Mind Matters Natural and Artificial Intelligence News and Analysis

TagRobert J. Marks

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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…

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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…

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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:…

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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…

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Abstract virtual binary code illustration on blurry modern office building background. Big data and coding concept. Multiexposure

Why the Unknowable Number Exists But Is Uncomputable

Sensing that a computer program is “elegant” requires discernment. Proving mathematically that it is elegant is, Chaitin shows, impossible

In this week’s podcast, “The Chaitin Interview IV: Knowability and Unknowability,” Walter Bradley Center director Robert J. Marks interviewed mathematician Gregory Chaitin on his “unknowable number.” That’s the topic of this series, based on the fourth podcast. Last week, we tried getting to know the unknowable number. Today, let’s look at the question of how we know that the number is unknowable — instead of merely non-computable. Lots of things are non-computable but we do not expect that to be true of numbers. Let’s see what’s happening here, as Chaitin offers a walk through his proof that it really is unknowable: https://episodes.castos.com/mindmatters/Mind-Matters-127-Gregory-Chaitin.mp3 This portion begins at 09:43 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks:…

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matrix made up of math formulas and mathematical equations - illustration rendering

Getting To Know the Unknowable Number (More or Less)

Only an infinite mind could calculate each bit

In this week’s podcast, “The Chaitin Interview IV: Knowability and Unknowability,” Walter Bradley Center director Robert J. Marks interviewed mathematician Gregory Chaitin on his discovery of the “unknowable number.” How can a number that is unknowable exist? Some numbers go on indefinitely (.999999999… ) but we can describe them accurately even if they don’t seem to come to an end anywhere. Some numbers, like pi (π), are irrational — pi goes on and on but its digits form no pattern. However, what does it mean to say that a number exists if it is unknowable? How do we even know it exists? That’s the topic of this series, based on the fourth podcast between Dr. Marks and Gregory Chaitin. Note:…

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hand of scientist holding flask with lab glassware in chemical laboratory background, science laboratory research and development concept

A Question Every Scientist Dreads: Has Science Passed the Peak?

Gregory Chaitin worries about the growth of bureaucracy in science: You have to learn from your failures. If you don’t fail, it means you’re not innovating enough

In this week’s podcast, “The Chaitin Interview III: The Changing Landscape for Mathematics,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on many things, including whether the great discoveries in science are behind us — not due to lack of creativity or ability on the part of scientists — but to the growing power of corporate and government bureaucracies to stifle research. But then a question arises: Could science, succumbing to the swamp of bureaucracy, be losing that inventive edge? https://episodes.castos.com/mindmatters/Mind-Matters-126-Gregory-Chaitin.mp3 This portion begins at 24:56 min. A partial transcript, Show Notes, and Additional Resources follow. Gregory Chaitin: What did an airplane engineers say once in a speech I heard? He said, “In the…

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Castaway in bureaucracy

Gregory Chaitin on How Bureaucracy Chokes Science Today

He complains, They’re managing to make it impossible for anybody to do any real research. You have to say in advance what you’re going to accomplish. You have to have milestones, reports

In last week’s podcast, “The Chaitin Interview III: The Changing Landscape for Mathematics,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on how Stephen Wolfram’s software has taken much of the drudgery out of math. At the same time, in Chaitin’s view, a threat looms: A new, more bureaucratic, mindset threatens to take the creativity out of science, technology, and math: https://episodes.castos.com/mindmatters/Mind-Matters-126-Gregory-Chaitin.mp3 This portion begins at 19:45 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: I was sitting down tallying, I think, the intellectual giants that have introduced new mathematical ideas, brand new. I was thinking of people like Claude Shannon, Lotfi Zadeh, yourself… I don’t know if we…

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Online education concept

How Stephen Wolfram Revolutionized Math Computing

Wolfram has not made computers creative but he certainly took a lot of the drudgery out of the profession

In last week’s podcast, “The Chaitin Interview III: The Changing Landscape for Mathematics,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on many things mathematical, including why math or engineering geniuses (Elon Musk came to mind, of course) can’t just follow the rules. This week, we look at Stephen Wolfram’s new program that checks your hard math. What can — and can’t — it do for mathematicians? https://episodes.castos.com/mindmatters/Mind-Matters-126-Gregory-Chaitin.mp3 This portion begins at 13:22 min. A partial transcript, Show Notes, and Additional Resources follow. Gregory Chaitin: Now, there is what I regard as a piece of AI, so it might be interesting to talk about it. My friend Stephen Wolfram (pictured), the system he’s created,…

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SpaceX Concept Spacecraft in orbit of the Earth. SpaceX Elon Musk Mars programm 3d render

Why Elon Musk and Other Geniuses Can’t Afford To Follow Rules

Mathematician Gregory Chaitin explains why Elon Musk is, perhaps unexpectedly, his hero

In last week’s podcast, “The Chaitin Interview III: The Changing Landscape for Mathematics,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on many things mathematical, including why great books on math, advancing new theorems, aren’t written much any more. This week, we look at why geniuses like Musk (whose proposed Mars Orbiter is our featured image above) simply can’t just follow the rules, for better or worse: https://episodes.castos.com/mindmatters/Mind-Matters-126-Gregory-Chaitin.mp3 This portion begins at 7:57 min. A partial transcript, Show Notes, and Additional Resources follow. Gregory Chaitin: Look at Elon Musk (pictured). He’s my great hero. He’s a wonderful engineer and he’s a wonderful entrepreneur and he doesn’t follow the rules. Robert J. Marks: He doesn’t,…

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black mathematics board with formulas

Why Don’t We See Many Great Books on Math Any More?

Decades ago, Gregory Chaitin reminds us, mathematicians were not forced by the rules of the academic establishment to keep producing papers, so they could write key books.

In our most recent podcast, “The Chaitin Interview III: The Changing Landscape for Mathematics,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on many things mathematical, including whether math is invented or discovered. This time out, Chaitin talks about why he thinks great books on math, advancing new theorems, aren’t written much any more: https://episodes.castos.com/mindmatters/Mind-Matters-126-Gregory-Chaitin.mp3 This portion begins at 02:49 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: You don’t hear the word “scholarship” very much anymore in academia. Gregory Chaitin: And people don’t write books. In the past, some wonderful mathematicians like G. H. Hardy (1877–1947, pictured in 1927) would write wonderful books like A Mathematician’s Apology (1940)…

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Red and Blue Spiral Fractal Background Image, Illustration - Vortex repeating spiral pattern, Symmetrical repeating geometric patterns. Abstract background

Mathematics: Did We Invent It Or Did We Merely Discover It?

What does it say about our universe if the deeper mathematics has always been there for us to find, if we can?

In this week’s podcast, “The Chaitin Interview III: The Changing Landscape for Mathematics,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin (pictured) on how math presents us with a challenging philosophical question: Does math image deep truth about our universe? Or do we just make up these math rules in our own minds to help us understand nature? https://episodes.castos.com/mindmatters/Mind-Matters-126-Gregory-Chaitin.mp3 This portion begins at 00:39 min. A partial transcript, Show Notes, and Additional Resources follow. Gregory Chaitin: Deep philosophical questions have many answers, sometimes contradictory answers even, that different people believe in. Some mathematics, I think, is definitely invented, not discovered. That tends to be trivial mathematics — papers that fill in much-needed gaps because…

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Spiral Background.

Hard Math Can Be Entertaining — With the Right Musical Score!

Gregory Chaitin discusses with Robert J. Marks the fun side of solving hard math problems, some of which come with million-dollar prizes

In last week’s podcast,, “The Chaitin Interview II: Defining Randomness,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on his method of describing true randomness:. If no theory is simpler than the data you are trying to explain, then the data is random. They also discussed the work of true randomness but also on how Ray Solomonoff (1926–2009), another algorithmic information theory founder, who pursued the “shortest effective string of information that describes an object.” But now, for a lighter touch, we learn that a musical comedy was made of Fermat’s Last Theorem. https://episodes.castos.com/mindmatters/Mind-Matters-125-Gregory-Chaitin.mp3 This portion begins at 19:24 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: If you…

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Real Php code developing screen. Programing workflow abstract algorithm concept. Lines of Php code visible under magnifying lens.

How did Ray Solomonoff Kickstart Algorithmic Information Theory?

He started off the long pursuit of the shortest effective string of information that describes an object

In last week’s podcast,, “The Chaitin Interview II: Defining Randomness,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on how best to describe true randomness but also on what he recalls of Ray Solomonoff (1926–2009), described in his obit as the “Founding Father of algorithmic information theory.” https://episodes.castos.com/mindmatters/Mind-Matters-125-Gregory-Chaitin.mp3 This portion begins at 10:30 min. A partial transcript, Show Notes, and Additional Resources follow. Gregory Chaitin (pictured): Ray Solomonoff was interested in prediction but I was more interested in looking at a given string of bits and asking, does it have structure or not, and the incompleteness results regarding this question. For example, most strings of bits have no structure, according to this definition. They…

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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…