Mind Matters Natural and Artificial Intelligence News and Analysis

CategoryMathematics

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Abstract futuristic stripe line printed circuit board pattern with gear wheel and math fornula on blue color background. Math science engineered drawn project plot concept

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…

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Abstract Computer generated Fractal design. 3D Illustration of a Beautiful infinite mathematical mandelbrot set fractal.

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…

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.

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…

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PI

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…

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Abstract background architecture lines. modern architecture detail

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…

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Big Bang in Space, The Birth of the Universe 3d illustration

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…

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Full body gold dragon in infinity shape pose with 3d rendering include alpha path.

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…

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Croupier behind gambling table in a casino

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…

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African American smiling woman math teacher stands at black board with pointer.

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…

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Casino BlackJack

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…

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hotel

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…

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flock of bees flying near the beehive

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…

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baseball players hitting

Why Giving the Second Best Guy a Chance Is a Smart Move

Business prof Gary Smith explains…

Gary Smith, author of The AI Delusion, has some interesting advice for those who think that a star athlete wins only on performance: It doesn’t quite work that way: A study by two business school professors, Cade Massey and Richard Thaler, found that the chances that a drafted player will turn out to be better than the next player drafted in his position (for example, the first quarterback drafted compared to the second quarterback drafted) is only 52%, barely better than a coin flip.Yet, teams pay much more for early draft picks than for later picks. Even leaving salary aside, teams that trade down (for example giving up the first pick in the draft for the 14th and 15th pick)…

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abstract illustration of Indian celebrating mathematics day Jayanti or Ramanujan Srinivasa holiday

A Brilliant Mathematician’s Last Letter Continues To Matter

Sadly, Ramanujan’s life was cut short by various health issues

One of the most remarkable mathematicians in history was Srinivasa Ramanujan (1887–1920) whose life was cut short by tuberculosis. In an interesting essay, psychiatrist Ashwin Sharma asks us to look at ways that his last letter helps us understand our universe better: A cryptic letter addressed to G.H. Hardy on January 12th, 1920, will be remembered as one of the most important letters in Scientific history. Written by Srinivasa Ramanujan, a self-taught mathematical genius who, laying on his deathbed, left hints of a new and incredible mathematical discovery. Unfortunately, the letter was to be his last, dying three months later at 32. Ramanujan’s discovery took over 80 years to solve, and with it came answers to some of the most…

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Girl solving mathematical addition

No, Civilization Has NOT Won the War on Math. Not Yet Anyway…

The war on math is now coming down to the race — not the ideas — of mathematicians

Legal scholar Jonathan Turley muses on the latest assault on math teaching in schools: We previously discussed the view of University of Rhode Island and Director of Graduate Studies of History Erik Loomis that “Science, statistics, and technology are all inherently racist.” Others have agreed with that view, including denouncing math as racist or a “tool of whiteness.” Now, as part of its “decolonization” efforts, Durham University is calling on professors in the math department to ask themselves if they’re citing work from “mostly white or male” mathematicians. According to the Telegraph and The College Fix a guide instructs faculty that “decolonising the mathematical curriculum means considering the cultural origins of the mathematical concepts, focusses, and notation we most commonly use.”  It adds: “[T]he question of whether we have allowed…

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matching keys made of circuits & led lights, encryption & crypto

New Clue in the Problem That Haunts All Cryptography?

A string that has no description shorter than itself is a good bet for cryptography. If the hacker doesn’t know it, he can’t use shortcuts to guess it.

A central problem in all computer security (branch of cryptography) is the one-way problem. Cryptography should function as a one-way street: You can go north but you can’t go south. So if a hacker doesn’t have the code to go north, he can’t go anywhere. Which is where the computer security expert would like to leave the hacker… Is there such a thing as a one-way function in mathematics? Mathematician Erica Klarreich says, probably yes, and explains what it looks like: To get a feel for how one-way functions work, imagine someone asked you to multiply two large prime numbers, say 6,547 and 7,079. Arriving at the answer of 46,346,213 might take some work, but it is eminently doable. However,…

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Math Equations of Artificial intelligence AI deep learning computer program technology - illustration rendering

Does Information Have Mass? An Experimental Physicist Weighs In

Physicist Melvin Vopson argues that information has mass; Eric Holloway replies that, if so, it must come from outside the universe. Meanwhile…

It’s generally held that information does not have mass. However, physicist Melvin Vopson, reflecting on the work of Rolf Landauer (1927–1999), offers a somewhat alarming view: Not only does information have mass but that — at the rate we humans output it now — its energy will outweigh Earth. Yesterday, Eric Holloway offered a response to that claim: Let’s accept that creation of information can indeed increase the amount of energy and mass in a system. But, according to the conservation of energy, the energy in a closed system remains constant. So, if Vopson is correct we now have a mystery because his theory is in tension with the conservation of energy. The only solution is that the system is…

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Two stingrays are swimming on the blue sea near the underwater rocks and white sand.

Did Researchers Teach Fish To “Do Math”?

Some test fish learned how to how to get food pellets but the difficulty, as so often, lies with interpretation

University of Bonn researchers think that they may have taught fish to count. They tested the fact that many life forms can note the difference in small quantities between “one more” and “one less,” at least up to five items. Not much work had been done on fish in this area so they decided to test eight freshwater stingrays and eight cichlids: All of the fish were taught to recognize blue as corresponding to “more” and yellow to “less.” The fish or stingrays entered an experimental arena where they saw a test stimulus: a card showing a set of geometric shapes (square, circle, triangle) in either yellow or blue. In a separate compartment of the tank, the fish were then…

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scoring during a basketball game - ball in hoop

Luck Matters More Than Skill When You’re at the Top

What? Shouldn’t it be the other way around? No, because… Prof. Gary Smith explains

With basketball fever at a high pitch… when LA Times sportswriter Jim Alexander talked to Pomona College business prof Gary Smith about what it takes to win, he got a different answer than some might have expected. If you are really good, it takes luck to win, Smith explained. What? Shouldn’t it be the other way around? No, because… “You can take the four best golfers in the world – any sport, but let’s do golf because it’s head-to-head,” Smith said in a phone conversation this week. “And they play a round of golf and see who gets the lowest score, and it’s pretty much random. Nobody’s going to win every single time. One guy might win more than 25…

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Science and research of the universe, spiral galaxy and physical formulas, concept of knowledge and education

Unexplained — Maybe Unexplainable — Numbers Control the Universe

For example, brilliant physicist Richard Feynman called 1/137, the fine structure constant, “a magic number that comes to us with no understanding by man”

In Carl Sagan’s Contact, the extraterrestrials embedded a message in the irrational number pi (the circumference of a circle divided by its radius). But some other numbers are critical to the structure of our universe too — and why they are critical does not make obvious sense. ➤ Perhaps the most fundamental and mysterious one is the fine structure constant of the universe: A seemingly harmless, random number with no units or dimensions has cropped up in so many places in physics and seems to control one of the most fundamental interactions in the universe. Its name is the fine-structure constant, and it’s a measure of the strength of the interaction between charged particles and the electromagnetic force. The current…