Saturday, June 27, 2015


1. Color Illusions: Yet another example of the "fallibility" of sense perception

2.  A quantitative take on coffee (WaPo)

3. The inscrutable problem of grouping people (quanta)

4. Man-made earthquakes (newyorker)

5. Amazing collection of interesting maps (

Wednesday, June 24, 2015

Apparent Mathematical Paradoxes

Paradoxes involving summations of divergent series have invaded my Facebook stream.These include, for example claims like, \[1 + 2 + 3 + ... = -1/12,\] \[1 -1/2 + 1/3 -1/4 + ... = 0,\] which involve reasoning that jumps from "infinity = infinity + something" - which is fine, to conclude that therefore, "something = 0" - which is faulty.

Such problems are great at teaching subtleties that can usually be shoved under the rug.

Here is yet another example that a student (Nathan C.) recently brought up. We know that \(i^2 = -1\). Consider the following steps and identify the misstep. \[i^2 = \sqrt{-1} \sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1 \neq -1\]

Monday, June 22, 2015

More Thoughts on Modeling

Let me relate a story to set up my thinking for this post:

During my PhD at Michigan, I worked on some approximate models for polymer dynamics called slip-link (SL) models. This (SL) model was more accurate than the standard theoretical model. As expected, it took greater computational resources to numerically simulate my model (a few hours) than the standard theory (a few seconds).

There are some applications where this trade-off between accuracy and speed is desirable.

Of course, there are other richer models, which are more accurate and even more computationally expensive than mine.

One of the people on my committee asked me: "If computational speed were infinite, would anyone care about your model?" I don't remember exactly how I responded; my guess: some mumbled gibberish.

But this is indeed a profound question that touches upon what I said recently about seeking too much accuracy in models. If I can numerically compute the most accurate model available for something, should I waste my time with alternatives?

Let's set aside the hypothetical "if computer speed were infinite" part of the question, and work under the premise that such computers were indeed available.

Should we then simply use ab initio quantum mechanics, or perhaps, the standard model of physics (whatever that is!) to study everything?

But seriously, if you want to study migration of birds, or mechanical properties of a spaceship, or the ups and downs of a business cycle, would you really want to study it in terms of quarks?

As Douglas Adams pointed out in the Hitchhikers Guide, our computer model may give us the "Answer to the Ultimate Question of Life, the Universe, and Everything", and yet we may be unable to comprehend it.

Misquoting Richard Hamming, "the purpose of modeling is usually insight, not numbers."

Thursday, June 18, 2015

Rational Links

About a month ago Massimo Pigliucci announced his "resignation" from the skeptic and atheist movement (SAM). His parting remarks were cutting and brutally honest.
[SAM is ] a community who worships celebrities who are often intellectual dilettantes, or at the very least have a tendency to talk about things of which they manifestly know very little; an ugly undertone of in-your-face confrontation and I’m-smarter-than-you-because-I-agree-with [insert your favorite New Atheist or equivalent]; loud proclamations about following reason and evidence wherever they may lead, accompanied by a degree of groupthink and unwillingness to change one’s mind that is trumped only by religious fundamentalists; and, lately, a willingness to engage in public shaming and other vicious social networking practices any time someone says something that doesn’t fit our own opinions, all the while of course claiming to protect “free speech” at all costs.
I will certainly miss both his insights, his ability to take other skeptics to task, and his deep understanding of the power and limitations of science.

Listen to this debate with Michael Shermer, for instance. He skillfully tears down the his opponents' arguments.

Tuesday, June 16, 2015

Cloudy Puzzle: Solution

We want to establish a mathematical relationship between the key variables of the problem: \(h, R, \theta\) and \(\phi\).

Consider the triangle PRQ; we can write, \[\tan\left(\frac{\pi}{2} - \theta \right) = \frac{QP}{RQ} = \frac{QP}{OQ - OR}.\] 
Recognizing that the length of the segment OP is \(R+ h\), we can further write:
\[\tan\left(\frac{\pi}{2} - \theta \right) = \frac{(R+h) \cos \phi}{(R+h) \sin \phi - R}.\]
Using the variable \(a = h/R\), we can try to solve for \(\phi\), which in this case leads to a quadratic equation. Let us limit \(\theta < \pi/2\), since all other cases are trivial extensions.

If we let,  \[A = a^{2} \tan^{2} \theta + a^{2} + 2 a \tan^{2}\theta + 2 a + \tan^{2} \theta,\] and \[B = a \tan \theta + \tan \theta - 1,\] then the two solutions are: \[\phi = -2 \tan^{-1} \left( \frac{a + 1 \pm \sqrt{A}}{B} \right).\]
Due to the geometry, acceptable values of \(\theta\) and \(\phi\) lie in the region \([0, \pi/2]\). This constraint may be used to cull out the extraneous root.

It is useful to consider the limit of \(a = h/R \ll 1.\) If we take the limit as it goes to zero, then  \[\phi = -2 \tan^{-1} \left( \frac{ \pm|\tan \theta| + 1}{\tan \theta - 1} \right).\]

Monday, June 8, 2015

Cloudy Puzzle

When I was walking to my office today, I was struck with this potentially interesting applied math problem.

I was looking at the sky, due east, I saw a fat and juicy rain cloud hanging at an angle of about 60 degrees to the horizon. The question that came to my mind was:

"How far away is the place where this particular rain cloud is directly over-head?"


To make the problem tractable, let us assume (i) wind effects are negligible, (ii) the earth is a perfect sphere, (iii) the local topography is perfectly flat (no mountains or valleys).

These simplifications allow me to draw a conceptual picture like:

The inner circle represents the earth with radius \(R\). You are at the point R in the picture. You observe the cloud at position P, which makes an angle of \(\theta\) with the horizon denoted by the line H1-H2.

Let us assume that this rain cloud is at a height \(h\) from the surface of the earth, so that the length of the line segment OP is \(R+h\).

Clearly, the picture is not to scale!

S is the location where the cloud is directly overhead. We want to find the length of the arc RS.

To summarize, we know \(R, h,\) and \(\theta\). If we can somehow find the angle \(\phi\) then we can infer the length of the chord RS, which would be the distance we seek.

Tuesday, June 2, 2015

Pinker and Taleb

Interesting back and forth between two heavy-weights: Steve Pinker and Nassim Taleb. If you ignore the drama (or perhaps get interested because of it), the debate has spurred a fair amount of interesting commentary, which makes for fascinating reading.

1. David Roodman offers some insights in two blog posts. Here is how he sets up the debate:
Two intellectual titans are arguing over whether humanity has become less violent. In his 2011 book, Steven Pinker contends that violence is way down since the stone age, or even since the Middle Ages. He looks at murder, war, capital punishment, even violence against animals. 
But in a working paper released yesterday, Pasquale Cirillo and Nassim Taleb, the latter the author of The Black Swan: The Impact of the Highly Improbable, contend that Pinker has it wrong. Well, more precisely (lest I incite a riot with demagoguery) they challenge the notion that the great powers have enjoyed a distinctly long peace since World War II.
He sums up the state of the attack on Pinker's argument as being pretty flimsy.
I think if you are going use statistics to show that someone else is wrong, you should 1) state precisely what view you question, 2) provide examples of your opponent espousing this view, and 3) run statistical tests specified to test this view. Cirillo and Taleb skip the first two and hardly do the third. The “long peace” hypothesis is never precisely defined; Pinker’s work appears only in some orphan footnotes; the clear meaning of the “long peace”—a break with the past in 1945—is never directly tested for.
2. Dart-Throwing-Chimp breaks the gist of the debate using a more common metaphor. His position expresses skepticism to the question "can we use data to answer a question like that?"
Of course, the fact that a decades-long decline in violent conflict like the one we’ve seen since World War II could happen by chance doesn’t necessarily mean that it is happening by chance. The situation is not dissimilar to one we see in sports when a batter or shooter seems to go cold for a while. Oftentimes that cold streak will turn out to be part of the normal variation in performance, and the athlete will eventually regress to the mean—but not every time. Sometimes, athletes really do get and stay worse, maybe because of aging or an injury or some other life change, and the cold streak we see is the leading edge of that sustained decline. The hard part is telling in real time which process is happening. To try to do that, we might look for evidence of those plausible causes, but humans are notoriously good at spotting patterns where there are none, and at telling ourselves stories about why those patterns are occurring that turn out to be bunk. 
The same logic applies to thinking about trends in violent conflict. Maybe the downward trend in observed death rates is just a chance occurrence in an unchanged system, but maybe it isn’t. [...] Just as rare events sometimes happen, so do systemic changes.
3. Michael Spagat puts in his two cents
[...] the only channel that Cirillo and Taleb implicitly empower to potentially knock their war-generating mechanism off its pedestal is the accumulation of historical data on war sizes and timings. Since they focus on extreme wars, however, it will take a very long time before it is even possible for enough evidence to accumulate to seriously challenge their assumption of an unchanging war-generating mechanism.