Later On

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Archive for the ‘Math’ Category

So where is the number designated by “1”?

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The comic strip at the left ponders a well-worn question: Is the subject of mathematics real? or not? Or, to put it another way, are mathematical entities discovered? or invented?

My view is that mathematics has a kind of intermediate reality. The reality of mathematics, unlike, say, the reality of the Moon, is restricted to human culture. Within human culture, the number one is real, but if there were no humans, there would be no number one.

In other words, math is as real as a human language. Sounds exist within nature, and humans can make complex sounds, but language — those sounds together with their meaning — exists only insofar as there are people who understand the meaning of sounds. The meaning is not “out there” — where the sounds are, in the vibration of air — but “in here,” where the brain extracts the meaning conveyed. 

The meaning clearly exists in a sense, and indeed has consequences “out there” in the “real” world — the Industrial Revolution and the consequences (such as the climate change we now are experiencing) wold not have occurred without language. But once all those who understand some language are gone, the language is no more. There may be carvings in rocks or marks on vellum, but the meaning of those is absent, so the incisions and marks no long longer are language but just physical things, bereft of the meaning they once conveyed.

(For that matter, sound is not “out there.” What’s “out there” are vibrations in the air. Sound is the way our brain interprets air vibrations that have been fed to it as electrical impulses from the motion of tiny hairs in the liquid contained in the cochleae of our ears. Until that transition is made, there is no sound, only air vibrations. Thus a tree falling in a remote forest with no animals nearby will produce air vibrations but not sounds, because there’s no one to translate air vibrations to brain signals.)

So a sheep on a hillside is not “one” sheep unless it is observed by a person who has learned the human idea (the meme) of counting, and only such a person might observe that there are “zero” horses and “zero” cows on the hillside.

Math, like language, like music, like fashion, and like religion, is a cultural construct, a set of memes. Math has the reality of memes (as does, say Don Quixote or unicorns) but it is “in here,” not “out there.”  And even “in here” there are problems, as Kurt Gödel pointed out.  

And yet, consider this poem by Clarence R. Wylie Jr.:

Paradox

Not truth, nor certainty. These I forswore

In my novitiate, as young men called

To holy orders must abjure the world.

‘If…,then…,’ this only I assert;

And my successes are but pretty chains

Linking twin doubts, for it is vain to ask

If what I postulate be justified,

Or what I prove possess the stamp of fact.

Yet bridges stand, and men no longer crawl

In two dimension. And such triumphs stem

In no small measure from the power this game,

Played with the thrice-attentuated shades

Of things, has over their originals.

How frail the wand, but how profound the spell!

Written by Leisureguy

25 August 2022 at 11:32 am

Can computer simulations help fix democracy by curtailing gerrymandering?

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Harry Stevens has a good article (gift link, no paywall) in the Washington Post on a good approach to help even mathematically illiterate judges — that is, the great majority of judges — to understand when gerrymandering has been done. The article begins:

After the release of the 2020 Census, legislatures across the country redrew their states’ congressional district maps, just like they do every decade. And, just like every decade, aggrieved citizens sued them for gerrymandering — the process whereby politicians craft district boundaries to ensure their own parties’ victory.

But this time around, something has changed. A technological revolution, decades in the making, has added a sharp new arrow to those citizens’ quiver of legal arguments. Known as algorithmic redistricting, the technology has persuaded judges to throw out gerrymandered maps in several states, including New York and Ohio. And it will be part of a case before the Supreme Court in October that could play a role in the 2024 election and the future of voting rights.

Here is how it works. . .

Continue reading. (gift link, no paywall)

It’s a very good article, with interactive graphics to explain the approach clearly.

Written by Leisureguy

22 August 2022 at 9:36 pm

Prediction of China’s 9/11: “China’s ENTIRE economy will crash by September 11, 2022”

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That’s the prediction made in the video below: that China’s economy will collapse 34 days after August 7, 2022. Perhaps by coincidence, that date is 9/11/2022. I have marked my calendar. In the meantime, it’s an interesting report and worth watching.

Written by Leisureguy

8 August 2022 at 9:54 am

The mathematical power of 3 random words

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Mary Lynn Reed, Professor of Mathematics, Rochester Institute of Technology, writes in The Conversation:

It’s hard to imagine that three random words have the power to both map the globe and keep your private data secure. The secret behind this power is just a little bit of math.

What3words is an app and web-based service that provides a geographic reference for every 3-meter-by-3-meter square on Earth using three random words. If your brain operates more naturally in the English measurement system, 3 meters is about 9.8 feet. So, you could think of them as roughly 10-foot-by-10-foot squares, which is about the size of a small home office or bedroom. For example, there’s a square in the middle of the Rochester Institute of Technology Tigers Turf Field coded to brilliance.bronze.inputs.

This new approach to geocoding is useful for several reasons. First, it’s more precise than regular street addresses. Also, three words are easier for humans to remember and communicate to one another than, say, detailed latitude and longitude measurements. This makes the system well suited for emergency services. Seeing these advantages, some car manufacturers are starting to integrate what3words into their navigation systems.

Ordered triples

Here’s how three random words in English or any other language can identify such precise locations across the whole planet. The key concept is ordered triples.

Start with the basic assumption that the Earth is a sphere, recognizing that this is an approximate truth, and that its radius is approximately 3,959 miles (6,371 kilometers). To compute the surface area of the Earth, use the formula 4πr2. With r = 3,959 (6,371), this works out to approximately 197 million square miles (510 million square kilometers). Remember: What3words is using 3-meter-by-3-meter squares, each of which contains 9 square meters of surface area. So, working in the metric system, Earth’s surface area is equivalent to 510 trillion square meters. Dividing 9 into 510 trillion reveals that uniquely identifying each square requires around 57 trillion ordered triples of three random words.

An ordered triple is just a list of three things in which the order matters. So “brilliance.bronze.inputs” would be considered a different ordered triple than “bronze.brilliance.inputs”. In fact, in the what3words system, bronze.brilliance.inputs is on a mountain in Alaska, not in the middle of the RIT Tigers Turf Field, like brilliance.bronze.inputs.

The next step is figuring out how many words there are in a language, and whether there are enough ordered triples to map the globe. Some scholars estimate there are more a million English words; however, many of them are very uncommon. But even using only common English words, there are still plenty to go around. You can find many word lists online.

The developers at what3words came up with a list of 40,000 English words. (The what3words system works in 50 different languages with independently assigned words.) The next question is determining how many ordered triples of three random words can be made from a list of 40,000 words. If you allow repeats, as what3words does, there would be 40,000 possibilities for the first word, 40,000 possibilities for the second word, and 40,000 possibilities for the third word. The number of possible ordered triples would then be 40,000 times 40,000 times 40,000, which is 64 trillion. That provides plenty of “three random word” triples to cover the globe. The excess combinations also allow what3words to eliminate offensive words and words that would be easily confused for one another.

Passwords you can actually remember

While the power of three random words is being used to map the Earth, the U.K. National Cyber Security Centre (NCSC) is also advocating their use as passwords. Password selection and related security analysis are more complicated than attaching three words to small squares of the globe. But a similar calculation is illuminating. If you string together an ordered triple of words – such as brilliancebronzeinputs – you get a nice long password that a human should be able to remember far more easily than a random string of letters, numbers and special characters designed to meet a set of complexity rules.

If you increase your word list beyond 40,000, you’ll get . . .

Continue reading.

Written by Leisureguy

7 July 2022 at 10:51 am

A Mirror of Nature

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In Introspection Mike Edmunds has an interesting essay on implications of the Antikythera mechanism. The essay begins:

THE ANTIKYTHERA MECHANISM, an astronomical calculator found in a first-century BCE shipwreck, has proven to be mechanically more sophisticated than anything known from the subsequent millennium. While many are amazed at such a discovery, a more appropriate response would be admiration, for the mechanism fits well into its historic context. Indeed, the ancient scholar Cicero offered contemporary accounts of similar devices, which he saw as embodying the peak of human ingenuity.1 But the significance of the Antikythera mechanism extends beyond the elegance and complexity of its design. It may also represent a major development in our understanding of the universe.

Astronomical Mechanisms

IN THE SPRING of 1900, sponge divers working near the Greek island of Antikythera came across the wreck of a Roman cargo ship. Among the remains was a corroded shoebox-sized case with more than thirty bronze gear wheels in interlaced trains.2 Those who attempted to reconstruct it over the next century would learn that it included annular dials on its front and large spiral dials on the back that represented the day in the year, the lunar month in the 235-month Metonic cycle, the lunar phase, the position of the sun and moon in the zodiac, and whether the month might contain a lunar or solar eclipse. Irregularities in lunar motion were incorporated by means of an ingenious pin-and-slot variable-speed device. Predicted eclipses and the lunar calendar were based on observed cycles passed down to the Greeks from the Babylonians. The device itself was probably constructed in Rhodes sometime between 150 and 160 BCE, though both the date—it might be as early as 205 BCE—and the source are subject to debate. Inscriptions on the device strongly suggest that its front face also displayed the positions in the zodiac of the known planets: Mercury, Venus, Mars, Jupiter, and Saturn.

The significance of the Antikythera mechanism, as it came to be called, only began to be more broadly realized after the publication of Derek de Solla Price’s paper “Gears from the Greeks” in 1974. Price found the mechanism so sophisticated that it might “involve a completely new appraisal of the scientific technology of the Hellenistic period.”3 Just four years earlier, Germaine Aujac had written a perceptive, although largely forgotten, review of several kinds of mechanical devices that could have influenced Greek views of the universe.4 Price may well have been unaware of Aujac’s article. He does not reference it and Aujac does not mention the Antikythera mechanism. It would take another thirty years before the publications of the international Antikythera Mechanism Research Project (AMRP) and of Michael Wright prompted more general awareness of the artifact as confirming the reality of such complex machines in the ancient world.5

The devices Aujac wrote about were sphaerae—or sphéropée in the original French—mechanisms depicting the sky moving around the earth, with or without the planets. Sphaerae could be three-dimensional terrestrial or celestial globes and armillary spheres, but also two-dimensional circular constructions like the Antikythera mechanism. According to Aujac, by combining observation with the theory and construction of sphaerae, the Greeks were bringing models of the earth and heavens closer to their real equivalents. In his writings, Ptolemy acknowledged the existence of sphaerae, although he seems to have thought that they were admired more for their craftsmanship than for their value as physical models.6 James Evans and Christián Carlos Carman have argued that geared technology may slightly predate, and might actually have inspired, the mathematical developments around 200 BCE in Greek planetary theory such as eccentrics and epicycles.7

A Mechanical Universe

FROM MY OWN perspective, the deeper question concerning sphaerae is to what extent the development of this technology prompted the Greeks and Romans into a new worldview.8 The technology may have affected not only mathematics, but also the idea that the universe itself is in some sense mechanical—and long before the so-called scientific revolution of the Renaissance. For Samuel Sambursky, the question is

whether these models are only convenient means of illustration, devices adapted to our needs for an ordered description, or whether they represent to a greater or lesser degree some faithful image of a physical reality corresponding to them.9

If meant as a faithful image of reality, there are several themes present in such an image. The first would be the . . .

Continue reading.

Written by Leisureguy

5 July 2022 at 12:19 pm

The Riddle That Seems Impossible Even If You Know The Answer

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The key is that if you start with the box labeled with your number, then either you arrive at a box that contains that number or you open boxes endlessly. Given that there are a finite number of boxes, that’s impossible. So by starting with the box labeled with your number, you know that you are in the loop with a box that contains your number. The only issue is whether that loop is 50 boxes or fewer.

Written by Leisureguy

4 July 2022 at 10:37 am

Posted in Math

Ingenious numerals

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And there’s more to this than just the system of numerals. Some links from the video description:

► SCRIPT w/ SOURCES: https://docs.google.com/document/d/1f…
► CORRECTIONS:
https://docs.google.com/document/d/1m…
► WORLD ANVIL:
https://www.worldanvil.com/about
—————
ARTIFEXIAN ON THE INTERWEB:
► TWITTER:
https://www.twitter.com/artifexian
► PODCAST:
https://www.youtube.com/channel/UC21W…
► REDDIT:
https://www.reddit.com/r/artifexian/

Written by Leisureguy

29 June 2022 at 11:29 am

Posted in Daily life, Math

Manny Brot in The Case of the Missing Fractals

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Written by Leisureguy

24 June 2022 at 4:43 pm

Posted in Humor, Math, Video

Gödel’s theoren explained

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Kevin Drum takes a solid swing at explaining Gödel’s theorem — what it is and what it means — in a post that begins:

Bob Somerby wants to know if the logician/mathematician Kurt Gödel is a genius or a charlatan. The answer is “genius,” but it’s hard for non-mathematicians to understand his seminal theorem or why it matters. Bob is relying on Rebecca Goldstein’s biography of Gödel, and this is a mistake since it’s a biography, not a mathematical treatise.

But it’s dex night, so I’ll take a crack at it. Fair warning: you really need to have at least a little bit of background in math to understand this. There’s just no way around it. However, you don’t need much as long as you’re willing to tolerate a bit of mathematical symbology. Here goes.

1. Mathematical symbols

Although most of us don’t think of it this way, mathematics is actually a formal logical system of symbol manipulation.¹ For this to work, it must be possible to express all mathematical statements in a formal symbolic language. And it is! Take this statement, for example:

For every number there is a number that’s one higher

In mathematical symbology it looks like this:

∀ x ∃ x + 1

(For all numbers x there exists x + 1)

There is a symbol for anything you can say in the language of mathematics. If you’re interested, a complete list is here—though there are some complicated nuances for certain kinds of expressions. Basically, though, there’s a symbol for everything, although non-mathematicians are unfamiliar with most of them.

2. Gödel numbering

Gödel’s initial insight was that

Continue reading.

Written by Leisureguy

21 June 2022 at 8:24 pm

Posted in Daily life, Math

The Art of John Edmark

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I recently posted a video of John Edmark discussing his work. Here is another talk on Edmark”s work.

Written by Leisureguy

11 June 2022 at 1:12 pm

Posted in Art, Daily life, Math, Science, Video

Creating the never-ending bloom

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Another video on John Edmark’s work.

Written by Leisureguy

8 June 2022 at 8:39 pm

Posted in Art, Math, Science, Video

Penrose-tiling a bathroom

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Cool idea. Penrose tiling should be more common. Image is from a post by Lior Pachter that describes the project. I blogged earlier a video on Penrose tiling.

Written by Leisureguy

27 May 2022 at 12:32 pm

Posted in Daily life, Math

The Most Powerful Computers You’ve Never Heard Of

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Fascinating video. 

Written by Leisureguy

3 May 2022 at 12:00 pm

Cumulative excess deaths from COVID-19

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The above chart is from Kevin Drum’s post looking at “excessive death” statistics, which probably provide a better measure of Covid deaths than the count of deaths explicitly attributed to Covid (since in many localities there’s a strong pressure not to list Covid as the cause of death). 

Read his post for the full explanation and more charts.

Written by Leisureguy

19 April 2022 at 9:30 am

The Antikythera Cosmos

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A very interesting video on the Antikythera Mechanism. This is via a Vice article by Becky Ferreira, which begins:

In the early 1900s, divers hunting for sponges off the coast of Antikythera, a Greek island in the Aegean Sea, discovered a Roman-era shipwreck that contained an artifact destined to dramatically alter our understanding of the ancient world.

Known as the Antikythera Mechanism, the object is a highly sophisticated astronomical calculator that dates back more than 2,000 years. Since its recovery from the shipwreck in 1901, generations of researchers have marveled over its stunning complexity and inscrutable workings, earning it a reputation as the world’s first known analog computer.

The device’s gears and displays cumulatively demonstrated the motions of the planets and the Sun, the phases of the lunar calendar, the position of Zodiac constellations, and even the timing of athletic events such as the ancient Olympic Games. The device also reflects a very ancient idea of the cosmos, with Earth at the center.

While some of the calculator’s mysteries have been solved over the past century, scientists at University College London’s Antikythera Research Team present, for the first time, “a radical new model that matches all the data and culminates in an elegant display of the ancient Greek Cosmos,” according to a study published on Friday in Scientific Reports.

Led by Tony Freeth, a mechanical engineer at UCL and a leading world expert on the mechanism, the interdisciplinary team called the artifact “an ancient Greek astronomical compendium of staggering ambition” and “a beautiful conception, translated by superb engineering into a device of genius,” in the study.

“This is such a special device,” said Adam Wojcik, a materials scientist at UCL and a co-author of the study, in a call. “It’s just so out-of-this world, given what we know, or knew, about contemporary ancient Greek technology. It’s unique and there’s nothing else that remotely approaches it for centuries, or maybe a millennia afterwards.”

“However, it exists and all the scholarship points to the fact that it is ancient Greek,” added Wojcik, who has been fascinated by the artifact since he was a child. “There’s no question about it and we just have to accept that there is so much about what they could do that we just don’t know and we can’t fathom. The mechanism is a window on that.”

Understanding the clockwork instrumentation of the Antikythera Mechanism has been a longstanding challenge for scientists because only a third of the artifact survived its multi-millennia entombment under the Mediterranean waves. The remains of the calculator include 82 fragments, some of which contain complex gears and once-hidden inscriptions, which were wedged between front and back display faces during the bygone era in which the artifact was fully intact.

As new experimental techniques emerged, research teams have been able to explain the purpose and dynamics of the Antikythera Mechanism’s back face, which includes a system of eclipse predictions. In particular, the use of surface imaging and high-resolution X-ray tomography on the artifact, described in a 2006 study also led by Freeth, revealed scores of never-before-seen inscriptions that helpfully amount to a user’s guide to the mechanism.

Now, Freeth and his colleagues believe they have tackled the missing piece of the puzzle: the complicated gearworks underlying the front “Cosmos” display of the calculator. Virtually nothing from this front section survived, and “no previous reconstruction has come close to matching the data” that does exist, according to the study.

The new paper “has synthesized other people’s work, and dealt with all the loose ends and the uncomfortable nuances that other people just simply ignored,” Wojcik said. “For example, there are certain features in the surviving bits—holes and pillars and things like that—which people have said: ‘well, we’ll just ignore that in our explanation. There must be a use for that but we don’t know what it is so we’ll just ignore it.’”

“Effectively, what we’ve done is we’ve not ignored anything,” he added. “So the enigmatic pillars and holes, all of a sudden, now make sense in our solution. It all comes together and it fits the inscriptional evidence.” . . .

Continue reading.

Written by Leisureguy

12 April 2022 at 10:33 pm

Bertrand’s Paradox

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Fascinating.

But wait! There’s more!

Written by Leisureguy

17 March 2022 at 12:14 pm

Posted in Math, Video

Correlation vs. Causation

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We are quite frequently told that correlation does not equal causation — perhaps a little too frequently. (Not so frequently observed is that, although correlation does not imply causation, causation does indeed result in a correlation.) 

But a simple reminder is not so good as specific examples, and that’s the benefit of the Spurious Correlations site (pointed out to me by Montreal Steve). On the site the charts are interactive — for example, hovering over a data point will display values — but this example screenshot is not interactive:

Written by Leisureguy

6 March 2022 at 6:24 am

Posted in Daily life, Humor, Math, Science

Algorithms are designing better buildings

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Silvio Carta, Head of Art and Design, University of Hertfordshire, writes in The Conversation:

When giant blobs began appearing on city skylines around the world in the late 1980s and 1990s, it marked not an alien invasion but the impact of computers on the practice of building design.

Thanks to computer-aided design (CAD), architects were able to experiment with new organic forms, free from the restraints of slide rules and protractors. The result was famous curvy buildings such as Frank Gehry’s Guggenheim Museum in Bilbao and Future Systems’ Selfridges Department Store in Birmingham.

Today, computers are poised to change buildings once again, this time with algorithms that can inform, refine and even create new designs. Even weirder shapes are just the start: algorithms can now work out the best ways to lay out rooms, construct the buildings and even change them over time to meet users’ needs. In this way, algorithms are giving architects a whole new toolbox with which to realise and improve their ideas.

At a basic level, algorithms can be a powerful tool for providing exhaustive information for the design, construction and use of a building. Building information modelling uses comprehensive software to standardise and share data from across architecture, engineering and construction that used to be held separately. This means everyone involved in a building’s genesis, from clients to contractors, can work together on the same 3D model seamlessly.

More recently, new tools have begun to combine this kind of information with algorithms to automate and optimise aspects of the building process. This ranges from interpreting regulations and providing calculations for structural evaluations to making procurement more precise.

Algorithmic design

But algorithms can also help with the design stage, helping architects to understand how a building will be used by revealing hidden patterns in existing and proposed constructions. These can be spatial and geometrical characteristics such as the ratio of public to private areas or the natural airflow of a building. They can be patterns of use showing which rooms are used most and least often.

Or they can be visual and physical connections that show what people can and can’t see from each point of a building and enable us to predict the flow of people around it. This is particularly relevant when designing the entrances of public buildings so we can place services and escape routes in the best position.

Algorithms can also be used to extend the capability of designers to think about and generate . . .

Continue reading. Much more, and it’s interesting.

See also this earlier post.

Written by Leisureguy

23 February 2022 at 10:44 am

Aspiring to a Higher Plane: Going Deeper into Abbot’s “Flatland”

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Ian Stewart has a very interesting article in The Public Domain Review (that’s their mis of fonts) about Abbot’s novel Flatland (at the link, available as an ebook in various formats free of charge). The article includes various illustrations from the book, and I’ll not include those, but instead offer the initial text of the article:

Edwin Abbott Abbott, who became Headmaster of the City of London School at the early age of 26, was renowned as a teacher, writer, theologian, Shakespearean scholar, and classicist. He was a religious reformer, a tireless educator, and an advocate of social democracy and improved education for women. Yet his main claim to fame today is none of these: a strange little book, the first and almost the only one of its genre: mathematical fantasy. Abbott called it Flatland, and published it in 1884 under the pseudonym A. Square.

On the surface — and the setting, the imaginary world of Flatland, is a surface, an infinite Euclidean plane — the book is a straightforward narrative about geometrically shaped beings that live in a two-dimensional world. A. Square, an ordinary sort of chap, undergoes a mystical experience: a visitation by the mysterious Sphere from the Third Dimension, who carries him to new worlds and new geometries. Inspired by evangelical zeal, he strives to convince his fellow citizens that the world is not limited to the two dimensions accessible to their senses, falls foul of the religious authorities, and ends up in jail.

The story has a timeless appeal, and has never been out of print since its first publication. It has spawned several sequels and has been the subject of at least one radio programme and two animated films. Not only is the book about hidden dimensions: it has its own hidden dimensions. Its secret mathematical agenda is not the notion of two dimensions, but that of four. Its social agenda pokes fun at the rigid stratification of Victorian society, especially the low status of women, even the wives and daughters of the wealthy.

Flatland’s inhabitants are . . .

Continue reading. There’s more, the illustrations are nifty, and you can get a free copy of the ebook.

Written by Leisureguy

25 January 2022 at 1:27 pm

How A.I. Conquered Poker

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In the NY Times Magazine, Keith Romer describes how poker has now been solved. (Gift link, no paywall.)

Last November in the cavernous Amazon Room of Las Vegas’s Rio casino, two dozen men dressed mostly in sweatshirts and baseball caps sat around three well-worn poker tables playing Texas Hold ’em. Occasionally a few passers-by stopped to watch the action, but otherwise the players pushed their chips back and forth in dingy obscurity. Except for the taut, electric stillness with which they held themselves during a hand, there was no outward sign that these were the greatest poker players in the world, nor that they were, as the poker saying goes, “playing for houses,” or at least hefty down payments. This was the first day of a three-day tournament whose official name was the World Series of Poker Super High Roller, though the participants simply called it “the 250K,” after the $250,000 each had put up to enter it.

At one table, a professional player named Seth Davies covertly peeled up the edges of his cards to consider the hand he had just been dealt: the six and seven of diamonds. Over several hours of play, Davies had managed to grow his starting stack of 1.5 million in tournament chips to well over two million, some of which he now slid forward as a raise. A 33-year-old former college baseball player with a trimmed light brown beard, Davies sat upright, intensely following the action as it moved around the table. Two men called his bet before Dan Smith, a fellow pro with a round face, mustache and whimsically worn cowboy hat, put in a hefty reraise. Only Davies called.

The dealer laid out a king, four and five, all clubs, giving Davies a straight draw. Smith checked (bet nothing). Davies bet. Smith called. The turn card was the deuce of diamonds, missing Davies’s draw. Again Smith checked. Again Davies bet. Again Smith called. The last card dealt was the deuce of clubs, one final blow to Davies’s hopes of improving his hand. By now the pot at the center of the faded green-felt-covered table had grown to more than a million in chips. The last deuce had put four clubs on the table, which meant that if Smith had even one club in his hand, he would make a flush.

Davies, who had been betting the whole way needing an eight or a three to turn his hand into a straight, had arrived at the end of the hand with precisely nothing. After Smith checked a third time, Davies considered his options for almost a minute before declaring himself all-in for 1.7 million in chips. If Smith called, Davies would be out of the tournament, his $250,000 entry fee incinerated in a single ill-timed bluff.

Smith studied Davies from under the brim of his cowboy hat, then twisted his face in exasperation at Davies or, perhaps, at luck itself. Finally, his features settling in an irritated scowl, Smith folded and the dealer pushed the pile of multicolored chips Davies’s way. According to Davies, what he felt when the hand was over was not so much triumph as relief.

“You’re playing a pot that’s effectively worth half a million dollars in real money,” he said afterward. “It’s just so much goddamned stress.”

Real validation wouldn’t come until around 2:30 that morning, after the first day of the tournament had come to an end and Davies had made the 15-minute drive from the Rio to his home, outside Las Vegas. There, in an office just in from the garage, he opened a computer program called PioSOLVER, one of a handful of artificial-intelligence-based tools that have, over the last several years, radically remade the way poker is played, especially at the highest levels of the game. Davies input all the details of the hand and then set the program to run. In moments, the solver generated an optimal strategy. Mostly, the program said, Davies had gotten it right. His bet on the turn, when the deuce of diamonds was dealt, should have been 80 percent of the pot instead of 50 percent, but the 1.7 million chip bluff on the river was the right play.

“That feels really good,” Davies said. “Even more than winning a huge pot. The real satisfying part is when you nail one like that.” Davies went to sleep that night knowing for certain that he played the hand within a few degrees of perfection.

The pursuit of perfect poker goes back at least as far as the 1944 publication of “Theory of Games and Economic Behavior,” by the mathematician John von Neumann and the economist Oskar Morgenstern. The two men wanted to correct what they saw as a fundamental imprecision in the field of economics. “We wish,” they wrote, “to find the mathematically complete principles which define ‘rational behavior’ for the participants in a social economy, and to derive from them the general characteristics of that behavior.” Economic life, they suggested, should be thought of as a series of maximization problems in which individual actors compete to wring as much utility as possible from their daily toil. If von Neumann and Morgenstern could quantify the way good decisions were made, the idea went, they would then be able to build a science of economics on firm ground.

It was this desire to model economic decision-making that led them to game play. Von Neumann rejected most games as unsuitable to the task, especially those like checkers or chess in which both players can see all the pieces on the board and share the same information. “Real life is not like that,” he explained to Jacob Bronowski, a fellow mathematician. “Real life consists of bluffing, of little tactics of deception, of asking yourself what is the other man going to think I mean to do. And that is what games are about in my theory.” Real life, von Neumann thought, was like poker.

Using his own simplified version of the game, in which  . . .

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Written by Leisureguy

18 January 2022 at 3:44 pm

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