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# Gregory Chaitin’s New Books About Math Make It Actual Fun

He is a favorite podcast guest of ours and, it turns out, a fan of Mind Matters
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Gregory Chaitin has been busy recently. He has produced two new short books, which can be downloaded free:

Building the World out of Information and Computation (2021), which he summarizes as

According to Pythagoras: All is Number, God is a Mathematician. Modern physics is in fact based on continuous mathematics, dierential and partial dierential equations, validating Pythagoras’ vision. In this essay we shall instead discuss a neo-Pythagorean ontology: All is Algorithm, God is a Programmer. In other words, can there be discrete computational models of the physical world? This is sometimes referred to as digital philosophy. There are in fact two books on digital philosophy, both in Italian: • Ugo Pagallo, Introduction to Digital Philosophy, from Leibniz to Chaitin • Andrea Vaccaro and Giuseppe Longo, Bit Bang: The Birth of Digital Philosophy Let’s review the development of digital philosophy. We’ll start, as is often the case, with Leibniz.

and

Philosophical Mathematics. Infinity, Incompleteness, Irreducibility (2023), which makes key themes in mathematics accessible. Here’s one that he offers in Chapter 2:

Let us give a beautiful example of the power of pure thought, a proof that there are infinitely many prime numbers. A prime is, of course, a non-negative integer that has no proper divisors, only itself and 1. Our proof is taken from Euclid’s Alexandrian compendium of classical Greek mathematics, and it is what is called a reductio ad absurdum, a reduction to an absurdity.

Suppose on the contrary that there are only finite many prime numbers. Multiply all of them together and add one, giving us the number M. M isn’t exactly divisible by any of those primes because it always leaves the remainder 1. Hence M must itself be a prime, contradiction!

Readers may remember Chaitin best from his podcasts with our Walter Bradley Center director Robert J. Marks, grouped as “Randomness, Information Theory, and the Unknowable” (December 30, 2021).

But in the math world, he is best known for “his discovery of the celebrated Omega number, which proved the fundamental unknowability of math.” (Penguin)

Dr. Marks offers, “Only a handful of individuals in history have made such significant contributions that they’ve given rise to a new field of study. Gregory Chaitin, a co-founder of algorithmic information theory and a remarkable individual, is one of them. I cover Chaitin’s work in my graduate course on information theory. Not only is his work genius, but is more mind-blowing than any science fiction you will ever read.”

Dr. Chaitin is a fan of Mind Matters News

It turns out that the esteem is mutual. Dr. Chaitin is a regular reader of Mind Matters News. In Chapter 4 (p.35f), he reproduces the transcript of our four-part interview with him. Needless to say, we are honored.

He has also endorsed books written by some of our authors and colleagues:

Some of the podcasts with Chaitin, with transcripts and notes:

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. 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. He 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.”