Non-fiction Book of the Month - How Not to Be Wrong

Non-fiction Book of the Month - How Not to Be Wrong

Jordan Ellenberg's How Not to be Wrong shows how a little mathematical knowledge can reveal all in our complicated world.

Posted on 1st June 2015 by Jordan Ellenberg
I would always say that I have a 'writing brain'. I like abstract concepts and ridiculous debates about words and whether language can ever truly express the true nature of our existence. All that sort of thing is, to me, incredibly fun and, to everybody I am just about friends with, incredibly annoying. But I also loved maths when I was younger. The neatness of it all, balancing equations until it all made sense. There's often a definite answer to maths problems and, if there doesn't appear to be, then it probably means we just haven't found it yet.

All of this, to bring us back around to the topic, is why I enjoyed How Not to be Wrong. Any book that combines big, abstract ideas with a solid grounding in logic was always going to appeal to me. In particular, the problem of how an infinite sum of 1+2+4+8+16+... can equal -1 is one that's going to wrattle my head for quite some time.

Jordan Ellenberg writes about maths brilliantly. His enthusiasm for his subject is infectious and he is able to break maths down into parts which anybody can understand, regardless of their previous mathematical knowledge.

Taking maths and expanding it into how it effects life, economics and our own human nature, How Not to be Wrong will have you looking at the world in a whole new way.

Read the opening pages below and, what's more, Jordan Ellenberg will be at our Trafalgar Square bookshop, Wednesday 3rd at 7pm, to discuss the book.

A few years ago, in the heat of the battle over the Affordable Care Act, Daniel  J. Mitchell of the  libertarian Cato Institute posted a blog entry with the provocative title: “Why  Is Obama Trying to Make America More Like Sweden when Swedes Are Trying to Be Less Like Sweden?”
Good question!  When you put it that way,  it does seem pretty perverse. Why, Mr. President, are we swimming against the current of history, while social welfare states around the world—even rich little Sweden!—are cutting back on expensive benefits and high taxes? “If Swedes have learned from  their mistakes and are now trying to reduce the size and scope of government,” Mitchell writes, “why are American politicians determined to repeat those mistakes?”
Answering this  question will require an extremely scientific chart. Here’s what the world looks like to the Cato Institute:

The x-axis represents Swedishness, (Here “Swedishness” refers to “quantity of social services and taxation,” not to other features of Sweden such as “ready availability of herring in dozens of different sauces,” a condition to which all nations should obviously aspire.)  and the y-axis is some measure of prosperity. Don’t worry about exactly how we’re quantifying these things. The point is just this: according to the chart, the more Swedish you are, the worse off your country is.

The Swedes, no fools, have figured this out and are launching their northwestward climb toward  free- market prosperity. But Obama’s sliding in the wrong direction.

Let me draw the same picture from the point of view of people whose economic views are closer to President Obama’s than to those of the Cato Institute. See the next image.

This picture gives very different advice about how Swedish  we should be.  Where do we find peak prosperity? At a point more Swedish than America, but less Swedish  than Sweden. If this picture is right, it makes perfect sense for Obama to beef  up  our  welfare  state while  the Swedes trim theirs down.

The  difference between the two pictures is the difference between linearity and nonlinearity, one of the central distinctions in mathematics. The Cato curve is a line; (Or a line segment, if you must. I won’t make a big deal of this distinction.)  the non-Cato curve, the one with the hump in the middle,  is not. A line is one kind of curve, but  not the only kind, and lines enjoy all kinds of special properties that curves in general  may not. The  highest point on a line segment—the maximum prosperity, in this example—has to be on one end or the other. That’s just how lines are. If lowering  taxes is good  for prosperity, then lowering  taxes even more is even better. And if Sweden wants to de-Swede, so should we. Of course, an anti-Cato think tank might posit that the line slopes in the other direction, going southwest to northeast. And if that’s what the line looks like, then no amount of social spending is too much. The optimal policy is Maximum Swede.
Usually, when someone announces  they’re a  “nonlinear  thinker” they’re about to apologize for losing something you lent them. But non- linearity is a real thing! And in this context, thinking nonlinearly is crucial, because  not all curves  are lines.  A moment of reflection will tell you  that the real curves of economics look like the second  picture, not the first. They’re nonlinear. Mitchell’s reasoning is an example of false linearity—he’s assuming, without coming right out and saying so, that the course of prosperity is described by the line segment in the first picture, in which case Sweden stripping down its social infrastructure means we should do the same.
But as long as you believe there’s such a thing as too much welfare state and such a thing as too little, you know the linear picture is wrong. Some principle more complicated than “More government bad, less government good”  is in effect. The generals who consulted Abraham Wald faced the same kind of situation: too little armor meant planes got shot down, too much meant the planes couldn’t fly. It’s not a question of whether adding more armor is good or bad; it could be either, depending on how heavily armored the planes are to start with. If there’s an optimal answer, it’s somewhere in  the middle, and deviating from it in either direction is bad news.
Nonlinear thinking means which way you should go depends on where you already are.
This insight isn’t new. Already in Roman times we find Horace’s famous remark “Est modus in rebus, sunt certi denique fines, quos ultra citraque nequit consistere rectum” (“There is a proper measure in things. There are, finally, certain boundaries short of and beyond which  what is right cannot exist”). And further back still, in the Nicomachean Ethics, Aristotle observes that eating either too much or too little is troubling to the constitution. The optimum is somewhere in between; because the relation between eating and health isn’t linear, but curved, with bad outcomes on both ends.
The irony is that economic conservatives like the folks at Cato used to understand this better than anybody. That second  picture I drew  up there? The extremely scientific one with the hump in the middle? I am not the first person  to draw  it. It’s called the Laffer curve, and it’s played a central role in Republican economics for almost forty years.  By the middle of the Reagan administration, the curve had become such a commonplace of economic discourse  that Ben Stein ad-libbed it into his famous  soul-killing  lecture in Ferris Bueller’s Day Off:

Anyone  know  what  this is? Class?  Anyone?  . . . Anyone?  Anyone seen this before? The  Laffer  curve.  Anyone  know  what this says? It says that at this point on the revenue curve,  you will get exactly the same  amount of revenue as at this point. This is very controversial. Does anyone  know what Vice President Bush called this in 1980? Anyone? Something-doo economics. Voodoo economics.

The  legend  of the Laffer curve  goes like this: Arthur Laffer, then an economics professor at the University of Chicago,  had dinner one night in 1974  with Dick Cheney, Donald Rumsfeld, and Wall Street Journal editor Jude  Wanniski at an upscale hotel restaurant in Washington, DC.  They were tussling over President Ford’s tax plan, and eventually, as intellectuals do when  the tussling gets heavy, Laffer commandeered a napkin (Laffer disputes the napkin  portion of the story, recalling  that the restaurant had classy cloth napkins that he would  never have vandalized with an economic  doodle.)  and drew  a picture. The picture looked like this:

The  horizontal axis  here  is level  of  taxation, and  the vertical axis represents the amount of revenue the government takes in from  taxpayers. On  the left edge of the graph,  the tax rate is 0%; in that case, by definition, the government gets no tax revenue. On  the right, the tax rate is 100%; whatever income you have, whether from a business  you run or a salary you’re paid, goes straight into Uncle Sam’s bag.
Which is empty. Because if the government vacuums up  every cent of the wage you’re paid to show up and teach school, or sell hardware, or middle-manage, why bother doing  it?  Over on  the right edge of  the graph,  people don’t work at all. Or, if they work,  they do so in informal economic niches  where  the tax collector’s hand can’t reach.  The government’s revenue is zero once again.
In the intermediate range in the middle of the curve,  where  the government charges us somewhere between none of our income and all of it—in other words, in the real world—the government does take in some amount of revenue. That means the curve recording the relationship between tax rate and government revenue cannot be a straight line.  If it  were, revenue would  be maximized at either the left or right edge of the graph; but it’s zero both places.  If the current income tax is really close to zero, so that you’re on  the left-hand side of  the graph, then raising taxes increases the amount of money the government has available to fund services and programs, just as you might intuitively expect. But if the rate is close to 100%, raising taxes actually decreases government revenue. If you’re to the right of the Laffer peak, and you want to decrease  the deficit without cutting spending, there’s a simple and politically peachy solution: lower the tax rate, and thereby increase the amount of taxes you take in. Which way you should go depends  on where you are.
So where are we? That’s where  things get sticky. In 1974,  the top income tax rate was 70%, and the idea that America was on the right-hand downslope of the Laffer curve held a certain appeal—especially for the few people lucky enough to pay tax at that rate, which only applied to income beyond the first $200,000. (Somewhere between a half million and a million dollars a year in today’s income.) And the Laffer curve had a potent advocate in Wanniski, who brought his theory into the public consciousness in a 1978 book rather self-assuredly titled The Way the World Works.(Like I’m one to talk.) Wanniski was a true believer, with the right mix of zeal and political canniness to get people to listen to an idea considered fringy even by tax-cut advocates. He was untroubled by being called a nut. “Now, what does ‘nut’ mean?” he asked an interviewer. “Thomas Edison was a nut, Leibniz was a nut, Galileo was a nut, so forth and so on. Everybody who comes with a new idea to the conventional wisdom,  comes with an idea that’s so far outside the mainstream, that’s considered nutty.”
(Aside: it’s important to point out  here  that people with out-of-the- mainstream ideas  who compare themselves to Edison and Galileo are never actually right. I get letters with this kind of language at least once a month, usually from people who have  “proofs” of mathematical statements that have been known for  hundreds of years to be false. I can guarantee you Einstein did not go around telling people, “Look, I know this theory of general relativity sounds wacky,  but that’s what they said about Galileo!”)
The Laffer curve,  with its compact visual representation and its agreeably counterintuitive sting, turned out to be an easy sell for politicians with a preexisting hunger for tax cuts. As economist Hal Varian put  it, “You can explain it to a Congressman in six minutes and he can talk about it for six months.” Wanniski became an advisor first to Jack Kemp,  then to Ronald  Reagan, whose experiences as a wealthy movie star in the 1940s formed the template for his view of the economy four decades  later. His budget director, David  Stockman, recalls:

I came  into the Big Money  making  pictures during  World  War  II,” [Reagan] would  always say. At that time the wartime income surtax hit 90  percent. You  could  only  make  four  pictures and  then you were  in the top bracket, he would  continue. So we all quit work- ing after about four  pictures and went off to the country. High  tax rates caused  less work.  Low  tax rates caused  more.  His  experience proved it.


These days it’s hard  to find a reputable economist who thinks we’re on the downslope of the Laffer curve.  Maybe that’s not surprising, considering  top incomes are currently taxed at just 35%, a rate that would have seemed absurdly low for most of the twentieth century. But even in Reagan’s day, we were probably on the left-hand side of the curve.  Greg Mankiw,  an economist at Harvard and a Republican who chaired the Council of Economic Advisors under the second President Bush, writes in his microeconomics textbook:

Subsequent history failed  to confirm Laffers conjecture that lower tax rates would  raise tax revenue. When Reagan  cut  taxes after he was elected, the result  was less tax revenue, not more.  Revenue from personal income taxes (per  person,  adjusted for inf lation) fell by 9 percent from  1980  to 1984,  even  though average  income (per  per- son, adjusted for inf lation) grew  by 4 percent over  this period. Yet once the policy was in place, it was hard  to reverse.


Some sympathy for  the supply-siders is now in order.  First of all, maximizing government revenue needn’t be the goal of tax policy.  Milton Friedman, whom we last met during World  War II doing classified military work for the Statistical Research Group, went on to become a Nobel-winning economist and advisor  to presidents, and a powerful advocate for low taxes and  libertarian philosophy. Friedman’s famous  slogan on taxation is “I am in favor of cutting taxes under any circumstances and  for any  excuse,  for any  reason,  whenever it’s possible.” He didn’t think we should be aiming for the top of the Laffer curve, where government tax revenue is as high as it can be. For Friedman, money obtained by the government would eventually be money spent by the government, and that money, he felt, was more often spent badly than well.

More  moderate supply-side thinkers, like Mankiw, argue that lower taxes can increase the motivation to work hard  and  launch  businesses, leading  eventually to a bigger, stronger economy, even if the immediate effect of the tax cut is decreased government revenue and bigger deficits. An economist with more redistributionist  sympathies would   observe that this cuts both ways; maybe the government’s diminished ability to spend means it constructs less infrastructure, regulates fraud less stringently, and generally does less of the work that enables free enterprise to thrive.
Mankiw also points out that the very richest people—the ones who’d been paying 70% on the top tranche of their income—did contribute more tax revenue after Reagan’s tax cuts. (Whether the increased tax receipts are because the rich started working harder once less encumbered by income tax, as supply-side theory predicts, is more difficult to say for certain.)   That leads to the somewhat vexing possibility that the way to maximize government revenue is to jack up  taxes on  the middle class, who have no choice but to keep on working,  while slashing rates on the rich; those guys have enough stockpiled wealth to make credible threats to withhold or offshore their economic activity, should their government charge  them a rate they deem too high. If that story’s right, a lot of liberals will uncomfortably climb in the boat with Milton Friedman: maybe  maximizing tax revenue isn’t so great after all.
Mankiw’s final assessment is a rather polite, “Laffer’s argument is not completely without merit.” I would  give Laffer more credit than that! His drawing made the fundamental and incontrovertible mathematical point that the relationship between taxation and revenue is necessarily nonlinear. It doesn’t, of course, have to be a single smooth hill like the one Laffer sketched; it could  look like a trapezoid

but if it slopes upward in one place, it has to slope downward somewhere else. There is such a thing as being  too Swedish. That’s a statement no economist would  disagree with. It’s also, as Laffer  himself pointed out, something that was understood by many social scientists before  him. But to most people, it’s not at all obvious—at least, not until you  see the picture on the napkin. Laffer understood perfectly well that his curve didn’t have the power to tell you whether or not any given economy at any  given  time was  overtaxed or  not. That’s why he didn’t draw any numbers on the picture. Questioned during congressional testimony about the precise location of the optimal tax rate, he conceded, “I cannot measure it frankly, but  I can tell you what the characteristics of it are; yes, sir.” All the Laffer curve says is that lower taxes could, under some circumstances, increase  tax revenue; but figuring out  what those circumstances are requires deep, difficult, empirical work, the kind of work that doesn’t fit on a napkin.
There’s  nothing wrong  with the Laffer curve—only with the uses people put it to. Wanniski  and the politicans who  followed his panpipe fell prey to the oldest false syllogism in the book:

It could be the case that lowering  taxes will increase government revenue;

I want it to be the case that lowering  taxes will increase  government revenue;

Therefore, it is the case that lowering  taxes will increase  government revenue.