Mind Matters Natural and Artificial Intelligence News and Analysis

CategoryMathematics

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solving algebra equation on whiteboard in classroom

How Eccentric Mathematician Kurt Gödel Opened the World

Science writer: As often happens, few people understood the significance of what had just happened. The one exception was John von Neumann.

Albert Einstein, Jogalekar tells us, considered it a privilege to walk home with Gödel every day. Why?: In an exceptionally elegant essay, science writer Ashutosh Jogalekar (no stranger to controversy) talks about the huge difference Kurt Gödel (1906–1978) made by eliminating the idea that some single, simple explanation would put an end to all questioning about the nature of the universe in favor of some simple materialism. In a review of Stephen Budiansky’s biography of Gödel, Journey to the Edge of Reason (Harvard 2021), Jogalekar explains how Gödel dashed such hopes: In September 1930, a big conference was going to be organized in Königsberg. German mathematics had been harmed because of Germany’s instigation of the Great War, and Hilbert’s decency…

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Close up of math formulas on a blackboard

Is Our “Number Sense” Biology, Culture — or Something Else?

It’s a surprisingly controversial question with a — perhaps unsettling — answer

British science writer Philip Ball, author of How to Grow a Human, offers an even-handed account of a controversy on the origin of our ability to understand numbers (numeracy). Numeracy is the beginning of mathematics, the most abstract of all human pursuits. It isn’t possible to get very far in mathematics without some ability to abstract. Ball cites as an example the difference between 152 and 153. Many life forms, competing for a pile of food items, can distinguish between 2 and 3. But distinguishing between 152 and 153 clearly requires abstraction. It’s the same principle as the chiliagon, a geometric figure like a triangle except that it has 1000 sides. A triangle can be envisioned concretely. A chiliagon can…

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Father And Son Competing In Video Games At Home

Why Did Video Gamers Uncover Fraud More Easily Than Scientists?

Video gamers are subject, a psychologist tells us, to much more rigorous constraints than scientists

In a recent article at The Atlantic, King’s College psychologist Stuart Ritchie, author of Science Fictions: How Fraud, Bias, Negligence and Hype Undermine the Search for Truth (2020), has noted a curious fact: Video gamers are much quicker to spot fraud than scientists. The video game fraud he focuses on involved a gamer’s claim that he had finished a round of Minecraft in a little over 19 minutes, a feat he attributed, Ritchie tells us, to “an incredible stretch of good luck.” “Incredible” is the right choice of word here. “Dream,” as the player was known, later admitted — in the face of skepticism — that he had “inadvertently” left some software running that improved his game — thus disqualifying…

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Space and Galaxy light speed travel. Elements of this image furnished by NASA.

No Free Lunches: Bernoulli is Right, Keynes is Wrong

What the Big Bang teaches us about nothing

Jacob Bernoulli made a now obvious observation about probability over three-and-a-half centuries ago: If nothing is known about the outcome of a random event, all outcomes can be assumed to be equally probable. Bernoulli’s Principle of Insufficient Reason (PrOIR) is commonly used. Throw a fair die. There are six outcomes, one for each face of the cube. The chance of getting five pips showing on the roll of a die is therefore one sixth. If a million lottery tickets are sold and you buy one ticket, the chances of winning are one in a million. This reasoning is intuitively obvious.  The assumption about the die is wrong if the die is loaded. But you don’t know that. You know nothing. So Bernoulli’s PrIOR…

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Statue of Saint Anselm and the towers of the Cathedral of Aosta, the Cattedrale di Aosta de Corso Pere-Laurent in Aosta. Aosta Valley. Italy. Europe

Gödel Says God Exists and Proves It

Here is a line-by-line explanation of his proof

Kurt Gödel, an intellectual giant of the 20th century, offered a mathematical proof that God exists. Those who suffer from math anxiety admire what the theorem (shown below) claims to do, but have absolutely no idea what it means. Our goal is to explain, in English, what Gödel’s existence of God proof says. Gödel’s proof shows the existence of God is a necessary truth. The idea behind the truth is not new and dates back to Saint Anselm of Canterbury (1033-1109). Great scientists and philosophers, including Descartes and Leibniz, have reconsidered and refined Anselm’s argument. Gödel appears to be the first, however, to present the argument using mathematical logic. Lexicography In any development of a mathematical theory, there are foundational axioms…

giant abacus
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…

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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…

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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.…

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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…

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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…

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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…

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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…

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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…

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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…

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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…

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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…

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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…

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,…

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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…

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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…