Later On

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Archive for the ‘Math’ Category

A map of mathematics

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Fascinating to explore if you like mathematics. Scroll, click, and read.

Written by LeisureGuy

15 February 2020 at 11:50 am

Posted in Math

The Dismal Kingdom: Do Economists Have Too Much Power?

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Paul Romer writes in Foreign Affairs:

Over the past 60 years, the United States has run what amounts to a natural experiment designed to answer a simple question: What happens when a government starts conducting its business in the foreign language of economists? After 1960, anyone who wanted to discuss almost any aspect of U.S. public policy—from how to make cars safer to whether to abolish the draft, from how to support the housing market to whether to regulate the financial sector—had to speak economics. Economists, the thinking went, promised expertise and fact-based analysis. They would bring scientific precision and rigor to government interventions.

For a while, this approach seemed a sure bet for steady progress. But several decades on, the picture is less encouraging. Consider, for example, the most basic quantitative indicator of well-being: the average length of a life. For much of the last century, life expectancy in the United States increased roughly in tandem with that in western Europe. But over the last four decades, the United States has been falling further and further behind. In 1980, the average American life was a year longer than the average European one. Today, it is two years shorter. For a long time, U.S. life expectancy was still rising but more slowly than in Europe; in recent years, it has been falling. A society is hardly making progress when its people are dying younger.

Binyamin Appelbaum makes this point in his new book, The Economists’ Hour. That book and another recent one—Transaction Man, by Nicholas Lemann—converge on the conclusion that the economists at the helm are doing more harm than good.

Both books are compelling and well reported, and both were written by journalists—outsiders who bring historical perspective to the changing role of economists in American society. Appelbaum tracks their influence across a wide range of policy questions since the 1960s. The language and the concepts of economics helped shape debates about unemployment and taxation, as one would expect. But they also influenced how the state handled military conscription, how it regulated airplane and railway travel, and how its courts interpreted laws limiting corporate power. Together, Appelbaum writes, economists’ countless interventions in U.S. public policy have amounted to no less than a “revolution”—well intentioned but with unanticipated consequences that were far from benign.

Lemann chronicles another, related revolution. In the first half of the twentieth century, especially after the calamity of the Great Depression, the conventional wisdom held that the power of corporations must be held in check by other comparably sized organizations—churches, unions, and, above all, a strong national government. But in the decades that followed, a new generation of economists argued that tweaks to how companies operated—more hostile takeovers, more reliance on corporate debt, bigger bonuses for executives when stock prices increased—would enable the market to regulate itself, obviating the need for stringent government oversight. Their suggestions soon became reality, especially in a newly deregulated financial sector, where they precipitated the emergence of junk bonds and other questionable innovations. Like Appelbaum, Lemann concludes that economists’ uncritical embrace of the market changed U.S. society for the worse.

Voters, too, have their doubts, in the United States and beyond. In the run-up to the 2016 Brexit vote, Michael Gove, then the British justice secretary, was asked to name economists who supported his position that the United Kingdom should leave the European Union. He refused. “People in this country have had enough of experts,” he snapped. “I’m not asking the public to trust me. I’m asking the public to trust themselves.” A majority of the British electorate followed his cue and voted to leave the EU, the warnings of countless economists be damned.

Economists should take that outcome as an admonition warranting a major course change. Writing in 2018, the economists David Colander and Craig Freedman proposed one such correction. Over the course of the twentieth century, they contended, economists had built more and more sophisticated models to guide public policy, and many succumbed to hubris in the process. To regain the public’s trust, economists should return to the humility of their nineteenth-century forebears, who emphasized the limits of their knowledge and welcomed others—experts, political leaders, and voters—to fill in the gaps. Economists today should recommit to that approach, even if it requires them to publicly expel from their ranks any member of the community who habitually overreaches.

ESCAPE FROM THE BASEMENT

Appelbaum’s book begins with a revealing anecdote from the 1950s about Paul Volcker, at the time a young economist working in the bowels of the Federal Reserve System and disillusioned about his career prospects. Among the Fed’s national leadership were bankers, lawyers, and a hog farmer from Iowa—but no economists. In 1970, William McChesney Martin, Jr., then chair of the Federal Reserve’s Board of Governors, could still explain to a visitor that although economists asked good questions, they worked from the basement because “they don’t know their own limitations, and they have a far greater sense of confidence in their analyses than I have found to be warranted.”

But Martin was on his way out, and as Appelbaum shows in the chapters that follow, economists were emerging from the basement—not just at the Fed but also across the government. To take just one example, consider the rapid spread of cost-benefit analysis as the tool of choice for assessing health and safety regulations. When the U.S. Congress created the Department of Transportation in 1966 and told it to make motor vehicles safer, lawmakers did not ask regulators to weigh the potential costs and benefits of proposed new rules: after all, no one could possibly determine the value of a human life. The economists Thomas Schelling and W. Kip Viscusi disagreed, arguing that people did in fact place a dollar value on human life, albeit implicitly, and that economists could calculate it.

Regulators initially rejected this approach, but as complaints about burdensome safety regulations grew louder, some began to waver. In 1974, the Department of Transportation used a cost-benefit analysis to reject a proposed requirement that trucks be fitted with so-called Mansfield bars, designed to prevent the type of accident that had killed the actress Jayne Mansfield in 1967. The cost of installing the bars on every truck, regulators calculated, would exceed the combined value of the lives that the bars would save. Soon, every participant in the conversation about safety regulations was expected to state and defend a specific dollar value for a life lost or saved.

Unfortunately, asking economists to set a value for human life obscured the fundamental distinction between the two questions that feed into every policy decision. One is empirical: What will happen if the government adopts this policy? The other is normative: Should the government adopt it? Economists can use evidence and logic to answer the first question. But there is no factual or logical argument that can answer the second one. In truth, the answer lies in beliefs about right and wrong, which differ from one individual to the next and evolve over time, much like people’s political views.

In principle, it is possible to maintain a clear separation between these two types of questions. Economists can answer such empirical questions as how much it would cost if the government required Mansfield bars. It is up to officials—and, by extension, up to the voters who put them in office—to answer the corresponding normative question: What cost should society bear to save a life in any particular context?

In practice, however, voters can provide only so much in the way of quantifiable directives. People may vote for an administration that promises safer cars, but that mandate alone is not specific enough to guide decisions such as whether to require Mansfield bars. Lacking clear guidance from voters, legislators, regulators, and judges turned to economists, who resolved the uncertainty by claiming to have found an empirical answer to the normative question at hand. In effect, by taking on the responsibility to determine for everyone the amount that society should spend to save a life, economists had agreed to play the role of the philosopher-king.

In Appelbaum’s account, this arrangement seems to have worked out surprisingly well in setting standards for automobile safety. Economists in the mold of Schelling and Viscusi seem to have channeled as best they could the moral beliefs of the median voter. When regulators first rejected Mansfield bars, in 1974, they put the value of a life at $200,000, but in response to pressure from voters demanding fewer traffic fatalities, economists and regulators gradually adjusted that number upward. Eventually, as the estimated value of the human lives lost to car accidents began to exceed the cost of installing Mansfield bars, regulators made the bars mandatory, and voters got the outcome they wanted.

Unfortunately, this outcome may have been possible only because, although the moral stakes were high, the financial stakes were not. No firm faced billions of dollars in gains or losses depending on whether the government mandated Mansfield bars. As a result, none had an incentive to use its massive financial resources to corrupt the regulatory process and bias its decisions, and the “don’t ask, don’t tell” system of using economists as philosopher-kings worked reasonably well.

The trouble arose when the stakes were higher—when the potential gains or losses extended into the tens of billions or hundreds of billions of dollars, as they do in decisions about regulating the financial sector, preventing dominant firms from stifling competition, or stopping a pharmaceutical firm from getting people addicted to painkillers. In such circumstances, it is all too easy for a firm that has a lot riding on the outcome to arrange for a pliant pretend economist to assume the role of the philosopher-king—someone willing to protect the firm’s reckless behavior from government interference and to do so with a veneer of objectivity and scientific expertise.

Simply put, a system that delegates to economists the responsibility for answering normative questions may yield many reasonable decisions when the stakes are low, but it will fail and cause enormous damage when powerful industries are brought into the mix. And it takes only a few huge failures to offset whatever positive difference smaller, successful interventions have made.

One such failure is . . .

Continue reading.

Written by LeisureGuy

12 February 2020 at 5:03 pm

The Hairy Nobel

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I got into the previous videos after watching this one:

Written by LeisureGuy

10 February 2020 at 11:44 am

Posted in Math, Science, Video

An equation with unusual resonance and range of applicability

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Fascinating. I particularly like the more in-depth view of the Mandelbrot set.

Written by LeisureGuy

4 February 2020 at 8:39 pm

Posted in Math, Science

Use a soroban — and teach your kids

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I used a soroban — a Japanese abacus — for some years, doing all my home bookkeeping on it. It really is quite fast and easy once you practice a bit and learn the basic movements.

The Japanese abacus has 1 bead above the central bar, this bead representing a vlue of 5, and 4 beads below the central bar, each representing a value of 1. Place value is assumed so the rightmost column represents units, and with all beads moved to the central bar, value 9 is represented (1 5-bead and 4 1-beads). The next column represents tends, and can prepresent a maximum of 90; the next column represents hundreds, and the maximum value is 900; and so on.

The Chinese abacus has two beads above the bar and five below, and the beads are ore rounded. Here’s a good soroban. Wooden beads are more pleasing than plastic (no surprise).

This instructional book is excellent, and you can also find YouTube tutorials. Just switching to the soroban in daily life will quickly produce profiency.

Full disclosure: I did learn and use the soroban long before spreadsheets were around. But I will say it’s enjoyable.

Written by LeisureGuy

28 January 2020 at 8:10 pm

Posted in Daily life, Math

Secrets of Math From the Bee Whisperer

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Scarlett Howard has taught honeybees how to add, subtract and understand zero. Their ingenuity suggests that all animals may have more mathematical talent than we thought. – photo credit: Anne Moffat for Quanta

Susan D’Agostino writes in Quanta:

carlett Howard teaches math to honeybees. She began with a few hives on a concrete balcony at RMIT University in Melbourne, when she was a doctoral candidate in zoology. Today, at the University of Toulouse, where she is a postdoctoral fellow, her lessons take place in a small field with approximately 50 hives.

It might seem a little strange — bees are insects, after all; what do they know about mathematics? A lot, it turns out. These eusocial flying insects can add, subtract and even comprehend the concept of zero.

“You can see their decision-making process in their movements and flight patterns,” Howard said. While deciding which of two answers is correct, they often fly toward one of the solutions before seeming to think better of it and flying off toward the other.

Howard teaches one bee at a time, placing it next to an apparatus known as a Y maze, a covered box shaped like a block letter Y. The bee enters the bottom leg of the Y and sees a mathematical question, expressed in shapes and colors. In the arithmetic lessons, blue shapes mean “add 1” to the given number of shapes, and yellow shapes mean “subtract 1.” To answer the question, the bee chooses from one of two possible solutions posted at the entrances to the Y’s upper arms. The bee will find a reward — sugar water — in the arm associated with the correct answer, and a punishment — tonic water, which bees find bitter — in the arm with the incorrect answer.

To teach bees about zero, she first trained them to understand the concept of “less than.” As with the addition and subtraction problems, she offered reinforcements for correct choices. Once an individual bee demonstrated it understood “less than,” she advanced that bee to the testing phase of her experiment, where it would decide if any number of shapes is less than zero shapes — a number the bee had never encountered before. Each bee had only one chance to answer. The bees often identified “zero shapes” as smaller than any number of shapes, and Howard concluded that they must possess an innate understanding that zero is smaller than any positive integer.

For each experiment, Howard trains and tests approximately 100 random bees from the thousands in her hives. Handling them is simple enough. After each correct choice, the bee flies back to the hive on its own, to offload its sweet reward. Then, at some point, it’ll come back. That’s because bees are central place foragers, meaning they will remember the experiment and return to it for additional resources. To prepare for her next pupil, Howard changes the stimuli on the Y maze. She has hundreds, possibly thousands of stimuli printed and laminated.

“They’re laminated so we can clean them with ethanol, because bees will scent-mark,” said Howard. “They’ll do anything to cheat the tests. They’re smart! They’ll mark the correct answer. Bees are not as simple as we used to think they are. Or even as some people still think they are.”

Quanta Magazine recently spoke with Howard about her research. The interview has been condensed and edited for clarity.

What first inspired you to research bees’ mathematical abilities? Were you a fan of the bugs?

I had always been really scared of bees. But when I was at university in Australia, Adrian Dyer, who works on bees’ cognitive abilities, told me, “Bees can do really cool things. They can recognize human faces and navigate mazes.” I thought, “Really? Is that true? I want to see that for myself.” So I pitched the idea of working on bees’ cognitive abilities to my potential Ph.D. supervisors.

We thought, “We can do something either really high-risk/high-reward, or we can do something less risky but less interesting.” We tried something risky first — whether bees could understand zero at the same level as some primates and birds do.

Most animals know, for example, whether or not they have “some food” or “zero food” in front of them. Do bees really possess more than this basic understanding of zero?

Bees are able to place zero within a numerical continuum. They know that zero is less than 1, it’s less than 2 and it’s less than 3. They also know that zero is more “less than 6” than it is “less than 1.”

Many animals have difficulty with zero. The number 1 might have been the lowest number they’d ever seen. When we got positive results from that experiment showing that they knew zero is lower than any positive integer, it was really exciting to see.

Of course, different bees have different processes of learning. Some do really well from the beginning. Some are really quite bad. You see this moment where they start to get things more and more right. You don’t want to anthropomorphize them too much, but it’s really incredible to watch how they learn.

Wait — some bees are better at math than other bees? . . .

Continue reading.

Written by LeisureGuy

22 January 2020 at 2:29 pm

Posted in Daily life, Math, Science

Queuing theory is counter-intuitive: What happens when you add a new teller?

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This is from October 2008, but I just stumbled across it. John Cook writes at his consulting company website:

Suppose a small bank has only one teller. Customers take an average of 10 minutes to serve and they arrive at the rate of 5.8 per hour. What will the expected waiting time be? What happens if you add another teller?

We assume customer arrivals and customer service times are random (details later). With only one teller, customers will have to wait nearly five hours on average before they are served. But if you add a second teller, the average waiting time is not just cut in half; it goes down to about 3 minutes. The waiting time is reduced by a factor of 93x.

Why was the wait so long with one teller? There’s not much slack in the system. Customers are arriving every 10.3 minutes on average and are taking 10 minutes to serve on average. If customer arrivals were exactly evenly spaced and each took exactly 10 minutes to serve, there would be no problem. Each customer would be served before the next arrived. No waiting.

The service and arrival times have to be very close to their average values to avoid a line, but that’s not likely. On average there will be a long line, 28 people. But with a second teller, it’s not likely that even two people will arrive before one of the tellers is free.

Here are the technical footnotes. This problem is a typical example from queuing theory. Customer arrivals are modeled as a Poisson process with λ = 5.8/hour. Customer service times are assumed to be exponential with mean 10 minutes. (The Poisson and exponential distribution assumptions are common in queuing theory. They simplify the calculations, but they’re also realistic for many situations.) The waiting times given above assume the model has approached its steady state. That is, the bank has been open long enough for the line to reach a sort of equilibrium.

Queuing theory is fun because it is often possible to come up with surprising but useful results with simple equations. For example, for a single server queue, the expected waiting time is λ/(μ(μ – λ)) where λ the the arrival rate and μ is the service rate. In our example, λ = 5.8 per hour and μ = 6 per hour. Some applications require more complicated models that do not yield such simple results, but even then the simplest model may be a useful first approximation.

Related postServer utilization: Joel on queueing

Continue reading for comments.

BTW, I’ve noticed in several branch banks they have two tellers active…

Written by LeisureGuy

30 December 2019 at 3:10 pm

Posted in Business, Daily life, Math

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