This is from 2015 FYI. Here's a more in-depth summary: https://www.forbes.com/sites/kevinknudson/2015/11/10/everyth...
The formula for pi buried in a hydrogen atom
21–26 of 26 posts
Re: The formula for pi buried in a hydrogen atom
#22This brings to mind Richard Hamming's wonderful essay "The Unreasonable Effectiveness of Mathematics", which really blew my mind ten years ago: "But if you do not like these two examples, let me turn to the most highly touted law of recent times, the uncertainty principle. It happens that recently I became involved in writing a book on Digital Filters [8] when I knew very little about the topic. As a result I early a…
> rather the probability of a random physical constant having a leading digit of 1, 2, or 3 is approximately 60%, and of course the leading digits of 5, 6, 7, 8, and 9 occur in total only about 40% of the time
This is a part of the arguments used both pro/con of Tau over Pi. Constants that start with 6 are "weird" because they are uncommon. They are uncommon because "we" like to just halve them and use constants that start with 3 and lots of factors of two when working with them. It's a weirdness entirely of mathematics' own creation. Neither is more "fundamental", people just likes 3s sometimes too much.
Re: The formula for pi buried in a hydrogen atom
#23This brings to mind Richard Hamming's wonderful essay "The Unreasonable Effectiveness of Mathematics", which really blew my mind ten years ago: "But if you do not like these two examples, let me turn to the most highly touted law of recent times, the uncertainty principle. It happens that recently I became involved in writing a book on Digital Filters [8] when I knew very little about the topic. As a result I early a…
> rather the probability of a random physical constant having a leading digit of 1, 2, or 3 is approximately 60%, and of course the leading digits of 5, 6, 7, 8, and 9 occur in total only about 40% of the time This is a part of the arguments used both pro/con of Tau over Pi. Constants that start with 6 are "weird" because they are uncommon. They are uncommon because "we" like to just halve them and use constants that…
He's talking about physical constants, to which Benford's law applies (since we expect the first digits to be distributed the same when stated in different units, e.g. meters and feet, we expect there to be the same number of constants that start with 1 in meters as those that start in 3, 4 or 5 in feet (since 1m=3ft, 2m=6ft), etc').
In any unit you choose, about 30% of rivers on Earth and about 30% of public company market caps will start with a 1. This has nothing to do with people's preferences and everything to do with scale invariance.
Re: The formula for pi buried in a hydrogen atom
#24I really love the idea that pure mathematics and nature are the same thing. On a philosophical level, our daily lives are the expression of the differences and inefficiencies of our systems compared to an optimal end-state. Also, sometimes I think that religion and science have more in common than people think.
I also love the idea that computation (ie rule/recipe-following) is fundamental to nature/reality, but I don't think it's pure-maths that is the bedrock.
Pure maths (as a tendency, not strictly) focuses on formal systems that are on the consistent end of the spectrum (as those are the ones that can be used to build more maths on top of). Nature probably doesn't (or needn't) have this restriction/tendency! Formally consistent might be higher performing in many cases (more stable over long time-spans, like dna), but it is not the end of the story as to what is ultimately possible in rule-following systems! Any sense of determinism demands this, even if the system makes its own rules as the universe/reality - the ultimate all, with nothing outside of it - must. Whether this fully gives way to a properly random (like an oracle of rand) system at the quantum level is still an open question.
It is possible to expand ones view of nature/ultimate-reality to include other rule-systems (such as including inconsistent or entirely 'complete' systems.) This can have more coverage (completeness) of any particular model, or even be totally incomplete and inconsistent! These 'degenerate' (according to maths) formal systems might sometimes be the ones that happen to work in nature/reality. Once you do this extension, now you're talking about computation, admitting a broader class of rule-following than pure maths. You can argue that the two are equivalent because sure, anything can be translated between them - i'm more talking about tendencies here. The computational universe includes lots of formal systems that are abhorrent or impractical to mathematical understanding, so mathematicians avoid them, but nonetheless they are still rule-following! Nature/evolution probably uses/finds those rules where they are the best performing solution (through its exploration of possibilities).
As to 'optimal end-states', I'm afraid you've lost me there. I gave up teleology (the idea that the universe or nature or evolution have any obvious goals) a long time ago, how could anything like that (high level) feed-in to reality? There is no hint of this in all of science. What self-organises and self-replicates is what persists. This self-support might even reach down into physics!
To me, religion is nearly the opposite to all of this: An explanatory system that says explanation-of-reality='fixed string' with no update method available. Very incomplete (unless ridiculously long, which current examples of religion are not) and inconsistent: The fixed string might assign True and False at once, or neither - to new knowledge that falls outside the original scope of its meaning.
So no, I don't think that this pi-result suggests that religion and science have a commonality here. Pi is awesome (even though very constructible) and we should probably expect to find it [0], especially in places like this (pure QM) that involve geometry.
Re: The formula for pi buried in a hydrogen atom
#25Earlier quoted context omitted.
> rather the probability of a random physical constant having a leading digit of 1, 2, or 3 is approximately 60%, and of course the leading digits of 5, 6, 7, 8, and 9 occur in total only about 40% of the time This is a part of the arguments used both pro/con of Tau over Pi. Constants that start with 6 are "weird" because they are uncommon. They are uncommon because "we" like to just halve them and use constants that…
He's talking about physical constants, to which Benford's law applies (since we expect the first digits to be distributed the same when stated in different units, e.g. meters and feet, we expect there to be the same number of constants that start with 1 in meters as those that start in 3, 4 or 5 in feet (since 1m=3ft, 2m=6ft), etc'). In any unit you choose, about 30% of rivers on Earth and about 30% of public company…
I certainly read it as applying to both. Certainly Benford's law is demonstrable in many areas, but there's also an interesting sort of "Benford's paradox" at play that when free choice is given between units options, people seem to "prefer" the versions of constants with the lower starting digit. An interesting question of whether one seems more "natural" than another simply because Benford's law so often applies to similar situations.