The other day I brought up Eulering, the use of extremely complicated math to debunk a common sense concept.

It’s a tough situation, because lots of our common sense concepts are genuinely false, and when they are it sometimes takes exceptionally rigorous analysis – ie math – to figure it out. On the other hand, there’s always a concern that unscrupulous people can use the language of math to confuse, obscure, and cast doubt upon otherwise obvious ideas. There’s no good way for the less mathematically gifted to figure out what to do (short of having mathematician friends whom we trust absolutely) so I suggested a “bag of tricks” approach.

One such trick: when somebody Eulers you, you try to Feynman them.

Richard Feynman was definitely good at math. But he was also good at using his non-mathematical intuitions to back up his mathematical genius. There’s a passage he wrote about his mental process – thanks to Douglas Knight for getting me the link:

I had a scheme, which I still use today when somebody is explaining something that I’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with terrific theorem, and they’re all excited. As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) -disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn’t true for my hairy green ball thing, so I say, “False!”

In other words, if you’re bad at abstraction, you can hold a lot of abstractions in your head simultaneously by mapping them on to an isomorphic concrete system. Then – as long as you’re sure the system is really isomorphic – you can draw conclusions based on the features of that system.

I’ve been using IQ as an example recently, and I feel bad about it because it’s so controversial and politicized. So let’s switch about something else instead.

How about race?

Everyone is talking about Nicholas Wade’s new book A Troublesome Inheritance: Genes, Race and Human History . I have not read it myself. I have, however, read a lot of reviews of it, which tend to sort themselves nicely into uncritical praise and untempered denunciation.

Intellectuals wading through Piketty and picketing Wade. — Scott Alexander (@slatestarcodex) May 11, 2014

A common talking point of the second sort of review is that race doesn’t exist. There seem to be two arguments along these lines:

1. Race does not separate nicely into a number of obvious clusters. Between Europeans and Asians, for example, there’s more of a gradient of kind-of-European-looking-kind-of-Asian-looking folk with kind-of-European-kind-of-Asian genes. In fact, it’s not even obvious how many clusters there are, and different people who talk about race discuss different numbers of races. The traditional classification is Caucasoid, Mongoloid, and Negroid, but it’s hard to argue that Aboriginal Australians don’t deserve their own group. A lot of people subdivide Asians into North Asians vs. Pacific Islanders, and a lot of others subdivide Africans into Khoisan and everyone else. If somebody wanted to say there are three hundred thirty three different races, there is not a shred of genetic or physiological evidence to stop them.

2. The within-race genetic differences are much greater than the between-race genetic differences. That is, a given black person and a given white person could be more genetically similar than two black people. Aside from the very obvious things like skin color genes, there are very few genes that reliably mark someone as part of a particular race, and you have to look at very subtle patterns in a lot of different genes at a time before you can find any hint of “races” in raw genetic data.

Therefore, race does not exist. Therefore, Wade’s talking points about how maybe the Chinese are more collectivist because there is some kind of gene for collectivism in the “Asian” “race” doesn’t even make sense.

Let me quote a couple of these reviews so I can establish these are indeed the arguments they’re making. From Why Evolution Is True:

What we do know is that most genes don’t show striking frequency differences among groups, and that “races” are delineated by combining information from many genes, each of which shows relatively small differences among populations. I’ll add once again that there is no unanimity on how many “races” there are, which is really a semantic question.

From Violent Metaphors:

To begin with, Wade can’t provide a clear definition of “race.” He tries to rely instead on loose associations rather than definitive characteristics, which forces him to conclude both that physical traits define race but that the traits can vary from person to person: “races are identified by clusters of traits, and to belong to a certain race, it’s not necessary to possess all of the identifying traits”. With such a shifty, casual footing, it’s no surprise that Wade’s conclusions are unsound. He can’t keep the number of races straight… It gradually became clear that this understanding [of races as meaningful biological categories] was not scientifically sound. Groupings of people by skin color did not produce the same result as groupings of people by skull shape, nor of blood type. Furthermore, as scientists began to study human variation with the tools of genetics (in the process creating my fields, anthropological genetics and human population genetics), it became apparent that human genetic variation does not divide humans into a few discrete groups. There are virtually no sharp boundaries, either with physical features or with patterns of genetic diversity, that show where one population “ends” and the next “begins”. These observations have led the majority of physical anthropologists, human biologists, and human geneticists in recent decades to conclude that the racial groups we recognize are social categories constructed in a specific cultural and historical setting, even if we consider physical features when categorizing people. These social categories can have biological consequences (for example, someone who experiences the stress of racism may be more likely to develop high blood pressure and hypertension than someone who does not).

From American Scientist:

Is Wade right? Are there human races? Is the variation seen between different cultures and locations best explained by genetic differences between human populations? And have anthropologists been turning a blind eye to the evidence in front of them? There is no shortage of scientific information, and it gives a clear answer: no. Wade’s claim that races really do exist is based partly on genetic sampling of geographically distant populations. These samples appear to show clustering into distinct groups by gene variants, also known as alleles. But sampling geographically distant parts of a continuum and ignoring the regions between the samples can provide apparent clustering that does not actually prove the existence of discrete groups.

With all this talk of clustering and sampling and continua and discrete groups, it sounds like math is going to show up here, and sure enough:

Much of [Wade’s] assertion that biological races exist within humans is contingent upon both his uncritical acceptance and misrepresentation of the significance of STRUCTURE type analyses of human genetic diversity. STRUCTURE is an algorithm designed to infer population structure (cluster individuals into ancestry groups) within a species (Pritchard, Stephens, and Donnelly 2000.) STRUCTURE produces for individuals an estimate of the probability that a randomly chosen genetic marker (e.g. single tandem repeats, STR, or single nucleotide polymorphisms, SNP) from that individual originated from one of a set of ancestral groups. The number of ancestral groups, K, is chosen to produce a best estimate of these probabilities, which are averaged over all genetic markers to assign a membership coefficient, namely a fraction of each individual’s ancestry to one of the ancestral groups (Feldman 2010.) The ancestral groups are not specified in advance, and the population membership of individuals is removed prior to analysis. Rosenberg et al. (2005) showed that the results of STRUCTURE style analyses are dependent upon whether allele frequencies are correlated or uncorrelated across populations, the number of loci used, the number of clusters specified, and the sample size. At very small numbers of loci and individuals examined, the results can be strongly influenced by random factors; thus, we have more confidence in the results of larger studies (more loci and more individuals.) For example, in their simulation with 993 loci and 1,048 individuals, the correlated model returned cluster coefficients of 0.51, 0.76, 0.84, 0.86, and 0.86 for K = 2, 3, 4, 5, and 6 respectively, and the uncorrelated model returned cluster coefficients of 0.49, 0.75, 0.80, 0.63, and 0.64 for K = 2, 3, 4, 5, and 6 respectively. Thus, the correlated model states that ancestry five clusters are just as valid as ancestry six clusters, and the uncorrelated model suggests that four clusters are better than five or six. One method used to quantify whether human populations can be thought of under the first definition is the use of Wright’s population subdivision statistic. Wade (p. 20) specifically discusses this where he quotes Henry Harpending and Alan Rodgers, who are supposedly speaking for Sewall Wright concerning the significance of his FST statistic. FST is the population subdivision statistic and can be calculated as: FST = (HT – HS)/HT Where FST is the average for multiple loci, HT is the average of the expected heterozygosity in the total population over loci, and HS is the average expected heterozygosity over subpopulations. Actually Wright never gave an explicit value for which FST would be considered great enough to indicate the existence of geographical/biological races. The Sewall Wright quote that Wade refers to via Harpending and Rodgers is: “We will take F = 0.25 as an arbitrary value above which there is very great differentiation, the range of 0.15 to 0.25 as indicating moderately great differentiation. Differentiation is by no means negligible if F is as small as 0.05 or even less as bought out in the preceding chapter”. In the non-human literature, the value of FST that has been used to describe subspecies or biological races is FST > 0.250, Wright’s value for very great differentiation (Smith et al. 1997; Templeton 2002.) Subsequent studies of multiple loci, including whole genome analyses, have generally shown human FST values at much less than Wright’s critical value.

Unless we are experts on “algorithms designed to infer population structure”, we are now well and truly Eulered. And to make it worse, the other side has equal and opposite math, complete with pretty diagrams. How do we Feynman our way out of this one?

What if we compare race to its closest analog, culture?

National cultures are complicated, because they might be enforced by national governments and show obvious clines at borders for historical reasons. So let’s talk about cultures within the United States.

It is generally believed that we can talk about the United States as being made up of different cultures. For example, Southern seems to be a culture. Midwestern seems to be another culture. Yankees are probably a third, and the West gets a fourth.

We have various stereotypes about these cultures – for example, Midwesterners as wholesome farmers, Westerners as rugged, independent types. Yankees are liberal and well-educated. Southerners are welcoming and friendly but also pretty racist.

We will probably never be able to agree on exactly how many cultures there are. If I say “Californian” is a culture, and you say it’s just part of the American West culture, and she says actually California has multiple cultures – Silicon Valley, LA, Central Valley, etc – there is no objective criteria by which we can say who is right.

If we really wanted to, we could ask people a bunch of questions about their politics, religion, philosophy, food preferences, art preferences, and so on, throw them in a statistical algorithm, and ask it to divide the US into a certain number of clusters. It could probably do so, giving us a set of clusters more or less like the ones we naively imagine. But this wouldn’t be especially interesting and it couldn’t solve the “fundamental” question of how many cultures there “are”.

It should be terribly obvious that almost all variation in people’s cultural traits is within-culture rather than between-culture. Do you play the piano? Speak Chinese? Eat meat? Vote straight Libertarian? Have gay sex? Go to the zoo? Certainly there is more variation among individuals within California in all of these areas than there is between the average Californian and the average New Yorker.

(to put this another way, if I wanted information about whether or not you played the piano, or whatever, I would gain only a tiny bit of knowledge when I learned you were a Californian, compared to a huge amount of knowledge when I learned which Californian you were)

Finally, a given person from Culture A may certainly be much more culturally similar to a given person from Culture B than they are to another Culture A member.

For example, I come from California and Ozy is from Florida – opposite ends of the country! – but we have similar interests, aptitudes, and preferences along many axes. On the other hand, my next door neighbor growing up in California was a high school cheerleader who is now a hairstylist pregnant with her umpteenth child – totally different from me in every respect.

There were two arguments against race being a real concept: it didn’t cluster nicely, and within-group variation was greater than between group variation. And both of these are equally true of culture. Any mathematical argument considering races as clusters of genes can be used equally well considering cultures as clusters of memes, and will likely return the same results.

Yet I can’t imagine someone saying “culture doesn’t exist” or “culture isn’t real”.

And more important, groups can vary in terms of culture; culture can have explanatory power; cultural stereotypes can be correct. I am pretty sure Westerners really are more rugged, Midwesterners more aw-shucks whitebread types, Southerners more racist.

(and everything really is bigger in Texas)

So in the debate between Wade and his critics on whether “algorithms designed to infer population structure” prove or disprove the existence of race, my opinion is that the mathematical question is totally irrelevant to whether you are allowed to make claims like “Asians are genetically more collectivist than white people”.

That claim might not be true. But if it’s false, it will be false for the usual reasons things are false, rather than because one of its terms totally fails to refer.

On the other hand, this framing also suggests that race doesn’t have any extra reality beyond culture. All debates are bravery debates because trade-offs, and if I’m misinterpreting the state of popular opinion on this one it may be that the most important thing this analogy can do is debunk a bunch of people who think that race is real and “scientific” in a way that culture isn’t.

Because we usually understand concretes more completely than we understand abstracts (especially if we are bad at math) we can convert an abstract into an isomorphic concrete and then apply the same arguments. If it disproves something that we know is true, the original argument proves too much and can be discarded.