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Taghalting problem

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…

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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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The Chaitin Interview IV: Knowability and Unknowability

What does it mean for something to be unknowable? Is creativity non-computable? Do all things have a level of consciousness? Jump into today’s podcast, where Robert J. Marks continues his discussion with Gregory Chaitin about mathematical theory and philosophy. Show Notes 00:23 | Introducing Gregory Chaitin 00:40 | What is unknowability? 06:07 | Does non-computable mean unknowable? 09:43 | A…

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For Computers, Smart Is Not the Same Thing as Fast

In response to a reader’s good question …

In a recent article, I argued that computers are not, and never can become smarter. An insightful reader wrote to ask, “What if smartness is defined by speed?” This is a good point. The debate revolves around the definition of “smart.” and if we define “smart” as “fast”, then since computers are certainly getting faster they will necessarily become smarter. Such a definition has intuitive appeal. Think of the world’s best chess player versus a beginner. One of the big distinctions is the chess expert will choose a good move more quickly than a beginner, and in general will play faster than a beginner. As such, play speed demonstrates a certain level of intelligence on the part of the player.…

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Human Ingenuity vs. the Computer’s Halting Problem

In a dialogue with a friendly skeptic, I suggested an explanation he found astounding but it’s the only possible one

When studying computer science a student invariably learns about the infamous halting problem. The problem states there is no general algorithm that can determine for every deterministic computer program whether that program will halt or not. This struck me as absurd when I first learned of the problem. Surely a whizkid like myself could design a simple algorithm to track the program’s memory and catch when it started repeating itself and determined it would not halt. Once convinced the problem was indeed provably unsolvable, I then thought the problem must show that humans are not computers. This is because it seems intuitive that for every program, if I watch it enough and think about it carefully enough, I should be…

Schrodinger's cat

Does Schrödinger’s Cat Think Quantum Computing Is a Sure Thing?

It might lead to more security, though not to thinking computers

Some hope that a move to quantum computing—qubits instead of bits, analog instead of digital—will work wonders, including the invention of the true thinking computer. In last week’s podcast, futurist George Gilder and computer engineer Robert J. Marks looked at, among other things, what’s really happening with quantum computing: https://episodes.castos.com/mindmatters/Mind-Matters-108-George-Gilder.mp3 (The quantum computing discussion begins at 15:04.) Robert J. Marks: What’s your take on quantum computing? It seems to me that there’s been glacial progress in the technology. George Gilder (pictured): I think quantum computing is rather like AI, in that it moves the actual problem outside the computational process and gives the illusion that it solved the problem, but it’s really just pushed the problem out. Quantum computing is…

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Computer Science Explains Why Communism Can’t Work

Successful communism is not only morally and practically flawed, it is mathematically impossible

Communism has been the target of many criticisms. The strongest deal with the mismatch between central planning and individuals’ desires for their lives and with the horrific human rights record of communist nations. Some scholars place the toll in human life due to communism at above 100 million in the 20th century. Those are criticisms of the practicality and ethics of communism. But is it also intrinsically flawed at a fundamental mathematical level? It turns out that the answer is yes. The basic idea behind central planning is this: If the central government makes most decisions that, in a freer society, individuals or small communities would make for themselves, more efficiency will follow—and, as a result, more prosperity. It doesn’t…


Why Your Computer Will Never Talk to You

As a jokester recently demonstrated, even “shirts without stripes” is a fundamental, unsolvable problem for computers

At first, “shirts without stripes” might not seem like much of an issue but it turns out that many important and interesting problems for computers fundamentally reduce to this “halting problem.” And understanding human language is one of these problems.

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Artificial Intelligence Must Be Possible! Really…?

Many arguments for strong artificial intelligence depend on an ideological commitment to explicit, unproven theories about the universe

Not only is it valid to ask whether artificial intelligence is impossible but the argument can be pursued on a scientific basis with quantifiable, empirical evidence.

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Why I Doubt That AI Can Match the Human Mind

Computers are exclusively theorem generators, while humans appear to be axiom generators

My primary reason for doubting that AI can match human intelligence is that the difference between mind and machine is a difference of kind, not of quantity. Understanding the distinction will help us exploit the abilities of each to their maximum potential.

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If Computers Thought Like Fruit Flies, They Could Do More

But even with more sophisticated buzz, there remain "non-computable" things that a computer cannot be programmed to think

Recently, researchers discovered that fruit flies use a filter similar to a computer algorithm to assess the odors that help them find fruit, only the flies’ tools are more sophisticated: When a fly smells an odor, the fly needs to quickly figure out if it has smelled the odor before, to determine if the odor is new and something it should pay attention to,” says Saket Navlakha, an assistant professor in Salk’s Integrative Biology Laboratory. “In computer science, this is an important task called novelty detection. Computers use a Bloom filter for that, Navlakha, an integrative biologist, explains: When a search engine such as Google crawls the Web, it needs to know whether a website it comes across has previously…


Human intelligence as a halting oracle

Jonathan Bartlett proposes to model the human mind as a halting oracle
A common objection to Bartlett’s idea is that humans cannot be halting oracles because we embed any unsolvable math problem as the halting condition for a loop and a human cannot tell us whether the loop will halt or not. This objection misses the fact that there is a range of oracles between plain Turing machines and a complete halting oracle. Read More ›