Annals of Perversion2

The Rise of Innumeracy

Now I want to start this with an observation that some of you may find obvious, but bear with me. It’s interesting how many of the men who defined modern science (and math), lived and worked n the 17^th and 18^th centuries. Consider: Isaac Newton (1643-1727), Antoine-Laurent Lavoisier (1743-1794), often considered the father of modern chemistry, whose career overlapped with Adam Smith (1723-1792), the founder of modern economics.

Not by any means the only important men of science. Consider Pascal, Fermat, and Huygens, responsible for probability theory and statistics; Gottfried Wilhelm Leibniz (1646-1716), who also developed calculus. And yes, there were important chemists before Lavoisier; men like Boyle (1627-1691), along with Dalton and Priestley. But that doesn’t contradict my argument.

And yes, I’m quite aware that going back for centuries, educated men had argued that the earth was spherical, and so forth and so on. But not only did the men I mentioned prove (and in some cases discard) earlier ideas, but with calculus and probability, they transformed scientific reasoning.

By the end of the 18^th century, not only was the concept of testing hypotheses by experiment and observation being practiced, but the mathematical tools needed to make sense out of the data were understood and employed.

The Paradox of Innumeracy

But after 1800, these developments were checked, perverted, and often overwhelmed by the rise of innumeracy. My boyhood gave me both a good deal of experience with that, and the realization about how complicated it was, that it went far deeper than the inability to do simple calculations.

Our farm was basically a ranch. We had about 1200 head (of cattle) and around 800 or so sheep; the cultivated parts were to produce hay and corn to aid in feeding them during the winter. We had a fellow who was on site, and part of his job was counting the cattle, as, unlike sheep, they could wander off, especially if there was a break in the fence—or even be stolen.

So Charley kept track of them. He was a keen observer, and very conscientious. So he’d ride around (on horseback) and count them using a simple principle: four vertical marks and a diagonal one.

Innumeracy at its most basic level, I suppose, but since we spent a lot of time together, I begin to understand how innumeracy affected thought processes. And yeah, you’re thinking, why is he going on and on about something everyone understands? Because until you experience the results, have to deal with the consequences, you don’t. He was a conscientious worker and a terrific observer (try differentiating one cow from another), but he couldn’t make decisions based on the data because for him, the number 12 was like one of Plato’s forms.

Decisions: did some cow figure out how to slip through the fence as opposed to there’s a break in the fence and we need to find it right now (a lot of the fencing separated pasture from forest and swamp).

And hence the paradox. As the mathematical tools enabling us to make decisions became more and more sophisticated, more and more technically educated people became innumerate. That is, not only were they unable to apply quantitative reasoning to resolve actual problems, but they refused to admit the possibility. Being an educator (in theory anyway), I started seeing that. I wasn’t alone.

The Triumph of Innumeracy 1: Systemic Failure

Around 1980, when I got stuck with being the chair of the English Department, both our both English and math departments had to teach a one year remedial course in both math and writing, because roughly ten percent of our entering students couldn’t demonstrate basic abilities in those two areas.

Not just our entering students, either. At one point I had to deal with a transfer student (a third year major who’d transferred in from a big state university), and had failed the departmental test, which required writing a 250 word essay in an hour, with minimal requirements for coherence and grammar.

The majors I was responsible for were definitely outraged about the remedial math course. It was pointless, irrelevant to being a major. So I examined the test. It was sobering: it barely got past things like fractions and questions more or less on the order of “If Fred had ten apples and Sally has eleven apples, and Fred give Brenda half of his apples, how many do Fred and Sally have left?”

I assumed our situation was atypical. But it wasn’t.

Fast forward until today. The University of California at San Diego (usually regarded a one the country’s outstanding public universities) found about the same percentage of its entering students were like ours. Harvard was having to offer remedial math. We’re not talking about mastering what in high schools was called Algebra 2, we’re talking about the concepts supposedly mastered in the first seven or eight years of schooling, things like fractions and percentages.

Alarming numbers of students couldn’t solve simple problems like the following: if eggs cost three dollars a dozen, how much is one egg? And if you went one step further, gave them a simple equation they were totally lost. For example, if x² plus 3x plus eight equals 26, what was the value of x?

Forget calculus and probability: these people couldn’t manage algebra one, so expecting them to understand how numbers can be represented by graphs using a simple xy axis, basic things like means and the median, visual aids like pie charts, isn’t going to happen.

Now the above may sound like I’m blaming university math departments, and initially, back in the 1960s, I did. But that’s hardly fair. The basic problem started way back there in primary and secondary education, revealed a systemic failure that had gradually crept in, so by the time it was realized, it was too late.

So what happened in the 19^th century was the emergence of men who came up with crackpot explanations that revealed higher levels of innumeracy, ideas that then became increasingly influential as successive generations of supposedly educated people steadily progressed towards more subterranean levels of innumeracy. A remark that explains why I’m now jumping around, all the way back to Thomas Malthus.

The Triumph of Innumeracy 2: Malthus

I’ve talked about Malthus before, and nowadays just about everyone is familiar with his claim that the population was increasing geometrically while the food supply was only increasing arithmetically. And yes, I’m aware that he put it differently. In my view because he was so innumerate he probably didn’t grasp the math. Now you would think that since his Essay on the Principle of Population appeared in 1798, by now it should be pretty obvious that he was wrong on several different levels. But the Malthusians are still around, like the Marxists, and I believe one reason is the rise of innumeracy. Malthus’s argument sounds mathematical (or scientific or logical). So it appeals to people who don’t understand any of the three.

Here’s the problem, which is in two parts. First, just restricting ourselves to the good parson’s country, although there had been attempts to calculate the wealth and population, most notably by Gregory King in 1688, since there was no census in the modern meaning of the term until 1801 (in the UK) Malthus had no actual evidence about the size of the population even in his own country.

The obvious point: You can’t make a comparison involving two quantities unless you have actual numbers.
Because in order to claim that the population was increasing, it takes more than one data point (duh!). When Malthus was writing, not only did those numbers not exist, but he had contemporaries who argued the opposite. Since he had no data to support his idea of the population increase, he had no way to calculate the rate of change.

What was worse was that at the same time, what evidence there was about the food supply, painted a picture quite the opposite of what Malthus was claiming.

Now a comparison involving numbers is simply an equation in which you make an argument about the relationship between two ratios: a/b if greater than, less than, equal to a1/b1. But the values in the equation Malthus was proposing weren’t just unknowns, they were unknowable. Sure, it sounds impressive. If you’re innumerate.

There’s another point here worth mentioning. Just because the notion of a progression on the order of 2, 4, 8, 16,32, 64 . . . is mathematically possible doesn’t mean it actually describes any actual events. Earlier, I mentioned the rise of actual chemistry. But one of the most basic things you noticed in observation was a saturation curve. That is, if you started adding salt to water, at first the salt dissolved very quickly, but then the rate slowed down, until eventually, all the salt you added just sank to the bottom. And if left out in the sun, the process reversed itself: the water evaporated and finally all you had was salt.

And by the time Malthus wrote, it was possible to calculate the rate of growth and decline.

I would go so far as to say that what Malthus was suggesting—basically a rate of increase that was a straight line, doesn’t actually exist in the observable world. Sure, if you graph the rate of increase, there is a part of the curve that can resemble a straight line, but it has a limit (which is where basic calculus comes in). People making arguments based on straight line projections (like the one Malthus was making) are pretty much like people who look at flight paths and can’t understand why they’re not straight lines (helpful hint: because the shortest distance between two points on a sphere is not a straight line, which is why spherical trig exists). I know! Next I’m going to say there’s no tooth fairy.

But since the innumerate don’t understand any of this, Malthus’ fantasies are alive and well. The innumerate simply can’t understand how numbers can describe reality—provided they’re actually accurately measuring things.

That leads me to remark I made in my last post about America’s remarkable wealth. So now I’ll explain why we know that and why it’s not understood.

Economics and Incomprehension

When I was an undergraduate, the basic economics course was a sophomore course, and in order to take it, the student had to have completed calculus, which in those days was a one year course that met five days a week, with course credit assigned in proportion. I managed to survive the course, became a member of the “elect,’ (as Calvin called them).

Because by definition, the calculus requirement eliminated the vast majority of undergraduates, since the percentage of undergraduates who were required to take the course comprised a distinct minority; basically those in math, physics, and so forth, as well as economics. In those bygone days, the number of majors in those fields was way less than the number of majors in English and history alone. And over the decades, although the numbers of students shifted dramatically, if anything, the proportion of students taking calculus diminished.

Except for a few oddballs like me, I doubt many undergraduates are inclined to take calculus unless they’re forced to to it by their degree requirements. But to be fair, the calculus requirement had benefits, probably the most important being that if you survived it, statistics and probability theory was easy. In fact, I did those on my own, just as earlier I had done spherical trig. And as I was deeply interested in the relationship between important artists and their society, I got into two sub-fields of economics: development theory and economic history.

So, when I read the first sentence in Professor Douglas North’s preface to his Economic Growth of the United States, 1790-1860, where he observes that “We were an industrial nation second only to Britain in manufacturing,” I understood both what he was saying and why a good many historians didn’t understand that he was making an important point based on an enormous amount of research.

As far as the second point goes, here’s the basic problem. When economic historians used the words industrialization and manufacturing, they are using them in a very specific sense. The terms have a stipulated meaning, and those meanings are almost invariably not the ordinary sense of the term in common speech. Assuming they mean anything at all to most people. The sunk cost fallacy?

There’s an additional complication. When most of us read Professor North’s sentence, we assume by an industrial nation he means one with lots of products like automobiles and airplanes, and the complicated components that go into them: steel, glass, plastic, rubber, and so forth.

Now obviously there weren’t any automobiles or airplanes in 1860, so we assume, reasonably enough, that he means their antecedents, such as locomo­tives and steam engines, which in turn demanded the production of steel and the fabrication of it into machinery, as well as products like the rails on which trains ran, the ores out of which the iron and steel were made, and the coal that was required to heat the smelters, which in turn involved mining.

And that’s assuming we even think about this at all.

But when economic historians speak of the Industrial Revolution and of industrialization, when we look at the basis for those claims, the first thing that stands out is that their use of the word “industry” and “industrial” is quite different from what one would logically assume, or certainly from what we would assume today. Of course the word has different connotations for different people, but I think it’s fair to say that when we think of an “industrial” power in connection with warfare, what comes to mind are weapons, explosives, and machines of various kinds. And on reflection, that capability implies establishments making various forms of metals, chemicals, and the like.

But the difficulty with the terms economists use is not just a matter of definition. That’s because economists, like chemists and physicists, rely heavily on mathematics, and not just the kind people use to balance a checkbook. I wouldn’t go so far as to say that you can’t really understand what they’re talking about unless you know calculus, but it definitely helps.

By the 1830s, both the United States and Great Britain were conducting censuses, and those not only gave population figures (and demographic information), but data that enabled the measurement of national wealth.

So what economic historians meant when they spoke of an “industrial revolution” was that the share of the national wealth represented by manufactures exceeded the wealth generated by agriculture, fishing, forestry, and mining.

The somewhat misleading part was in the definition of “manufactures,” basically any product that created from the basic materials. So, for example, taking metallic ores and turning them into iron or steel, turning wheat into flour, making cloth out of cotton, were all considered manufactures.

But that was because economic historians were confronted with a problem: the data being compiled to measure the aggregate value of what our country produced (for example) was divided into five categories: Raw Materials, Crude Foods, Manufactured Foods, Semi-Manufactures, and Manufacturers. So only the last deals with goods turned out in a manufacturing process.

So if we want to find out the extent to which we were an “industrial” nation, all we have to do is add up the value of the fifth category and compare it to the total, since the end point of the process of industrialization is that this category is the largest share of the country’s output.

And when we look at the most significant items in these categories, we can draw some interesting conclusions. But to get to the point, actually two points, the first being that when you dive into the census categories that Professor North helpfully gives you, and then compare them with Great Britain’s at the same time, we were in a pretty distant second place.

But basically, it’s the second point that’s crucial, and actually it’s pretty simple. The share of the national wealth created by manufactures was greater than the share created by the production of the unrefined categories I listed above.

But here’s where innumeracy gets involved and it’s both extremely important and often misunderstood. The easiest way to present the data (and not just if you’re an economist) is to draw a circle on the board, label it 1688 (or 1740 or something), say the circle represents the national wealth, and the shaded part is the percentage that’s agrarian. Then you draw another circle representing the national wealth in say, 1860 (or 1890), and indicate the way the proportions have shifted.

Easy to visualize. Except that the second circle has to be way bigger in order to make a proper comparison between the national wealth at the two periods. That’s where the math comes in—and hence the confusion. Because as everyone knows, we can easily calculate the area of a circle by using a simple formula in which the area equals pi*r², where pi is a constant roughly equal to 3.14, and r² square is the square of the radius. A simple formula you remember from high school, right?

But here’s what’s important: as r increases, the increasingly small slice of the pie represented by agrarian products gradually becomes much bigger than the “larger” slice in the original.

And here’s an excellent example of how the failure to grasp this leads to serious error. By 1860 cotton was a major factor in our country’s wealth, and, as Professor North explains, we had an absolute comparative advantage in producing it. As I mentioned in my last post, a good deal of what is currently believed about the 1861-1865 war is incorrect, and one of these assumptions is that the South’s cultivation of cotton either required slave labor or was heavily dependent on it.

In 1850, there were slightly over six million acres of cotton. But by 1895, there were almost 24 million acres under cultivation for cotton. In fact, this fourfold expan­sion was almost the same as the expansion in the cultivation of grain. Of course the country was bigger, but the whopping increase really makes it impossible to argue that slavery was particularly relevant to the profitability of growing cotton.

But although innumeracy precludes making comparisons using numbers—the fundamental work of economists, the problem I outlined with Malthus—it also precludes any serious understanding of much else. And in a curious way the Malthusians help to empower Marxian notions, or to hide the effects.