### Black Lives MAFFer

Janet Bloomfield had a guest post up from a black mathematician by the name of Dr. Jonathan David Farley a few days ago, in which Dr. Farley attempted to explain (blacksplain?) away the fact that NO black mathematician has ever been awarded a Fields Medal.

The Fields Medal is, of course, the most prestigious award handed out in the field of mathematics, and is considerably harder to win as an achievement that any of the Nobel Prizes. (That includes the "pretend" Nobel Prize for Economics, which is in fact the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel. It ain't the real deal, Buster.)

That would be because, unlike the annual Nobel Prize, the Fields Medal is handed out every four years. And unlike the "Nobel Prize" in Economics, you actually have to be a truly great mathematician to be awarded one of these.

You have to be under 40 to qualify. You have to have solved some of the most knotty, challenging, fiendishly complex, mind-bendingly arcane mysteries within your field- and in case you are wondering, yes, mathematics can get seriously weird. I'm talking about things like figuring out how to navigate around non-spherical topologies (such as toroids)- in plain English, this means "how to drive on a planet shaped like a donut without getting lost". Or dealing with the Riemann Conjecture. Or, say, proving Fermat's Last Theorem.

Now, Dr. Farley basically attempted to explain the lack of black Fields Medalists by resorting to attacks on the well-known and highly controversial (to some minds, anyway) arguments about the wide disparities in average racial intelligence between blacks and whites. He followed these up with anecdotal attacks on people that he felt had denied him a possible shot at a Fields Medal.

Here is what he actually wrote, so we are clear:
John Derbyshire, a columnist for the National Review, wrote an essay last week implying that black people were intellectually inferior to white people: “Only one out of six blacks is smarter than the average white.” Derbyshire pulled these figures from a region near his large intestine.
One of Derbyshire’s claims, however, is true: that there are no black winners of the Fields medal, the “Nobel prize of mathematics”. According to Derbyshire, this is “civilisationally consequential”. Derbyshire implies that the absence of a black winner means that black people are incapable of genius. In reality, black mathematicians face career-retarding racism that white Fields medallists never encounter. Three stories will suffice to make this point.
The first involves Saunders Mac Lane, one of the most influential algebraists of the last century. He co-authored, with Garrett Birkhoff, a text that enthralled me as a first-year undergraduate. I first encountered lattice theory, which for a long time I loved more than anything in life, in that book. In 1951, Mac Lane was president of the Mathematical Association of America (MAA). Vanderbilt University hosted an MAA conference, and three black mathematicians wished to attend the conference’s banquet. They were barred and Mac Lane refused to take a stand: Vanderbilt University was in racially segregated Tennessee, and he did not want to offend his hosts.
The second story involves one of the few black mathematicians whom white mathematicians acknowledge as great – or, I should say, “black American mathematicians”, since obviously Euclid, Eratosthenes and other African mathematicians outshone Europe’s brightest stars for millennia. His name was David Blackwell. I first met Blackwell in 1995, in the common room of Berkeley’s maths department, one of the few times two black people had ever been in the room. Blackwell obtained his PhD in mathematics when he was only 22.
While he had a fellowship to work at the Institute for Advanced Study, the American home of Einstein and the other-worldly logician Kurt Gödel, nearby Princeton University refused to allow Blackwell to attend lectures because he was black. Although he later became the first black member of the National Academy of Sciences (with a colleague saying, “he would come into a field that had been well studied and find something really new that was remarkable”), the University of California at Berkeley’s maths department would not hire Blackwell on account of his race (a European later asked Blackwell to join the statistics department).
Now, the moment I read a couple of those claims, my BS detector popped straight up and started yammering. There is something deeply suspect about these arguments, and a mathematician- especially a highly accomplished one like Dr. Farley- should know better than to make them.

Let us start with the first: that John Derbyshire, well-known pessimist, curmudgeon, and remarkably astute scholar of the human condition, pulled the statistic about "one in six blacks being smarter than the average white" out of his arse.

Well now. The average IQ for black Americans is about 85, give or take a couple of points depending on which dataset you're using. The average IQ for white Americans is about 103, again depending on which dataset you use.

Let us assume, for simplicity's sake- it's never this simple in real life, obviously- that the standard deviation of the distribution of both black and white IQs is 15 points. That is, after all, the well-known and repeatedly recorded gap between the two races.

What does the distribution of IQs look like under these conditions?

 Brought to you courtesy of the R statistical computing package- go get it, it's totally FREE
OK, so that's obviously fairly remarkable, if you know how to interpret a graph like that (not difficult), but it's not in itself particularly revealing.

To check Derb's figures, we need to do a little MAFF ourselves. Again, not terribly taxing.

So. If you open up any standard high-school mathematics textbook, you will note that it is possible to calculate the cumulative density function, or distribution, of a Gaussian distribution given a mean and a standard deviation.

For the anoraks among you, this is the actual formula:

$F(x|\mu,\sigma^2)=\int_{-\infty&space;}^{x}\frac{1}{\sqrt{2&space;\sigma&space;^2&space;\pi}}&space;e&space;^{\&space;\frac{\left(&space;x-\mu&space;\right&space;)&space;^2}{2\sigma^2}}=\frac{1}{2}\left&space;[&space;1&space;+&space;erf&space;\left&space;(&space;\frac{x&space;-&space;\mu}{\sigma&space;\sqrt{2}}&space;\right&space;)&space;\right&space;]$

(I do so love LaTeX...)

It just so happens, of course, that we have three Gaussian distributions right there in front of us. So we can therefore calculate the percentage of the population, under each scenario, that is above the "average" intelligence level of 100, given those distributions.

We do this by simply subtracting from 1 the cumulative density of each distribution, at the point 100.

Easily enough done. What do we get if we do this?
• For the population as a whole, where the average is 100 and the S.D. is 15, exactly 50% of the population is smarter, and 50% is dumber, than the average IQ of 100- which is not exactly a surprise
• For whites, though, given an average IQ of 103, 58% of whites are smarter than the 100 mark
• For blacks, only 16% are smarter- and that's after rounding up
Hey- you know else 16% is? It's a little less than 1 over 6.

And if you subtract from 1 the cumulative density of a Gaussian distribution centred at 85 with an S.D. of 15 at the point 103, you get: 11.5%.

That's right. Just 11.5% of all black Americans are smarter than the average white American.

So actually, Dr. Farley has a point, just not the one he thought he did. Derb was, if anything, understating his case.

We can extend the analysis done above a little bit further, by the way.

Consider, for instance: what are the general, white-specific, and black-specific probabilities of hitting genius-level IQ? This is generally considered to be an IQ score of 130 and above, but it depends somewhat on exactly which assessment method you're using.

(The usual caveat about genius-level IQs applies; just because someone has a nosebleed IQ doesn't mean he isn't an idiot in many respects. Jimmy Carter supposedly has an IQ of around 150; Ronald Reagan had an IQ of about 140. Odoofuss, of course, has an IQ of roughly 116. The defence rests.)

Now then: if you use the same method to figure out what percentage of people would be over genius level, here are the results:
• General population: 2.275%
• White Americans: 3.593%
• Black Americans: 0.135%
Yeah. Less than a fifth of a percent of blacks register in the genius range.

Now I will be the first to admit that these intelligence tests are flawed, and that these comparisons are not nearly as cut-and-dried as a maths geek like me would like to believe. But they are still useful exercises nonetheless, and they illustrate precisely how different black America is from white America.

Then we get to the whole point about Euclid and Erastosthenes, and... oh dear.

Euclid was not "North African". He was Greek. The only way you can call him "North African" is if you argue that he was born in Alexandria- without bothering to consider that Alexandria was founded by the very European conqueror named, you guessed it, Alexander the Great.

Eratosthenes was born in Cyrene a couple of hundred years later. Now, the last time I checked, Cyrene was a Greek city at the time, even though it was situated in Libya. Furthermore, both of these mathematicians and great philosophers worked under Greek rule- that's what the Ptolemaic Dynasty was, a group of Greeks descended from one of Alexander the Great's top generals who controlled Egypt from the time of the conqueror's death to the coming of Julius Caesar.

Oh dear. Dr. Farley is already 0-2. Now what?

Without contending further with the points that he has raised, let us at least stop to consider that there are, in fact, clear and observable racial differences between different subgroups of humans, and that furthermore there is nothing wrong with concluding that some of these differences do determine the types and quality of societies that these subgroups will establish.

There is nothing in the least bit wrong with making any of these observations, as long as they are backed up by fact. There is everything wrong with subscribing to a simplistic "blank slate" theory of human behaviour that refuses to take into account the very real differences between us.

Such a theory cannot be intellectually rigourous, for it cannot be open to disproof. Therefore it is no actual theory at all, it is merely ideology masquerading as intelligent discourse.

One would think that a man with a PhD in mathematics would know better.

1. I nearly choked at the greek names being treated as blacks, or _anything_ other than greek

2. Perhaps the Dr. got an Affimative Action PhD.