Posts Tagged ‘Fermat’
“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.
“Everything we care about lies somewhere in the middle, where pattern and randomness interlace”*…
We tend dramatically to underestimate the role of randomness in the world…
Arkansas was one out away from the 2018 College World Series championship, leading Oregon State in the series and 3-2 in the ninth inning of the game when Cadyn Grenier lofted a foul pop down the right-field line. Three Razorbacks converged on the ball and were in position to make a routine play on it, only to watch it fall untouched to the ground in the midst of them. Had any one of them made the play, Arkansas would have been the national champion.
Nobody did.
Given “another lifeline,” Grenier hit an RBI single to tie the game before Trevor Larnach launched a two-run homer to give the Beavers a 5-3 lead and, ultimately, the game. “As soon as you see the ball drop, you know you have another life,” Grenier said. “That’s a gift.” The Beavers accepted the gift eagerly and went on win the championship the next day as Oregon State rode freshman pitcher Kevin Abel to a 5-0 win over Arkansas in the deciding game of the series. Abel threw a complete game shutout and retired the last 20 hitters he faced.
The highly unlikely happens pretty much all the time…
We readily – routinely – underestimate the power and impact of randomness in and on our lives. In his book, The Drunkard’s Walk, Caltech physicist Leonard Mlodinow employs the idea of the “drunkard’s [random] walk” to compare “the paths molecules follow as they fly through space, incessantly bumping, and being bumped by, their sister molecules,” with “our lives, our paths from college to career, from single life to family life, from first hole of golf to eighteenth.”
Although countless random interactions seem to cancel each another out within large data sets, sometimes, “when pure luck occasionally leads to a lopsided preponderance of hits from some particular direction…a noticeable jiggle occurs.” When that happens, we notice the unlikely directional jiggle and build a carefully concocted story around it while ignoring the many, many random, counteracting collisions.
As Tversky and Kahneman have explained, “Chance is commonly viewed as a self-correcting process in which a deviation in one direction induces a deviation in the opposite direction to restore the equilibrium. In fact, deviations are not ‘corrected’ as a chance process unfolds, they are merely diluted.”
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As Stephen Jay Gould famously argued, were we able to recreate the experiment of life on Earth a million different times, nothing would ever be the same, because evolution relies upon randomness. Indeed, the essence of history is contingency.
Randomness rules.
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Luck matters. A lot. Yet, we tend dramatically to underestimate the role of randomness in the world.
The self-serving bias is our tendency to see the good stuff that happens as our doing (“we worked really hard and executed the game plan well”) while the bad stuff isn’t our fault (“It just wasn’t our night” or “we simply couldn’t catch a break” or “we would have won if the umpiring hadn’t been so awful”). Thus, desirable results are typically due to our skill and hard work — not luck — while lousy results are outside of our control and the offspring of being unlucky.
Two fine books undermine this outlook by (rightly) attributing a surprising amount of what happens to us — both good and bad – to luck. Michael Mauboussin’s The Success Equation seeks to untangle elements of luck and skill in sports, investing, and business. Ed Smith’s Luck considers a number of fields – international finance, war, sports, and even his own marriage – to examine how random chance influences the world around us. For example, Mauboussin describes the “paradox of skill” as follows: “As skill improves, performance becomes more consistent, and therefore luck becomes more important.” In investing, therefore (and for example), as the population of skilled investors has increased, the variation in skill has narrowed, making luck increasingly important to outcomes.
On account of the growth and development of the investment industry, John Bogle could quite consistently write his senior thesis at Princeton on the successes of active fund management and then go on to found Vanguard and become the primary developer and intellectual forefather of indexing. In other words, the ever-increasing aggregate skill (supplemented by massive computing power) of the investment world has come largely to cancel itself out.
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After a big or revolutionary event, we tend to see it as having been inevitable. Such is the narrative fallacy. In this paper, ESSEC Business School’s Stoyan Sgourev notes that scholars of innovation typically focus upon the usual type of case, where incremental improvements rule the day. Sgourev moves past the typical to look at the unusual type of case, where there is a radical leap forward (equivalent to Thomas Kuhn’s paradigm shifts in science), as with Picasso and Les Demoiselles.
As Sgourev carefully argued, the Paris art market of Picasso’s time had recently become receptive to the commercial possibilities of risk-taking. Thus, artistic innovation was becoming commercially viable. Breaking with the past was then being encouraged for the first time. It would soon be demanded.
Most significantly for our purposes, Sgourev’s analysis of Cubism suggests that having an exceptional idea isn’t enough. For radical innovation really to take hold, market conditions have to be right, making its success a function of luck and timing as much as genius. Note that Van Gogh — no less a genius than Picasso — never sold a painting in his lifetime.
As noted above, we all like to think that our successes are earned and that only our failures are due to luck – bad luck. But the old expression – it’s better to be lucky than good – is at least partly true. That said, it’s best to be lucky *and* good. As a consequence, in all probabilistic fields (which is nearly all of them), the best performers dwell on process and diversify their bets. You should do the same…
As [Nate] Silver emphasizes in The Signal and the Noise, we readily overestimate the degree of predictability in complex systems [and t]he experts we see in the media are much too sure of themselves (I wrote about this problem in our industry from a slightly different angle…). Much of what we attribute to skill is actually luck.
Plan accordingly.
Taking the unaccountable into account: “Randomness Rules,” from Bob Seawright (@RPSeawright), via @JVLast
[image above: source]
* James Gleick, The Information: A History, a Theory, a Flood
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As we contemplate chance, we might spare a thought for Oskar Morgenstern; he died on this date in 1977. An economist who fled Nazi Germany for Princeton, he collaborated with the mathematician John von Neumann to write Theory of Games and Economic Behavior, published in 1944, which is recognized as the first book on game theory— thus co-founding the field.
Game theory was developed extensively in the 1950s, and has become widely recognized as an important tool in many fields– perhaps especially in the study of evolution. Eleven game theorists have won the economics Nobel Prize, and John Maynard Smith was awarded the Crafoord Prize for his application of evolutionary game theory.
Game theory’s roots date back (at least) to the 1654 letters between Pascal and Fermat, which (along with work by Cardano and Huygens) marked the beginning of probability theory. (See Peter Bernstein’s marvelous Against the Gods.) The application of probability (Bayes’ rule, discrete and continuous random variables, and the computation of expectations) accounts for the utility of game theory; the role of randomness (along with the behavioral psychology of a game’s participants) explain why it’s not a perfect predictor.
“I have had my results for a long time, but I do not yet know how to arrive at them”*…
Andrew Wiles gave a series of lectures cryptically titled “Modular Forms, Elliptic Curves, and Galois Representations” at a mathematics conference in Cambridge, England, in June 0f 1993. His argument was long and technical. Finally, 20 minutes into the third talk, he came to the end. Then, to punctuate the result, he added:
=> FLT
“Implies Fermat’s Last Theorem.” The most famous unverified conjecture in the history of mathematics. First proposed by the 17th-century French jurist and spare-time mathematician Pierre de Fermat, it had remained unproven for more than 350 years. Wiles, a professor at Princeton University, had worked on the problem, alone and in secret in the attic of his home, for seven years. Now he was unveiling his proof.
His announcement electrified his audience—and the world. The story appeared the next day on the front page of The New York Times. Gap, the clothing retailer, asked him to model a new line of jeans, though he demurred. People Weekly named him one of “The 25 Most Intriguing People of the Year,” along with Princess Diana, Michael Jackson, and Bill Clinton. Barbara Walters’ producers reached out to him for an interview, to which Wiles responded, “Who’s Barbara Walters?”
But the celebration didn’t last. Once a proof is proposed, it must be checked and verified before it is accepted as valid. When Wiles submitted his 200-page proof to the prestigious journal Inventiones Mathematicae, its editor divvied up the manuscript among six reviewers. One of them was Nick Katz, a fellow Princeton mathematician.
For two months, Katz and a French colleague, Luc Illusie, scrutinized every logical step in Katz’s section of the proof. From time to time, they would come across a line of reasoning they couldn’t follow. Katz would email Wiles, who would provide a fix. But in late August, Wiles offered an explanation that didn’t satisfy the two reviewers. And when Wiles took a closer look, he saw that Katz had found a crack in the mathematical scaffolding. At first, a repair seemed straightforward. But as Wiles picked at the crack, pieces of the structure began falling away…
How mistakes– first Fermat’s, then Wiles’– reinvigorated a field, then led to fundamental insight: “How Math’s Most Famous Proof Nearly Broke.”
* Karl Friedrich Gauss
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As we ponder proof, we might we might spare a thought for Josiah Wedgwood; he died on this date in 1795. An English potter and businessman (he founded the Wedgwood company), he is credited, via his technique of “division of labor,” with the industrialization of the manufacture of pottery– and via his example, much of British (and thus American) manufacturing.
Wedgwood was a member of the Lunar Society, the Royal Society, and was an ardent abolitionist. His daughter, Susannah, was the mother of Charles Darwin.
What’s (the) matter?…
On the heels of yesterday’s film recommendation, another… albeit somewhat different: Stanford physics professor, Leonard Susskind, one of the fathers of string theory, articulator of the Holographic Principle, and explainer of the Megaverse, has a gift for making science accessible… a gift that is on display in this lecture, “Demystifying the Higgs Boson“:
(email readers, click here)
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As we say “ahh,” we might spare a thought for Pierre de Fermat; he died on this date in 1665. With Descartes, one of the two great mathematicians of the first half of the Seventeenth Century, Fermat made a wide range of contributions (that advanced, among other fronts, the development of Calculus) and is regarded as the Father of Number Theory. But he is best remembered as the author of Fermat’s Last Theorem.* Fermat had written the theorem, in 1637, in the margin of a copy of Diophantus’ Arithmetica– but went on to say that, while he had a proof, it was too large to fit in the margin. He never got around to committing his proof to writing; so mathematicians started, from the time of his death, to try to derive one. While the the theorem was demonstrated for a small number of cases early on, a complete proof became the “white whale” of math, eluding its pursuers until 1995, when Andrew Wiles finally published a proof.
* the assertion that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two
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