by Dave Maier
The blog post screams: “If you think 2 + 2 always equals 4, you’re a racist oppressor.”
It then proceeds to attribute this ghastly sentiment to the Seattle Public School district, on the basis of a preliminary document for a proposed curriculum in “Math Ethnic Studies.” Other critics of this pseudo-educational abomination are cited; math, they agree, is an area “which all people should be able to view as objectively settled.” To doubt this is to fall prey to the worst postmodern relativism and skepticism. And so on, in familiar fashion.
I’m not here to defend the Seattle Public School district specifically, nor multiculturalism in general, nor postmodern relativism and skepticism, for that matter. But to respond to the first two with the same tired 90s-era pomo-bashing (“Apparently math is now subjective,” mocks one critic) is to combine sloppy interpretive procedure with half-baked folk philosophy. Let’s put the latter aside for now and start with the former.
If our opponents say A (“maybe 5 + 5 = 10 isn’t always true” – I’ve modified the example, as we’ll see below) in order to support their broader claim B (“we need explicitly multicultural Math Ethnic Studies to modify or supplement math class”), and B doesn’t sound any better to us than A does, it’s natural for us to take A to mean something crazy – as crazy, that is, as B itself sounds to us – with the hope of rhetorically linking the two and thus bring out the full craziness of B (as identical, again, with the seemingly manifest craziness of A). Look at how crazy multiculturalists are – they can’t even add properly!
In an earlier post I noted how tricky it is in general to get extreme charges (such as that of doubting obvious facts) to stick. It’s a natural temptation on our part, but there’s no reason to attribute this sort of full-on epistemic relativism to multiculturalists. They truly believe, I take it, that capitalism (or whatever) is the root of all evil; and they truly believe that the West takes other cultures to be not simply less advanced but somehow incorrect or backward. If anything, that’s not skeptical; it’s dogmatic.
The better interpretive procedure – and, not coincidentally, the more effective critique – is to make clear why it should seem that something like A would support something like B. This will make our attribution of A stick, even at the cost of requiring further work to show that not all is as it should be, and the risk – if that’s what it is – of leaving an out for them in the form of a rather less extreme claim (which, if they want to, they may insist was all along what they meant). The crazier we make A sound, the harder that is to do. Interpretive charity, that is, isn’t a matter of being nice; it’s a matter of getting things right. What are these people thinking?
Let’s try a thought experiment. Fleeing repression after the violent takeover of their town by neighboring Shelbyville, refugees from Springfield, including the Simpson family, come to live in our area and send their children to our public schools. Lisa Simpson is the smartest person in her family and a math whiz. Let’s show her the solved addition problem 5 + 5 = 10 and ask her if it’s always true. In reply, she says “no, five plus five never equals eight! Can’t you add??”
What just happened? It seems that Lisa, like all Springfieldians, has only eight fingers, so their school taught them base-eight arithmetic. (This is not actually true on the show; everyone there uses base-ten numbers, even though only God, whose hands we see in one memorable episode, has ten fingers.) This relativity of representation – which of course we do already teach in school, without multiculturalist prodding – should not, we feel, threaten what we naturally think of as settled mathematical facts. In fact, relativity in math goes deeper still: unlike alternate arithmetical bases, non-Euclidean geometrical systems, for example, aren’t simply saying the same thing in different ways, yet each is “equally valid.” Curiously, though, no one is screaming for such radical postmodern attacks on mathematical objectivity to be stricken from the curriculum (although as it happens, Riemannian geometry is a tad advanced even for high school seniors, except possibly those taking AP Physics, I’m not sure).
Let’s get back to Lisa Simpson. How next does she respond to our question? Specifically, what does she say? “Five plus five doesn’t equal eight; five plus five equals [___].” She can’t say “ten,” because that’s not a word she knows. She won’t say “one two base eight” because that’s not what our hypothesized octal speakers say (we don’t say “six plus six equals one two base ten”; we say “six plus six equals twelve”). What’s the octal word for (what we call) “ten,” that is: this many [# # # # # # # # # #], which they write “12” (that is, one eight and two ones)?
This may look trivial – after all, it’s certainly true that “one two base eight” perfectly well denotes that number – but I think that in a way it cuts to the heart of the matter: ours, that is, that of differing representations of “settled mathematical facts.” (How, for example, is the supposedly contentious number sentence 5 + 5 = 10 pronounced?) Our interest, I take it, is in the fact that while in most cases something is certainly settled, such that it would be perverse to put it in question (as anti-postmodernists rightly claim), it can be a surprisingly non-trivial matter to say “neutrally” which fact that is. It’s interesting that the issue of how to pronounce octal numbers never comes up in basic discussions of octal, or any that I could find for that matter. I did some Googling and not one source even mentioned it. They tell you how to convert from one to the other but they never tell you what the octal names are for numbers higher than eight. I doubt there’s any consensus on the matter, but if anyone knows I’d love to hear about it.
Why should this matter? After all, our natural thought is that Lisa isn’t really disagreeing with us, as she too will say, for example, that this many [# # # # #] plus that same number [# # # # #] equals this many [# # # # # # # # # #]. She simply says that same thing – we want to say – another way. The relativity here, that is, is completely painless, as there was never any thought that base 10 (that is, base ten; all bases are base “10”!) was the One True Base. But it’s not merely trivial either. After all, mathematically speaking, if all bases refer to the same numbers, the only differences between them are pragmatic; but that doesn’t mean they aren’t real. (Imagine a child trying to memorize the base-this many [# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #] times table, or multiplying 1001010010100100100101 x 10000101010010001111100 in binary.)
That’s not the only issue, since – and of course this is one thing multiculturalists (and other social scientists) are right about – we use mathematics in all kinds of ways outside the classroom (which is why we’re learning it in the first place, duh). In the context of a community of refugee immigrants perhaps not entirely welcome here (was the Shelbyville “repression” really all that bad, that we should now have those four-fingered #@$s on every block??), all kinds of weighty sociocultural significance can accrue even to which word one uses for this many [# # # # # # # # #], let alone how one adds. Yet none of this changes the fact that [# # # # #] and [# # # # #] make not [# # # # # # # #] but instead [# # # # # # # # # #] [I’m tempted to make a mistake here, just to see what it would mean to do so …].
Of course, nothing prevents our Math Ethnic Studies instructor from saying that. But surely, we think, that wasn’t the point of asking the question about whether 5 + 5 = 10 is “always right.” In fact, we should note, that question is only our anti-postmodern paraphrase. In the MES document itself we have instead “who gets to say when an answer is right?” – but since this appears alongside “Can you recognize and name oppressive mathematical practices in your experience?” and “Why/how does [sic] data-driven processes prevent liberation?” under the heading “Where does [sic] Power and Oppression show up in our math experiences?” we can see why that paraphrase may have looked appropriate.
Moving forward: let’s try a charitably weak construal of the multicultural idea, and then compare with it another, rather less attractive (if possibly thereby more realistic) one. Naturally we cannot simply attribute the former to Seattle on the basis of what looks like a brainstorming session rather than a full-fledged proposal; but let’s start here.
1) If you present the math we learn in school as if it had been engraved on stone tablets handed down on Mount Sinai rather than invented by actual historical human beings, we miss something.
2) Missing that something makes that which is actually contingent look necessary and reasonable alternatives look simply wrong; both of these are false, and this can be harmful in various ways, not all of them obvious, especially in math class, where our focus is naturally on learning math.
3) We need a multicultural (“ethnic studies”) corrective to even things out. This need not involve anything more controversial than straightforward cross-cultural anthropology as applied to mathematics.
Note that we ourselves recognize 1) as true in math class when we need to – that is, when what we’re missing is not that “ancient African math was Just As Good” or something, but some mathematical reason: other number bases, non-Euclidean geometry, the Axiom of Choice, whatever. This means that we can detach that issue from the explicitly multicultural one and address it separately. When we do, we see immediately what can go wrong.
If straightforward anthropology of the above sort were all there ever was to multiculturalist education, then the worst we could say of it, if proposed as part of a social studies curriculum, would be that it’s a bit ambitious given all the other things kids of that age need to know. (Indeed one of the linked critics of the actual proposal suggests, reasonably enough, that “a progressive college course” would be the natural place for such things.) No one would have to get bent out of shape about it at all, let alone warn that the very existence of truth itself would be placed in danger.
But alas, it is not so. At the extreme – let’s call it “dogmatic multiculturalism” – we have a trendy mishmash of only minimally consistent tendencies. The first I call the Two Minutes Hate: Euro-American colonialist capitalism cum imperialist patriarchy is the oppressive root of all evil, and all other cultures the virtuous victims of same. (I exaggerate to mark the extreme case; we don’t know if Seattle’s is like this, although the proposal as written – see above, and at the link – did make me wince a bit. Still, let’s not jump to conclusions.) This, apparently, is why we are to warn the kids that math (and, I take it, other technology) is not the enlightened mark of the “best” culture that the traditional curriculum presumably implies that it is, but instead a tool of the oppressor. Fight the power!
The other main thread of this sort of multicultural dogma I call Everyone Gets Trophies. In math and science classes we learn the Euro-American way of doing these things, which may have allowed us (them?) to get to the moon; but that doesn’t mean we should think of other cultures as backward savages, since they had math and science of their own. So described, this might simply be the benign relativity of anthropology, indeed in some senses a welcome corrective, but in dogmatically multiculturalist hands it can become the crude leveler of cultural relativism: we must agree that all cultures are equal, on pain of ethnocentrism and other non-woke things.
As may already be obvious, Everyone Gets Trophies sits uncomfortably with the Two Minutes Hate. Math is a diabolical tool of colonial oppression … but we invented it first! So we’re just as good as they are! Or maybe it goes: when we invented math it was used for good, but then they pushed it too far, twisted it to their own cruel ends. But that’s hard to make sense of too. Let’s say one takes the Manhattan Project to be the epitome of imperialist evil. (I don’t agree with this either, but bear with me.) The A-bomb would not have been possible without certain conceptual advances in theoretical physics. The extent to which scientists bear moral responsibility for the results of the use of technology by others is a real issue. But extend this thought too far away from the irradiated ruins of Hiroshima and Nagasaki, and it becomes facile. How did the moral rot enter mathematics, which was equally essential for nuclear technology? Not too far back, or now everyone is guilty; but where? Newton and Leibniz were white male Europeans; can we pin the blame on them? And, more importantly for math education, what exactly is it about the infinitesimal which allowed the imperialist demons to corrupt what up to that point had been “pure” mathematics? (Actually, come to think of it, this example isn’t as far-fetched as it looks: people said similarly bizarre things about the evil of infinitesimals at the time and since!)
If we can’t do that – make sense of the imperialist corruption of mathematics in mathematical terms – then we are left with the mere truism that every tool can be used for good or evil, which would not bear a tenth of the polemical weight our posited multiculturalists are trying to put on it. The dogmatic condemnation of the Two Minutes Hate is structurally incompatible with the cultural relativism of Everyone Gets Trophies. Ironically (or not), this means that (less dogmatic) multiculturalists can use each to temper the other, should they see the need. For after all, the same issues threaten to plague anthropology itself: strangely enough, if an anthropologist regards each of his commitments as Absolute Truth, he cannot see the native culture objectively but only through his own lens. But if he defers to the natives on matters about which (ex hypothesi) he knows better, or even differently, he’ll make the corresponding mistake. Anthropologists must find a way to bridge the gap rather than collapse it; it is the creativity that this effort requires (as in history and, yes, philosophy as well) that makes it so difficult – and so valuable, when done well. There’s no reason multicultural education couldn’t find that balance, even if it rarely does.
Most of what I have to say to back up my opening gripe about “half-baked folk philosophy” will have to wait (although it will surely be familiar to regular readers, if such there be). Here I’ll simply note that even after carefully disentangling the crazy but unattributable from the merely debatable in order better to judge the latter, it’s almost irresistibly tempting to snatch interpretive defeat from the jaws of victory.
Remember how the one critic demanded math be recognized as an area “which all people should be able to view as objectively settled.” Settled, yes; but “objectively” adds nothing we should want. If something is settled in the sense that everyone sees it as true, then we’re good to go. That’s as settled as it is rational for a proposition to get, even in math. To “write it in stone” is to commit not simply to a fact – we already did that! – but also to a method of representation considered as transparent – which is entirely gratuitous, both pragmatically (I hope obviously) and also theoretically. All I want is the fact, however you get it to me. To demand more is the dogmatism I warned about at the outset: it doesn’t add anything, and only further skepticism can result.
Part of my metaphysical gripe here is the traditional pragmatist idea that it doesn’t make sense to demand metaphysical grounding for what we already believe (only doubt can provoke inquiry). Here, the reflexive anti-postmodernist demand for objectivity makes it sound like this is something over and above the agreed-on facts, such that to resist that metaphysical call were thereby to doubt the fact. This accounts for critics’ mockery of the alleged “racism” of simple addition, but is – to be charitable – misleading at best.