Posts Tagged ‘Mathematics’
“If I am to be remembered, I hope it will not be primarily for my specialized scientific work, but as a generalist; one to whom, enlarging Terence’s words, nothing human and nothing in external nature was alien.”*…
Interdisciplinary artist, writer, and musician Ross Simonini with 47 thoughts on the glory of looking– and living– beyond a specialty…
1. I was raised to believe that I was made to do one thing. Find that one pursuit that fills my life with meaning and empty all my energy into it. This is the realization of human potential: to excel with rigorous focus on a refined lifelong mission. This and only this will bring us to our greatest success and fulfillment.
For me, this was not something I even had to be told—though I was, many times, by many people—because I implicitly understood that this kind of teleology was woven into the fibers of my world. I also knew that rejecting a singular pursuit would be an insult to my very existence. Without this unifying reason for being alive, I would wander aimlessly into the barren void of nihilism. I’d heard about great artists who refused to create, who stepped away from their work to fritter away their time on leisure, and I knew this was a life of tragedy.
Likewise, I understood that sliding your attention across interests is a way to waste your gift. The more hours you put into a skill, the more skilled you become—right? To treat your gift with the proper deference, you must exhaust yourself into it.
Within this paradigm, the most unfortunate people are those who do not have a single, clear vocation. These types float from job to job without a trajectory; they are vagabonds who have given up on greatness.
This may sound a little dramatic, but somewhere inside me, these beliefs are there—and as a lifelong generalist, I spend every day rubbing up against them.
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16. Let’s talk about mastery. Everyone wants to be a master, even if they are disgusted by the monstrous implications of the word. Mastery suggests dominance over something, but every true master knows that they are merely a supplicant at the mercy of their field, which existed long before them and will exist long after them. Anyone who believes in their own mastery likely suffers from hubris. Work hard enough at something and you watch your dominance slip ever further away.
Mastery is an illusion, a notion of a fictional purity that cannot be understood or measured in terms of time. Just look at those young savants who excel wildly after only a few years spent on their craft. For them, mastery cannot be the result of time plus work, as we all assume it is. In fact, maybe the newness of their skills is precisely what gives their work its value.
But these little wonders are exceptions, right? The rest of us have to dedicate our lives to something to achieve greatness, and anyone who doesn’t do this will likely be middling in their work. Most writers I know are immediately suspicious when an actor publishes a novel. We delight in calling the person a moonlighter. Literature is our territory, and the only way to live here is to put in the time and labor.
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24. Isaiah Berlin, the political theorist, ethicist, philosopher, and historian, wrote a book called The Hedgehog and the Fox, in which he divides people into two types: hedgehogs, who see the entire world through one big thing, and foxes, who see the world as many things that cannot be reduced. According to Berlin, hedgehogs include Plato, Dostoyevsky, and Proust, while foxes include Aristotle, Shakespeare, and James Joyce.
“Everything I learned in my life, I learned because I decided to try something new,” said David Lynch (musician, filmmaker, painter, lamp maker, sculptor, writer, actor, and lecturer, mostly on meditation).
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29. Sometimes history hides generalism to preserve a specialized agenda. Isaac Newton, a figure whom we consider the father of modern math, physics, and reasoned thinking, was also a dedicated alchemist. Alchemy, a generalist practice in itself, was a precursor to modern chemistry. It involves spirituality, myth, belief, and metallurgy, but its inclusion of belief stands in direct conflict with the scientific rationalism Newton now represents. Subsequent generations of historians and scientists buried Newton’s dedication to the occult, willfully ignoring the blow it deals to their obsessive, single-minded materialism. But Newton’s own records tell a different story. He wrote over a million words on alchemy in his lifetime, and his study of the subject helped inspire some of his most paradigm-shifting discoveries.
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31. A filmmaker must understand aspects of sound design, photography, storytelling, music, acting, props, environment, finance, writing, and dialogue. In this way, some jobs are naturally suited to the generalist. A skilled homemaker, for example, understands everything from cooking to cleaning to healing to sociology. Acting, too, is a fairly generalist vocation. The practice of writing, what I am doing right now, is extremely broad, without consistent subject matter, form, or even mediums.
Generalism can be an approach of the neophyte or of the seasoned worker. Some entry-level positions (assistant, secretary, intern) are, in fact, compilations of micro-jobs, and some high-level positions—
managers, CEOs, directors, business owners, presidents—are positions of vast, nonspecific oversight. Sometimes the highest perch has the widest perspective.…
39. A generalist must engage with both sides of any argument: skepticism and belief, optimism and pessimism. So, for this essay, it would only be right to take a look at the dark side of generalism and the side effects of adopting it as a whole-life philosophy.
The glaring danger of general thinking in its extreme form is relativism, a sort of mushy non-position in which there are no universal standards: nothing can ever be condemnable or universally wrong. At the most dramatic levels, relativism might dismiss murder and genocide. It’s a slippery slope of open-mindedness.
Likewise, a generalist must contend with political centrism. In our bifurcated world, the center is one of the most reviled of all political positions, and a generalist will come to understand whether their own centrism is an evasion of choice or a refusal of unpalatable options.
Few things are more torturous than making decisions, and a mind will do anything to avoid such a relentlessly complex activity. Adherence to these vague philosophies, as I see them, can certainly be used as an excuse for escaping commitment. As a generalist, I must stay vigilant against this kind of laziness of mind and instead allow many fierce, contrary ideas to exist at once.
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42. Generalism is not a thing. It’s definitely not an ism or some kind of doctrine. The general approach defies the nature of ideologies, which are characterized by the limits they place on understanding the world. There is no system of generalism. The general philosophy is to love variety.
For this reason, generalists don’t exist—not in the way that, say, Marxists do—because they can’t identify as generalists. I can call myself intra-, cross-, multi-, inter-, and trans-disciplinary—which, for some, are all legitimate and distinct prefixes—but that does more to distinguish and alienate me from others than to connect me with a community. There is no lineage of generalists, as there is for microbiologists or flutists, because every generalist works with their own complex bouquet of interests.
Probably this whole essay is my attempt to give a sense of unity to my life. Maybe I have to write a manifesto on “the art of doing many things”because I fear that if our culture doesn’t have a catchy keyword for my role, I’ll just fade away. So here I am, reducing generalism to a single, branded snap, just like a specialist.
After all, generalists are, in moments, great specialists. Likewise, a deep specialist can approach their niche from an ever-growing number of perspectives. A man with a repetitive job can endlessly engage with his work from fresh angles. And, of course, it’s all relative. A single task looked at from another angle is a plentiful cornucopia of individuated micro-tasks.
Some long-term generalists focus exclusively on a single activity for a number of years before moving on to the next. Rather than doing many things simultaneously, they do them sequentially.
Pure generalism and pure specialism are just intellectual games. Our minds drift between unified oneness and individuality without ever settling into either. Binary thinking is for computers.
These two states of being are not roles we need to inhabit but rather nodes to be considered. One situation requires diligent focus, but another benefits from a more diffuse form of attention. Certain qualities of engagement can occur only when you do multiple things at once. This is the value of the glance.
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47. Generalism is not the opposite of specialism. It includes specialism. Everyone gets to experience both. Or maybe both approaches lead to the same place. Maybe the study of quantum physics brings a mind to the same conclusions as basketry. Maybe it’s like meditation: You can sit in open awareness and experience everything until you reach an unprejudiced understanding of life. Or you can unflinchingly focus on a single mantra for decades, repeating it with each breath, and as you plunge deeper toward a single infinite point, you discover that everything is already right there.
Eminently worth reading in full: “In praise of generalism” from @thebeliever.net.
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As we widen our irises, we might send carefully-calculated birthday greetings to Pierre de Fermat; he was born on this date in 1601. While he is remembered as one of the two great mathematicians of the early 17th century (with Descartes), Fermat was (like Descartes) driven by wider interests. Fermat was a trained lawyer, who served as a councilor at the Parlement de Toulouse, one of the High Courts of Judicature in France. He was fluent in six languages and praised for his written verse in several of them; his advice was eagerly sought regarding the emendation of Greek texts… which is to say that mathematics was but one of his interests, and more a hobby than a profession at that. Still, Fermat made foundational contributions to analytical geometry, probability, number theory and calculus.
“Patience’s design flaw became obvious for the first time in my life: the outcome is decided not during the course of play but when the cards are shuffled, before the game even begins. How pointless is that?”*…
As Simone de Rochefort explains, Patience– or as we tend to know it, solitaire— illustrates the way in which some of humanity’s oldest toys are our most complex…
… last year, I got addicted to Solitaire.
Why me.
During the dark final days of 2024, I was averaging 12 wins per day in Sawayama Solitaire, one of the Solitaires created by developer Zachtronics. Sawayama Solitaire is a variant of Klondike — the one that’s been bundled into every version of Windows since 1990.
Some games of Sawayama Solitaire felt impossible. Some were absurdly easy. Most of them were a satisfying detangling of cards that had me immediately pressing that “new game” button once I got the win.
How was the most basic card game on Earth owning my life like this?
I think it’s because we don’t understand playing cards.
In 1969, as protests raged against the Vietnam War and counterculture made waves across the nation, a magician [and dear friend of Ricky Jay] named Persi Diaconis went to college.
Diaconis had been a professional magician since age 14, and was skilled in sleight-of-hand tricks. But it was probability that fascinated him.
He went on to take a degree in statistics. He became a world-renowned mathematician. In 1992, he proved that it takes seven riffle shuffles to truly randomize a 52-card deck, alongside fellow mathematician Dave Bayer. His research on card shuffling has implications for scientific fields as far-flung as the study of glass melting and the creation of magnets.
He doesn’t know how Solitaire works.
“One of the embarrassment of applied probability is that we can not analyze the original game of solitaire,” he wrote in the abstract for an academic talk called “The Mathematics of Solitaire,” given at the University of Washington in 1999. The talk has been given several times over the years, and is currently viewable on YouTube. One of his most recent appearances, in 2024, reiterates that despite all the technical advances we’ve made in science and mathematics, the complexity of cards is still somewhat a black box.
“What’s the chance of winning, how to play well, how do various changes of rules change the answers?” Diaconis wrote. “Surely you say, the computer can do this. Not at present, not even close.”
It’s not hard to see the relationship between magic and math. Cards contain limitless possibilities. In fact, math tells us there are more combinations of cards in a 52-card deck than there are atoms on Earth.
Writing for Quanta Magazine, Erica Klarreich asked mathematician Ron Graham what that means in practice. He told her, “If everyone had been shuffling decks of cards every second since the start of the Earth, you couldn’t touch 52 factorial,” the number of possible arrangements of a 52-card deck. Klarreich goes on: “Any time you shuffle a deck to the point of randomness, you have probably created an arrangement that has never existed before.”
So that’s nuts…
More amazement at “No one understands how playing cards work,” from @polygon.com.
And here:
* David Mitchell, Cloud Atlas
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As we shuffle along, we might spare a thought for Christiaan Huygens; he died on this date in 1695. A mathematician, physicist, engineer, astronomer, and inventor, he was a key figure in the Scientific Revolution. In physics, Huygens made seminal contributions to optics and mechanics, while as an astronomer he studied the rings of Saturn and discovered its largest moon, Titan. As an engineer and inventor, he improved the design of telescopes and invented the pendulum clock, the most accurate timekeeper for almost 300 years. A talented mathematician and physicist, his works contain the first idealization of a physical problem by a set of mathematical parameters, and the first mathematical and mechanistic explanation of an unobservable physical phenomenon.
Relevantly to the piece above, Huygens also contributed to the development of probability theory and statistics. In 1665 he visited Paris and encountered the work of Fermat and Pascal, which led him to write what was, at the time, the most coherent presentation of a mathematical approach to games of chance in De Ratiociniis in Ludo Aleae (On reasoning in games of chance)– a work contains early game-theoretic ideas.
“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality”*…
As Gregory Barber explains, two new notions of infinity challenge a long-standing plan to define the mathematical universe…
It was minus 20 degrees Celsius, and while some went cross-country skiing, Juan Aguilera, a set theorist at the Vienna University of Technology, preferred to linger in the cafeteria, tearing pieces of pulla pastry and debating the nature of two new notions of infinity. The consequences, Aguilera believed, were grand. “We just don’t know what they are yet,” he said.
Infinity, counterintuitively, comes in many shapes and sizes. This has been known since the 1870s, when the German mathematician Georg Cantor proved that the set of real numbers (all the numbers on the number line) is larger than the set of whole numbers, even though both sets are infinite. (The short version: No matter how you try to match real numbers to whole numbers, you’ll always end up with more real numbers.) The two sets, Cantor argued, represented entirely different flavors of infinity and therefore had profoundly different properties.
From there, Cantor constructed larger infinities, too. He took the set of real numbers, built a new set out of all of its subsets, then proved that this new set was bigger than the original set of real numbers. And when he took all the subsets of this new set, he got an even bigger set. In this way, he built infinitely many sets, each larger than the last. He referred to the different sizes of these infinite sets as cardinal numbers (not to be confused with the ordinary cardinals 1, 2, 3…).
Set theorists have continued to define cardinals that are far more exotic and difficult to describe than Cantor’s. In doing so, they’ve discovered something surprising: These “large cardinals” fall into a surprisingly neat hierarchy. They can be clearly defined in terms of size and complexity. Together, they form a massive tower of infinities that set theorists then use to probe the boundaries of what’s mathematically possible.
But the two new cardinals that Aguilera was pondering in the Arctic cold behaved oddly. He had recently constructed them, along with Joan Bagaria of the University of Barcelona and Philipp Lücke of the University of Hamburg, only to find that they didn’t quite fit into the usual hierarchy. Instead, they “exploded,” Aguilera said, creating a new class of infinities that their colleagues hadn’t bargained on — and implying that far more chaos abounds in mathematics than expected.
It’s a provocative claim. The prospect is, to some, exciting. “I love this paper,” said Toby Meadows, a logician and philosopher at the University of California, Irvine. “It seems like real progress — a really interesting insight that we didn’t have before.”
But it’s also difficult to really know whether the claim is true. That’s the nature of studying infinity. If mathematics is a tapestry sewn together by traditional assumptions that everyone agrees on, the higher reaches of the infinite are its tattered fringes. Set theorists working in these extreme areas operate in a space where the traditional axioms used to write mathematical proofs do not always apply, and where new axioms must be written — and often break down.
Up here, most questions are fundamentally unprovable, and uncertainty reigns. And so to some, the new cardinals don’t change anything. “I don’t buy it at all,” said Hugh Woodin, a set theorist at Harvard University who is currently leading the quest to fully define the mathematical universe. Woodin was Bagaria’s doctoral adviser 35 years ago and Aguilera’s in the 2010s. But his students are cutting their own path through infinity’s thickets. “Your children grow up and defy you,” Woodin said…
More on the fascinating state of play at: “Is Mathematics Mostly Chaos or Mostly Order?” from @GregoryJBarber in @quantamagazine.bsky.social.
* Albert Einstein
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As we get down with Gödel, we might send insightful birthday greetings to John Allen Paulos; he was born on this date in 1945. A mathematician, he is best known as an advocate for– and a skilled teacher of– mathematical literacy. His book Innumeracy: Mathematical Illiteracy and its Consequences (1988) was a bestseller, and A Mathematician Reads the Newspaper (1995) extended the critique. Paulos was a regular columinst for both The Guardian and ABC News. And in 2001 he created and taught a course on quantitative literacy for journalists at the Columbia University School of Journalism– an exercise that stimulated further programs at Columbia and elsewhere in precision and data-driven journalism.
Happy 4th of July to readers in the U.S… but are we commemorating the right day?
“We must not forget that the wheel is reinvented so often because it is a very good idea”*…
… but when was it first discovered? And, and given its obvious and ubiquitous utility, why there (and not somewhere else)? Kai James offers an answer…
Imagine you’re a copper miner in southeastern Europe in the year 3900 B.C.E. Day after day you haul copper ore through the mine’s sweltering tunnels.
You’ve resigned yourself to the grueling monotony of mining life. Then one afternoon, you witness a fellow worker doing something remarkable.
With an odd-looking contraption, he casually transports the equivalent of three times his body weight on a single trip. As he returns to the mine to fetch another load, it suddenly dawns on you that your chosen profession is about to get far less taxing and much more lucrative.
What you don’t realize: You’re witnessing something that will change the course of history – not just for your tiny mining community, but for all of humanity.
Despite the wheel’s immeasurable impact, no one is certain as to who invented it, or when and where it was first conceived. The hypothetical scenario described above is based on a 2015 theory that miners in the Carpathian Mountains – in present-day Hungary – first invented the wheel nearly 6,000 years ago as a means to transport copper ore.
The theory is supported by the discovery of more than 150 miniaturized wagons by archaeologists working in the region. These pint-sized, four-wheeled models were made from clay, and their outer surfaces were engraved with a wickerwork pattern reminiscent of the basketry used by mining communities at the time. Carbon dating later revealed that these wagons are the earliest known depictions of wheeled transport to date.
This theory also raises a question of particular interest to me, an aerospace engineer who studies the science of engineering design. How did an obscure, scientifically naive mining society discover the wheel, when highly advanced civilizations, such as the ancient Egyptians, did not?…
Read on to find out: “How was the wheel invented? Computer simulations reveal the unlikely birth of a world-changing technology nearly 6,000 years ago,” from @us.theconversation.com.
* “We must not forget that the wheel is reinvented so often because it is a very good idea; I’ve learned to worry more about the soundness of ideas that were invented only once.” – David Parnas
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As we roll along, we might we might send a “Alles Gute zum Geburtstag” to man at the center of the question of the invention of another foundational “technology”: the polymathic Gottfried Wilhelm Leibniz, the philosopher, mathematician, inventor (of, among other things, an early calculator) and political adviser.
Leibnitz was important both as a metaphysician and as a logician, but who is probably best remembered for his independent invention of the calculus; he was born on this date in 1646. Leibniz independently discovered and developed differential and integral calculus, which he published in 1684; but he became involved in a bitter priority dispute with Isaac Newton, whose ideas on the calculus were developed earlier (1665), but published later (1687). Scholars largely agree that, in fact, Leibnitz and Newton independently developed “the greatest advance in mathematics that had taken place since the time of Archimedes.”
“I didn’t study theology out of piety. I studied it because I wanted to know.”*…
Beatrice Marovich on a discipline declining…
People often assume that theology is only for true believers: those who want to defend the existence of God against the skepticism of secular outsiders. But there’s an old open secret in the field: theologians often have a complicated relationship with belief, and some theologians are even non-believers. I’ve always been a secular—or non-religious—person. That’s the “tradition” I was raised in. But I’m also a theologian.
I knew that it was a risk, going into the field of theology. There are conversations I’ve been shut out of because I’m not religious enough. And I’m often marked as a troubling outsider by scholars who see themselves taking a purely secular approach to the interdisciplinary study of religion. But as a graduate student, and even early in my career as a faculty member at a small liberal arts college, I believed the field of theology was opening up, and becoming more complex. It felt, to me, as if there were a creative disintegration happening that might make more room for scholars like me. But after more than a decade in the field, I’ve come to feel that something else is happening instead. It feels like the field is dying.
People are still doing theology in public (if, by doing theology we mean talking about gods, spirits, and other divine powers). But the field I was trained in as a scholar—academic theology—feels like it’s dying. It’s a field that’s often philosophical, but always theoretical. Because of this, theology can verge quickly into the abstract, and the speculative. Theologians might make use of anthropological, sociological, and historical studies of religion. But they tend not to feel beholden to any of those disciplines. Indeed, theologians are often wading into explicitly interdisciplinary conversations about science, politics, gender, and race (among other things). In its lack of clear focus, theology might be the most undisciplined discipline in the American academy today. And that undisciplined discipline feels like it’s dying. At least to me.
But is theology really dying? Or is this just the feeling I have, as I’m being squeezed out of the field? Or, perhaps I’m I fixated on the mortality of this collective project because I’ve been writing, thinking, and teaching about death. When I looked at enrollment numbers at seminaries and theological schools, the numbers aren’t necessarily damning. At least not yet. They don’t necessarily confirm my feeling, or my mood. Neither did Sean Larsen’s 2020 State of Theology study, funded by the Templeton Religion Trust. There were people, in that study, who remained optimistic about the discipline’s prospects. And while Ted Smith’s 2023 book The End of Theological Education does acknowledge that the institutions that built theology in America are collapsing, he remains optimistic about what the church can do for the future of theology.
I needed to know if others shared my feeling, or mood. So, I decided to have a conversation with my colleagues. I reached out to people in my network, to see who felt compelled to weigh in. I had three questions for them: Is academic theology really dying? If so, how do you feel about this death? And, finally, If you could save one thing from the sinking ship that is academic theology, what would it be? This essay is a kind of report: it’s what my colleagues told me.
What you’ll read here does reflect a bias: these are voices from within my network. Nevertheless, I think their words are worth sharing. Whether or not academic theology is really dying, it may still be worth thinking about its mortality. If I’ve learned any lesson from writing and thinking about death, it’s that when we acknowledge that it’s there, when we remember that we’re always living in death’s shadows, we take what’s in front of us much more seriously. We can see the full fragility of things, and we can try—against the odds—to resist entropy and protect what we think is worth saving, inheriting, or carrying on into the future. And we can think about what we’re ready to let go of. Because all things, in time, do die. It’s only a question of when…
[Marovich examines the state of the field v ia a recounting of highlights from her conversations with colleagues…]
… I conducted these interviews in the spring of 2024, in what feels to me (now) like a different world. What David Kline so succinctly described as the “institutional frameworks for intellectual life” seem more fragile and threatened than ever, as the Trump administration rapidly defunds education and research, and attacks media outlets. And we can’t forget, of course, about the many threats that Artificial Intelligence—in the form of Large Language Models like ChatGPT—poses to these fragile frameworks for intellectual life. I’m aware that it may seem small-minded and naïve to worry about my own obscure little academic discipline, when the whole structure is falling apart. So, it does seem important for me to clarify that I have spent (and will continue to spend) many hours grieving, as if in anticipation, what feels like the evaporation of intellectual possibilities—intellectual life itself!—in America. I am torn up about all of this. And yet, simultaneously, I do remain concerned about the strange little ecosystem that comprises my corner of the world.
As I think over these conversations with my colleagues, I find myself torn between letting go and holding on—or, perhaps better said, trying to hold space. I agree with Hanna Reichel when they suggest that letting go of the growth mindset is painful and difficult for Americans, perhaps more than anyone else. And this contributes to so much of the damage that American life does to the planet we share with others. I recognize that this is a problem. And I am compelled by Colby Dickinson’s suggestion that perhaps learning to die—learning an ars moriendi—might be the best thing that theology could do right now. So much of what is good about theology is probably already in diaspora, as Amaryah Armstrong has suggested. I do have a certain kind of faith that much of the power of theology will live on, in some shape and form, wherever it goes.
And yet Sameer Yadav’s point about academic theology existing as a kind of “nowhere” space strikes me as so deeply true. That nowhere space has given me so much room to explore, it’s opened dimensions of life to me that I would never have seen, and it’s introduced me to so many incredible people—living and dead. I am grateful for this community, and I feel like I owe it something. I feel compelled to somehow preserve that generative and undisciplined nowhere space for others. Like Meg Mercury, I would like to see this nowhere space open up and expand, for those people who don’t feel as if they belong in traditional religious structures. And yet, I also recognize that the cash value of this sort of space—for the church and for the academy—is more or less zero. The odds that it will survive, even if (as David Congdon noted) there is some educational New Deal that revives higher education, are slim. But perhaps this is one of the reasons why I felt compelled to speak with my colleagues, and write this piece, in the first place. Perhaps it was a gesture at letting go. Or perhaps it was a little leap of faith—a little gesture towards expanding space and time for this nowhere community to find new forms of shelter in which to gather…
On doing hospice care for an academic discipline: “Is Theology Dying?” from @beamarovich.bsky.social in The Other Journal.
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As we ponder the preservation of perspicacity, we might send controversial birthday greetings to a man whose experience illustrates (one episode in) the long history of theology’s peril, Bernard Lamy; he was born on this date in 1640. A French Oratorian and mathematician, he was was also an important theologian… whose teachings were judged alternately either controversial or irrelevent at the series of institutions to which he was forced continually to move throughout his career.
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