Of Course, 2 + 2 = 4 is Cultural. That Doesn’t Mean the Sum Could be Anything Else.
By: Keith Devlin @profkeithdevlin
The following statement appeared on Twitter recently:
“The idea of 2 + 2 being 4 is cultural and because of western imperialism/colonization, we think of it as the only way of knowing.”
It was actually part of a discussion among academic scholars about educational issues that K-12 teachers face all the time. But as sometimes happens, the Twitter-thread got derailed by trolls, most of whom who had no idea what they were talking about (they rarely do), and for the most part spouted nonsense about straw men of their own creation.
Though the Twitter storm flared up rapidly and then slowly died away, it got me thinking about the issues the tweet was addressing, not from the standpoint of K-12 education in a multicultural setting, on which the tweeter was focusing, but the world of college and university mathematics I am part of. With my usual readership of MAA readers in mind, then, let me ask you this question:
What do you make of the statement that the identity 2 + 2 = 4 is cultural?
I’ll come back to this. But first let me use the example of the Twitter spat to set the scene.
A storm in a twittercup
It’s a common practice in various academic disciplines to question how we come to know what we consider to be self-evident – even the most basic ideas that we regularly take for granted. Discussions are often couched in terms of simple examples, some of which come to be used repeatedly to illustrate and codify the issues. In the case of mathematics, 2 + 2 = 4 is one of them.
In their mammoth three-volume tome Principia Mathematica, after over 350 pages of detailed logical deductions, Whitehead and Russell managed to prove that 1 + 1 = 2.
For instance, back at the start of the Twentieth Century, Alfred North Whitehead and Bertrand Russell produced a mammoth, three-volume work examining the logical foundations of mathematics, titled Principia Mathematica, in which they used the even simpler identity 1 + 1 = 2 as an illustrative example, taking over 350 pages to establish its truth by logical deduction from first principles.
The goal was not to check if the identity is correct in a real world sense. That’s obviously true. The issue was to determine the logical correctness of mathematics. The motivation was that Russell had shown some seemingly obvious mathematical facts led to contradictions. They proved 1 + 1 = 2 to demonstrate that the basic identities can be formally proved, and how it could be done.
Fortunately for Whitehead and Russell (and for the rest of us), there was no Twitter in 1910, and the two scholars carried out their work entirely within the ivory-towered world of Cambridge University, so there was no outcry. But to anyone outside such rarified (and somewhat exclusive) worlds, catching a glimpse of an academic discussion using a simplistic example can act as a red rag to a bull – or, in the case of the 2 + 2 =4 identity, many bulls, though that was by no means for the first time that Twitter has lost its collective mind over something it did not understand. (George Orwell’s inclusion of that very identity 2 + 2 = 4 in his dystopian, futuristic novel 1984 probably made the rag an even brighter red in the current US political climate.)
The key word in the 2 + 2 = 4 tweet that the Twitter trolls missed, and as a result lost their marbles, was the second one: idea.
In that particular context, the word packed a substantial punch. The underlying issues being discussed were (and are): what is the nature of mathematics, who gets to set the rules for how mathematics is done, who determines which people can participate, what constitutes correct–or good–math, what role does it play in society, and whose society are we talking about anyway?
The identity 2 + 2 = 4 is frequently used for that discussion precisely because it is noncontroversial in all human cultures that have counting numbers and arithmetic up to 4. Which is pretty well all cultures today. (Though you don’t have to go back very far in time to find anthropological studies of remote societies that organized counting in different ways.) Indeed, thinkers and writers have been using that very identity to illustrate an “obvious truth” since at least to the Sixteenth Century, often contrasting it to the “obvious falsity” of 2 + 2 = 5.
What makes the identity noncontroversial is that, if you count things in the world, four is what you get when you combine two with two. Being forced to think about the big idea of doing mathematics today in terms of that ridiculously simple, noncontroversial, universal example brings the much deeper underlying issues dramatically to the fore.
Before I look at some of the issues that tweet was really getting at, which will bring me back to my question about whether 2 + 2 = 4 is cultural, let me say it once more, just to be clear. The red-rag statement in the original tweet absolutely is not about whether, when you have two objects in one hand and two in the other, you have four objects altogether. By spouting off about that, the majority of Twitter trolls simply demonstrated their ignorance – to say nothing of a total lack of common sense. Social media can do that to people.
Is mathematics cultural?
So, do you agree with the statement that 2 + 2 = 4 is cultural?
I know from experience that there are mathematics professionals with Ph.D.s in math who react strongly against that statement. Indeed, from within the culture of contemporary professional mathematics, the claim that the identity is cultural is nonsense. Within that culture, it’s a universal fact, both theoretical and empirical. I’ve been in that culture my entire adult life, and I absolutely view 2 + 2 = 4 that way.
Why is that? In particular, where does the empiricism come in?
Well, anyone who engages in research in mathematics eventually acquires an overpowering sensation that it is a process of discovery – discovery of eternal truths about an abstract realm.
Why does that happen? An answer is to be found by reflecting on our knowledge of how the human brain evolved, and how it works. The sensation of discovery of an abstract realm is an inevitable consequence of sufficient immersion in mathematical thought. After all, what we perceive as reality comes down to neural activity, whether we are contemplating what we see with our eyes or experience with our other senses. Consequently, a human-created world of abstraction is inevitably going to seem real. We are creating a similar pattern of neural activity initiated by our own thoughts rather than by the inputs from our external senses. Similar patterns, similar sensation of “reality.” [That’s why, when I get the inevitable question after giving a public talk, “Is mathematics discovered or invented?” I answer “Yes; it’s both.”]
The ancient Greek philosopher Plato (ca.425–ca.345 BCE) proposed reasoning about issues in a decontextualized manner using imaginary idealized objects in what is today called an abstract Platonic Realm. This form of reasoning is known as Platonism. Most mathematicians report that the sensation of working on a mathematical problem is one of reasoning in that manner.
But regardless of how it comes about, that sense of mathematics being a process of discovery of truths in an eternal “Platonic” realm seduces you into thinking everything is empirical.
Having spent my entire adult life within that culture, I get that. But that’s the point. It is a view from within a culture. How well does it stand up to further analysis?
In the case of 2 + 2 = 4, if I seek some absolute certainty, I can fall back on my worldly experience. If I have two objects in my left hand and two more in my right, altogether I have four objects. There is absolutely nothing controversial about that. The scholars in the original Twitter discussion were certainly not questioning that. The Twitter commentators who claimed otherwise were, in the majority of cases I suspect, simply posturing that such was the issue in order to score some points in whatever culture war they are engaged in. But at best they just made themselves look silly. (And at worst malicious.)
But let’s take it a step further.
I learned the 2 + 2 = 4 fact early in my childhood. A bit later, as a young adult in high school, I acquired the knowledge that 0.999… = 1. Here there was no way to check it out by physical means. But I could (and did) follow a simple logical argument that convinced me.
But that worked only because I bought into the idea that you could arrive at truth by way of logical reasoning based on a set of initial assumptions (“axioms”). Later on in my life, when I was older and wiser, I came to realize the truth: It’s a fact only because previous generations of mathematicians decided it should be, and chose axioms accordingly. (With good reasons, to be sure – at least within a mathematical culture that sought, successfully, to achieve various ends, such as providing accurately predictive, numerical descriptions of the Solar System and other physical systems.)
Despite my knowing the real story, however, I continue to view 0.999… = 1 as an empirical fact (about the abstract Platonic realm of mathematics) with the same status as 2 + 2 = 4. In other words, I believe it.
On the other hand, I absolutely do not believe that
1 + 2 + 3 + 4 + 5 + … = –1/12
Yet, I know that the chain of decisions, of which “0.999… = 1” is but one of the early ones, leads irrevocably to that jolting conclusion. (So it’s actually not “on the other hand,” it’s on the same hand that I had already accepted!)
Since that chain of decisions was seductively rational, and formally correct logically, my only explanation for having bought the package is that it is a package – a cultural creation that begins innocently enough, with the reification to an abstract equation of the empirical fact that if I have two objects in my left hand and two in my right, I have four objects altogether.
[ASIDE: Incidentally, with regards to reification of real-world situations to abstract equations, I should note that in this essay I’m not looking at the issue of language, either natural language or the symbolic language of mathematics. Those are, of course, cultural constructions, and the structures of those languages influence the mathematics that gets done, but I want to focus on the mathematics that the linguistic constructs refer to.]
Having bought into that math package, and found myself facing an identity that’s highly counter-intuitive, I certainly can’t allow the logical chain to break down at an arbitrary point midway through the progression. The only justifiable jump-off point is when infinity creeps in, as it does with 0.999…, but most of we professional mathematicians would not have a job if that were our exit point. (Hmmm?) So I’m stuck with the entire package, as are all other mathematicians.
The truths in that package range from some that we can check out in our everyday lives (2 + 2 = 4), to others that accord with our intuitions and are consistent with things we can observe and measure in the world (0.999… = 1), and to some that, for all that we can tell a really good explanatory story that professional mathematicians (and only those!) can follow, fly in the face of everything we know first-hand about the world (1 + 2 + 3 + … = –1/12).
That package is a product of our culture. How can it be anything other? I think a lot of the talking-past-one-another you see on the “all mathematics is cultural” issue results from confusing “universality of truth” (of mathematical ideas) with the cultural basis of those ideas.
The reason we mathematicians think that mathematics does not depend on culture is that we are in that culture. Indeed, it’s probably more general than mathematicians. I suspect that most people in modern, industrialized societies think mathematics is a culture-free discipline.
I should add that it’s not by any stretch of the imagination Western Culture. It has roots in pretty well all cultures that developed anything we would regard as a civilization supported by engineering and technologies, going back at least 10,000 years to the introduction of counting numbers by the Sumerians.
Modern industrialized societies are built on science, engineering, and technologies that depend on the mathematical view of the physical world developed by Galileo (1564–1642, Image A) in Italy. But Galileo built on mathematics introduced into Western Europe by Leonardo of Pisa (a.k.a. Fibonacci, ca.1170–ca.1245, Image B). Leonardo in turn built on work of al-Khwarizmi (ca.780–ca.850, Image C) in Baghdad, who drew on work in India by Brahmagupta (598–670, Image D). Thus, today’s “Western Culture” is the cumulative result of advances in at least three major civilizations. (While each is a justifiable intellectual giant, the four individuals are, of course, essentially icons for the thinkers in their culture.)
Incidentally, the reference to “imperialism/colonialization” in the original tweet surely contributed to its red-flag effect. The tweet can be read at face value (as I did at first glance) as assuming counting-numbers arithmetic is an invention of western societies, which it was not. But the tweeter emphatically did not make that assumption. (I checked.)
In fact, counting numbers and their arithmetic were around long before modern western societies came on the scene, and many cultures can lay claim to helping advance that mathematics. Rather the reference is, I believe, an acknowledgment of the fact that modern western society, with its heavy dependence on technology, puts mathematics center stage, and makes it the only obligatory school subject besides the students’ native language.
Whose culture?
As I mentioned, the 2 + 2 = 4 tweet that got me thinking about the role of culture in my world of professional mathematics was part of a discussion thread about K-12 math education, with a particular slant towards making the subject more welcoming to students from diverse backgrounds.
K-12 educators have different aims from those of us in the more discipline-focused world of higher education. Two primary goals of K-12 education are bringing the next generation into the culture they are growing up in and making them familiar with aspects of that culture that society has decided merit passing on. Among those aspects of the culture to be taught are a number of “scholastic disciplines”, including (parts of) mathematics.
In that context, a half-century of research has been conducted into the role of culture in mathematics education, and a lot has been written. I may be wrong, but I suspect that many college and university mathematics educators are not familiar with much, if any, of that literature. I became aware of it only well into my career, when I served a term on the Mathematical Sciences Education Board back around the time of the Millennium, and found myself around the table with some of the nation’s leading K-12 mathematics education specialists.
Among the scholars whose writings I consulted, the one I found most helpful (given my perspective as a mathematician) is Alan J. Bishop. For a quick (and hence, highly selective) introduction to his thinking, I recommend his 1988 paper Mathematics Education in Its Cultural Context, published in Educational Studies in Mathematics, May, 1988, Vol. 19, No. 2, pp. 179-191 (Kluver). It’s just twelve pages long.
Alan J. Bishop is Emeritus Professor of Education at Monash University in Australia. In 1977, while engaged in research on spatial abilities and visualization, he spent a sabbatical leave in Papua New Guinea, where he began to think about the process of mathematical enculturation and how it is carried out in different countries. His subsequent book, Mathematical Enculturation: A Cultural Perspective on Mathematics Education, published in 1988, developed a ground-breaking new conception of mathematics as being a cultural product. In 2015, the International Commission on Mathematical Instructions (ICMI) awarded him the Felix Klein Medal for his work.
In particular, Bishop explains what he means by the statement “mathematics is a product of human culture,” and how it results in a body of knowledge that is universal across all modern cultures.
He observes that there are six activities that all modern human societies engage in:
counting, locating, measuring, designing, playing, and explaining.
“Mathematics, as cultural knowledge, derives from humans engaging in these six universal activities in a sustained, and conscious manner,” he says.
He goes on to elaborate on this thesis. I’ll leave you to check it out for yourself, as I have, but I suspect that like me, your mind will have already ticked off some of the key ways those six activities can give rise to mathematics as you read through the list. It’s not unlike the rational-reconstruction explanation I gave for the evolutionary acquisition of mathematical thinking capacity in my book The Math Gene (Basic Books, 2001).
So do check out Bishop’s argument. (If you are at a college or university, you likely have institutional access to the source website that will avoid you paying for the article.) Many of his educational points seem to me to be more relevant to K-12 education, but I find they raise questions for the professional mathematical community – at least if we view our community as an integrated part of society as a whole.*
[* Not all mathematicians do view it that way, of course, and society benefits from having mathematicians and scientists who focus exclusively on the nuts-and-bolts of their disciplines. I was such a mathematician in my younger days. If that describes you, I’m surprised you’ve got through this essay far enough to read these words. The younger me probably would not have.]
I list below some of the questions Bishop’s article raised for me when I first read it.
What do you think?
The archive reading room at the Riccardiana Library in Florence, Italy houses some of the original medieval texts that were the foundation of Western (European) Civilization, which was built on mathematics, science, engineering, and the development of technologies, together with the concept of rationalism. But many of the key ideas presented in those texts were developed much earlier, in different parts of the world.
Do we have different cultural perspectives on mathematics?
What counts as a culture here? And whose culture is it?
Have those perspectives changed over time? Have there been debates, historical or in our lifetime, about what classifies as “admissible” or “good” mathematics? (Remember Bishop Berkeley! And the fights over negative numbers and over imaginary numbers. And George Cantor’s theory of infinite sets and their arithmetics. And Whitehead and Russell, who took some 350 pages to prove 1 + 1 = 2. And the different axiomatizations of set theory that give rise to mutually contradictory results in “solid” subjects such as real analysis and algebra. And non-standard analysis. And more recently experimental mathematics.)
If you are an academic who has done mathematical work outside academia, have you experienced a different approach, different priorities, and a different notion of what constitutes an acceptable solution or a good one, compared to mathematical praxis in university mathematics?
Within your academic institution, do the Physics Department, the Biology Department, and the Department of Statistics agree with the Mathematics Department on what the core curriculum should be, or how it should be taught to their students?
Is the driving goal of mathematics the search for mathematical truth or is it something else, with the formulation of axioms and the proof of theorems just way-stations.
Who are the gatekeepers who determine who gains accredited entry into the mathematical community?
Who determines where any available funding goes to support mathematical research?
Is consideration of the ethical aspects of mathematics something mathematicians should be concerned about, or is that the job of others?
I could go on, but hopefully you get the point. The idea that there is a single body of knowledge or a single way of thinking that we call “mathematics” is a myth.
Sure, within some of the mathematical cultures, it can all seem clean and well-defined. That’s particularly true in the various branches of Pure Mathematics, and for the mathematics done by physicists and engineers. But in those cases, the domain on which the mathematics is done is to a large extent defined by what the available mathematics can do! So, of course it all seems uncontroversial if you are within that culture. Things seem less clear cut in the more messy worlds of biology and statistics, and even more so when mathematics is applied in many important areas in today’s social and political worlds.
[ASIDE: That last is something I have first-hand experience in. First, the research that brought me from the UK to Stanford in 1987, trying to use mathematics to develop theories of information and communication to make sense of the role emerging information technologies were playing and would play in work and society, and to guide the development and use of new information technologies. And second, as a consequence of the Stanford work, trying to use mathematics in the IT and the construction industries, and then on projects for various branches of the US Department of Defense. Those projects were all super messy. And none of us involved ever had any certainty about the mathematics we were using or developing. In many cased the only justification for calling it mathematics was that it used symbolic notations and was being developed by mathematicians who set out to develop mathematics! Though we did give talks at academic conferences and occasionally published papers in peer-reviewed journals, success was largely determined by whether our work proved useful to the sponsors.]
But here’s the rub. The very fact that those questions can legitimately be asked (and in some cases not easily answered) indicates that mathematics is cultural. All of it. Even the seemingly clean and well-defined stuff. Any sense of certainty exists only when you are deep into the culture it is a product of.
You may, of course, disagree. I get that. But that would surely indicate that we occupy different mathematical cultures.
FOOTNOTE (added 8/3/2020): After this post was published, there was some discussion on social media of the –1/12 result for the sum of all the counting numbers. There is a great 15-minute video by mathematician Ed Frenkel where he discusses this. It provides a deeper dive into the process of increasing generalization and abstraction I referred to, starting with 2 + 2 = 4. He mentions how this process is somewhat validated by the fact that such “wild” results are found to be correct in physics. You can find the video here.