Science

Stigler’s law and Avogadro’s number

Posted on by John

Stigler’s law says that no scientific discovery is named after its original discoverer. Stigler attributed his law to Robert Merton, acknowledging that Stigler’s law obeys Stigler’s law.

Avogadro’s number may be an example of Stigler’s law, depending on your perspective. An episode of Engines of our Ingenuity on Josef Loschmidt explains.

The Italian, Romano Amadeo Carlo Avogadro, had suggested [in 1811] that all gases have the same number of molecules in a given volume. Loschmidt figured out [in 1865] how many molecules that would be.

You could argue that Avogadro’s constant should be named after Loschmidt, and some use the symbol L for the constant in honor of Loschmidt. Jean Perrin came up with more accurate estimates and proposed in 1909 that the constant should be named after Avogadro. Loschmidt made several important contributions to science that are now known by other’s names.

As I’d mentioned in an earlier post, there are some fun coincidences with Avogadro’s number.

  1. NA is approximately 24! (i.e., 24 factorial.)
  2. The mass of the earth is approximately 10 NA kilograms.
  3. The number of stars in the observable universe is 0.5 NA.

Type R error

Posted on by John

Andrew Gelman added a couple more types of error to the standard repertoire of type I and type II errors. He suggests using type S error to describe a result that gets a sign backward, reporting that A is bigger than B when in fact B is bigger than A. He also suggests using type M error for results that get the magnitude of a result wrong.

Maybe we could add to this list type R for reification error: treating an abstraction as if it were real, forgetting that a model is a model and stretching it beyond its limits.

Related links:

Bing Crosby science

Posted on by John

In a recent interview, Gary Taubes calls picking data that support your conclusion “Bing Crosby science.” This comes from a song by Bing Crosby that begins “You’ve got to accentuate the positive, eliminate the negative.”

Taubes uses the phrase to refer specifically to epidemiology, though it applies to all science. He credits “a Scottish researcher” with coining the phrase, but doesn’t say any more about who this researcher was.

Related post: Amputating reality

Why experts exaggerate

Posted on by John

Seth Roberts writes this morning:

How can you tell when an expert is exaggerating? His lips move.

Some people will misunderstand his point. Roberts is not saying experts exaggerate their conclusions per se. He’s saying experts exaggerate their confidence in their conclusions.

If an expert says that playing a harmonica decreases your risk of influenza by 10%, she’s probably not making that figure up out of thin air (though I am). There probably was some data that implied the 10% figure. It’s not that the data suggested 5% and the scientist said “Let’s call it 10%.” But the quality and quantity of the data may not justify rushing out to buy a harmonica.

One reason experts exaggerate their confidence is that they may be at a loss for words to explain their degree of uncertainty to a popular audience. Journalists can understand “Harmonica playing is good for you” though they probably cannot understand confidence intervals, Bayes factors, or the differences between retrospective versus prospective experiments. (The experts may not really understand these things either.)

Another reason for exaggeration is that you don’t get the attention of the press by making tentative claims. This creates an incentive to suppress uncertainty. But even if experts were transparent regarding their uncertainty, there would still be a selection bias: experts who are sincerely more confident are more likely to be heard.

There are numerous other reasons experts may be wrong, some psychological and some statistical.

I liked the first comment on Roberts’ post:

I tended to rate my colleagues partly by how often the words “I don’t know” passed they lips. Often = good.

Related posts

Cartoon guide to the uninteresting

Posted on by John

If you’re not interested in a subject, do cartoons make it more palatable?

My guess is that cartoons may help keep your attention if you’re moderately interested in a subject. If you’re fascinated by something, cartoons get in the way. And if you’re not interested at all, cartoons don’t help. The cartoons may help in the sweet spot in between.

No Starch Press has given me review copies of several of their Manga Guide books. The first three were guides to the universe, physics, and relativity. I’ve reviewed these here and here. Recently they sent a copy of the newest book in the series, The Manga Guide to Biochemistry.

I’m much more interested in physics than biology, so I thought this would be a good test: Would a manga book make it more interesting to read about something I’m not very interested in studying? Apparently not. It didn’t seem that the entertaining format created much of an on-ramp to unfamiliar material.

It seemed like the information density of the book was erratic. Material I was familiar with was discussed in light dialog, then came a slab of chemical equations. Reading the book felt like having a casual conversation with a lawyer who periodically interrupts and asks you to read a contract.

Someone more interested in biochemistry would probably enjoy the book. Please understand that the title of this post refers to the fact that I find biochemistry uninteresting, not the book. If I had to study a biochemistry book, the Manga Guide to Biochemistry might be my first choice. At times I’ve found biochemistry interesting in small doses, describing a specific problem. But it would be nearly impossible for me to read a book on it cover to cover.

O’Reilly’s “Head First” series is similar to the Manga guide series, though the former has more content and less entertainment. I enjoyed the first Head First book I read, Head First HTML with XHTML & CSS. Maybe I enjoyed it because the subject matter was in the sweet spot, a topic I was moderately interested in. The cartoons and humor helped me stick with a dry subject.

When I tried another Head First book, I was hoping for more that same push to keep going through tedious content. The books clearly had the same template though with different content. What was interesting the first time was annoying the second time, like hearing someone tell a joke you just heard. So at least for me, the Head First gimmick lost some of its effectiveness after the first book.

Fourier’s personal heat problem

Posted on by John

Joseph Fourier is perhaps best known for his work studying heat conduction. He developed what we now call Fourier series as part of this work.

I recently learned that Fourier had a personal problem with heat.

Even though Fourier conducted foundational work on heat transfer, he was never good at regulating his own heat. He was always so cold, even in the summer, that he wore several large overcoats.

Source: The Physics Book

Physical constants and factorials

Posted on by John

The previous post mentioned that Avogadro’s constant is approximately 24!. Are there other physical constants that are nearly factorials?

I searched SciPy’s collection of physical constants looking for values that are either nearly factorials or nearly reciprocals of factorials.

The best example is the “classical electron radius” re which is 2.818 × 10-15 m and 1/17! = 2.811 × 10-15.

Also, the “Hartree-Hertz relationship” Eh/h equals 6.58 × 1015 and 18! = 6.4 × 1015. (Eh is the Hartree energy and h is Plank’s constant.)

Here’s the Python code I used to discover these relationships.

from scipy.special import gammaln
from math import log, factorial
from scipy.optimize import brenth
from scipy.constants import codata

def inverse_factorial(x):
    # Find r such that gammaln(r) = log(x)
    # So gamma(r) = x and (r-1)! = x
    r = brenth(lambda t: gammaln(t) - log(x), 1.0, 100.0)
    return r-1

d = codata.physical_constants
for c in d:

    (value, unit, uncertainty) = d[ c ]
    x = value
    if x < 0: x = abs(x)
    if x < 1.0: x = 1.0/x
    r = inverse_factorial(x)
    n = round(r)
    # Use n > 6 to weed out uninteresting values.
    if abs(r - n) < 0.01 and n > 6:
        fact = factorial(n)
    if value < 1.0:
        fact = 1.0/fact
    print c, n, value, fact

Avogadro’s number

Posted on by John

Avogadro’s number NA is the number of atoms in 12 grams of carbon-12. It’s about 6.02 × 1023. (Update: Avogadro’s number is now exactly NA = 6.02214076 × 1023 mol-1 by definition. More details here.)

Here are a few fun coincidences with Avogadro’s number.

  1. NA is approximately 24! (i.e., 24 factorial.)
  2. The mass of the earth is approximately 10 NA kilograms.
  3. The number of stars in the observable universe is 0.5 NA.

The first observation comes from here. I forget where I first heard the second. The third comes from Andrew Dalke in the comments below, verified by WolframAlpha.

For more constants that approximately equal factorials, see the next post.

Related posts

The most powerful people are right

Posted on by John

From Seth Roberts:

If you ignore data, the answer to every hard question is the same: the most powerful people are right. That way lies stagnation (problems build up unsolved because powerful people prefer the status quo) and collapse (when the problems become overwhelming).

Science is far more political than I had imagined before starting a career in science. Data trumps politics eventually, but it may take a long time.