Mind Matters Natural and Artificial Intelligence News and Analysis

# CategoryMathematics

## Can We Rewire Our Brains To Be More Fluent in Math?

An artsy who flunked math — but later became an electrical engineering prof — says yes

Barbara Oakley, a self-confessed math phobe, nonetheless became a professor of electrical engineering at Oakland University in Michigan, as well as an author. In 2014, she offered some secrets: at Nautilus. Be warned: Her secrets are not “Forget homework!” or “Math is a tool of oppression!” No, this is quite a different message. It’s about neuroplasticity, the ways our brains adapt to our circumstances, to give us the tools we need. But to adapt, the brain needs practice: Japan has become seen as a much-admired and emulated exemplar of these active, “understanding-centered” teaching methods. But what’s often missing from the discussion is the rest of the story: Japan is also home of the Kumon method of teaching mathematics, which emphasizes Read More ›

## More Hard Math Does Not Necessarily Mean More Useful Solutions

It is sometimes tempting to overemphasize the math and underemphasize the relevance

Math is said to be the language of science in that most (but definitely not all) scientific models of the world involve mathematical equations. The Pythagorean theorem, the normal distribution, Einstein’s energy-mass equivalence, Newton’s second law of motion, Newton’s universal law of gravitation, Planck’s equation. How could any of these remarkable models be expressed without math? Unfortunately, it is sometimes tempting to overemphasize the math and underemphasize the relevance. The brilliance of the models listed above lies not in mathematical pyrotechnics but, if anything, in their breath-taking simplicity. Useful models help us understand the world and make reliable predictions. Math for the sake of math does neither. Examples of mindless math are legion. I will give three very different examples. Read More ›

## Researchers Are Zeroing In on Animal Number Sense

We’re beginning to find out more about how animals that don’t really “think” much can keep track of numbers, when needed

University College cognitive psychology prof Brian Butterworth, author of Can fish count? (Basic Books, 2022), talks about animal number sense in a recent article in Psyche: He offers many examples of animals counting single digit numbers but then helpfully addresses the question of how they do it. We are talking here about a variety of very different types of neurological equipment — insects vs. amphibians, for example. Neuroscientists are beginning to pinpoint specific brain functions associated with counting for specific tasks: Female túngara frogs benefit by mating with the male that can produce six croaks in one breath, over the male that can manage only five, because this is an indicator of respiratory fitness. Naturally, the male will try to Read More ›

## The War on Math Is Becoming an Entrenched Ground War

If math skills are rooted in white supremacy, as alleged, one current solution is tests that don’t require math skills

When we first started talking about the war on math, many readers may have thought we were joking. No. The war on 2 + 2 = 4 is getting some pushback but it continues. The basic idea is that the rules of math are rooted in white supremacy. Last December, the question was mooted at USA Today, “Is math racist?” The context was proposed changes to math education: After Ebri switched to emphasizing real-world problems and collaboration, her students, most of whom are Black, improved their scores on Florida’s math exam in 2020-21 – even with 1 in 3 learning from home. But other, bolder recommendations to make math more inclusive are blowing up the world of mathematics education. Schools Read More ›

## Step Away From Stepwise Regression (and Other Data Mining)

Stepwise regression, which is making a comeback, is just another form of HARKing — Hypothesizing After the Results are Known

There is a strong correlation between the number of lawyers in Nevada and the number of people who died after tripping over their own two feet. There are similarly impressive correlations between U.S. crude oil imports and the per capita consumption of chicken — and the number of letters in the winning word in the Scripps National Spelling Bee and the number if people killed by venomous spiders. If you find these amusing (as I do), there are many more at the website Spurious Correlations. These silly statistical relationships are intended to demonstrate that correlation is not causation. But no matter how often or how loudly statisticians shout that warning, many people do not hear it. When there is a Read More ›

## The Vector Algebra Wars: A Word in Defense of Clifford Algebra

A well-recognized, deep problem with using complex numbers as vectors is that they only really work with two dimensions

Vector algebra is the manipulation of directional quantities. Vector algebra is extremely important in physics because so many of the quantities involved are directional. If two cars hit each other at an angle, the resulting direction of the cars is based not only on the speed they were traveling, but also on the specific angle they were moving at. Even if you’ve never formally taken a course in vector algebra, you probably have some experience with the easiest form of vector algebra — complex numbers (i.e., numbers that include the imaginary number i). In a complex number, you no longer have a number line, but, instead, you have a number plane. The image below shows the relationship between the real Read More ›

## Don’t Worship Math: Numbers Don’t Equal Insight

The unwarranted assumption that investing in stocks is like rolling dice has led to some erroneous conclusions and extraordinarily conservative advice

My mentor, James Tobin, considered studying mathematics or law as a Harvard undergraduate but later explained that I studied economics and made it my career for two reasons. The subject was and is intellectually fascinating and challenging, particularly to someone with taste and talent for theoretical reasoning and quantitative analysis. At the same time it offered the hope, as it still does, that improved understanding could better the lot of mankind. I was an undergraduate math major (at Harvey Mudd, not Harvard) and chose economics for the much the same reasons. Mathematical theories and empirical data can be used to help us understand and improve the world. For example, during the Great Depression in the 1930s, governments everywhere had so Read More ›

## Mathematics Can Prove the Existence of God

Atheist biologist Jerry Coyne finds that difficult to believe but it’s really a matter of logic

In a recent post, atheist biologist Jerry Coyne takes issue with a commenter who asserts that God exists in the same sort of way mathematics exists. Here’s the analogy the commenter offered, as quoted by Coyne: Think of numbers for example, or mathematical equations, these are metaphysical things, that have not been created, however were discovered. The number 7 was the number 7 before anything at all came into existence. This is also true concerning the nature of God. He is not some material being that has come into existence, he is like a number that has always existed, (and by the way nobody will deny this logic with the number, however when someone mentions God a problem occurs). Jerry Read More ›

## Recent Research: Imaginary Numbers Are Part of the Real World

If we try to leave them out of quantum mechanics, our description of nature becomes faulty

Imaginary numbers, beginning with the square roots of minus numbers, are part of the world in which we live, even though we can’t quite picture them. Try it. The square root of 1 is 1 (1 × 1 = 1). But what’s the square root of -1? It can’t be -1 because if we multiply -1 × -1, we still get 1. The two minus numbers cancel each other out. That’s why the square root of -1 is written as i. Now, here’s the odd part: Imaginary numbers are not just a conundrum; they are part of a science description of the world in which we live in: Though imaginary numbers have been integral to quantum theory since its very Read More ›

## Some Infinities Are Bigger Than Others But There’s No Biggest One

Georg Cantor came up with an ingenious proof that infinities can differ in size even though both remain infinite

When a child is asked “what is bigger than infinity,” the response is often “Infinity plus one.” No. Infinity plus one is still infinity. But we can show that the number of points on the interval zero to one is a bigger infinity than the counting numbers are. The first clue is the fact that we can’t count the number of points on a line interval. Try labeling the points on a line as points 1, 2, 3, etc. No matter what scheme you come up with, there will always be some points on the line segment that are not included in your count. Georg Cantor (1845–1918) came up with an ingenious argument to show that the infinite number of Read More ›

## 4. How Almost Any Number Can Encode the Library of Congress

That’s a weird, counterintuitive — but quite real — consequence of the concept of infinity in math

We are used to dealing with simple numbers, like ½ and 2. Most numbers are not that simple. Most numbers, like 0.847859028378490… go on forever and ever without repeating or showing any pattern. Note that such numbers, called irrational numbers, have an infinite number of digits. And there are a lot of them. The number 0.847859028378490… for example differs from the number 0.847859023378490… (See if you can spot the difference.) If two numbers differ only at the billionth decimal and are otherwise the same, they are different numbers. Because an irrational number is infinitely long — and we have seen in the first three posts that weird things happen with infinity — we’d expect something weird to happen with irrational Read More ›

## 3. In Infinity, Lines and Squares Have an Equal Number of Points

We can demonstrate this fact with a simple diagram

In previous posts, we have established that two sets are of the same size if there is a one-to-one correspondence between the elements of both sets. Applying this principle to Cantor’s theory of infinity leads us to the weird but valid conclusion that the number of points on a line segment is the same as the number of points in a square. To show that this is true, here is a picture of a unit length line segment and a unit square. Let’s choose a point on the line segment. Let’s say 0.6917381276543… . It’s shown with a big blue dot on the line segment on the left. If this point corresponds to an irrational number, it goes on forever Read More ›

## 2. Infinity Illustrates That the Universe Has a Beginning

The logical consequences of a literally infinite past are absurd, as a simple illustration will show

The size of a set is how many elements it contains. The set of letters {A,B,C} and the set of girls {Shirley, Goodness, Mercy} both have a cardinality of three. In a previous post, we showed that the infinities of counting numbers and even numbers are the same. Many subsets of the counting numbers have the same infinite size as the counting numbers. For example, consider the counting numbers and the set of numbers divisible by 10. and The size of the two sets is the same if there is a one-to-one mapping from one set to another. Here, 1 maps to 10, 2 maps to 20, 3 to 30, etc. This continues forever. The two sets are the same Read More ›

## 1. Why Infinity Does Not Exist in Reality

A few examples will show the absurd results that come from assuming that infinity exists in the world around us as it does in math

Does infinity exist in reality? There are, surprisingly, scientists who think infinity is a possibility even though they are unable to point to any example of infinity in reality. The great mathematician David Hilbert claimed that “the infinite is nowhere to be found in reality.” Nevertheless, the mathematical theory of infinity developed by Georg Cantor is beautiful. Hilbert was in awe of Cantor’s beautiful theory and said “No one shall drive us from the paradise which Cantor has created for us.” An assumption of the infinite leads to weird counterintuitive results. In this and the following four articles, various ludicrous properties of the infinite are explored. We’ll see, for example, that the entire Library of Congress is encoded somewhere in almost every Read More ›

## Can Casinos Ban Customers Who Might Get TOO “Lucky”?

Sal Cordova was good enough at card counting that his photo was circulated and casino nabbed his driver’s licence…

In a recent podcast, “When the house can’t win the game, it will change the rules” (June 9, 2022), Walter Bradley Center director Robert J. Marks continued his discussion with mathematician, computer scientist, and engineer Salvador Cordova on the mathematics of gambling — who wins, who loses, and why. Last week, we looked at the struggle between the casino and the “advantage player” who knows very well how the system works and spots its weaknesses. But now, what about banning a suspiciously “lucky” would-be customer outright? Here’s what happened to Sal Cordova: https://mindmatters.ai/wp-content/uploads/sites/2/2022/06/Mind-Matters-News-Episode-190-Sal-Cordova-Episode-2-rev1.mp3 This portion begins at roughly 11:10 min. A partial transcript and notes, Show Notes, and Additional Resources follow. Sal Cordova: One of the better things is that Read More ›

## Reviving the Relational View of Mathematics

Unfortunately, some textbooks teach number rules rather than relationships, so students may not know why the rule matters

While helping a friend’s teenage son with math, I was perusing the textbook used. I was dismayed by the presentation of the topic of translating graphs. More than that, I believe the issue reflects some general problems with how mathematics is typically presented to high school students. Specifically, the text addressed how to do graph transformations for exponential functions. That is, if you have a function with the form y = a ⋅ bx (where a and b are constants), how would you create a new equation whose graph was moved up, down, left, or right? The method the book proposed, while technically correct, misses a huge opportunity to help students. The book presents a general form for transforming exponential Read More ›

## Casinos: How Nerds Gamble and Win, Using the Law of Large Numbers

The American Physical Society created Las Vegas’s worst week in history and Don Johnson cleaned out Atlantic City. How?

In last week’s podcast, “The house always wins in the long run” (June 2, 2022), Walter Bradley Center director Robert J. Marks interviewed mathematician, computer scientist, and engineer Salvador Cordova on the world of gamblers and how they try to improve their odds by physically manipulating dice (dice sliding ) and cards (false shuffling). Meanwhile, the house is relying on the Law of Large Numbers, which — being a mathematical law — wins out in the end. Sure, the Law may always win — but perhaps anyone can play it. Where we left the matter last time, in the first portion of this episode, Cordova talked about how “advantage players” try to make it work. In this second segment of Read More ›

## Math Fun: Hilbert’s Hotel Manager Copes With Infinity With Poise!

What, exactly, happens when a would-be guest shows up at a fully booked hotel — with infinite rooms?

Hilbert’s Hotel is a thought experiment that the great mathematician David Hilbert (1862–1943) developed to help us see the “counterintuitiveness of infinity.” He asks us to imagine a hotel which is “full” — except that because it is infinite, it can always create one more room. Mathematician Marianne Freiberger explains: Suppose that your hotel has infinitely many rooms, numbered 1, 2, 3, etc. All rooms are occupied, when a new guest arrives and asks to be put up. What do you do? It’s easy. Ask the guest in room 1 to move to room 2, the guest in room 2 to move into room 3, the guest in room 3 to move into room 4, and so on. If there Read More ›

## Claim: Honeybees, “Like Humans” Can Tell Odd vs. Even Numbers

Ants, fruit flies, and even plants can also calculate but it does not follow that they are conscious of what they are doing

Recently, researchers, using sugar water, taught honeybees to distinguish odd from even numbers: Our results showed the miniature brains of honeybees were able to understand the concepts of odd and even. So a large and complex human brain consisting of 86 billion neurons, and a miniature insect brain with about 960,000 neurons, could both categorize numbers by parity. Scarlett Howard, Adrian Dyer, Andrew Greentree and Jair Garcia, “Honeybees join humans as the only known animals that can tell the difference between odd and even numbers” at Phys.org (April 29, 2022) The paper is open access. That should, of course, be a hint that bees are probably using a much less complex process than humans. Bees would be useful for this Read More ›