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 expressed as the sum of two primes. If one had a halting problem or a halting oracle, if you will, you could solve Goldbach’s Conjecture very easily by looking for a single counterexample or showing that no counter example exists forever. Do you think that something like Chaitin’s number can be calculated to a precision that would allow for the proof or disproof of something like Goldbach’s Conjecture, which is a relatively short program?
Gregory Chaitin: It’s relatively short as computer programs go, but there are a lot of programs up to that size. It grows exponentially. The calculations get quite horrendous. The algorithms that extracts, given the n bits of Omega, that tells you for each of the programs up 10 bits and size, which one holds in which one doesn’t. If you do the obvious algorithm, its runtime is worse than super exponential. It grows as the busy beaver function of n.
It looks tough. By the way, there’s another interesting example of a famous mathematical problem called the Twin Prime Conjecture: that there are infinitely many prime numbers that are two consecutive odd numbers.
Robert J. Marks: Yes.
Gregory Chaitin: It certainly looks to be the case. In fact, there are very good estimates that seem to be empirically validated formulas that tell you the distribution of twin primes, how many there are. It can’t be proven but there’s a formula which gets more and more accurate, the more twin primes are calculated. In one way, it looks like there are infinitely many but we know the distribution, at least in the sense that an empirical scientist knows anything. But the question of whether there are infinitely many twin primes is not equivalent to a halting problem.
Robert J. Marks: Yes, it can’t be solved with a Turing oracle code.
Note: “An oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, which is able to decide certain decision problems in a single operation. The problem can be of any complexity class. Even undecidable problems, like the halting problem, can be used.” – Sadika Amreen, Reazul Hoque, “Oracle Turing Machines,” Tickle College of Engineering
Gregory Chaitin: Yeah, there is no finite counterexample. You can’t have a program searching for a counterexample. Then if the program doesn’t hold, you know there is no counterexample. Omega doesn’t solve all problems.
Now there are extended versions of Omega. There’s a hierarchy of Omegas that look at more and more abstract mathematical questions. Knowing the bits of Omega, wouldn’t allow you to solve the Twin Prime Conjecture positively or negatively but there’s a thing called the Jump of the Omega number that sort of would. You have Omega, Omega prime, Omega double prime…
Robert J. Marks: Omega prime assumes that you have a halting oracle?
Gregory Chaitin (pictured): Exactly. That’s correct.
Robert J. Marks: You have this Omega prime that includes a halting oracle but if you have a halting oracle, then you have to have a meta halting oracle that looks at regular computer programs with the regular halting oracle. The Turing halting problem becomes more and more problematic. But in each case you have more and more sophisticated things that you can do with Omega, which is just astonishing.
Gregory Chaitin: But you would need an oracle to do those things.
Robert J. Marks: Yes. Those are probably not computable. I’m wondering if there’s any way to get a handle on what Omega prime is… ?
Next: The Jump of the Omega Number
Don’t miss the stories and links from the previous podcasts:
From Podcast 5:
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. Gregory Chaitin reflects on the fact that if he had to do practical work 60 years ago, there wouldn’t be practical research today based on the Omega number.
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? Chaitin explains, after you have computed some of the numbers, all the rest looks totally random and unstructured and you’ll never know them.
Is Chaitin’s unknowable number a constant? One mathematics team has succeeded in the first 64 bits of a Chaitin Omega number. Gregory Chaitin explains that it’s more complicated than that; it depends on whether you use a universal Turing machine that allows for very concise programs.
From Podcast 4:
Can mathematics help us understand consciousness? Gregory Chaitin asks, what if the universe is information, not matter? Some philosophers see the universe as created by mathematics, not matter. Gregory Chaitin prefers to see it as created by information. God is then a programmer.
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. Creativity is what we don’t know. Once it is reduced to a formula a computer can use, it is not creative any more, by definition.
The paradox of the first uninteresting number. Robert J. Marks sometimes uses the paradox of the smallest “uninteresting” number to illustrate proof by contradiction — that is, by creating paradoxes. Gregory Chaitin: You can sort of go step by step from the paradox of the smallest “uninteresting” number to a proof very similar to mine.
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. Gregory Chaitin walks readers through his proof of unknowability, which is based on the Law of Non-Contradiction.
Getting to know the unknowable number (more or less). Only an infinite mind could calculate each bit. Gregory Chaitin’s unknowable number, the “halting probability omega,” shows why, in general, we can’t prove that programs are “elegant.”
From Podcast 3:
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. Robert J. Marks and Gregory Chaitin discuss the reasons high tech companies are leaving Silicon Valley for Texas and elsewhere.
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 Chaitin’s view, a key problem is that the current system cannot afford failure — but the risk of some failures is often the price of later success.
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. Gregory Chaitin also discusses the amazing ideas early mathematicians developed without the software-based methods we are so lucky to have today.
Why Elon Musk, and others like him, can’t afford to follow rules. Mathematician Gregory Chaitin explains why Elon Musk is, perhaps unexpectedly, his hero. Very creative people like Musk often have quirks and strange ideas (Gödel and Cantor, for example) which do not prevent them from making major advances.
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. Chaitin himself succeeded with significant work (see Chaitin’s Unknowable Number) by working in time spared from IBM research rather than the academic rat race.
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? Gregory Chaitin, best known for Chaitin’s Unknowable Number, discusses the way deep math is discovered whereas trivial math is merely invented.
From the transcripts of the second podcast: 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. The musical Fermat’s Last Tango features the ghost of mathematician Pierre de Fermat pestering the math nerd who solved his unfinished Last Conjecture.
Chaitin’s discovery of a way of describing true randomness. He found that concepts f rom computer programming worked well because, if the data is not random, the program should be smaller than the data. So, Chaitin on randomness: The simplest theory is best; if no theory is simpler than the data you are trying to explain, then the data is random.
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. Gregory Chaitin reminisces on his interactions with Ray Solomonoff and Marvin Minsky, fellow founders of Algorithmic Information Theory.
Gregory Chaitin’s “almost” meeting with Kurt Gödel. This hard-to-find anecdote gives some sense of the encouraging but eccentric math genius. Chaitin recalls, based on this and other episodes, “There was a surreal quality to Gödel and to communicating with Gödel.”
Gregory Chaitin on the great mathematicians, East and West: Himself a “game-changer” in mathematics, Chaitin muses on what made the great thinkers stand out. Chaitin discusses the almost supernatural awareness some mathematicians have had of the foundations of our shared reality in the mathematics of the universe.
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. Gödel’s incompleteness theorems showed that we cannot devise a complete set of axioms that accounts for all of reality — bad news for positivist atheism.
You may also wish to read: Things exist that are unknowable: A tutorial on Chaitin’s number (Robert J. Marks)
Five surprising facts about famous scientists we bet you never knew: How about juggling, riding a unicycle, and playing bongo? Or catching criminals or cracking safes? Or believing devoutly in God… (Robert J. Marks)
- 00:27 | Introducing Gregory Chaitin and Chaitin’s number
- 01:32 | Chaitin’s number or Chaitin’s constant?
- 07:16 | Must the halting problem be solved for smaller programs in order to get Chaitin’s number?
- 09:50 | The usefulness of philosophy and the impractical
- 17:17 | Could Chaitin’s number be calculated to a precision which would allow for a proof or disproof of something like Goldbach’s Conjecture?
- 19:20 | The Jump of the Omega Number
- Gregory Chaitin’s Website
- Unravelling Complexity: The Life and Work of Gregory Chaitin, edited by Shyam Wuppuluri and Francisco Antonio Doria
- Elements of Information Theory by Thomas Cover and Joy Thomas
- “On the length of programs for computing finite binary sequences,” by Gregory Chaitin, published when he as a teenager. (Journal of the ACM (JACM) 13, no. 4 (1966): 547-569).
- Chris Calude, professor at the University of Auckland
- Gottfried Wilhelm Leibniz, German Enlightenment philosopher, mathematician, and political adviser
- Stephen Wolfram, computer scientist and physicist
- Goldbach Conjecture
- Collatz Conjecture
- Legendre’s Conjecture
- Elon Musk, engineer and entrepreneur
- G.H. Hardy, English mathematician
- Ray Solomonoff, one of the founders of algorithmic information theory
- Hector Zenil, computational natural scientist
- Marvin Minsky, cognitive and computer scientist
- George Gilder, economist and co-founder of the Discovery Institute
- Twin Prime Conjecture
- Georg Cantor, mathematician
- BBC 4’s Dangerous Knowledge