Posts Tagged ‘paradox’
“In reality, there is no such thing as not voting: you either vote by voting, or you vote by staying home and tacitly doubling the value of some Diehard’s vote.”…
… but even when we do vote, as Alex Stone reports, there’s an issue: how we vote…
Mathematically speaking, democracy is next to impossible.
That may sound like the premise of a Silicon Valley billionaire’s six-hour podcast, but it’s actually a statement about a remarkably stubborn problem: How do you translate a group of individual preferences into a single choice?
In 1785, the French mathematician and philosopher Nicolas de Condorcet [see also here] discovered that a majority can sometimes vote itself into a logical loop. In elections with three or more candidates, each voter might rank the candidates in a perfectly clear and consistent way — yet when all the votes are tallied, every candidate loses in a head-to-head contest against one of their rivals. Changing the rules only gets us so far: As the economist Kenneth Arrow later proved, any method for combining individual rankings into a collective order must sacrifice at least one seemingly reasonable democratic requirement.
Now three researchers have shown that Condorcet’s 240-year-old paradox has a shape, and it’s suitably peculiar: a Klein bottle, a surface with no distinct inside or outside. The result gives mathematicians a geometric way to explore why seemingly reasonable rules for collective decision-making can become impossible to reconcile.
When there are only two choices, the will of the people is easy to divine: Barring a tie, the candidate preferred by a majority wins. Add a third contender, however, and things get weird. Suddenly, even agreeing on what it means to win becomes a problem.
You might simply elect whoever receives the most votes, as in a typical American election. But suppose Alice receives 35% of the vote, Bob 34%, and Carol 31%. Alice wins. Yet if Carol’s supporters all prefer Bob to Alice, then 65% of the electorate would rather have Bob. By one perfectly reasonable definition of majority rule, the wrong candidate has won.
Before anyone reaches for a tinfoil hat, let’s be clear: Every ballot is counted correctly. The question is what should count as winning.
One solution is to ask voters to rank the candidates, then choose whoever would defeat every rival in a head-to-head contest. But consider the following scenario:
- One-third of voters rank Alice over Bob over Carol.
- One-third rank Bob over Carol over Alice.
- One-third rank Carol over Alice over Bob.
Now compare the candidates in pairs:
- Alice beats Bob, because the first and third groups prefer Alice to Bob.
- Bob beats Carol, because the first and second groups prefer Bob to Carol.
- Carol beats Alice, because the second and third groups prefer Carol to Alice.
Every voter has a perfectly coherent ranking, yet together they have produced a result that runs in circles.
This is the paradox that Condorcet, a passionate believer in democracy, encountered in the late 18th century. As the secretary of the French Academy of Sciences, he’d been looking for ways to make sure the group was choosing new members fairly. (More enlightened than most, Condorcet would also question the democratic arithmetic of excluding half the population, explicitly defending women’s political rights in a 1790 essay. A few years later, during the Reign of Terror, the revolution he had supported put him in a prison cell; he was found dead two days after his arrest.)
Over the centuries, mathematicians, political scientists, and reformers have devised an astonishing variety of voting systems in hopes of breaking Condorcet’s impasse. Yet every method brings its own vulnerabilities. Change the system and you may well change the winner, even when the voters and their preferences remain exactly the same.
In 1951, the economist Kenneth Arrow showed how deep the problem ran. He sought a voting system that stayed consistent while meeting just a few simple democratic requirements. Voters should be free to rank the candidates in any order. No single voter should unilaterally dictate the outcome of an election. (Read: no kings.) If everyone prefers one option to another, the group’s ranking should reflect that preference. And the ranking of two candidates should depend only on how voters compare those two: Where voters place Carol on their ballots shouldn’t affect whether the group ranks Alice above Bob.
Each requirement sounds reasonable enough on its own. But, Arrow showed, no method can satisfy them all. “Quite often the will of the people doesn’t exist,” said Iain McLean of the University of Oxford.
Arrow’s impossibility theorem, as this result is now known, extended Condorcet’s paradox beyond majority rule, establishing a fundamental limit on how individual votes can be combined into a group decision. Different voting systems can change which of Arrow’s requirements fail, but no single system can ever preserve them all. “You have to make a choice as a system designer as to which of the conditions it’s most harmless to fail,” McLean said.
In 1980, to better understand this phenomenon, the mathematician and economist Graciela Chichilnisky turned to topology, an area of math in which shapes are considered equivalent if one can be stretched into another without tearing. In a footnote, she mused that Condorcet’s paradox in particular could be thought about as a topological problem.
Now mathematicians have proved her intuition true…
[Stone unpacks the (admittedly rather heady) math; he concludes…]
… So where does that leave democracy?
Topology makes the paradox tangible, but choosing a voting system still means picking your poison. Mathematics can tell us what the trade-offs entail. Our priorities determine which ones are worth making. And that’s something no theorem can settle for us. “It’s not really a mathematical debate,” McLean said. “It’s a debate about values.”
A new topological model connects a centuries-old voting paradox to a surface with no distinct inside or outside: “How Voting Is Like a Klein Bottle,” from @quantamagazine.org.
Tuesday’s election is largely a “binary choice” affair– do vote!
* David Foster Wallace, Consider the Lobster and Other Essays
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As we celebrate suffrage (with all of its shortcomings), we might recall that it was on this date in 1897 that the first Library of Congress building opened its doors to the public; the library had previously been housed in the Congressional Reading Room in the U.S. Capitol.
“All is change”*…
The Wikipedia article on the Ship of Theseus has changed entirely since its inception.
A paradoxical thought experiment made meta: “The Same Boat,” from Greg Ross in his wonderful blog, Futility Closet.
* Euripedes, Heracles (and others, e.g., Marcus Aurelius, echoing Heraclitus)
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As we contemplate conundra, we might recall that it was on this date in 1971 that the mix was completed on David Bowie’s “Changes,” which had been recorded nine days earlier at Trident Studios in London. Co-produced by Bowie and Ken Scott, it featured Rick Wakeman on piano and the musicians who would later become known as the Spiders from Mars—Mick Ronson, Trevor Bolder and Mick Woodmansey.
“Changes” is widely considered to be one of Bowie’s best songs and has appeared on many best-of lists. His biographers have suggested that the track is a manifesto of his entire career, predicting a constant change of musical styles: the first verse focuses on the compulsive nature of artistic reinvention and distancing oneself from the rock mainstream; the second verse concerns clashes between children and their parents, urging them to allow their children to be themselves as teenagers. Bowie frequently included the song in his live performances; it was the final song he performed on stage before his death in 2016.
See Bowie perform it here.
“We are the first generation to feel the effect of climate change and the last generation who can do something about it”*…
One of the issues that vexes coordinated response is a paradox that lies at the heart of the phenomenon: Earth’s climate is chaotic and volatile. Climate change is simple and predictable. How can both be true? Joseph Howlett explains…
The Earth’s atmosphere is nothing but freely roaming molecules. Left alone, they would drift and collide, and eventually even out into a mixture that’s dynamic, yet stable and broadly unchanging.
The sun’s rays complicate things. Energy enters the Earth system in daily cycles, the bulk of it going to whichever half of the planet is tilted toward the sun (and experiencing summer). The molecules in that half acquire more energy than others, which sets the global atmosphere steadily swirling. Depending on the season and location, molecules in our atmosphere might traverse warm land, then cold seas. They might encounter a mountain range that forces them to high altitudes, where the air pressure is low and water condenses. Then they might become part of large-scale phenomena, such as currents, atmospheric rivers, turbulent jet streams and continental fronts.
These phenomena are erratic. They interact at every scale and manifest as weather, from clear sunny days to blustery blizzards and the anomalous events — from hurricanes and polar vortices to hailstorms and tornadoes — that are happening with increasing intensity. Any thought of stability is illusory; no patch of molecules dances in isolation.
The result, from seemingly simple inputs of molecules and energy, is emergent, incalculable chaos. Some individual molecule in the room you are sitting in is careening about blindly and colliding with its immediate neighbors. Zoom out — block to city, field to landscape, region to continent — and patterns appear and intermix. Complexity abounds and compounds. Nothing in the atmosphere is untethered from the rest of the global picture.
We live with this unpredictable mess of an atmosphere every day. We tote around unopened umbrellas, or refresh weather apps and watch our weekend plans dissolve. Anticipating conditions any further out than a week or two is a fool’s errand. The Earth is a complex dynamical system — an interwoven mass of moving parts, each of which requires a different branch of science to understand. Even with advanced knowledge, sophisticated algorithms and modern instruments, it defies and eludes us.
Yet this engine of chaos is now under our influence. It is incontrovertible fact that we are changing the Earth’s temperature by adding more carbon dioxide to the atmosphere. We know exactly how we are changing it — that when we double the proportion of carbon dioxide in the thin layer that rests over the surface of the Earth, the planet will become 2 to 4 degrees Celsius warmer, overall, than it is today. This conclusion has remained essentially unchanged since 1896, when the Swedish scientist Svante Arrhenius arrived at an estimate of 2 to 5 degrees. (Using an extraordinarily simplified picture of Earth, he made a number of mistakes that, in the end, balanced out.) Some details may remain uncertain, some chaos untamable, but the basic conclusion is a matter of unwavering scientific agreement — 97% is a rare degree of consensus on almost any subject. We are nearly as sure of this as we are of the causes of infectious disease, or how stars form, or the fact that life evolves through natural selection.
oth things are true: The climate system is vastly complex, and we’re certain about what we are doing to it. How can we be so confident in a hundred-year projection when we can’t predict the weather with any reliability more than a week out?
“How can it be that both are true?” said Nadir Jeevanjee, an atmospheric physicist at NOAA’s Geophysical Fluid Dynamics Laboratory, a leading institution for cutting-edge simulations of the atmosphere. “It’s a huge tension that’s lurking behind the whole conversation.”
It turns out that complexity can be a veil concealing more basic truths. An enormously complicated system can yield simple answers. You just have to ask a simple enough question…
Read on for Howlett’s fascinating– and important– explanation: “The Climate Change Paradox,” from @quantamagazine.bsky.social.
And for a reminder that this matters (as though we need one…): “Human-Caused Warming Tripled the Death Toll of European Heat Waves This Summer, New Report Shows,” from @insideclimatenews.org.
* Barack Obama
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As face reality, we might recall that on this date in 1988, the #1 song in the U.S. was Bobby McFerrin‘s “Don’t Worry, Be Happy,” the first a cappella song to reach the top of the Billboard Hot 100 chart, a position it held for two weeks.
(Produced by Colossal Pictures, Directed by Drew Takahashi)
“Dada was a bomb… can you imagine anyone, around half a century after a bomb explodes, wanting to collect the pieces, sticking it together and displaying it?”*…
Marcel Duchamp was hugely influential in the revolutionary developments in the arts in the early 20th century. After helping establish Cubism, he turned to what he called “Readymades,” “found objects” which he selected and presented as art. By far the most famous of these was the piece he entitled “Fountain.” Damon Young and Graham Priest recount the stir that ensued… and unpack the work’s philosophical comment, making a case for why it resonates to this day…
In 1917 a pivotal event occurred for art and philosophy: Marcel Duchamp unveiled his artwork Fountain in Alfred Stieglitz’s New York studio. This was simply a porcelain urinal, signed ‘R. Mutt’.
Fountain was notorious, even for avant-garde artists. It has become one of the most discussed works of art of the 20th century. The Society of Independent Artists rejected it, though every artist who paid the exhibition fee was supposed to have their work shown. For almost a century, it has remained a difficult artwork. The philosopher John Passmore summed up Fountain as: ‘a piece of mischief at the expense of the art world’, though many have taken it very seriously.
No doubt there was some tomfoolery involved – Duchamp did not choose a urinal randomly. Yet there is more to Fountain than nose-thumbing. What makes this artwork so striking is its philosophical contribution.
Commentators often highlight the influence of Fountain on conceptual art, and this most ‘aggressive’ readymade, as Robert Hughes put it, has certainly had an enduring legacy. In 2004, it was voted the most important 20th-century work by hundreds of art experts. From Andy Warhol to Joseph Beuys to Tracey Emin, this urinal inspired artists to reconsider the traditional artwork. Instead of paintings and sculptures, art was suddenly Brillo boxes, an unmade bed, or a light-bulb plugged into a lemon: ordinary objects, some readymade, removed from their original contexts and placed on display in art galleries. The art critic Roberta Smith sums it up this way: ‘[Duchamp] reduced the creative act to a stunningly rudimentary level: to the single, intellectual, largely random decision to name this or that object or activity “art”.’ As we will see, Duchamp’s choice was not random at all, but Smith’s description points to the broader shock that Duchamp’s work prompted: if this can be art, then anything can.
Since then, scholars have discussed Fountain to demonstrate a shift away from aesthetics to thought. As the philosopher Noël Carroll notes, it’s possible to enjoy thinking about Duchamp’s work without actually looking at it, which cannot be said for Henri Matisse’s vivid paintings or Barbara Hepworth’s dignified stone sculptures.
These traditional ideas, as we will see, are all important to Fountain. But they do not go far enough. They treat Fountain as art, but of a mocking sort: a kind of intellectual heckling that nudged artists to taunt and scoff more academically at their own field. Our explanation of the artwork’s power is much more controversial: we believe that Fountain is art only insofar as it is not art. It is what it is not – and this is why it is what it is. In other words, the artwork delivers a true contradiction, what’s called a dialetheia. Fountain did not simply usher in conceptual art – it afforded us an unusual and intriguing concept to consider: a work of art that isn’t really a work of art, an everyday object that is not just an everyday object…
Marcel Duchamp’s ‘Fountain’ is not just a radical kind of art; it’s a philosophical dialetheia: a contradiction that is true: “It is and it isn’t,” from @damonyoung.com.au and Graham Priest @aeon.co. Eminently worth reading in full.
We might note that it’s not altogether clear that the dialetheia which the authors celebrate was what Duchamp had in mind. In any case (in line with the quote at the top) Duchamp, a father of Dada, was not entirely pleased with the influence that his work had:
This Neo-Dada, which they call New Realism, Pop Art, Assemblage etc. [Duchamp is referring to Andy Warhol and Roy Lichtenstein], is an easy way out and lives on what Dada did. When I discovered readymades I thought to discourage aesthetics. In Neo-Dada they have taken my readymades and found aesthetic beauty in them. I threw the bottle-rack [here] and the urinal in their faces as a challenge and now they admire them for their aesthetic beauty…
– Duchamp in a 1962 letter to Hans Richter
And as this is the centenniel of Dada’s “child,” Surrealism, we might peruse “The Small Magazines That Birthed Surrealism.”
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As we ponder paradox, we might raise a glass in celebration of National Cartoonists Day, observed on this day each year. The date was chosen to recognize the first appearance (in color) of the mischievous cartoon character “The Yellow Kid” in the New York World newspaper (on May 5, 1895).
“The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth.”*…
Zeno shows the Doors to Truth and Falsity (Veritas et Falsitas). Fresco in the Library of El Escorial, Madrid (source)
As Joel David Hamkins explains, an ancient puzzle leads ultimately to a remarkable observation on the malleable nature of infinite sums…
The Greek philosopher Zeno of Elea (c. 490–430 BC) argued in antiquity that all motion is impossible. It is simply impossible to walk through town or even across the room, to go from here to there. What? We know, of course, that this is possible—we walk from here to there every day. And yet, Zeno offers us his proof that this is an illusion—we simply cannot do it.
Zeno argued like this. Suppose it were possible for you to move from some point A to another distinct point B.
Before you complete the move from A to B , however, you must of course have gotten half way there.
But before you get to this half-way point, of course, you must get half way to the half-way point! And before you get to that place, you must get half way there.
And so on, ad infinitum.
Thus, to move from A to B , or indeed anywhere at all, one must have completed an infinite number of tasks—a supertask. It follows, according to Zeno, that you can never start moving—you cannot move any amount at all, since before doing that you must already have moved half as much. And so, contrary to appearances, you are frozen motionless, unable to begin. All motion is impossible.
Is the argument convincing? On what grounds would you object to it? Do you think, contrary to Zeno, that we can actually complete infinitely many tasks? How would that be possible?
It will be no good, of course, to criticize Zeno’s argument on the grounds that we know that motion is possible, for we move from one point to another every day. That is, to argue merely that the conclusion is false does not actually tell you what is wrong with the argument—it does not identify any particular flaw in Zeno’s reasoning. After all, if it were in fact an illusion that we experience motion, then your objection would be groundless…
Learning from an enigma– plus “the most contested equation in middle school” and more: “Zeno’s paradox,” from @JDHamkins.
* Niels Bohr
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As we interrogate infinity, we might send-well-groomed birthday greetings to Frank Joseph Zamboni, Jr.; he was born on this date in 1901. An engineer and inventor, he is best known for the modern ice resurfacer, seen at work at hockey games and figure skating competitions (completing its rounds, Zeno notwithstanding); indeed, his surname is the registered trademark for these devices.
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