508 karma · joined May 1, 2014
> Bedford's Law is apparently just a fancy name for the phenomenon where 1's appear much more frequently than 9's in any random sample of numbers.
because, as my example shows, it's false for the case of random numbers sampled uniformly across five orders of magnitude. You're also right that it's possible to construct a distribution (such as taking -log(rand())) where this behavior is observed.
There are many distributions which do lead to the distribution of leading digits that follow Benford's law, just not uniform. Nor "uniform over several orders of magnitude".
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
sns.set_context('poster', font_scale=1.3)
n_values = 10000
lower_bound = 1
upper_bound = 100000
nums = np.array(
[int(str(x)[0]) for x in np.random.randint(lower_bound, upper_bound, size=n_values)]
)
fig, ax = plt.subplots(figsize=(12, 8))
sns.distplot(nums, kde=False, bins=list(range(1, 11)), ax=ax)
ax.set_xticks(list(range(1, 11)))
fig.tight_layout()
And you'll see it's essentially a uniform distribution of leading digits.> "...And I agree we should use ads to promote our own products, but I'd still block companies that compete with our core from gaining any advantage from us. I'd also keep blockingGoogle but otherwise wouldn't extend the block to anyone else."
> "These actions may look anti-competitive, but these companies are all foreign (China, South Korea, Japan)."
> Explicitly mentions Google.
> Google has a bigger ad network than Facebook so how is that a monopoly?
Do you find this statement anti-competitive: "I'd still block companies that compete with our core"?
More details here: https://en.wikipedia.org/wiki/Oil_drop_experiment#Millikan's...
which states that, worldwide, just over 5,000 Americans renounced citizenship in 2017. There are also about 9 million Americans living abroad and ~1 million living in Europe. So, no, I don't think that even if all 5,000 were in Europe that 0.5% renounced their citizenship would count as "many".
This is a tangential point, but they weren't living in an appreciably younger solar system. 65 million / 4.5 billion is ~1.5%. So, the solar system (and the Earth) was 98.5% of its current age.
Which would probably burn that into your brain pretty well.
100% yes.
> It is not that the fine structure constant has some particular status as "more fundamental" than others.
Agreed, as I wrote above "as the fine-structure constant is one of fundamental constants of the universe", it's in a class of fundamental constants (or one of the irreducible values to be plugged in to use your phrasing).
> If constant #1 can be computed given the theory and constant #2, then they are not independent, and the choice between them is arbitrary; a convention.
Sure. You can choose one input over the other once you have made at least one other measurement.
Are you arguing something like, "there are X irreducible inputs to the Standard Model. For any particular input, a, you might be able to swap it out for a different one, g, so that you still have X irreducible inputs, but now they are a different set. Therefore, because we swapped a for g, a is not a fundamental constant"?
Do you take issue with the phrasing of the passages in this wikipedia article (https://en.wikipedia.org/wiki/Dimensionless_physical_constan...)? Specifically:
1) > ...physicists ... reserve the use of the term fundamental physical constant solely for dimensionless physical constants that cannot be derived from any other source.
2) > Fundamental physical constants cannot be derived and have to be measured.
And 3) its classification of the fine-structure constant as a fundamental physical constant?
This whole thread started because the top comment said that a physics theory predicted the value of the fine-structure constant. Which is wrong, as the fine-structure constant is one of fundamental constants of the universe and one _whose value is not predicted by any theory_.
At this stage, two claims are in tension.
The first is your claim, that QED predicts the fine-structure constant once you measure the magnetic moment of the electron using several thousand Feynman diagrams.
The second claim is Feynman, himself, literally writing in his book titled _QED_, that we have no idea how to predict the value of this constant.
Could it be that Feynman overlooked the fact that he, himself, predicted the value of the fine-structure constant? He thought that not being able to predict its value was such an unsolved problem as to call it "one of the greatest damn mysteries of physics"? That "all good theoretical physicists put this number up on their wall and worry about it"?
The long and the short of it is that I think you're missing something much deeper. Yes, _once you measure_ something that has a tight coupling with the fine-structure constant, you now know the value of the fine-structure constant. But, before you made that measurement you DO NOT KNOW and further CANNOT PREDICT the value of the fine-structure constant. If you could, you'd be able to claim your own Nobel prize.
> There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly! — Richard Feynman, Richard P. Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129. ISBN 978-0-691-08388-9.
[my emphasis added]
"Second place for me goes to Matlab... I can't say anything about Matlab..."
References:
- https://unu.edu/media-relations/releases/global-income-inequ...
- https://ourworldindata.org/global-economic-inequality#the-la...