Mind Matters Natural and Artificial Intelligence News and Analysis


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Most Real Numbers Are Not Real, or Not in the Way You Think

Typical real numbers contain an encoding of all of the books in the US Library of Congress

Pick a random real number between zero and one. The number you choose, with probability one, will contain an encoding of all of the books in the US Library of Congress. This sounds absurd, but real numbers require infinite precision and every time you deal with the infinite, things get absurd. Infinities, including the infinite number of digits to express almost every real number, don’t exist. Curiously then, real numbers are not real. How do we choose a random number between zero and one? The easiest way to explain is using binary decimals. The binary number 0.1000… with zeros forever denotes the number ½ or, in base 10 notation, 0.5. The binary decimal 0.01000… with zeros forever is the number…

Young serious female latin math school teacher wearing glasses holding writing equation on whiteboard in classroom. Hispanic university college tutor, graduate student learning, teaching during class.

Can We Add New Numbers to Mathematics?

We can work with hyperreal numbers using conventional methods. It could start in high school

Sometimes mathematics is moved forward by the discovery of new formulas and solutions to problems. However, sometimes mathematics grows by adding new kinds of numbers to the number system. In the early days of mathematics, it was thought that whole numbers were the only kind that existed. Sure, there were fractions, but fractions are merely ratios of whole numbers. It was thought that every possible number could be written in terms of whole numbers. These numbers were called rational numbers because they could be written as a ratio. There is a story about a Greek philosopher, Hippasus who discovered, roughly 2500 years ago, that certain numbers (specifically the square root of two) could not be written in terms of ratios…

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How Bayes’ Math Rule Can Counter Unreasonable Skepticism

Mathematics is much more interesting if we know a bit about the players and their positions

Yesterday, we discussed the importance of Bayes’ rule in statistical reasoning. We used the example of a person who goes for a battery of screening tests and comes up positive for HIV. Let’s say she is surprised (and alarmed) because she is not at any known risk for HIV. But, it turns out, the risk of false positives for the test is several times greater than the incidence of HIV in the population. In that case, it is reasonable for her to suspect—on a statistics science basis, not just wishful thinking—that the test is a false positive. The formula we used is part of Bayesian reasoning, originally developed by an eighteen-century British clergyman and mathematician Thomas Bayes (1702–1761), but now…

high risk cholesterol test results

Can an 18th Century Statistician Help Us Think More Clearly?

Distinguishing between types of probability can help us worry less and do more

Thomas Bayes (1702–1761) (pictured), a statistician and clergyman, developed a theory of decision-making which was only discussed after his death and only became important in the 20th century. It is now a significant topic in philosophy, in the form of Bayesian epistemology. Understanding Bayes’ Rule may be essential to making good decisions. Let’s say that you are a generally healthy person and have no symptoms of any illness and no specific risk factors for any illness. Acting on a friend’s suggestion, you get screened for a variety of diseases, just to be sure. Of the diseases you test for, the HIV test comes back positive. You read on the package that the test is 99.6% accurate. Are you more likely…

Educated school kid lifting world globe chalk doodle drawing on green chalkboard for education concept

To Fix Math Education, See It as a Program That Needs an Update

As a computer programmer, I’ve seen this problem in my work: The basic idea is still sound but “fixes” have made it too complex

In this series we are looking at ways that math education can be reformed. In contrast to some other math reform efforts, we are not trying to fundamentally rewrite what math education is doing but to simply admit that we can do better and see where that takes us. (See Part 1, Part 2, and Part 3.) Here in Part 4, let’s look at specific content issues that, I will argue, we could improve when we do a curriculum revision. Mathematics is an old subject. We have inherited quite a bit of mathematical thought. We must educate future generations so as to make sure that this hard-won knowledge is not lost. But one of the biggest impediments to our task…

Bored student girl during math lesson

Helping Students See How Math Benefits Them in the Long Run

To keep them motivated, we need to answer the “Why bother?” question honestly and directly

In this series we are looking at ways that math education can be reformed. In contrast to some other math reform efforts, we are not trying to fundamentally rewrite what math education is doing but to simply admit that we can do better and see where that takes us. (See Part 1 and Part 2.) Here in Part 3, we will concentrate on making the curriculum more “conscious” of what students are supposed to be learning in mathematics. One of the primary complaints students have about higher mathematics is that they don’t see where they are going to use the information later in their jobs. There are a number of ways of answering this question but first I want to…

Maths class

Straight Talk About Fitting the Math Curriculum to the Student

We need to avoid pushing too much too soon, lest students come to see themselves as “bad at math” when they are just not ready for it

In this series we are looking at ways that math education can be reformed (Part 1 here). In contrast to some other math reform efforts, we are not trying to fundamentally rewrite what math education is doing but to simply admit that we can do better and see where that takes us. This article, Part 2, will concentrate on improving math education by better identifying where students are when we encounter them. Mathematics curriculum is generally developed with a goal of “fitting it all in.” That is, educators assume that people learn at a relatively fixed pace. They then pace the lessons so as to fit all of them into the curriculum in the right amount of time. However, this…

Jellybean Candy in a Jar

The Wisdom of Crowds: Are Crowds Really Wiser Than Individuals?

According to the theory, with a large number of guessers, the median number is very likely to be close to the true value

Statistician Sir Francis Galton went to a country fair in 1907 where a prize was to be awarded to the person who made the most accurate guess of the butchered weight of an ox that was on display. Galton collected and analyzed the 787 guesses and, not surprisingly, found that some guesses were far too high and others were much too low. However, the average guess (1,197 pounds) was only 1 pound lower than the actual weight (1,198 pounds). The average was more accurate than the guesses of the vast majority of both the amateurs and the experts. In the 1980s, a finance professor named Jack Treynor (1930–2016) performed a similar, and now legendary, experiment with jelly beans. Professor Treynor…

Girl solving mathematical addition

How Can We Really Fix the Way Math Is Taught?

First, we must understand why we teach math in the first place

Many people recognize that there are problems with modern mathematics curricula. However, the solutions proposed by current would-be math reformers are, I fear, worse than the cure. Some reformers want to stop having kids memorize their arithmetic facts, some want kids to just use computers to solve their problems, others think that the way we teach mathematics is racist, and still others seem to want to just greatly reduce the quantity of math education altogether. In this first part of a four-part series of short posts, I want to look at the most basic question: Why do we teach math? Earlier this month, an article by Yoree Koh in the Wall Street Journal took a look at the “Movement to…

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…

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…

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…

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…

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 ›
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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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,…

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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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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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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