View from
The Left

On the “Sokal Squared” Trio

(Mike Nayna)

“Now, this is not a review of the book, nor a critique of them personally but, rather, a critique of their function as public intellectuals.”

As the culture war rages on—and now with “cancel culture” and whether “2+2=4” is true (yes, really) being central topics of discussion—there are particular public figures who are bound to rise in prominence, not in the least because their public profiles were built on the culture war. Among the most salient, in my view, are the three authors of the Sokal Squared hoax, named after the original hoax by American physicist Alan Sokal. The three of them—Helen Pluckrose, James Lindsay, and Peter Boghossian—have devoted much of their public careers to criticizing cancel culture and what they call “grievance studies,” most recently in Pluckrose and Lindsay’s newly released book Cynical Theories. Now, this is not a review of the book, nor a critique of them personally but, rather, a critique of their function as public intellectuals. I think the three of them are interesting because they are representative of a larger phenomenon that seems inevitably tied to culture warring. To be sure, cancel culture is a problem in some sectors of the Left, and the type of scholarship these three writers criticize—which is overwhelmingly left-aligned—has real issues, and many on the Left have pointed this out (a fact, by the way, which they often ignore). Now, I even believe some of their specific critiques are warranted. Yet, I think who makes certain critiques—and why—is an essential piece of information.

To critique their positions on these matters, even if there is some agreement, is necessary for the Left because the direction in which the criticism is pushing matters. Importantly, I believe their criticism of disciplines such as critical and feminist theory is sloppy and unserious. In many ways, they fall into the exact form of ideologically motivated reasoning that they (sometimes correctly) accuse scholars of these disciplines of using. Of course, this is a strong claim, so I will do my best to substantiate it in what follows. There are three threads that run through Boghossian, Lindsay, and Pluckrose’s general argument that I think merit the most interest. First of these is their approach to what they group under the umbrella term “grievance studies.” Second and third—which are related—is their treatment of questions of truth and reality in themselves, and of epistemology, the way we know whether we are arriving at truth and reality.

Fields of study like feminist philosophy and critical theory (whether it be race, gender, or others), and importantly, anything related to intersectionality, all fall under “grievance studies.” Their attack on these disciplines is what originally put them in the public spotlight, particularly through the publication of various papers in gender studies journals. This is the reason behind the “Sokal Squared” moniker. Their successful publication has been described by them and by others as proof that these fields are little more than glorified sophistry. Let me preface everything that follows by saying that I do think that is an apt description of some works in the humanities. I think, however, the scope of their critique is too broad and goes much further than what the results of their hoax would allow. As an example, take on of the most publicized papers they published entitled “Human reactions to rape culture and queer performativity at urban dog parks in Portland, Oregon.” The paper makes very bold claims about themes like rape culture as inferred from human reactions toward canine interactions. In particular, it claims to look at how humans react to dogs fighting and “humping” (initiating sex with) other dogs, in relation to the sex of the dog being humped. These claims are, of course, all false because everything claimed in the paper is false. The upshot is meant to be that since the claims in the paper—ridiculous as they may be—support a particular grievance narrative, the journal editors were keen to publish them because their purpose is advancing that narrative, rather than rigorous and scientific pursuit of truth.

I can even accept that this might have formed part of the motivation for publishing. Also, I do think that had the review process been more rigorous, the paper would have never been published. It would have been useful to have some way of verifying that the observations did, in fact, take place (they did not). Moreover, the article contains some rather unique pieces of writing, such as “the gendered status of the a-/moral paradox in human interpretations of domesticated canine behavior.” It seems clear that the way “a-/moral” is spelled with a hyphen and a slash serves no linguistic purpose and its meant to mock the liberal use of punctuation by many postmodern or poststructuralist authors. But I think there is a deeper issue. It is much easier to make fake grandiose claims if one simply creates an entire fake dataset to support such claim. As I said previously, the reviewers ought to have had some way to ascertain the data; however, putting that aside, had the data been real, I find it very difficult to argue that the findings would not have been at least worthy of some attention. The paper claims that the fake author conducted observations every day for one full year (excluding days of heavy rain) for a period of between two and seven hours per day. Further it is claimed that on some occasions, the incidents had a frequency of up to one every three to five minutes. The number of data points collected under these assumptions would have been staggering—more than enough for a statistically valid sample. Moreover, according to the supposed results, 97% of humping in which the dog being humped was male, there was an intervention by humans to stop it, compared to only 32% of the time when the dog being humped was female. These results—with a sample of that size—would have been notable. In one case, there is an almost one to one correspondence between intervention, something virtually unheard of in any social science study (admittedly this should have raised some red flags). Additionally, a difference in proportions of 97% to 32% with a sample that large would unquestionably have a statistical significance way beyond that which is expected in any kind of social science study.

A further complication with Pluckrose, Lindsay, and Boghossian’s arguments is that it is not hard to find academic studies in the exact fields that they criticize that fly in the face of what they claim. Take, for example, the paper “Causally Interpreting Intersectionality Theory” by Liam Kofi Bright, Daniel Malinsky, and Morgan Thompson, published in Philosophy of Science in 2016. This paper does have some criticism for the field’s reliance on subjective experience and its suspicion of quantitative approaches. Yet, it takes specific claims made by intersectionality theory and organizes them into claims that make specific predictions, which can then be tested using statistical methods. In other words, it shows that intersectionality theory, as it exists currently, is entirely compatible with the scientific methods that the authors of the Sokal Squared hoax claim these theories have abandoned in favor of advancing political agendas. Now, it is certainly not as straightforward as testing, for example, whether two spheres of different weight fall at different rates from a tower, as Galileo famously did. However, this added complexity should—under no circumstances—be considered a hinderance to the potential scientific status of such theories. Yet, it seems to me that it is precisely this that is at the heart of the trio’s disenchantment with theories like intersectionality. In what follows I will try to explain why I believe that to be the case.

As mathematicians and philosophers pointed out, however, whether “2+2=4” is true depends entirely on the arithmetical axioms, rules, or assumptions under which we are operating.

As for the other two matters, one of Lindsay’s (and to a lesser extent Boghossian’s) latest crusades is a good starting point. It begun—as many discussions do nowadays—on Twitter, with Lindsay defending the seemingly simple assertion that 2+2=4 is an obvious immutable mathematical proposition. An article published in New Discourses, a new project the trio continues to, even goes as far as to say that we cannot build an apartment complex or bake a cake if we cannot accept this simple fact (now, this particular piece was not written by any of them, but I find it hard to believe that they would publish it if they did not support the basic argument). This essay—in line with much of Lindsay, Pluckrose, and Boghossian’s past writing—argues that a failure to accept such a basic fact amounts to nothing short of the destruction of Western Civilization. As mathematicians and philosophers pointed out, however, whether “2+2=4” is true depends entirely on the arithmetical axioms, rules, or assumptions under which we are operating. Unless one is a mathematical Platonist, who believes that mathematical objects and structures really exist somewhere, then mathematics is nothing more than a set of mental constructs or—more plausibly in my opinion—a set of rules that follows from a finite set of axioms which we accept prima facie. Here, I want to make a precision before moving on: I take Lindsay’s crusade over “2+2=4” to be about more than this particular arithmetic operation. If it were exclusively about it, it would be a rather pedantic and uninteresting point. I take it instead to be more generally about the inviolability and universality of the rules of mathematics—and whether holding that to be true is somehow a political statement.

Of course, it is undeniably true that under the standard set of assumptions and axioms of arithmetic, two plus two really equals four. Classical arithmetic is all that 99.99% of the population ever needs to use and is probably what professional mathematicians, logicians, and philosophers work with most of the time. Because of this, we do not need to think about, let alone explicitly state the axioms we are working with when we deduce that we will need four eggs to bake double the amount of cake that typically needs only two. But there is such a thing as “inconsistent mathematics,” based on using what are called “paraconsistent logics” (more on that later) as the foundation of mathematics, instead of classical logic. These might lead to some results that would seem strange to us. However, the commonality or practicality of usage of a rule should not be confused with its universality or its immutability. Lindsay would tell us that such basic facts underpin Western Civilization and science, and that alternative, non-male, non-Western “ways of knowing” add nothing to scientific understanding. He is plainly wrong on both counts.

Let us first look at this through the lens of the Western philosophical canon. His claim—even if he is not explicitly saying it—that suggests that “2+2=4” is what philosophers call an analytic statement. This identifies a statement that is true by virtue of its meaning. A classic example of this is “all bachelors are unmarried.” The definition of a bachelor is an unmarried man; therefore, the sentence “all bachelors are unmarried” is true in virtue of the meanings of the terms that it contains. Analytic truths are logical truths that follow from the rules of the language that we are using such as identity or synonymy, including formal languages such as mathematics. Another way of understanding this is that the negation of an analytic statement results in a logical falsehood. This can be readily seen through the negation of the previous example (i.e. “not all bachelors are unmarried”) which is a clear contradiction. Statements such as this should not be confused with others that may seem trivially true such as “the daytime sky is blue.” While this may seem just as obvious as “all bachelors are unmarried,” a key difference is that nothing about the definition of “daytime sky” implies that it must be blue. It just happens to be the case based on the composition of the atmosphere of the Earth. And, in fact, if we were on Mars or Venus, it would not be true. Statements of this kind, which require some form of empirical evidence, are called synthetic.

The so-called analytic/synthetic distinction was a central feature of much of Western philosophy, especially—and perhaps most notably—with the Logical Empiricists of the early 20th century, like Rudolf Carnap. It also seems to be central to Lindsay’s worldview, with his insistence on simple, universal, undeniable truths. Yet, not even the most mainstream Western philosophy would take this view so lightly. The 1951 paper “Two Dogmas of Empiricism” by American philosopher W.V.O. Quine argues that the sharp distinction between analytic and synthetic statements is—in fact—unwarranted. He also contends that there is no logical truth that cannot potentially be subject to revision. In his concluding remarks, Quine takes on something even more fundamental than 2+2=4: namely, the Law of the Excluded Middle (LEM). This principle from Aristotle—and which is arguably central to much of the Western philosophical tradition—states that a proposition is either true or false, but it cannot be neither. It is closely related to the Law of Non-Contradiction (LNC) which states that a proposition “P” and its negation “it is not the case that P,” or not(P), cannot both be true. Quine posits that some of the results coming from quantum mechanics might force us to revise the LEM. This paper is one of the most highly cited works of philosophy and has even been called “the most important in all of twentieth century philosophy.” If anything is part of standard mainstream Western thought, it is “Two Dogmas of Empiricism.” Now, before anyone accuses Quine of being a postmodern neo-Marxist, a bit of historical trivia is relevant. Quine was notably a social conservative—and one of the signatories of an open letter urging the University of Cambridge not to grant an honorary doctorate to Jacques Derrida arguing that there was a lack of rigor and seriousness in his work.

Quine, of course, was not a postmodern neo-Marxist. He was, however, a logical pluralist, as Quine scholar and University of North Carolina professor Gillian Russell explained in a lecture delivered for the Edinburgh University Philosophy Society. Logical pluralism, as she explains, is the view that there is no such thing as a single universal logical system. Instead, there are several distinct logics which are—in some sense—correct, (e.g. are most appropriate in particular circumstances, or capture different correct senses of inference). It is worth noting that logical pluralism is a mainstream position among logicians and completely within the Western tradition which Lindsay so adamantly claims to defend, yet this is something that his worldview does not admit. The subject of that same lecture would perhaps be even more appalling to anyone who holds this simplistic and naïve worldview, namely logical nihilism, or the view that there is, in fact, no logic, and that all our perceived logical rules are contingent and arbitrary. Now, Russell herself does not defend this view, but she nevertheless presents it as a philosophical position that should be considered.

Logical pluralism is a good way to connect with the other side of this epistemological coin, so dearly held by the Sokal Squared trio. What I have in mind here is their stringent opposition to what they call alternative “ways of knowing,” which they often characterize specifically as antithetical to reason and science. Pluckrose has called the notion “racist & sexist bullshit”; Lindsay has mocked the idea on several occasions; and, in characteristically dramatic fashion, Boghossian has named it among the steps toward the destruction of Western civilization. Why is logical pluralism relevant? As I stated before, the LNC and the LEM are two central features of what we might call classical logic. Yet, there are systems of logic that, as Quine suggested, do away with these principles. Consider a system of paraconsistent logic. Giving a full account of what this means is beyond the scope of this essay; but, briefly, a system of logic is paraconsistent if a contradiction does not entail that every proposition is true—on paraconsistency we may accept a contradiction without logic forcing us to accept everything. Logically, this move is made by including true contradictions in the set of logical possibilities. A true contradiction is a case of a proposition P and its negation not(P) being both true. Once we accept this, we need to do away with the idea that Law of Non-Contradiction as a fundamental logical truth. Again, further elaboration on this is beyond my present purpose, but there are several rigorous academic resources on these subjects for anyone who wishes to know more about them.

Just like Priest has drawn on East Asian philosophy to guide his work on different areas of formal logic, is it not possible that one may find analogous cases of useful “ways of knowing” in African, Mesoamerican, or any other forms of non-western philosophical traditions?

Graham Priest, a professor of logic at City University of New York, is one of the pioneers of these systems of logic in the Western philosophical tradition. Priest has explicitly defended the fact that Eastern ways of knowing—to be consistent with Lindsay et al.’s terminology—are much more useful in understanding these systems of logic. In particular he has cited Buddhist philosophy as a strong influence in his work on logic. The reason is that Buddhism and Eastern philosophical works, such as the Daodejing, treat contradictions differently than it is treated by some of the central figures in modern Western logic, such as Bertrand Russell, Gottlob Frege, and their contemporaries. This distinct—and perhaps, to us, alien—way of treating contradiction is, nevertheless, formalizable in the same way as our more familiar views, as Priest’s work shows. But the important upshot here is that none of this suggests some form of relativism. Paraconsistent logics are still self-contained systems in which theorems can be written and proved, just like with the classical logic familiar to most people. Notably, insofar as they allow for true contradictions, they are useful in the realm of self-referential paradoxes, some of which are essential for the foundation of mathematics. Notably, Priest and others like Chris Mortensen, professor emeritus at the University of Adelaide, have even worked on the field of inconsistent mathematics, which takes non-classical logics and develops an arithmetic the same way one might develop classic mathematics from classical logic. Mortensen has worked with systems, which, at their core, have theorems like (0=2) and not(0=2). As strange as this may sound, Mortensen has argued that classical mathematics are a special case of inconsistent mathematics. Moreover, Priest has defended this mathematical pluralism, while  explaining why none of this amounts to relativism of the “anything goes” kind. Just like Priest has drawn on East Asian philosophy to guide his work on different areas of formal logic, is it not possible that one may find analogous cases of useful “ways of knowing” in African, Mesoamerican, or any other forms of non-Western philosophical traditions? It seems to me that it is an entirely reasonable possibility.

Given what we just saw, I think it is perfectly fair to argue that this simplistic picture of science, reason, and the West is ridiculous. On the one hand, we saw how the Western philosophical tradition—even leaving aside postmodernism, critical theory, and other bogeymen—is at least skeptical of this notion of universal immutable truths. Further, we saw that there can be some value in looking beyond the West for different forms of approaching science and reason.

Now, the idea of The Truth keeps coming up, which leads me to a final point which has been previously made by both Lindsay and Boghossian. In particular, they claim that the so-called culture war really comes down to what one’s position about The Truth is. In a 2019 essay by Boghossian in American Mind, he argues that to be on the right side of the culture war—their side—is to endorse the correspondence theory of truth: the view that a claim is true only if it has correspondence to reality. Boghossian, however, adds a further claim to his definition of the correspondence theory of truth. He states that this theory also posits that truth, as it corresponds to reality, is knowable through science and reason. In addressing these contentions, I will draw once again from Liam Bright, who has previously discussed this. As Bright notes, the correspondence theory of truth—as understood by philosophers who work on these matters professionally—does not say anything about how we arrive at the truth, so Boghossian’s addition about reason and science is not part of the standard definition. Yet, it is a way for Boghossian to contrast it with something that, by now, should be identifiable as a favorite opponent of the trio: alternative ways of knowing.

In his previously cited essay, he contrasts the correspondence theory with the idea that, for example, heterosexual cis-gender white males might not be able to see the full truth about the world by virtue of their privilege, whereas underprivileged people are more attuned to the social realities of the world. First, we should note that it is quite odd that the theory of truth on which the balance of the world presumably hangs is not even defined properly. This shows lack of rigor, especially given that, as Bright notes again, the Stanford Encyclopedia of Philosophy article that Boghossian himself links in his definition makes no mention of science or reason in the definition because they are not a part it. One has to wonder if he read the article, and if he did not, what else he failed to read. But more importantly, the correspondence theory of truth is not, in any way, opposed to these so-called alternative ways of knowing. One could imagine gaining knowledge by means of divine revelation which—despite being completely independent of science and reason—would, nevertheless, correspond to reality. Now, I should clarify that I do not believe in divine revelation, but the example should serve to show why the “way of knowing” that Boghossian characterizes as anathema to the correspondence theory is actually wholly compatible with it. It is entirely possible that only underprivileged people, through their lived experience, had access to some social facts, but that this knowledge acquired through lived experience still corresponded to reality. I do not endorse this position either; however, once again, the example should show why Lindsay and Boghossian’s claims about the correspondence theory do not actually bear out.

What ultimately makes matters worse, in my view, is that like many contemporary reactionary figures, they claim that their project is not political—and that they are simply defending reason and science. They claim they are not reactionaries, and I do think this is true. Yet, they claim their projects are apolitical, which is explicitly stated in the New Discourses project. But this claim is hard to defend in the face of the rather sui generis view of science, reason, and Western civilization to which they cling. What is more, even if they are not reactionaries, in many ways they end up doing their work for them. When they criticize groups based on a particular political affiliation, it is overwhelmingly the Left—and a rather crudely defined Left at that. They characterize figures like Robin DiAngelo, author of the best-selling book White Fragility as part of the Left. And while it is impossible to have actual numbers, I am quite confident that saying “no one on the Left agrees with Robin DiAngelo” is much closer to the truth than “everyone on the Left agrees with Robin DiAngelo.” I truly welcome good faith criticism of the Left, and I even think some of Linsday et al.’s really is that. However, I also think it is important to recognize the majority of their project as what it is: a political project against the Left under the guise of a dispassionate defense of a rather particular conception of Western reason and science.

Néstor de Buen holds an M.A. in social sciences from The University of Chicago. He has previously written at Quillette.

7 thoughts on “On the “Sokal Squared” Trio

  1. “It begun—as many discussions do nowadays—on Twitter, with Lindsay defending the seemingly simple assertion that 2+2=4 is an obvious immutable mathematical proposition” this is not correct. if you read the thread you link, you will see that it starts with nothing to do with maths at all, but truth claims more generally. he sums it up this way: ‘2+2 = whatever, all that matters is who is making the claim, and who benefits from it being any particular value’. the general twitter reaction, and to a certain extent this article, validate this analysis. what 2+2 really equals doesn’t matter, we can play philosophical games all day long to deny anything, all that matters is that James is wrong, because if hes not, then he might well have a point. which is what makes this whole situation absurd, the people he criticises as having an orwellian ideology have made a political litmus test of 2+2, and have come down on the side of it equalling 5… as if 2020 could be any more surreal…

  2. Oh boy. This is a wild misrepresentation of their work, viewpoints, and intentions – all of which they have made clear. They’ve been consistently transparent and reflective.

    At best, this is a thin and superficial understanding of their work. It’s laden with strawmen and false premises. You should have waited until Cynical Theories is released, and then written a thoughtful, considerate response. Or you could have carefully read the work they’ve already published. You haven’t read Pluckrose’s writings in Areo, or Lindsay’s (extensive) analyses on New Discourses. This is just another lazy, ignorant, shallow rush to judgment.

  3. Did I just start reading an essay that claims that 2+2 4?
    Edit: sorry I got it wrong, the author is talking about “2+2” not 2+2. LOL

  4. Not sure I agree with your conclusions here. While you have brought up some points of nuance that I think are lost in the conversation, Pluckrose, Lindsay and Boghossian all seem more focused on their critics’ disingenuous attempts to explicitly and intentionally redefine terms. Sure, you can probably come up with some situational expressions wherein 2+2=5. However, in every rebuttal about a way in which you may come to such a conclusion the terms are either redefined or the situation changed and every case still uses logical outcomes. The problem that they (the collective you write about here) are addressing are arguments that would take those fancy manipulations of terms and conditions to absolutely state that 2+2=5. To use your example: bachelors are married. Both are patently false based on the proper and common definitions. Privilege has nothing to do with it. And in the case of the ethno-math peddlers their bailey is just that argument and when called on it they use arguments like yours to claim that no, in fact they were using some kind of alternative calculation wherein a rounding computation or clock math would produce 5 — or where a man named Bachelor is married; that P is also !P.

    For the same reason I like my fellow drivers to stay in their respective lanes and signal when they merge it is important that we also have clear definitions for the topics that we discuss – we need boundaries within our discourse. Just like it is okay for a race driver to exceed the speed limit on a closed course, we can wax philosophical on exceptions when it is appropriate. But when we have to move whole communities, states, countries and populations in a direction of common good then we need to have traffic lights and we need people to stay in the lines. The 2+2=5 crowd are quite literally trying to argue that it is acceptable for a student to give that answer in class. Furthermore, they expect that such an answer will come from someone who is not “privileged” and that as such, that person will have a reason for producing that answer and that answer will be correct for that person. However, if we are in fact talking about a child in a public school classroom (and this is the intent) then we all need that child to understand what 2 and 2 are and why they produce 4 and not 5. A child shouldn’t have to worry that their teacher really means 2.3 and 2.7 or some variation thereof. That alone would be nefarious for a whole slew of reasons that are also reasons why this discussion is both dangerous and alarming.

    I think we can both agree that there is value in common definitions and conclusions. I may not be as apocalyptic in my thinking as you have suggested Boghossian might be, but there is definitely something to be said for why we might want to teach basic logic and mathematics and require that our children “get it right” before moving on to more advanced or more abstract ways of thinking about things. After all, I am happy that the engineer who designed my house or the foreman who poured the concrete in the bridge I drive across both drafted and followed plans that relied upon a common understanding of math – in base 10 no less. I assume they used the imperial measurements for both but even if they used metric the math still has to come to the same conclusion. In such conditions P or !P. Maybe the painting of the bridge can include both, but the construction cannot rely on such semantics.

    Finally, I think you both hit the point of their papers and completely missed it. Your criticism of their point sounded as if you were reviewing commentary on SNL’s weekend update: this could be useful content if only they took it more seriously! That was their point.

  5. You never once mention 2+2=5, the actual root of the, as you call it, “crusade”. That’s dishonest framing right from the get-go. ’nuff read.

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.