Mind Matters Natural and Artificial Intelligence News and Analysis

CategoryMathematics

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A cute girl playing with wooden abacus. Educational concept for kids. Learing to count. Math for children. Math for children.

The Mystery of Numeracy: How DID We Learn To Count?

Some animals can do rough figuring but only humans count

At a time when many people see numbers, arithmetic, and mathematics as mere oppression, others are intrigued by the basic question, how did the human race learn to do math? There are some clues, as a new project called Quanta is trying to establish. First, while no animals use abstract numbers in the wild, some animals can do rough figuring: Although researchers once thought that humans were the only species with a sense of quantity, studies since the mid-twentieth century have revealed that many animals share the ability. For instance, fish, bees and newborn chicks can instantly recognize quantities up to four, a skill known as subitizing. Some animals are also capable of ‘large-quantity discrimination’: they can appreciate the difference Read More ›

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Hausschwein

Did an AI Disprove 5 Math Conjectures With No Human Help?

Yes and no. Read the fine print…

Mathematicians have thought that five long-standing conjectures in graph theory might be true but they have not been able to prove them: Wagner programmed a neural network to create random examples and use these measures to assess their suitability as a counterexample. The AI discarded the worst scoring ones and then replaced them with more random examples before starting again. In dozens of cases the AI was unable to find an example that disproved the theory, but in five cases it landed on a solution which showed that the conjecture must be false. Matthew Sparkes, “An AI has disproved five mathematical conjectures with no human help” at New Scientist (May 20, 2021 A subscription is required to read the whole Read More ›

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Education and maths concept

Will AI Change — or Eliminate — the Mind-Body Problem?

Can an artificial intelligence program that calculates really understand mathematics? Or is that a “hard ceiling” for AI?

In last week’s podcast, Walter Bradley Center director Robert J. Marks interviewed Concordia University philosopher Angus Menuge on the difficult mind–body problem: Dr. Menuge sees mind–body interaction as a transmission of information between two realms; our minds and bodies are one integrated system with a translation function… like developing and then writing down an idea. But what about artificial intelligence? We are told that artificial general intelligence (AGI) is now pushing towards a machine that can totally duplicate the functions of the human mind. But what if the mind is not simply a mechanical function of the brain? What if it is non-algorithmic and non-computable? This portion begins at 29:04 min. A partial transcript, Show Notes, and Additional Resources follow. Read More ›

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Circle of people. Teamwork. Business meeting. Negotiations, reaching consensus in disagreements. Joint problem solving. Conflict resolution through dialogue. Compromise. Cooperation and collaboration

Consensus Gives Us Information Only If We Are Free to Doubt

There are so many credentialed people on the internet with sufficiently differing views that it sometimes seems as if we could find an expert somewhere to support almost any harebrained idea. So how does a non-expert figure out the truth? Most of us lack the time, training, and inclination to investigate most subjects sufficiently so we are often urged to adopt the consensus opinion. While an individual expert may have wild and crazy ideas, the consensus will most likely be an average informed view. But it’s not that simple. Most of the time it is impossible for the public to determine the consensus opinion. What is usually labeled as consensus opinion is what media believe it to be. And the Read More ›

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Big brother

How Orwell’s 1984 Can Be Seen As an Argument for God’s Existence

Atheism is not only fundamental to the power of the Party in 1984 but is also its central weakness

University of Nebraska political science prof Carson Holloway (pictured) asks, “Does discrediting the existence of God promote enlightened thinking or a lack of objective reality?” Unpacking the social structure in George Orwell’s classic totalitarian dystopia, 1984 (1949), he observes that not only does the Party have the power of life and death but the atheistic Party faithful fear death as utter annihilation: Atheism is the moral basis of the Party’s unlimited hold on its own members because it makes them terrified of death as absolute nonexistence. Like any government, the Party in 1984 has the power to kill disobedient subjects. Party members, however, view death not just as the end of bodily life, but as a complete erasure of their Read More ›

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Purple transparent molecule model over purple

Could the World Above the Level of the Particle Be Superposed?

It’s been done with molecules. After that, there are barriers

In last week’s podcast,,” our guest host, neurosurgeon Michael Egnor, interviewed idealist philosopher of science and physicistBruce Gordon on how the quantum physics that underlies our universe makes much more sense if we have a non-materialist view of reality. Even then, it challenges our conventional view of how nature “must” work: We were introduced to the quantum eraser experiment, which showed that what happens at the level of individual particles depends on whether you choose to measure it or not and to non-locality, the Cheshire Cat’s science of being in no one particular place at any time. Particles can do that even if we can’t. But wait: Is it possible that things larger than particles can in fact do that Read More ›

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Mad wide smile with many teeth on black background. Deco element, card-, flyer- base, clip art

IS the Moon There If No One Looks? Or Is There No “There” There?

Elementary particles do not need to be in a particular place until they are observed and then that's where they are

In last week’s podcast,,” our guest host, neurosurgeon Michael Egnor, interviewed idealist philosopher of science and physicist Bruce Gordon on how the quantum physics that underlies our universe makes much more sense if we have a non-materialist view of reality. Even then, it challenges our conventional view of how nature “must” work: We were introduced to the quantum eraser experiment, which showed that what happens at the level of individual particles depends on whether you choose to measure or not. This segment looks at non-locality, the science of being in no one particular place. Elementary particles can do that too: https://episodes.castos.com/mindmatters/Mind-Matters-130-Bruce-Gordon.mp3 This portion begins at 12:36 min. A partial transcript, Show Notes, and Additional Resources follow. Bruce Gordon: So another Read More ›

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Quantum Wave

In Quantum Physics, “Reality” Really Is What We Choose To Observe

Physicist Bruce Gordon argues that idealist philosophy is the best way to make sense of the puzzling world of quantum physics

In last week’s podcast,,” our guest host, neurosurgeon Michael Egnor, interviewed idealist philosopher of science and physicist Bruce Gordon on how the quantum physics that underlies our universe makes much more sense if we have a non-materialist view of reality. Even then, it challenges our conventional view of how nature “must” work: https://episodes.castos.com/mindmatters/Mind-Matters-130-Bruce-Gordon.mp3 A partial transcript, Show Notes, and Additional Resources follow. Michael Egnor: When I was in college, I was a biochemistry major and I took some courses in quantum mechanics. It was noted in the course that when you look at the most fundamental properties of subatomic particles, matter seems to disappear. That the reality of the subatomic particles is that they’re mathematical concepts. It utterly fascinated me Read More ›

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3D rendering of abstract blocks of mathematical formulas located in the virtual space

What Is Math About? Is It Discovered or Invented?

Philosopher Edward Feser suggests that the velociraptor, an extinct birdlike dinosaur, might illustrate the problem

Pasadena City College philosopher Edward Feser (pictured) offers some thoughts that may be relevant to the current war on math. Pointing to a recently published article by mathematician James Franklin, he writes, What is mathematics about? The Platonist says that it is about a realm of abstract objects distinct from both the world of concrete material things and the human mind. The nominalist says that it is not really about anything, since mathematical entities are in no way real. The Aristotelian approach rejects nominalism and agrees with Platonism that mathematical entities are real. But it disagrees with the Platonist about the location of these entities. They are, for the Aristotelian, properties of concrete particular things themselves, rather than denizens of Read More ›

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Surprised nerd student

Fermat’s Last Tango: Lively Musical For Nerds

The ghost of Fermat and other giants from the Aftermath Club help (frustrate?) a mathematician’s effort to prove Fermat’s famous Last Theorem

If you are a nerd, the musical Fermat’s Last Tango (2001) is hilarious. Mathematician Pierre de Fermat proposed his last theorem around 1637. He wrote a note in the margin of a copy of Arithmetica, a book written by a 3rd-century Alexandrian mathematician, Diophantus. Fermat’s short scribble claimed that he could prove that a specific Diophantine equation had no solution. But whatever Fermat was thinking died with him in 1665. A proof of Fermat’s last theorem eluded mathematicians over 300 years until Princeton’s Andrew Wiles proved it in 1995. Fermat’s Last Tango is a fantasy account of Wiles’s life while he was working on the proof. The play is a musical sprinkled with nerdy inside jokes. For example, part of Read More ›

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random numbers

The “Jump” of Chaitin’s Omega Number

Gregory Chaitin explains, “For any infinity, there’s a bigger infinity, which is the infinity of all subsets of the previous step”

In last week’s podcast, “The Chaitin Interview V: Chaitin’s Number,” Walter Bradley Center director Robert J. Marks asked mathematician Gregory Chaitin (best known for Chaitin’s unknowable number) if the unknowable number could prove (or disprove) Goldbach’s Conjecture that every even number can be expressed as the sum of two primes. This task is harder than it first appears because even numbers go on indefinitely. A proof that Christian Goldbach (1690–1764) was right or wrong must show that even numbers must be like that, no matter how big they are or how many of them there are. This time out, Dr. Marks and Dr. Chaitin discuss what we can know about Omega numbers — and where famous mathematicians are buried. This Read More ›

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List of Prime Numbers below 100, Vintage type writer from 1920s

Could Chaitin’s Number Prove Goldbach’s Conjecture At Last?

Chaitin notes that the problem grows exponentially and the calculations get quite horrendous

In last week’s podcast, “The Chaitin Interview V: Chaitin’s Number,” Walter Bradley Center director Robert J. Marks continued his conversation with mathematician Gregory Chaitin, best known for Chaitin’s unknowable number. One thing they discussed was the usefulness of philosophy, with Chaitin saying that if he had had to do practical work 60 years ago, there wouldn’t be practical research today based on the Omega number. But then they turned to the question of whether the unknowable number could prove Goldbach’s famous Conjecture: This portion begins at 17:17 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks (pictured): The poster problem for the Turing halting problem, is Goldbach’s Conjecture, which says that every even number can be Read More ›

Logical Diagrams
Making business plan. Businessperson drawing diagrams. Many graphs and hand drawn diagrams.

Why Impractical Things Like Philosophy Are Actually Quite Useful

Chaitin argues that the human spirit is capable of doing both practical things and impractical things which may have practical consequences later

In last week’s podcast,, “The Chaitin Interview V: Chaitin’s Number,” Walter Bradley Center director Robert J. Marks continued his conversation with mathematician Gregory Chaitin, best known for Chaitin’s unknowable number. Last time, they looked at how Chaitin’s unknowable number relates to computer pioneer Alan Turing’s vexing halting problem in computer science. This time, they look at the way pure mathematics has a way of being highly practical: It creates a basis for new understanding, leading to technical breakthroughs: This portion begins at 09:50 min. A partial transcript, Show Notes, and Additional Resources follow. Gregory Chaitin: There are always going to be a few of us who like to do practical things. That’s part of my personality too, but there’s also, Read More ›

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Businesswoman protect wooden block fall to planning and strategy in risk to business Alternative and prevent. Investment Insurance ,Business risk control concept,

Chaitin’s Number Talks To Turing’s Halting Problem

Why is Chaitin’s number considered unknowable even though the first few bits have been computed?

In last week’s podcast,, “The Chaitin Interview V: Chaitin’s Number,” Walter Bradley Center director Robert J. Marks continued his conversation with mathematician Gregory Chaitin( best known for Chaitin’s unknowable number) on a variety of things mathematical. Last time, they looked at whether the unknowable number is a constant and how one enterprising team has succeeded in calculating at least the first 64 bits. This time, they look at the vexing halting problem in computer science, first identified by computer pioneer Alan Turing in 1936: https://episodes.castos.com/mindmatters/Mind-Matters-128-Gregory-Chaitin.mp3 This portion begins at 07:16 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: Well, here’s a question that I have. I know that the Omega or Chaitin’s number is based Read More ›

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Omega, the letter of a Greek alphabet. Greek numerals, mathematical eight hundred number concept. Abstract, digital, wireframe, low poly mesh, Raster blue neon 3d illustration. Triangle, line dot

Is Chaitin’s Unknowable Number a Constant?

One mathematics team has succeeded in the first 64 bits of a Chaitin Omega number

In this week’s podcast, “The Chaitin Interview V: Chaitin’s Number,” Walter Bradley Center director Robert J. Marks continued his conversation with mathematician Gregory Chaitin, best known for Chaitin’s unknowable number. In this segment, Dr. Marks and Dr. Chaitin discuss whether the unknowable number is really a number… or is it a constant? In earlier podcasts linked below, they have discussed a variety of topics ranging from gifted mathematicians of the past through how to understand creativity in a mathematical way—and more. https://episodes.castos.com/mindmatters/Mind-Matters-128-Gregory-Chaitin.mp3 This portion begins at 01:32 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks (pictured): I want to clear up something first of all. Stanford’s Thomas Cover and Joy Thomas wrote a book that Read More ›

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Evolving Abstract Visualization

Can Mathematics Help Us Understand Consciousness?

Gregory Chaitin asks, what if the universe is information, not matter?

In last week’s podcast, “The Chaitin Interview IV: Knowability and Unknowability,” Walter Bradley Center director Robert J. Marks interviewed mathematician Gregory Chaitin, best known for Chaitin’s Unknowable Number, on, among other things, consciousness. What can mathematics contribute to the discussion. Also, what does Chaitin think about panpsychism (everything is conscious”)? The discussion began with reference to David Chalmers’s 1996 book, The Conscious Mind: In Search of a Fundamental Theory, in which Chalmers coined the term “Hard Problem of Consciousness.” The term acknowledged what everyone knew, that human consciousness is a very difficult problem to understand, especially from a materialist perspective.Are there other approaches? Chaitin offers a look at the challenge panpsychism presents to materialism: https://episodes.castos.com/mindmatters/Mind-Matters-127-Gregory-Chaitin.mp3 This portion begins at 28:25 Read More ›

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child hands showing a colorful 123 numbers agains wooden table. Concept of Child education, learning mathematics and counting

The Fundamental Problem With Common Core Math

Intuition relies on skill, not the other way around

In 2010, a bold effort to reform math curriculum was adopted by the majority of the United States. Known as “Common Core Math,” the goal of this endeavor was to establish a common foundation of mathematics education across the country, and to help bolster not only students’ mathematical abilities, but also their mathematical intuition. The goal was to help students think about math more deeply, believing that this will help them work with mathematics better in later years. Before discussing problems with this approach, I want to say that I appreciate the idea of helping students think more deeply about mathematics. After years and years and years of mathematics education, many students wind up thinking about mathematics as merely a set Read More ›

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Teamwork and brainstorming concept with businessmen that share an idea with a lamp. Concept of startup

Why Human Creativity Is Not Computable

There is a paradox involved with computers and human creativity, something like Gödel’s Incompleteness Theorems or the Smallest Uninteresting Number

In last week’s podcast, “The Chaitin Interview IV: Knowability and Unknowability,” Walter Bradley Center director Robert J. Marks interviewed mathematician Gregory Chaitin, best known for Chaitin’s Unknowable Number, on a number of things, including whether computers can show creativity. Chaitin has thought a lot about that: https://episodes.castos.com/mindmatters/Mind-Matters-127-Gregory-Chaitin.mp3 This portion begins at 21:34 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: We’re talking, just in general, about the unknowable. Roger Penrose recently won a Nobel Prize for his work with Stephen Hawking on black hole theory. He also wrote a book called The Emperor’s New Mind: Concerning Computers, Minds and The Laws of Physics (1989) and he followed it up with The Shadows of the Mind: Read More ›

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group of school kids raising hands in classroom

Practicing the Basics: Teaching Math Facts in the Classroom

How to help students make deeper connections within mathematics with creative games.

Many people learn to hate math early on. One of the places where people learn to hate math first is in high-stakes speed testing for math facts. This has caused quite a bit of angst in mathematics education for people on both sides of this issue. On the one hand, some have advocated for getting rid of math facts memorization altogether. On the other hand, others have doubled-down, saying that we need speed tests in order to make sure that the cognitive load of arithmetic is limited for later mathematics work. While I fall more into the latter camp than the former, I do think that a more balanced approach to mathematics education may help students in the long run.  Read More ›

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abstract metallic number background

The Paradox of the Smallest Uninteresting Number

Robert J. Marks sometimes uses the paradox of the smallest “uninteresting” number to illustrate proof by contradiction — that is, by creating paradoxes

In this week’s podcast, “The Chaitin Interview IV: Knowability and Unknowability,” Walter Bradley Center director Robert J. Marks interviewed mathematician Gregory Chaitin on how he proved that the number that determines whether computer programs are elegant (in the sense of maximally efficient) is “unknowable.” As Dr. Chaitin explained in the segment published yesterday, any solution would be contradictory. Thus, his proof is a proof by contradiction. By way of illustrating the concept of proof by contradiction, Dr. Marks then offered his proof by contradiction that “all positive integers — numbers like 6 or 129, or 10 100 — are interesting.” This portion begins at 19:45 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: If [some Read More ›