God Plays Dice

Necessary but not sufficient: 596,000 google hits.

Sufficient but not necessary: 134,000 google hits.

Exercise for readers: explain the vast gap here. It seems like the two should be equally common, but when a friend of mine used "sufficient but not necessary" that sounded strange to me; that's what led me to Google, which shows that indeed this phrase is much less common than the reverse.

In 1736 Euler solved the following problem: the city of Königsberg is set on both sides of the Pregel river and on two islands between them. There are bridges connecting the various landmasses; is it possible to walk around the city in such a way that you cross each bridge exactly once? The answer is no; (the network of landmasses and bridges in) Königsberg didn't have an Eulerian path. In order to have an Eulerian path the graph corresponding to this network must have zero or two nodes of odd degree; that is, if we consider the number of bridges on each landmass, exactly zero or two of these numbers can be odd. In Königsberg all four degrees were odd.

But the bridges were bombed in World War II, the city was renamed Kaliningrad, and only five of them were rebuilt. These are bridges connecting each of the islands to each of the shores, and a brige connecting the two islands. As you can see, there are three bridges on each island and two on each shore of the river; two of these numbers are odd, so there exists an Eulerian path. It's still somewhat useless, because you have to start on one island and end on the other.

Here's a map of the route, from Microsiervos (in Spanish), and here are some pictures that some folks took when they were visiting Kaliningrad and actually doing this.

I also remember once seeing the analogous network for New York City (the relevant landmasses being Manhattan, Long Island, Staten Island, the Bronx, and New Jersey), which has a lot of bridges, with the question of whether that network had an Eulerian path. I don't remember the answer. I also think it wouldn't be as much fun; New York has lots of traffic and is much larger than Kaliningrad.

Through my logs, I came across a forum where people have pointed to a post on this blog.

They then veer off into saying things about religion. I suspect this may be due to the title of this blog.

I just want to state that "God Plays Dice" has nothing to do with the Judeo-Christian-Islamic-etc. deity. It is a reference to the following quote of Einstein, in a letter to Max Born:

Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secrets of the Old One. I am at any rate convince that He does not play dice."

(I'm copying this out of Gino Segre's Faust in Copenhagen; it's originally from Einstein's letter to Born, December 4, 1926, which is reprinted in The Born-Einstein Letters.) The "Old One" to whom Einstein is referring here was, as far as we know, not what is usually meant by "God"; I suspect that this is why the translator (Irene Born) chose this translation, although I don't know what Einstein said in the original German. To be totally honest, I don't know if the original was even in German.

The purpose of the title is that I feel that probability is an important tool for understanding the world, which Einstein may have been a bit skeptical about, at least in the case of quantum mechanics. And there's something of a tradition in the titling of math blogs of taking sayings of well-known mathematicians and "replying" to them. (By "tradition" I mean The Unapologetic Mathematician also does it, in response to Hardy's A Mathematician's Apology.)

Also, for some reason I had thought it was Bohr, not Born, that he wrote this to. I suspect this is because I've heard more things about Bohr than Born, and they sound similar.

I suspect the people at the forum in question won't read this, though. But making this post makes me feel like I've replied to them.

edited, 5:56 pm: I was wondering if there were any blogs whose titles riff on the quote that "A mathematician is a device for turning coffee into theorems" (usually attributed to Erdos, but supposedly actually due to Renyi). I found Tales from an English Coffee Drinker. The quote from Goethe, "Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different", also would be good as a source for a blog title.

Here's something interesting from a New York Times poll a couple weeks ago. People were asked what percentage of all Americans are black. Results include that 8 percent of whites, and 17 percent of blacks, guessed that more than 50 percent of all Americans are black. (It's question 80 in the poll.)

The actual figure, from the 2006 census estimates, is 12.4 percent. (If you had asked me, I would have probably said twelve percent, which is the figure I learned quite some time ago.)

Jordan Ellenberg, who linked to this poll, asks whether people are ignorant of what "50 percent" means, or whether they're ignorant of the actual makeup of the United States population. I'm not sure how to answer this.

But I'd be interested to know how people's guesses of the percentage of the population which is black are correlated with the percentage of the population in their immediate area which is black. People probably expect that the people around them are representative of the general population, because psychologically we may be wired that way; numbers, even numbers obtained from counting millions of people, just don't have the same psychological impact as the faces you see while walking down the street. (You might have to factor in some other things, though, such as people's choice of television shows, movies, etc.; subconsciously we might not be that good at distinguishing between people that we're seeing on television and people we're seeing in reality.)

Similar questions could be asked in other populations. For example, if you ask Philadelphians about the racial distribution of Philadelphia, what do they say? For black and white people, the answer is 44.3% black, 41.8% white, from this Census Bureau page with a ridiculously long URL. But most Philadelphians live in neighborhoods that are mostly black or mostly white, so I suspect you'd get a lot of extreme answers.

Although the extreme answers might not correspond to what people actually see day to day! There may be people living in mostly-white neighborhoods who think most Philadelphians are white, or people living in mostly-black neighborhoods who think most Philadelphians are black. But you might also see people living in mostly-white neighborhoods who feel like their neighborhood is one of the only places where white people live, and guess that the city is mostly black, or vice versa. (Note to people who know anything about Philadelphia -- I am not saying that such neighborhoods exist, or that I know which ones they are. I'm just saying I can imagine them.)

Yes, in my secret other life I want to study things like that.

Last week I wrote a post about population densities.

Take a look at the interesting graphs at Bill Rankin's Radical Cartography; they show how population density is related to:

• racial and ethnic groups (American Indians and Alaska Natives, not surprisingly, live at the lowest population densities; what surprised me was the large amount of Hispanic population at between 1 and 10 per square mile, which Rankin says might correspond to ranchers);


• age. Roughly speaking, people ages 18 to 39 or under 5 are overrepresented at "high" densities (above 4000 or so), and other ages are overrepresented at "low" densities (below that same cutoff). This is, I suspect, a reflection of people moving to the city when they leave their parents house, and then leaving the city when it's time for their kids to go to school.


• income is highest at suburban and central-city densities, with a valley in between. Not surprising; in general the central part of a city is rich, it's surrounded by poorer neighborhoods, and then eventually income starts going up again. Rural places are poor as well.


• gender -- there are more women at high density, which I can't explain.


• population and area -- I tried to make a plot like this but had some trouble, because I was just playing around with output from another web site and didn't have the raw data.

I'm currently watching Science Saturday at bloggingheads.tv, which this week features Peter Woit (Not Even Wrong) and Sabine Hossenfelder (Backreaction).

When the video started, I thought "hmm, Woit has an awful lot of yellow books behind him for a physicist".

Woit, it turns out, works in the mathematics department at Columbia, as I was reminded when he started talking about the different job situations in physics and mathematics about fifteen minutes in. Basically, Woit says that jobs in physics are scarcer compared to PhDs in physics than the analogous situation in mathematics, so physicists feel more pressure to "do everything right" -- in his view this means they feel unnatural pressure to work in string theory, which Woit sees as a bad thing. After all, what if string theory's wrong? Physics as a discipline should diversify.

Quite some time ago, the folks at 360 asked if there have been heads of state who were by training mathematicians. This is really two questions in one: people who were trained as mathematicians, and people who had a mathematical career before going into politics.

The first question doesn't seem that interesting, because it seems to include cases in which Politician X majored in math as an undergrad, then went to law school, became a lawyer, and then entered politics from the law, as so many do. That's not the question I want to answer.

For the second question, a bit of clicking around turns up this list, which inclues Alberto Fujimori (president of Peru), Paul Painlevé (prime minister of France), and Eamon de Valera (president of Ireland). Painlevé in particular made a name for himself as a mathematician; the other two appear to have at least taught it in some capacity at some point.

I had thought that Henri Poincaré had been in politics, but it appears that I was confusing him with his cousin Raymond. Borel served in the French National Assembly. I haven't done any sort of systematic sampling, but it seems like mathematician-politicians are particularly prevalent in France, that wonderful country where they name streets after mathematicians. (Here in the United States, for example in my native city of Philadelphia, we name streets after mathematical objects, namely the positive integers.)

One interesting close call is Einstein. The story has it that he was offered the presidency of Israel in 1952. Of course Einstein was a physicist, but given the title of this blog I feel I can mention him.