Mind Matters Natural and Artificial Intelligence News and Analysis

CategoryMathematics

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Infinite random numbers, original 3d rendering background, technology and science concepts

Chaitin’s Discovery of a Way of Describing True Randomness

He found that concepts from computer programming worked well because, if the data is not random, the program should be smaller than the data

In this week’s podcast, “The Chaitin Interview II: Defining Randomness,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on randomness. It’s a subject on which Chaitin has thought deeply since his teenage years (!), when he published a journal paper on the subject. How do we measure randomness? Chaitin begins by reflecting on his 1969 paper: https://episodes.castos.com/mindmatters/Mind-Matters-125-Gregory-Chaitin.mp3 This portion begins at 1:12 min. A partial transcript, Show Notes, and Additional Resources follow. Gregory Chaitin: In particular, my paper looks at the size of computer programs in bits. More technically you ask, what is the size in bits of the smallest computer program you need to calculate a given digital object? That’s called the program…

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White, grey and pink mandelbrot fractal.

How Kurt Gödel Destroyed a Popular Form of Atheism

We don’t hear much about logical positivism now but it was very fashionable in the early twentieth century

In this week’s podcast, “The Chaitin interview I: Chaitin chats with Kurt Gödel,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin. Earlier, we noted his comments on the almost supernatural awareness that the great mathematicians had of the foundations of reality in the mathematics of our universe. Yesterday, we heard Chaitin’s recollection of how he (almost) met the eccentric genius Kurt Gödel (1906–1978). One way that Gödel stood out from many of his contemporaries was that he believed in God. He even wrote a mathematical proof of the existence of God. https://episodes.castos.com/mindmatters/Mind-Matters-124-Gregory-Chaitin.mp3 This portion begins at 17:16 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: One of the things…

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mandelbrot set abstract mathematics computer generated

Gregory Chaitin’s “Almost” Meeting With Kurt Gödel

This hard-to-find anecdote gives some sense of the encouraging but eccentric math genius

In this week’s podcast, “The Chaitin interview I: Chaitin chats with Kurt Gödel,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin. Yesterday, we noted his comments on the almost supernatural awareness that the great mathematicians had of the foundations of reality in the mathematics of our universe. This time out, Chaitin recounts how he (almost) met the eccentric genius Kurt Gödel (1906–1978): https://episodes.castos.com/mindmatters/Mind-Matters-124-Gregory-Chaitin.mp3 This portion begins at 12:42 min. A partial transcript, Show Notes and Additional Resources follow. Robert J. Marks: You mentioned that you read the article about Gödel’s Incompleteness Theorem in Scientific American I also know that you had a near brush with Gödel and I’ve heard the story from you. But…

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Silhouette of human with universe and physical, mathematical formulas

Gregory Chaitin on the Great Mathematicians, East and West

Himself a “game-changer” in mathematics, Chaitin muses on what made the great thinkers stand out

In this week’s podcast, “The Chaitin interview I: Chaitin chats with Kurt Gödel,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on the almost supernatural awareness that the great mathematicians had of the foundations of reality in the mathematics of our universe: https://episodes.castos.com/mindmatters/Mind-Matters-124-Gregory-Chaitin.mp3 This discussion begins at 8:26 min. A partial transcript, Show Notes and Additional Resources follow. Robert J. Marks: There are few people who can be credited without any controversy with the founding of a game changing field of mathematics. We are really fortunate today to talk to Gregory Chaitin (pictured) who has that distinction. Professor Chaitin is a co-founder of the Field of Algorithmic Information Theory that explores the properties of…

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Schoolchildren writing on chalkboard in classroom during math lesson

Antiracism In Math Promotes Racism and Bad Math

If you are scratching your head over how math might be racist, you are not alone

Recently, a conglomeration of California education associations got together to work on a series of resources for mathematics teachers. The goal? Eliminate racism in mathematics classes by promoting Equitable Math. If you are struggling to imagine how mathematics could be racist, you are not alone. I am certain there exist racist teachers, and probably teachers who exhibit racist expectations of their students. I would support any reasonable action to get rid of or reform such teachers. But that is not the primary goal of these resources. The website, equitablemath.org, instead believes that the very way that mathematics is commonly taught is not just racist, but is specifically white supremacist. While I consider myself to be somewhat of a mathematics reformer…

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Educational kids math toy wooden board stick game counting set in kids math class kindergarten. Math toy kids concept.

Yes, There Really Is a War on Math in Our Schools

Pundits differ as to the causes but here are some facts parents should know

The Oregon Department of Education (ODE) recently encouraged teachers to register for training that encourages “ethnomathematics,” an education trend that argues, “among other things, that White supremacy manifests itself in the focus on finding the right answer”: “The concept of mathematics being purely objective is unequivocally false, and teaching it is even much less so,” the document for the “Equitable Math” toolkit reads. “Upholding the idea that there are always right and wrong answers perpetuate objectivity as well as fear of open conflict.” … An associated “Dismantling Racism” workbook, linked within the toolkit, similarly identifies “objectivity” — described as “the belief that there is such a thing as being objective or ‘neutral’” — as a characteristic of White supremacy. Instead…

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Flying cranes

Random Evolution Doesn’t Produce Algorithmic Functions in Animals

A bird does not fly just because it has wings; it needs a “flight” program in its brain. Explanations of the evolution of flight do not account for that.

In a recent article “Evolution and artificial intelligence face the same basic problem,” Eric Holloway addressed the conundrum faced by artificial intelligence theorists: How can “a random process with no insight into the environment… increase information about that environment within evolving DNA sequences and/or artificial intelligence programs. By what mechanism can randomness ‘know’ anything?” Dr. Holloway’s challenge goes to the heart of the problem with the materialist worldview regarding origins, evolution, and ultimately intelligence. Software vs. hardware in your body Imagine you knew absolutely nothing about roller skates. Then you awoke this morning to find your ankles and feet permanently installed into roller skates. Instantly, everything you understood about walking and running is worthless. Getting onto your feet at all…

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colorful numbers background

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…

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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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probability likelihood

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…

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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…

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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…

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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…

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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…

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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…

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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…

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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…