[ttctciyf]

I've long admired Chaitin for his original thinking and especially his ability to clearly convey his ideas about foundations, complexity and information in concise and digestible short proofs.

I'm a little surprised, however, to see him here proselytizing for a particular side in the constructivism debate. I associate him more with what he has described as a "quasi empirical" approach to mathematics[0] where the adoption of new axioms (such as for example the axiom of choice) is justified by their resulting in new, interesting mathematics.

But here, it seems his goal is to arrive at a somewhat Platonic conclusion, that either the reals are valid numbers or (seemingly he prefers) not.

My lay and naive take would be: if you adopt these rules (this Formal Axiomatic System) then you can have Big Fun in the playground of ever more esoteric and complex infinite cardinals, or if you adopt this other FAS you get to discover which results can and can't be obtained under a strict constructivist regime, and whichever FAS you choose it's just the same process of choosing axioms and applying valid deductive steps to arrive at a result you find interesting, with no more "existence" implied than the thoroughly non-Platonic existence of a solution to a problem, which can be demonstrated by solving it: if you take such-and-such steps then such-and-such result will follow.

I suppose the point being made is that avoiding axioms which imply the "existence" of the reals is more useful for doing physics, but that seems non-obvious in a field which for the last 200 years has seemingly sought to progressively make more and more phenomena intelligible by means of differential equations from infinitesimal calculus!

0: see, for example: https://arxiv.org/pdf/math/0303352, 1.9 Is Mathematics Quasi-Empirical

[exac]

> According to Pythagoras everything is number, and God is a mathemati- cian. This point of view has worked pretty well throughout the development of modern science. However now a neo-Pythagorian doctrine is emerging, according to which everything is 0/1 bits, and the world is built entirely out of digital information. In other words, now everything is software, God is a computer programmer, not a mathematician, and the world is a giant information-processing system, a giant computer [Fredkin, 2004, Wolfram, 2002, Chaitin, 2005]

¯\_(ツ)_/¯ For some reason I get the same vibe from this as people referring to LLM inference using gendered pronouns instead of "it".

[cgio]

Arguably starts with Wheeler and “it from bit”. Zuse also, who predates Wolfram. For me it’s a little bit like calling the egg you hold in your hand the centre of the Universe while rolling on a skateboard.

[nyc111]

> However now a neo-Pythagorian doctrine is emerging, according to which everything is 0/1 bits, and the world is built entirely out of digital information

Can this be true? DNA is the information system of living creatures and as far as I know, it is not coded with 0s and 1s. So, how can we justify that "the world is built entirely out of digital information"?

[atoav]

For all I, a Victorian everyman, know the world is built from small pistons, gears and pulleys.

Every age has their technology which they will project onto the world. A century ago one may have started to talks about everything being electrical wires and switches.

[d_tr]

It doesn't matter whether the model assumes bits, pulleys, elves or whatever, as long as it does a better job at describing physical reality than whatever exists at the time anyway. People will try and very probably succeed in providing alternative formalism anyway.

All that matters is whether it facilitates reasoning towards the goal.

[Diogenesian]

I think footnote 16 on pg 12 clarifies his view.

[lioeters]

From page 12:

> Why should we believe in real numbers, if most of them, it turns out,[^15] are maximally unknowable like Ω? [^16]

The footnotes:

> [^15]: See the chapter entitled The Labyrinth of the Continuum in [Chaitin, 2005]

> [^16]: In spite of the fact that most individual real numbers will forever escape us, the notion of an arbitrary real has beautiful mathematical properties and is a concept that helps us to organize and understand the real world. Individual concepts in a theory do not need to have concrete meaning on their own; it is enough if the theory as a whole can be compared with the results of experiments.

---

The reference [Chaitin, 2005] in footnote 15 links to..

Meta Math! The Quest for Omega - http://arxiv.org/abs/math/0404335

> This book presents a personal account of the mathematics and metamathematics of the 20th century leading up to the discovery of the halting probability Omega. The emphasis is on history of ideas and philosophical implications.

Irreducible Complexity in Pure Mathematics - http://arxiv.org/abs/math/0411091

> By using ideas on complexity and randomness originally suggested by the mathematician-philosopher Gottfried Leibniz in 1686, the modern theory of algorithmic information is able to show that there can never be a "theory of everything" for all of mathematics.