Ship of Theseus

Sadly, Futurama is ending, again. Since the entire series is available on Netflix, I’m trying to catch up on any episodes I may have missed. What I love about the show, in addition to the trenchant parodies of sci-fi, are some of the most advanced math ever to appear on mainstream TV. For example, the main plot of the episode Prisoner of Benda hinges on an new mathematical theorem created by the writer Ken Keeler, who has a Ph.D. in applied mathematics from Harvard.

In fact, many of the show’s writers have math and science backgrounds, and sprinkle in references liberally. Sometimes, classic philosophical puzzles also feature prominently. For example, in the Six Million Dollar Mon, Hermes incrementally upgrades his body with cybernetic implants, while Zoidberg keeps the discarded human parts for a ventriloquist’s dummy. The writers cleverly bring up the logical paradox usually called the Ship of Theseus: If you take a single plank from a boat and replace it with an identical one, almost everyone would agree that it is the “same” boat as before. But if you keep repeating this process, you can end up with a boat that consists of none of the “original” planks. Even worse, if you take all the discarded planks and build a second, identical boat, which of the two is the “real” one? Far from being a game of semantics, this process really occurs during biological processes like DNA replication, prion diseases,  and the ability of amoebae to move by breaking down and rebuilding actin filaments in their cytoskeleton.


In our digital world, computer files are copied and the originals overwritten almost constantly. We readily say that the file is the same, referring to the information as opposed to the physical bits. More speculatively, the transporter in Star Trek essentially obliterates the person being transported and recreates him or her somewhere else. Yes, very few people in the Star Trek universe refuse to use the transporter for this reason (see also the duplicating machine in the film “The Prestige“).

Perhaps this is the way we should think about life, or replicating crystals, in general. We should focus on the self-replicating information, as opposed to the particular physical realization.


I guess it is never good practice to try to play the Devil’s advocate for someones profoundly stupid remark – it would have been smarter for Mr. Bell to have donned full Yankee regalia and strolled the Boston Commons at night – but I just finished reading Jared Diamond’s (most recent) excellent book, and I think it sheds some light on the situation. Among other topics, the book discusses the sharp transitions that marked the evolution of human societies from a Band, to Tribe, to Chiefdom, finally State, in order of increasing population. In a great example of “more is different,” each classification represents a marked “regime change” (in the physics sense, as well as political). For example, a band is made up of about one or two dozen families, so it is still small enough for everyone to know everyone, decisions to be made by group consensus, and there is no need (or surplus resources to spend on) non-food-producing bureaucrats. In such a world, there is no need to legislate conduct between strangers, since the only strangers are hostile outsiders. In contrast, States and Chiefdom need to have some centralized executives, tax collectors and other officials, as well as laws for dealing with disputes among citizens. The increase population makes it possible for some of the members to be specialists in something other than food production.

In any case, it seems to me that a large part (but not all) of the Blue-State/Red-State divide in the US can be explained simply as a matter of population density. In Boston, the population is large enough that when a dangerous terrorist is on the loose, it is simply prudent to order all civilians to stay indoors, and let a specially trained squad of police with infrared-sensing helicopters apprehend the bad guy. Maintaining such a well-equipped force is only practicable for large cities with high population densities that produce enough surplus to be able to afford such things. Suppose instead that the suspect was at large somewhere in rural Arkansas. In that case, it might actually make sense to gather up a posse of AR-15 wielding citizens.

This extends to matters beyond guns, of course. Should the government take a more active role in supporting the poor? In cities, the job of taking care of so many (unknown) people in need seems like something only a centralized government could manage. In contrast, people in rural areas might prefer to take care of there own, since their needs are known to them – much more so than a distant bureaucrat. Having spent time in places like Mobile, Alabama and Manhattan, Kansas, I can say for sure than these problems look differently than they do in New York or Miami. The biggest proponents of gun control are, not surprisingly  the mayor of big cities, who see guns as implements of crime. In rural areas, guns may be in only protection if the nearest police station is many miles away.

Tax Day

In the US, April 15th is a sort of holiday, the deadline to file Federal Income Tax returns. In the minds of some, tax brackets are an important (only?) instance of a nonlinear function. That is, since we, as a society have decided to increase the rate that income is taxed goes up as the amount increases, we have a progressive tax system. However, this leads to complications (in addition to the complications of calculating the various deductions and credits to politically favored groups, such as homeowners) when deciding how much to withhold when both spouses in a couple work and file jointly. Professional tax preparers know that when the formula for withholding is applied separately to each income, the total withheld will generally be too low when the incomes and tax liabilities are combined. This is a great example of Jensen’s inequality, in which the average of a function will be the function of the average input value only if the function is linear: <f(x)> = f(<x>) holds if f is a linear function of x, but NOT in general for some other functionality. Specifically, if f is a convex function (the second derivative is positive, <f(x)> is greater than f(<x>). In the case of the IRS, imagine that the function f is the tax liability for x adjusted gross income (after deductions, etc.). In a couple filing jointly, the total withheld will be: f(x1) + f(x2), but the actual tax liability is f(x1+x2), which is GREATER than  f(x1) + f(x2), since the marginal tax rate increases for larger incomes. This means that too little was withheld and the couple will still owe money come tax day. Put another way, combining incomes puts more of the couple’s money into a higher tax bracket. For a while this meant couples actually paid more taxes – the so-called “marriage penalty” – but congress changed the tax structure to mitigate this effect. However, when calculating the withholding, each member of the couple is still considered individually.

For more, see the excellent and accessible book: The Flaw of Averages.

Quorum Sensing in Grasshoppers

Quorum sensing is just one example of how “more is different” rules in nature. The behavior of several species can undergo drastic changes under certain conditions, which can include a density of other individuals of the same species in close proximity. Slime mold cells called Dictyostelium, which live as individual amoebae under normal circumstances, can aggregate into a single slug-like creature when food is scarce, but only if there are enough cells around. Each dictyostelium senses the presence of its pals by chemical signals. Fascinatingly, the lack of food can also alter the behavior of much larger organisms, turning innocuous grasshoppers into swarms of ravenous locusts, but only if enough other grasshoppers are around:

From the NYT article

“As the grasshoppers crowd together, something shifts. The insects, which normally live alone, begin bumping into one another. When grasshoppers touch one another’s hind legs, the contact sets off hormonal changes: The adults’ neutral brown coloring is replaced with a fearsome bright yellow, and they become “gregarious” group insects, coordinating their growth, behavior and egg laying.”