Mind Matters Reporting on Natural and Artificial Intelligence

CategoryMathematics

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man sitting and feeding birds

Pigeons Can Solve the Monty Hall Problem. But Can You?

The dilemma pits human folk intuition against actual probability theory, with surprising results

Animals often outperform humans. My son’s dog is more friendly than I could ever be. Cheetahs run faster, baby horses walk earlier, and elephants can lift more. Birds fly and humans can’t. Is there anything else birds can do better than humans? Yes. Apparently, pigeons learn to solve the Monty Hall problem more quickly. Let’s Make a Deal was a television game show first hosted by Monty Hall (1921–2017) in 1963. There have been various remakes since then. The basic idea is that there are three doors and a contestant’s job is to barter with Monty for the most valuable prize behind the doors. The Monty Hall problem, loosely based on the quiz show, was popularized by Marilyn vos Savant…

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Girl business woman sitting at a wooden table with a laptop and discussing a new project with her mentor boss teacher. The man is fooling around. New business development concept

Why Intelligent Women Marry Less Intelligent Men

Are they trying to avoid competition at home as well as at work? Or is there a statistical reason we are overlooking?

When I read that Ruth Bader Ginsburg’s late husband was a wonderful man but less accomplished than his wife, I was reminded of “Ivy,” one of the most impressive students I ever had the privilege to teach. Ivy excelled in her coursework, won a prestigious scholarship for postgraduate study in England, went to a top-five law school, clerked for a Supreme Court Justice, and is now a law professor at a great university. Like Ruth Ginsberg (1933–2020, pictured), Ivy married a man who is very nice but less intelligent than she. This is not an unusual situation. I made a list of the dozen most intelligent female students with whom I’ve kept in touch over the years. These women are…

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The Settlers of Catan

In Science, We Can’t Just “Settle” for Data Clusters

The board game, Settlers of Catan, offers a clear illustration of what can go wrong when we are duped by data clusters

Settlers of Catan is an incredible board game created by Klaus Teuber, a German game designer. It has been translated into dozens of languages and tens of millions of sets have been sold. The basic four-player board consists of 19 hexagons (hexes) representing resources: 3 brick, 4 lumber, 4 wool, 4 grain, 3 ore, and 1 desert. Players acquire and use resources based on dice rolls, card draws, trading, and the location of their settlements and cities. Part of the game’s seductive appeal is that there are many, many ways to arrange the 19 hexagons and successful strategies depend on how the hexagons are arranged. The rules are simple but winning strategies are complex and elusive. The official rules of…

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tropical hurricane approaching the USA.Elements of this image are furnished by NASA.

Female Hurricanes: How A Mass of Hot Air Became a Zombie Study

When a reporter first asked me about a study claiming that “Female Hurricanes are Deadlier than Male Hurricanes,” I was skeptical

It is once again hurricane season on the East Coast and social media are reliably abuzz with reports of a debunked study that claimed that hurricanes given feminine-sounding names are deadlier than those given masculine-sounding names. (Full disclosure: I was one of the debunkers.) When a reporter initially asked me about this study, titled “Female Hurricanes are Deadlier than Male Hurricanes,” I was skeptical. Hurricane names alternate between female and male on a schedule set before the hurricane season begins so any relationship between name and death toll was surely coincidental. I looked at the study and discovered that the authors were not arguing that female hurricanes are inherently deadlier but that sexist humans die because they don’t take female…

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man riding on self balancing board graffiti

Election Models: Predicting the Past is Easy — and Useless

You can seldom see where you are going by looking in a rear-view mirror
I told my students that I had a model that predicted the popular vote for the the last ten presidential elections (1980–2016) perfectly. Read More ›
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Groupe de Bonobos autour d'un hôtel à insectes

Can Animal Minds Rival Humans Under the Right Circumstances?

Are we just not being fair to animals, as some researchers think?

In 2007, Sue Savage-Rumbaugh, a psychologist and primatologist , published a paper in the Journal of Applied Animal Welfare Science with a remarkable citation: Sue Savage-Rumbaugh, Kanzi Wamba, Panbanisha Wamba, and Nyota Wamba, “Welfare of Apes in Captive Environments: Comments on, and by, a Specific Group of Apes,” Journal of Applied Animal Welfare Science 10:1 (2007): 7–19. What is remarkable about the paper is not the text but the authorship statement. Kanzi, Panbanisha, and Nyota Wamba are not co-author colleagues—they’re apes, bonobos to be specific. Dr. Savage-Rumbaugh (right) is a controversial scientist who believes that animals have intellectual powers that can, under the right circumstances, rival the human intellect. She included her ape subjects as co-authors on the paper because…

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3D rendering of technology background

Why is Bell’s Theorem Important for Conservation of Information?

Proving a negative is difficult. Demonstrating that there are no leafy green crows is hard to do without examining every crow. But there's another way.

Proving a negative is difficult. Think about it. For example, demonstrating that there are no leafy green crows is hard to do without exhaustively examining every crow in existence. On the other hand, proving there are no crows naturally emblazoned with the text of the King James Bible is a bit easier to do. Proving a negative is possible if the extremes are large enough. Such as result is known as a no-go theorem. One of the most profound no-go theorems can be found in quantum physics. Physicist John Bell (1928–1990) proved — entirely from first principles — that there is a fundamental difference between how particles interact classically compared with how they interact within quantum physics. In classical physics,…

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Blue glowing multiverse in space

Is Big Bang Theory’s Sheldon Right re the Multiverse?

Sheldon Cooper insists that in no universe would he dance with Penny

A collection of universes is called a multiverse. If there are enough universes in a multiverse, can almost anything happen? No. Common models of the universe aren’t big enough. The argument that anything can happen in a multiverse is nicely presented in a 2011 scene in the sitcom The Big Bang Theory (2007–2019) involving consummate nerd Sheldon Cooper and Penny, the girl next door (here). Penny: Morning, Sheldon! Come dance with me! Sheldon: No. Penny: Why not? Sheldon: While I subscribe to the Many Worlds theory, which posits the existence of an infinite number of Sheldons in an infinite number of universes, I assure you that in none of them I am dancing. Penny: Are you fun in any of…

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medical statistics and graphic charts with stethoscope

Making Sense of the Numbers Behind COVID-19

Media and politicians put statistics before us to sway our opinions. But what do they really mean?

Numbers can frighten or enlighten. The secret is making them explain themselves. Here’s a quick primer.

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Close-up view of the Difference Engine

Lovelace: The Programmer Who Spooked Alan Turing

Ada Lovelace understood her mentor Charles Babbage’s plans for his new Analytical Engine and was better than he at explaining what it could do

Turing thought that computers could be got to think. Thus he had to address Lovelace’s objection from a century earlier, that they could not be creative.

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American Earth Venn Diagram

Physicist Rejects Free Will — and Thus Fails Logic

If we accepted his argument for materialism, we would have to stop believing in it—a curious, self-refuting result

Carroll’s argument that man is wholly governed by physics is self-refuting. Because physics and logic share no commonality, materialists like Carroll implicitly assert that their own arguments lack logic. One might say that the only thing materialists get right is that their ideas are nonsense. If man is all physics, he can have no logic.

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If Then logic statement written in white chalk on a black chalkboard isolated on white

Gödel and God: A Surprising History

A thought-provoking account of master logician Gödel’s largely unknown proof of the existence of God

In an unsanitized, politically incorrect (but factual) history, Selmer Bringsjord talks about how the tormented genius Kurt Gödel took up a quest that dated back a thousand years to prove the existence of God by formal logic. His original version didn’t quite work but his editor’s version passed an important logic test.

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Math Shows Why the Mind Is Not Just a Formula

The Liar’s Paradox shows that even mathematics cannot be reduced to a fixed set of axioms

Gödel’s discovery brought back a sense of wonder to mathematics and to the rest of human knowledge. His incompleteness theorem underlies the fact that human investigation can never exhaust all that can be known. Every discovery builds a path to a new discovery.

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Blackboard inscribed with scientific formulas and calculations in physics and mathematics.

Bartlett’s Calculus Paper Reviewed in Mathematics Magazine

The paper offers fixes for long-standing flaws in the teaching of elementary calculus

Jonathan Bartlett tells us, “The review was mixed, but most importantly the reviewer didn’t disagree with the results, only their potential usefulness."

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Polly Want a … Statistician?

Ethology, the science of animal behavior, offers interesting data but the interpretations are too often witless

Can birds really do statistics? A reporter writing up the results of a study for a popular science magazine seems to think so. The researchers are (appropriately) more cautious. But what are the issues here?

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Photo by Gayatri Malhotra

Why Does Mathematics Interpret Reality?

In the latest issue of Communications of the Blyth Institute, Gordon Mullings presents his account of why mathematics and physics are connected

The amazing applicability of mathematics to the real world has caused many mathematicians, philosophers, and physicists to pause throughout history. How can something as abstract and ideal as mathematics apply to the real world?

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An abstract computer generated fractal design. A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales.

Are Divergent Series Really an “Invention of the Devil”?

The real villain in the piece is horrendously non-specific concepts of infinity. But that can be fixed

It turns out that hyperreal numbers (i.e., infinities that obey algebraic rules) resolve many of the paradoxes that previously plagued conceptions of divergent series. It is now possible to assign specific values to divergent series.

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Could Information Be — at Long Last — the Missing Dark Matter?

Materialist thinkers may need to see information as material, whether that approach fits information or not

There is no evidence that information is dark matter or that consciousness is physical but materialists understandably long for evidence that would make their theory more viable.

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Photo by Amanda Jones

Bridge: Why Shuffle the Deck Seven Times?

For years, competitive bridge players complained that computer shuffling of cards produced goofy results. Statisticians sided with the computers

Bridge is one of the few games where computer algorithms have not yet demolished the best human players but, despite claims to the contrary, algorithms do a much better job of random shuffling of the deck.

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Faith Is the Most Fundamental of the Mathematical Tools

An early twentieth century clash of giants showed that even mathematics depends on some unprovable assumptions

David Hilbert wanted all mathematics to be proved by logical steps. Kurt Gödel showed that no axiomatic system could be complete and consistent at the same time.

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