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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: As anticipated, the unknowable number itself declined to be interviewed, vanished from the studio, and left no forwarding address.

This portion begins at .40 min. A partial transcript, Show Notes, and Additional Resources follow.

Robert J. Marks: You wrote a book for Springer in 1999 called The Unknowable. And also there was a tribute book to you, Unravelling Complexity: The Life and Work of Gregory Chaitin, and you were solicited to be an author of one of the chapters, “Unknowability in Mathematics, Biology and Physics.” What is unknowability from your perspective?

Gregory Chaitin: I have a number called the “halting probability omega.” You can define it mathematically very simply… In general, you can’t prove that programs are “elegant.”

Note: “An elegant program is the smallest program to achieve an objective. You can also think of it in terms of images. If you have a big image, what is the most that you can compress that image. That would be the elegant program for the image.” – Robert J. Marks

Gregory Chaitin (pictured, at home): Now, the question is: Does that number exist? It exists in the mind of God. To an infinite mind, it is knowable. You could calculate each bit. But we’re not infinite minds. And you can argue that it doesn’t even exist, that it’s a fantasy object.

Mathematically, it doesn’t use very sophisticated math. There’s much wilder mathematics. But the numerical value of this number is maximally unstructured and maximally unknowable. So if you think about the kind of things you do normally in mathematics, this is a forever unknowable thing… the value of each individual bit in the numerical value of this number.

Note: Mathematicians recognize a variety of different types of numbers:

— – Whole Numbers (like 0, 1, 2, 3, 4, etc) [They can be plus or minus numbers.]

– Rational Numbers (like 3/4, 0.125, 0.333…, 1.1, etc ) [If they display a pattern, it goes on indefinitely.]

– Irrational Numbers (like π, √2, etc ) [They go on indefinitely without ever forming a pattern.] – Math Is Fun

Mathematicians recognize other types of numbers as well, for example imaginary and infinite numbers. But the ones listed above are called “real numbers.”

Gregory Chaitin: The real number, to an engineer, is a fantasy. The real number has infinite precision. You don’t work with infinite precision. We’re happy to get a few decimal digits.

burst set of random numbers glowing on a black background

Different problems need more precision. But infinite precision is nowhere to be found except in the imagination of mathematicians, who talk about a real number, which is determining the position of a point with infinite precision. But the mathematics is simpler. It makes for beautiful mathematics, and a lot of physics is based on partial differential equations and mathematics. It depends on the fantasy of infinite precision, real numbers. So, in a sense, it’s justified by its practical applications. But nowadays, not so much, because people don’t solve equations that much anymore to do engineering. They use computer simulations.

Robert J. Marks: Or they go to Mathematica.

Gregory Chaitin: Right, which is using finite precision mathematics, not infinite precision real numbers… People are more concerned now with calculating than they are with proving theorems. In that sense, I’m a dinosaur.

Robert J. Marks: Well, in fact, we have a name for them in engineering. We call them keyboard engineers. If they’re presented with a problem, they don’t go to the theory—which gives you depth and insight into what’s happening. They go to a keyboard and try to work things out just to get a surface sort of answer.

Gregory Chaitin: Instead of messing around with very complicated equations and trying to find a closed-form solution, you simulate the system and see how it behaves, and you change the design a little bit to see if it behaves better, more like what you want. So that’s a new paradigm that the computer enables us to do.

At the day-to-day level, that approach means that more basic questions, like the halting problem, don’t come up.

Gregory Chaitin: Will the program halt if given an arbitrarily large amount of time? Now computers will not last for an arbitrary amount of time. They’ll break or the earth will freeze or the sun will go nova. It’s like talking about the unicorns or flying horses.

Next: Why the unknowable number exists but is uncomputable

Note: If you would like to learn more about math with numbers that are not real numbers, you might enjoy Jonathan Bartlett’s “Yes, you can manipulate infinity in math!


Don’t miss the stories and links from the previous podcast with Gregory Chaitin:

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.

and

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.

Here are the stories, with links, to an earlier recent podcast discussion with Gregory Chaitin:

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.

and

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)

and

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)

Show Notes

  • 00:23 | Introducing Gregory Chaitin
  • 00:40 | What is unknowability?
  • 06:07 | Does non-computable mean unknowable?
  • 09:43 | A simple explanation
  • 21:34 | Is creativity non-computable?
  • 25:55 | Defining creativity
  • 28:19 | Panpsychism

Additional Resources

Podcast Transcript Download


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Getting To Know the Unknowable Number (More or Less)