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?

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, WolframAlpha…

What Euler would have accomplished with that is unbelievable. Euler and Gauss were wonderful at doing calculations, and they would do lots of calculations and then making conjectures based on the patterns they saw.

Well, if Euler or Gauss had had WolframAlpha or Mathematica, they would have done a lot more. Especially when you go to WolframAlpha, it begins to start feeling like an AI. Now it’s an AI which has a big team of people behind it who take information and curate it about the world, about physics, about chemistry, about economics, about geography. They curate it and they put it into the system.

Note: “Have you ever given up working on a math problem because you couldn’t figure out the next step? Wolfram|Alpha can guide you step by step through the process of solving many mathematical problems, from solving a simple quadratic equation to taking the integral of a complex function.” – Wolfram Alpha Blog, (July 17, 2019)

“Over the course of more than four decades, he has been a pioneer in the development and application of computational thinking—and has been responsible for many discoveries, inventions and innovations in science, technology and business.” – “About Stephen Wolfram”

Gregory Chaitin: It’s pretty amazing. It would have looked like magic, I think, to people. Well computers, almost any computer would have looked like magic just a few years ago.

I think this is genuine AI — but it’s not a human general intelligence and it’s not creative. It’s different. I think it’s an enormous achievement.

Robert J. Marks (pictured): Wolfram’s Mathematica and his other works are just astonishing in what they can do. But as you mentioned, they’re all algorithmic. The logical steps, much like the theorem checker, are something humans have placed in there which allow you to put in things like indefinite integrals and advanced calculus equations — and it gives you the solution. It’s really, really remarkable.

Gregory Chaitin: Wolfram is a genius. I rate him with Elon Musk. He’s a genius at different kinds of technology than Elon is. So WolframAlpha is an accomplishment of this man of genius who is just like Elon. Elon has an enormous team of very talented engineers. But he’s on top of the whole thing, making it work. Wolfram has wonderful mathematicians, wonderful software people working for him.

So this artificial intelligence of an inhuman kind that they’ve created is very powerful. It’s done by human beings, so I think we should be proud of that achievement. But it’s not creative.

Robert J. Marks: Yes. I think the creativity is the big thing.

Note: A test proposed to identify creativity in computers, should it happen, is the Lovelace Test — the computer demonstrates creativity that did not arise from its programming: “In ‘Thinking machines? The Lovelace test raises the stakes,’ Rensselaer philosopher and computer scientist Selmer Bringsjord argued that the iconic Turing test for human-like intelligence in computers is inadequate and easily gamed. That is, merely sounding enough like a human to fool people does not establish human-like intelligence in the product; it may point only to superior cunning in the creators. He pioneered the much more challenging Lovelace test, based on an observation from computer pioneer Ada Lovelace (1815–1852) that true creativity distinguishes humans from machines.”

Originally conceived by Selmer Bringsjord and named after computer pioneer Ada Lovelace, it was intended as a replacement for the troubled Turing Test. In any event, the test has never been passed by a computer.

Gregory Chaitin (pictured): The computer didn’t program itself. Wolfram worked very hard with all his people to make it capable of doing more and more and more. It wasn’t his software that made this thing evolve to what they can do now. It was all of them working very hard on it and Wolfram making sure they had a system that could be extended.

Because what often happens to software is — I know because I worked doing software for IBM — there comes a point where basically the software dies. Because what happens is, it’s so complicated that no one can understand it anymore. Which means, if you get bugs, it’s tough to debug it. And it’s also tough to make any enhancements.

So the fact that the mathematical language has gotten us all the way to WolframAlpha is something that Stephen worked very hard on, to have a system that could grow and be extendable that wouldn’t end up dropping him in a corner like most large corporate software does eventually. So far he’s achieved this remarkably.

Yeah, but this is a human being of genius with a very talented team of engineers, mathematicians. This is not software that reprogrammed itself.

Robert J. Marks: I think that AI in general is going to be a tool which we can use to better ourselves.

Gregory Chaitin: Absolutely, like a steam shovel, right? It doesn’t mean that human beings are obsolete.

Robert J. Marks: I read the chapter by Stephen Wolfram in your tribute book. He went to a library and he took a bunch of pictures of the notes of Leibniz. I tell you, boy, we’ve come a long way. These old mathematicians, they couldn’t compute e to the third power. They just couldn’t enter it. They had to go to their margins and work out all the details. It’s astonishing all of the work that they had to do that we don’t have to do today.

Gregory Chaitin: Exactly, and Leibniz made mistakes in some of his arithmetical calculations there in the manuscripts. He wasn’t good at that.

But you could say we don’t have anybody at the intellectual level of Leibniz. It depends how you rank it, because he was good at so many, he came up with fundamental new ideas in so many fields. Maybe it’s because he never married, never had children.

But he was off the scale which shows what human beings can achieve, Euler and Ramanujan and Cantor showed what human beings can achieve.

Note: The tribute book referred to above is Unravelling Complexity: The Life and Work of Gregory Chaitin (2020), edited by Shyam Wuppuluri (R N Podar Institute, India) and Francisco Antonio Doria (Universidade Federal do Rio de Janeiro, Brazil). “The revolutions that Gregory Chaitin brought within the fields of science are well known. From his discovery of algorithmic information complexity to his work on Gödel’s theorem, he has contributed deeply and expansively to such diverse fields.

This book attempts to bring together a collection of articles written by his colleagues, collaborators and friends to celebrate his work in a festschrift.”

Next: Gregory Chaitin on how bureaucracy chokes off science

You may also wish to read these earlier posts in the series of conversations with Gregory Chaitin (of Chaitin’s unknowable number):

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
• 00:39 | Is math discovered or invented?
• 02:49 | The pressure to publish papers
• 08:31 | A human-computer symbiosis?
• 13:22 | Computer software proofing mathematics
• 19:45 | Bureaucratic obstacles to genuine research

Additional Resources

• The Chaitin Interview Part I
• The Chaitin Interview Part II
• Gregory Chaitin’s Website
• Unravelling Complexity: The Life and Work of Gregory Chaitin, edited by Shyam Wuppuluri and Francisco Antonio Doria
• Proving Darwin: Making Biology Mathematical by Gregory Chaitin
• Henri Poincaré, 19th century French mathematician
• Georg Cantor, German mathematician
• A Mathematician’s Apology by G.H. Hardy
• Claude Shannon, mathematician, “the father of information theory”
• Lofti Zadeh, world-renowned computer scientist
• Andrew Wiles, English mathematician
• Karl Popper, Austrian-British philosopher
• William Sealy Gosset, statistician and chemist
• Elon Musk, engineer and entrepreneur
• Neuralink
• Kurt Gödel, Austrian-born mathematician
• Alan Turing, mathematician and philosopher
• Stephen Wolfram, computer scientist and physicist
• WolframAlpha
• Mathematica
• Leonard Euler, Swiss mathematician and physicist
• Carl Friedrich Gauss, German mathematician and physicist
• Srinivasa Ramanujan, Indian mathematician

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