Mind Matters Natural and Artificial Intelligence News and Analysis

# TagDavid Hilbert

## Why Just Anything Can’t Happen via Infinite Universes

We can see why not, using simple mathematical reasoning in this universe

Can anything happen if there are an infinite number of universes each with an infinite number of possibilities in each? Can you be bald in one universe and fully haired in another? Can you have two eyeballs in this universe and three in another? The answer is no. In a nutshell, the reason is that some infinities are bigger than other infinities. (And this is not a claim like infinity plus one is bigger than infinity. Infinity plus one is still infinity.) The number of points on a line segment from, say zero to one, is a bigger infinity than the number of counting numbers {1,2,3,…}. We can label the infinite number of universes in the multiverse as universe #1, Read More ›

## 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

In this week’s podcast, “The Chaitin Interview II: Defining Randomness,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on randomness. It’s a subject on which Chaitin has thought deeply since his teenage years (!), when he published a journal paper on the subject. How do we measure randomness? Chaitin begins by reflecting on his 1969 paper: https://episodes.castos.com/mindmatters/Mind-Matters-125-Gregory-Chaitin.mp3 This portion begins at 1:12 min. A partial transcript, Show Notes, and Additional Resources follow. Gregory Chaitin: In particular, my paper looks at the size of computer programs in bits. More technically you ask, what is the size in bits of the smallest computer program you need to calculate a given digital object? That’s called the program Read More ›

## Most Real Numbers Are Not Real, or Not in the Way You Think

Typical real numbers contain an encoding of all of the books in the US Library of Congress

Pick a random real number between zero and one. The number you choose, with probability one, will contain an encoding of all of the books in the US Library of Congress. This sounds absurd, but real numbers require infinite precision and every time you deal with the infinite, things get absurd. Infinities, including the infinite number of digits to express almost every real number, don’t exist. Curiously then, real numbers are not real. How do we choose a random number between zero and one? The easiest way to explain is using binary decimals. The binary number 0.1000… with zeros forever denotes the number ½ or, in base 10 notation, 0.5. The binary decimal 0.01000… with zeros forever is the number Read More ›

## Faith Is the Most Fundamental of the Mathematical Tools

An early twentieth century clash of giants showed that even mathematics depends on some unprovable assumptions

David Hilbert wanted all mathematics to be proved by logical steps. Kurt Gödel showed that no axiomatic system could be complete and consistent at the same time.