything to go. Thus we find ourselves in an Alice-in-Wonderland world where anything is possible because we can imagine, and even demand, that it be so. Without universal norms and truths, groups make things up any way they like. The philosopher Richard Rorty thus defined truth as what his peers (social group) would let him get away with.

To this, the Christian philosopher Alvin Plantinga replied, What if I don’t let you get away with that? Plantinga’s point was that this socially constructive view of knowledge is self-referentially incoherent. He’s right. If all knowledge is socially constructed, then the claim that all knowledge is socially constructed is itself socially constructed, which means that it is pretending to an objectivity and universality to which it has no rights.

Even so, Rorty and his disciples couldn’t care less about Plantinga’s comeback because self-referential incoherence is simply the logic of reductio ad absurdum, and a socially constructed knowledge owes no allegiance to logic. Sure, Plantinga’s social group might want to retain such logical standards. But others may not and are in their rights not to—at least from the perspective of the constructivists.

To see what's at stake with the social construction of knowledge, consider the following mini-lesson in epistemology (the branch of philosophy that studies knowledge) drawn from the game of baseball (I'm indebted to physicist John Archibald Wheeler for this lesson, though he probably got it elsewhere). The plate umpire in baseball is responsible for calling balls and strikes. In justifying the validity of his calls, he might take three approaches:

1. I calls 'em the way I sees 'em.

2. I calls 'em the way they is.

3. They ain't nothing till I calls 'em.

The first of these takes an empiricist or observational approach, basing knowledge on perception. The second takes a realist approach, seeing the mind as capable of accurately grasping, at least to some extent, the way the world really is. The last sees all knowledge as a human construction that owes nothing to independent observation or an objective reality.

So consider again biology. Biology, as it was sociologically constructed in the past, put a premium on sex differences, on intelligence as a genetic endowment (at least in part), etc. But as a social construction, biology could just as well have been put on a completely different basis. Rather than sex differences being fundamental, what if physiology were socially constructed to be peripheral, rendering sexual identity malleable and even volitional. Taken to its logical conclusion, which is where we are today, anyone can define oneself to have any sexual identity whatsoever.

There is a genre of horror fiction in which the antagonists can take any form whatsoever. They can assume any guise, walk past any barrier, infiltrate any system, operate with complete impunity, and wreak any havoc. Like the Greek god Proteus, they can be whatever they want to be. The social construction of knowledge allows anyone to be Proteus. But that's not all. Proteus's power was in transforming himself. Armed with the sociology of knowledge, one gets not only to remake oneself but also to remake the world to be whatever one desires it to be. Indeed, if knowledge is socially constructed, then reality itself, which is the object of knowledge, becomes socially constructed.

Such a view, in which reality can be whatever we decide it to be, is not far removed from the lunatic asylum. And yet, its advocates on the left have found a way not only to avoid the asylum but also to weaponize the social construction of knowledge so that rather than have this view of knowledge subjected to scrutiny and refutation, it is used to undermine, unmask, and deconstruct the verities on which we have banked for millennia. Given that all knowledge is socially constructed (and what else could it be, asks the social constructivist), then these verities are themselves socially constructed and so are merely relative to an outdated social setup that gave rise to them in the first place.

This view is self-referentially incoherent. It cannot be reasonably maintained. Yet it can be vigorously and loudly asserted to drown out voices of reason.

The War on Mathematics

Turning finally to the title of this essay, let’s consider the claims made by constructivists about mathematics. I’ll focus mainly on Laurie Rubel, who made headlines a few years back for her Twitter posts rejecting that “math is neutral because 2 + 2 = 4,” claiming that such thinking “reeks of white supremacist patriarchy,” and claiming that “the idea that math is objective or neutral IS A MYTH.” Here’s a mindbender: Can challenging that math is objective or neutral be done objectively or neutrally? If not, why accept the challenge?

At the time she made these Twitter posts, Rubel was a professor of secondary education at Brooklyn College in New York. Her main affiliation now is as a math education professor in Israel at the University of Haifa. Moving beyond Twitter, she has extended her war on 2 + 2 = 4 to more substantial academic forums. Consider a 2020 Springer anthology (with a thoroughly international cast of contributors—this war on math is widespread) titled Borders in Mathematics Pre-Service Teacher Education. “Pre-service” here refers to getting teachers while they’re still in training—young and malleable—before they become full-fledged in-service teachers.

Rubel co-authored a paper in this anthology titled “Queering Mathematics: Disrupting Binary Oppositions in Mathematics Pre-service Teacher Education.” Here’s the upshot (pp. 238–239), which I quote at length to make clear the mindset and motivation, combining as it does a constructivism of knowledge with an agenda to advance a leftist conception of justice:

Queering mathematics inquires about and questions boundaries, not only around social categories of gender and sexuality but also around mathematical categories (Sheldon, 2019). Sheldon and Rands (2013) and Sheldon (2019) highlight the possibility of queering mathematical concepts like time, infinity, space, measurement, place-value, and more. Mathematics is not universal, “the same for everyone,” but a human activity, enriched by the diverse intellectual activity bound in life. Authentic cultural mathematical practices are connected to lived experiences, to bodies, immediate needs, and desires (Greer et al., 2012; Mukhopadhyay & Roth, 2012), implying that mathematical activity can be multisensory, can involve or benefit from our hands, bodies, or eyes, and can involve direct interaction with the physical and social world. Queering mathematics, as well as new kinds of logics and ways of thinking about gender and sexuality, implies creating space for new ways of mathematical thinking. In other words, queering mathematics will support us to interpret existing questions in new ways, ask altogether new questions, challenge premises that seem no longer self-evident, develop new kinds of representations and arguments, see patterns that may have been invisible before, and will ultimately support us in solving both new and heretofore unsolved problems.

These are extraordinary claims, and they require, if not extraordinary evidence, at least some evidence. Indeed, the burden of evidence is on those who would deny that mathematics is universal. Proofs in geometry and number theory by the ancient Greeks (such as Pythagoras, Euclid, and Diophantus) have stood the test of time and are still valid. If anything can claim universality, it is mathematics. And if anything can’t claim universality, it is faddish theorizing by leftist thinkers like Rubel. The queering of mathematics, we are assured “will ultimately support us in solving both new and heretofore unsolved problems.” Really? When exactly? And in what field? Mathematics? I’m willing bet that the queering of mathematics will never lead to the resolution of any outstanding open mathematical problem deserving of a Fields Medal.

Rubel is hardly alone in her war on 2 + 2 = 4. A group of math teachers in Ontario is reported likewise as having declared war on 2 + 2 = 4, regarding this equation as an assertion of white supremacy. The Vanderbilt Peabody College of Education and Human Development lists among its faculty Luis Leyva. He achieved notoriety for a lecture titled “"Undergraduate Mathematics Education as a White, Cisheteropatriarchal Space and Opportunities for Structural Disruption to Advance Queer of Color Justice." Here is Leyva’s mission:

Using an interdisciplinary approach, Leyva’s research explores how interlocking systems of power, including racism and cisheteropatriarchy, impact classroom teaching and student support in undergraduate mathematics and STEM higher education. He aims to disrupt those systems and advance intersectional justice through education in math and science.

I find it curious, as a professional mathematician (UofChicago PhD, 1988), that the people weighing in against 2 + 2 = 4 are not themselves mathematicians but in education departments where they teach the teaching of mathematics. They embody the old expression, “Those who can, do; those who can’t, teach; and those who can’t teach, teach teachers.”

What Is the End Game?

What is the ultimate aim of this war on math? George Orwell was remarkably prescient in understanding the war on 2 + 2 = 4 and its purpose. In a word, the ultimate aim is tyranny over the life of the mind. At the start of this essay, we saw the character of Winston characterize 2 + 2 = 4 as the last bastion for a free life of the mind. It’s therefore precisely at that point that the totalitarian system in which Winston finds himself seeks to crush his spirit.

Thus later in 1984, O’Brien, the totalitarian inquisitor of Winston, subjects him to interrogation and torture. In the interrogation, O’Brien diagnoses Winston’s problem. As you read the following passage from that interrogation, ask yourself where you’ve seen all this before. Spoiler alert: you will be reading an exact statement of the social construction of knowledge, where the social entity that’s doing the knowledge construction is “the Party.” O’Brien is here speaking to Winston:

“You believe that reality is something objective, external, existing in its own right. You also believe that the nature of reality is self-evident. When you delude yourself into thinking that you see something, you assume that everyone else sees the same thing as you. But I tell you, Winston, that reality is not external. Reality exists in the human mind, and nowhere else. Not in the individual mind, which can make mistakes, and in any case soon perishes: only in the mind of the Party, which is collective and immortal. Whatever the Party holds to be the truth, is truth. It is impossible to see reality except by looking through the eyes of the Party. That is the fact that you have got to relearn, Winston. It needs an act of self-destruction, an effort of the will. You must humble yourself before you can become sane.”

What follows immediately after this passage is a torture scene in which O’Brien tortures Winston. I quote it at length not only because it is prophetic for our age but also because in points to a significant variation in the challenges we face today. Here’s the scene:

“Do you remember,“ [O’Brien] went on, “writing in your diary, ‘Freedom is the freedom to say that two plus two make four’?”

“Yes,” said Winston.

O’Brien held up his left hand, its back towards Winston, with the thumb hidden and the four fingers extended.

“How many fingers am I holding up, Winston?”

“Four.”

“And if the party says that it is not four but five—then how many?”

“Four.”

The word ended in a gasp of pain. The needle of the dial had shot up to fifty-five. The sweat had sprung out all over Winston’s body. The air tore into his lungs and issued again in deep groans which even by clenching his teeth he could not stop. O’Brien watched him, the four fingers still extended. He drew back the lever. This time the pain was only slightly eased.

“How many fingers, Winston?”

“Four.”

The needle went up to sixty.

“How many fingers, Winston?”

“Four! Four! What else can I say? Four!”

The needle must have risen again, but he did not look at it. The heavy, stern face and the four fingers filled his vision. The fingers stood up before his eyes like pillars, enormous, blurry, and seeming to vibrate, but unmistakably four.

“How many fingers, Winston?”

“Four! Stop it, stop it! How can you go on? Four! Four!”

“How many fingers, Winston?”

“Five! Five! Five!”

“No, Winston, that is no use. You are lying. You still think there are four. How many fingers, please?”

“Four! five! Four! Anything you like. Only stop it, stop the pain!”

[O’Brien stops the pain and comforts Winston.]

“You are a slow learner, Winston,” said O’Brien gently.

“How can I help it?” he blubbered. “How can I help seeing what is in front of my eyes? Two and two are four.”

“Sometimes, Winston. Sometimes they are five. Sometimes they are three. Sometimes they are all of them at once. You must try harder. It is not easy to become sane.”

Writing in the era of Stalin, Orwell saw brute force as the way to instill faith in the Party line. During the torture, O’Brien tells Winston it’s no use and he’s lying. But 1984 ends with Winston coming around, making his peace with 2 + 2 equaling anything. The last two sentences of 1984 read: “He had won the victory over himself. He loved Big Brother.”

This is where the present moment departs from Orwell’s prophetic vision. Today’s left does not torture us into believing that 2 + 2 could equal anything. Rather, it seeks to convince the young and teachable that 2 + 2 could equal anything by indoctrinating them into this belief. And the older people, like Winston, rather than being indoctrinated to accept that 2 + 2 could equal anything, are given strong incentives to give into or play along with the lie that 2 + 2 need not equal 4.

In his book Influence: The Psychology of Persuasion, the psychologist Robert Cialdini describes how American soldiers held captive by the Chinese during the Korean War were brainwashed. It was not the thorough-going brainwashing of a Manchurian Candidate. Rather, it was a subtle brainwashing that in place of brute force substituted compromise and complicity.

American soldiers were given rewards if they wrote essays putting a positive spin on China and a negative spin on America. They were told they didn’t need to believe what they were writing. But the more they praised China and decried America in their writings, the more they themselves were praised and the more benefits they received during their captivity. Those same soldiers, when they returned to America, remained in the thrall of China, thinking much more highly of it than their military peers who had not been subjected to Chinese brainwashing.

These soldiers made themselves complicit in a lie, though not by coercion but by choice. They thus exemplified the antithesis of Alexandr Solzhenitsyn’s point in his essay “Live Not By Lies.” There he wrote: “We find, neglected by us, the simplest, the most accessible key to our liberation: a personal nonparticipation in lies! Even if all is covered by lies, even if all is under their rule, let us resist in the smallest way: Let their rule hold not through me!”

For the purposes of this essay, Solzhenitsyn’s point can be restated more simply: Once we become complicit in a lie, those for whom we lie own us. An immediate corollary of this principle is that if you can swallow one truly big encompassing lie, all the other lies follow, and you won’t just be owned but you’ll be completely owned/compromised.

Thus, from the vantage of those promoting the war against 2 + 2 = 4, it doesn’t matter that failing to respect this and other mathematical truths will lead to bridges collapsing or wheels coming off airplanes (as has been in the news recently). Not that this matters to the left as it pushes for the queering of mathematics. As far as they’re concerned, faulty bridges and unsafe aircraft are a small price to pay for the injustices committed in the name of white supremacist mathematics. Better to queer mathematics and let justice prevail.

But this misses the point. It’s not that math is being devalued or that math education is being undermined (though, yes, that is happening). It’s that by agreeing to question 2 + 2 = 4, we have tacitly agreed to question everything, putting everything up for grabs. Everybody knows deep down that two plus two does equal four. If that truth can’t be asserted with confidence, forget about other truths, such as the truths of biology, and even the truths about colored doors.

But what if 2 + 2 = 4 can be asserted with confidence? As Winston put it in 1984 before he himself was compromised and lost his freedom: Freedom is the freedom to say that two plus two make four. If that is granted, all else follows. That’s right. It means there’s some domain where tyranny can stake no claim.