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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?

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 somebody has to publish. So you take some problem, you change the wording of the mathematical problem a little bit, then you solve it, and then you write a paper.

But other mathematics does seem to be discovered. That’s when you find some really deep, fundamental mathematical idea, and there it really looks inevitable. If you hadn’t discovered it somebody else would have discovered it.

One idea is that mathematics is in the mind of God or in the Platonic world of ideas. It’s all there, and all we do is discover it. But I think there’s a distinction. A famous mathematician, Henri Poincaré, said (it sounds better in French than in English) “There are problems that we pose and problems that pose themselves.”

Note: Henri Poincaré (1854–1912, pictured) was a mathematician, theoretical physicist and a philosopher of science famous for discoveries in several fields and referred to as the last polymath, one who could make significant contributions in multiple areas of mathematics and the physical sciences. – Stanford Encyclopedia of Philosophy

He wrote,

“What we call objective reality is, in the last analysis, what is common to many thinking beings, and could be common to all; this common part, we shall see, can only be the harmony expressed by mathematical laws. It is this harmony then which is the sole objective reality, the only truth we can attain; and when I add that the universal harmony of the world is the source of all beauty, it will be understood what price we should attach to the slow and difficult progress which little by little enables us to know it better.” – Introduction, The Value of Science (1905), p. 14, as translated from the French by George Bruce Halsted (1907).

Gregory Chaitin: So those are the two different kinds of math, invented and discovered. When you find some really fundamental new mathematical idea, you have this feeling that you’ve seen into the mind of God and it’s really fantastic…

If you hadn’t discovered it, someone else would have, because it’s so basic, it’s so beautiful that it’s got to be there in the mind of God or in the Platonic world of ideas. But everything is in the Platonic world of ideas, the Platonic world of mathematics.

But if you’re a mathematician at a university and you’re struggling to publish, I don’t know how many papers per year, you can’t work on such fundamental questions all the time. Because then you won’t publish enough papers, right?

Robert J. Marks: Yes.

Gregory Chaitin: So there is this pressure to invent stuff, minor variations on previous work. That is a shame I think, and I think it should be regarded as being invented, although one attitude is to say that it’s all in the Platonic world of mathematics.

Robert J. Marks: In engineering, we call these 3DB papers, three decibel papers, because three decibels is the minimal amount that by which can increase the volume of something and detect it. So there are landmark papers and then there’s lots of 3DB incremental papers …

Gregory Chaitin: Yeah. You guys have the same pressure to publish as everyone else, right?

Robert J. Marks: Exactly. Exactly.

Next: Chaitin asks, why don’t we see many great books on math any more?


You may also enjoy:

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 trying to frustrate the math nerd who solved his unfinished Last Conjecture.

Also, 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. 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

Podcast Transcript Download


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