Rife with Paradoxes

Calculus transformed the practice of mathematics and opened up insights into the workings of the natural world

In the seventeenth century the Jesuits placed themselves on the wrong side of history by opposing Galileo’s defense of Copernicus and heliocentrism. Less well known is their stalwart opposition at almost the same time to a much more arcane—but equally seminal—idea. In his book, Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, Amir Alexander describes the Jesuit condemnation of the use of indivisibles in mathematics and portrays it as an additional example of the suffocating effect of dogmatism and obsessive control in any context.

According to Alexander, Italy had led Europe’s emergence from the Dark Ages prior to the clerics’ opposition by reviving the long-dormant commercial economy, nurturing lively political experimentation, producing the first and wealthiest bankers, and leading the way in an artistic and cultural revival that transformed Europe. It had produced leaders in almost every field:

Prior to the seventeenth century, “As a land of creativity and innovation, it is fair to say, Italy had no peer,” according to Alexander.

But following the disruption and disorder brought about by the Reformation, the Society of Jesus, aka, the Jesuits, was formed to restore order and the authority of the Pope in Europe. In their worldview, no dissent from papal edicts could be permitted, no other sect or belief could be allowed a foothold, and any hint of political opposition needed to be forcefully stifled. They believed it was up to them to “expel the demons of strife and confer the light of truth upon the people,” according to Alexander.

As the Jesuits worked to re-establish the control of the Catholic Church over European society, they mounted a campaign to specifically condemn an increasingly popular mathematical idea promoted by a monk named Bonaventura Cavalieri and a growing number of other mathematicians: “The continuum is composed of a finite number of indivisibles” (i.e., A continuous line is composed of distinct and infinitely tiny parts). Why the insistence that this simple proposition never be taught or even mentioned? Because the Jesuits repeatedly determined that the concept was “dangerous and subversive, a threat to the belief that the world was an orderly place, governed by a strict and unchanging set of rules.”

According to Alexander:

Where the Jesuits insisted on clear and simple postulates, the new mathematicians relied on a vague intuition of the inner structure of matter; whereas the Jesuits celebrated absolute certainty, the new mathematicians proposed a method rife with paradoxes, and seemed to revel in them; and whereas the Jesuits sought to avoid controversy at all cost, the new method was mired in intractable controversies seemingly from its very inception. It was everything that the Jesuits thought mathematics must never be.

Eventually, the Jesuits found themselves on the wrong side once again as indivisibles, or infinitesimals, became one of the most important tools of modern science. They were a primary source of inspiration for Isaac Newton’s invention of calculus, which transformed the practice of mathematics and the mathematical sciences, and opened up insights into the workings of the natural world, according to Alexander.

As a result of the Jesuits’ victory over the Galileans in Italy in the matter of infinitesimals, Italian thought became characterized by stagnation and decay, in contrast to countries where the idea was embraced. In the years following, for example, “England exhibited a marked and ever-increasing openness to dissent and pluralism. Politically, religiously, and economically, England became a land of many voices, where rival views and interests competed openly, relatively free from state repression. And it was in this relative freedom that England discovered its path to wealth and power” and became the world’s first modern state.

Sometimes insisting on certainty is the most certain way to miss the truth.


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Getting Better, X

Facts show that every day the world is getting better for millions of people

When the media and our education systems are driven by ratings and self-interest, they can often fail to inform us effectively and accurately about the big picture. And our own confirmation bias can cause us to filter the facts we receive in order to support our existing worldview.

Having a false understanding of how the world is changing can create and reinforce a general feeling of discontent rather than understanding that, in many fundamental and important ways, the world is getting better for millions of people.

Here are two specific examples of major trends in today’s world that are largely misunderstood and often under-reported:

  • On average, 137,000 people escaped extreme poverty every day between 1990 and 2015.
    • However, 52% of people think global poverty is rising.
  • Child deaths have plummeted from 20 million a year to 6 million a year since 1960, and the fertility rate has fallen by half, driven by improvements in conditions for women and the health of children.
    • But 61% of people don’t know that child mortality is declining in poor countries.

How does this matter? When we understand them correctly, many of the statistics on global change tell us that it is possible to make the world better. Understanding the successes that are being realized should encourage us to work for more progress, rather than despairing and drawing inward. Fear and incrimination are poor motivators, but drawing strength from the success of effective policy initiatives can produce the energy and courage to achieve more.

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Credit Where Credit is Due

The Wright Brothers learned from many other aviation pioneers and worked for years to achieve their success

Wilbur and Orville Wright deserve every bit of the recognition they have received for inventing the first heavier-than-air craft capable of controlled flight. But it doesn’t diminish their achievement at all to understand that their work did not take place in a vacuum or emerge complete on December 17, 1903.

Because we prefer history to be simple and straightforward, stories often describe the Wrights as geniuses who simply built a plane and took off one day, rather than recognizing their years of research into the work of contemporary aviation pioneers, their dogged persistence as experimental engineers and their courage as the first real pilots.

In his book, First Flight: The Wright Brothers and the Invention of the Airplane, T. A. Heppenheimer lists a significant group of other inventors in mostly Britain and France who made incremental advances toward the ability to fly during the late nineteenth and early twentieth century:

The Wrights’ work was truly superior, particularly in the achievement of controlled flight, but they built upon much existing work and many important previous insights as they developed and refined the capabilities of their machine. In his 1899 letter to The Smithsonian Institution asking for the Smithsonian’s papers and a list of other works about flight, Wilbur Wright said he had built models “after the style of Cayley’s and Penaud’s machines.” They also repeated the experiments of Wenham and Phillips, confirming or correcting their claims.

The Wrights had what Malcolm Gladwell has called a “different kind of genius,” one characterized by “the ability to work hard and to extract joy and satisfaction from that kind of hard work.” This genius is much more imitable by others and more transferable to other fields than the common image of a brilliant eccentric. According to Heppenheimer, the Wrights’ genius was “their ability to raise and to address the most important issues,” using their mechanical skill and carefully obtaining their own data from their experiments. Unlike the other pioneers of aviation, they served as their own designers, builders and test pilots, and thus were able to apply their flight experience to their design and—over years of hard work—to introduce improvements to their own work and the work of others.

Mostly unknown to history—certainly in comparison to the legend of Kitty Hawk—they spent a year and a half at Huffman Prairie mastering control of their flying machine and improving its engine. There they made the first turn and the first circle in the air as they learned to fly in varying winds and to maintain complete control throughout their flights. Finally, by the fall of 1905, their plane could bank, turn circles, and make figure-eights.

To imply or believe that the Wrights’ work sprang forth from nothing and consisted of a few short flights on a sand dune serves only to undermine its genius, not magnify it.

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A Terrible Master

Don't be a slave to your technology

When I was in high school in the 1970s, everyone suddenly had their own handheld calculator. Mine allowed me to narrowly escape having to learn how to use a slide rule instead. But ever since, despite their convenience, there has been debate about whether using calculators causes us to forget how to perform basic math on our own.

Lots of current and future technological assistants (GPS, search engines, autonomous cars) prompt the same question: Do they make us dumber and too dependent?

In fact, this question has been around for as long as there has been “technology” that promised to make our lives easier. In his 2010 book, Hamlet’s Blackberry, William Powers related that when writing was invented, many feared that it would cause people to lose their ability to remember things! I’m glad people decided to keep on writing things down.

In folklore, John Henry fought against technology and in the end, “he hammered his poor heart to death.” I don’t want to be John Henry, choosing to die rather than change, but I also don’t want to blindly allow technology to change me without considering the consequences.

New technologies are going to continue to appear and be adopted. But incorporating new things into our lives does not excuse the cessation of rational thought, the abandonment of common courtesy and an absence of appropriate skepticism. Ignoring someone standing in front of me—be it stranger, friend or spouse—to stare at a screen, or accepting baseless accusations because they demand our attention is indefensible.

Whenever the next new thing comes along, we need to retain control and think about how we can use it responsibly, because technology makes a great slave but a terrible master.

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Genuinely Interested in Everyone

The machine says I'm awesome

When I add a post to this blog, I get an email that says:

“[Random WordPress blogger] thought [title of my post] was awesome. Maybe you should find out what they’re up to.”

The message has multiple links to that blogger’s site. It seems reasonable to assume that every other WordPress blogger gets those messages too.

I know a blogger who believed that those messages came from a human being who had read his content and been moved to write to him in response. In fact, those messages are generated and sent by a computer program that only simulates genuine interest in a post.

It’s an intoxicating technique, isn’t it? Who doesn’t want to believe that someone thinks their writing is awesome?

In his novel, Gnomon, Nick Harkaway described a future computer network called “The System” that claimed to provide psychological benefit via a similar effect:

“Many instances of [clinical paranoia] used to result from a horror of personal smallness; a deep, almost existential fear that the pattern of a given life had no meaning against the tide and chatter of the majority, or the vast indifference of the universe beyond. But part of what is remarkable about the System is that no one is insignificant to it. Every action, every choice, worry, question; every bold or idiotic inspiration can be acknowledged by the tranquil and endless machine. There is no silence into which the lonely fall. The System is quite genuinely interested in everyone.” [emphasis added]

When I suggested to my fellow blogger that a machine was generating some of the responses to his posts in order to simulate interest, he cast his eyes down and seemed hurt and resistant to the idea. But he already knew that humans—even ones who know him—can act in pretty much the same way. He had told me that people he knew regularly “liked” things he posted but later confessed that they hadn’t even read them. He had been disappointed when he found that they too were just simulating genuine interest.

Online communication has been accused of “bringing out the worst in people.” But it’s really just like other forms of marketing, with two exceptions: it can be effortlessly “personalized” and its reach is limitless. That’s how The System can seem to be genuinely interested in everyone. Coupons, sales, exclusive discounts (“You’ve been selected!”) and member privileges are great, but they’re not offered to me because the vendors are genuinely interested in me. They’re offered to manipulate me into buying something I wouldn’t otherwise. Companies ask, “Like us on Facebook” or “Follow us on Twitter,” simply to acquire my information and sell me something. It’s up to me not to be seduced by what may seem like “genuine interest.”

There’s nothing wrong with wanting someone to read what I write and maybe offer a comment of their own. But when “likes” and “followers” become my goal, I’ve failed to realize how much of that “genuine interest” is algorithmic, random or imaginary, and I’ve agreed to give away a part of myself and my integrity for little or nothing of value.

Long before blogging and social media appeared, British author Cyril Connolly said, “Better to write for yourself and have no public, than to write for the public and have no self.”

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Making the Mighty Sweat

Seymour Hersh exposed the My Lai massacre during the Vietnam war, for which he won the Pulitzer Prize. (Photograph: Wally McNamee/Corbis)

The New York Times said Seymour Hersh’s 2018 book, Reporter: A Memoir, is about “a life making the mighty sweat.” Most notably, Hersh won a Pulitzer Prize in 1970 for a series of articles he wrote as a freelancer about the previously unreported My Lai massacre during the Vietnam war. His reporting showed how war can corrupt and corrode the minds and souls of young men, not just the enemy’s, but America’s as well.

Hersh’s original article “triggered a new debate about the Vietnam war that would dominate front pages.” When newspapers and television networks finally picked up the story, it shocked the national conscience, and President Richard Nixon later claimed that the incident undermined his efforts to build support for the war.

Hersh pursued the My Lai story when he heard from a part-time political columnist for The Village Voice that a U.S. soldier was being court-martialed for leading a massacre of Vietnamese civilians, and the Army didn’t want it reported. In 2015, Hersh said, “Determined to understand how young men—boys, really—could have done this, I spent weeks pursuing them.”

In early October of 1969, Hersh got one of his former military sources to name Army Lieutenant William Calley as the soldier being charged and then overcame a series of obstacles to conduct an interview with Calley, who had been accused of the premeditated murder of “109 Oriental human beings” in South Vietnam.

Piecing together the story, Hersh interviewed Ronald Ridenhour, an ex-G.I. who had not participated in the attack but prompted an investigation by writing letters to the Pentagon, the State Department, the White House, and twenty-four congressmen describing the murders after discussing the operation with G.I.s who had been there; and Paul Meadlo, a soldier from rural Indiana who, on Calley’s orders, had fired into groups of women and children who had been rounded up amid the massacre.

But when Hersh began offering his account to primary media sources, he was shocked by the stonewalling he encountered. Multiple major publications refused to run his stories because they were fearful of government denials that American soldiers would murder dozens of innocent Vietnamese citizens.

Hersh had interviewed Calley, whose name would become synonymous with the murders; he’d interviewed Calley’s attorney, a retired Army judge named George Latimer; and he’d seen the Army charge sheet accusing Calley. But, Hersh said, “none of that mattered to the various editors to whom I brought the story—including an editor of Life magazine who, I later learned, had been told about My Lai months earlier by an American G.I. No one wanted to be the first to publish.”

“It was more than a little distressing; it was frightening,” Hersh said. “I had empirical evidence of a major American crime in Vietnam, plus an interview with the guy accused of leading it, and the mainstream media apparently wanted nothing to do with it.”

So Hersh syndicated the stories through Dispatch News Service, “a small antiwar news agency” in Washington. He was reviled by many members of the military and their families for disloyalty to the Army and for claiming that the atrocities committed by American troops were as heinous as those of the enemy.

His reports made it clear, however, that for a significant number of American military, the Vietnamese weren’t considered fellow human beings and didn’t deserve to live, whether they were combatants or not. According to U.S. soldier Michael Terry: “A lot of guys feel that [the South Vietnamese civilians] aren’t human beings; we just treated them like animals.” Episcopal chaplain Carl E. Creswell, who resigned from the Army soon after his tour with the Americal Division, said, “I became absolutely convinced that as far as the United States Army was concerned there was no such thing as murder of a Vietnamese civilian. I’m sorry, maybe it’s a little bit cynical. I’m sure it is, but that’s the way the system works.”

In a letter Hersh sent to Bob Loomis, his editor at Random House, he wrote: “Both the killer and the killed are victims in Vietnam; the peasant who is shot down for no reason and the GI who is taught, or comes to believe, that a Vietnamese life somehow has less meaning than his wife’s, or his sister’s, or his mother’s.”

Hersh ended up believing it was military and political leaders who were responsible for My Lai and the war, not the soldiers. He said to Loomis, “Calley is really no more at fault than anyone else there: he shouldn’t have been an officer, he shouldn’t have been sent to fight a war he could not comprehend, he shouldn’t have known the body count as the only standard of success, and he shouldn’t be on trial any more than the higher-ranking officers who did nothing about the slaughter afterwards, thus inducing that many more killings. Perhaps there is even less reason to try Calley than the top brass at the Pentagon, or maybe an American president or two, or three.”

Enlisting people to fight a war changes them. As the mother of Paul Meadlo said, “I sent them a good boy and they made him a murderer.” When will that prospect “make the mighty sweat”?

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No Other Solution

Solar and wind offer our only realistic chance of avoiding dangerous climate change

Change is hard. Changing from one energy infrastructure to another is hard. But having multiple good reasons for making the changes makes it easier.

According to The Conversation, “Solar photovoltaic and wind power are rapidly getting cheaper and more abundant—so much so that they are on track to entirely supplant fossil fuels worldwide within two decades, with the time frame depending mostly on politics.”

“The reality is that the rising tide of solar photovoltaics (PV) and wind energy offers our only realistic chance of avoiding dangerous climate change,” The Conversation said. “No other greenhouse solution comes close.”

The Union of Concerned Scientists reports that, according to a study by the Department of Energy’s National Renewable Energy Laboratory (NREL), the U.S. could generate 80 percent of its electricity from currently available renewable energy technologies—including wind turbines, solar photovoltaics, concentrating solar power, biopower, geothermal, and hydropower—by 2050.

Benefits of this increased sourcing from renewables include dramatically reduced global warming emissions, improved public health, job creation and other economic benefits. In an 80 percent renewables future, carbon emissions from the power sector would be reduced by 80 percent, and water use would be reduced by 50 percent.

The Australian think tank Future Crunch listed several examples of advances in the use of renewables already being made in other countries:

Nothing stays the same forever because we’re getting smarter all the time and looking forward rather than back—at new opportunities rather than lost traditions—can help us adapt to and thrive in the new reality.

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