Every pure mathematician has experienced that awkward moment when asked, “So what’s your research good for?” There are standard responses: a proud “Nothing!”; an explanation that mathematical research is an art form like, say, Olympic gymnastics (with a much smaller audience); or a stammered response that so much of pure math has ended up finding application that maybe, perhaps, someday, it will turn out to be useful. That last possibility is now proving itself to be dramatically true in the case of category theory, perhaps the most abstract area in all of mathematics. Where math is...

If greed is good, as Gordon Gekko proclaimed in the 1987 movie Wall Street, then economics ought to be a superlative science. After all, at the core of economic theory sits a greedy idealization of human nature known as Homo economicus. It’s a fictitious species that represents the individual economic agent, motivated by selfishness. H. economicus is completely rational, by which economists mean it’s out for itself. And selfishness is supposedly the smart strategy when competing for the resources needed to survive. As Gordon Gekko also mentioned, greed “captures the essence of the evolu...

Time after time, physicists have tried to explain time. Many claim to have succeeded. But they haven’t. Otherwise everybody would quit trying to explain it all over again. One of the most recent such efforts comes from the mathematician/cosmologist George F.R. Ellis. He thinks solving the time mystery involves figuring out the difference between the past and the future. That’s not as obvious as it sounds. Physical laws governing motion make no distinction between future and past. Equations describing the scattering of billiard balls on a pool table work equally well if all the balls retr...

Fight Club had its First Rule (don’t talk about Fight Club). The Transporter enforces Rule Number 1 (never change the deal). And NCIS Special Agent Leroy Jethro Gibbs observes Rule 1 (never mix the suspects together in the same room). Physics has the second law of thermodynamics. It’s weird when you think about it. Movies and TV shows always give their prime rule top billing. But the physics rule alleged by Sir Arthur Eddington to hold the “supreme position among the laws of Nature” is only Number 2. Nevertheless, scientists generally consider it the most unbreakable law of all. It i...

Science is not a democracy. Nature’s laws are not subject to the whims of popular vote. A scientific theory succeeds by providing logical explanations for puzzling phenomena and making correct predictions about the outcomes of new experiments. It doesn’t matter how many scientists believed in the theory beforehand (or even afterward, for that matter). In fact, revolutionary new theories are seldom very popular. As Max Planck, the founder of quantum theory, once noted, sometimes a theory doesn’t get widely accepted until its opponents die. Nevertheless, in certain scientific matters it�...

Fermat’s Last Theorem is so simple to state, but so hard to prove. Though the 350-year-old claim is a straightforward one about integers, the proof that University of Oxford mathematician Andrew Wiles finally created for it nearly two decades ago required almost unimaginably complex theoretical machinery. The proof was a dazzling demonstration of that machinery’s value, but one aspect of it troubled mathematicians: It relied on stronger axioms than mathematics normally requires, and ones far more complex than are needed to state the problem. Surely, many mathematicians thought, it was poss...

For all the deference to “laws” of nature that supposedly govern everything that happens, the truth is that randomness rules the world. Everywhere you look, randomness is at work, in all the processes described by the mathematics of probability. The temperature of the air and the capriciousness of the weather all depend on random collisions of molecules. Computers operate on the principles of information theory, which is rooted in quantifying probabilities. Time rushes onward and disorder replaces order by virtue of the probabilistic second law of thermodynamics. Randomness determines eve...

In olden days, before the Star Trek holodeck and movies like TRON and The Matrix, philosophers used to wonder whether life was but a dream. Nowadays they’re more concerned that reality could be just a computer simulation. Sure, that’s not very likely. But you can’t rule out the possibility. Computers simulate all sorts of things, and some scientists have seriously suggested that nature’s supposedly rock-solid reality is really just some smart alien teenager’s science fair project. Most people respond to that suggestion with a shrug. What does it matter? You have to row, row, row yo...

Medicare could waste billions of dollars, bankrupt small businesses and leave seniors without crucial equipment like oxygen tanks and wheelchairs, some economists warn, with a new auction-based purchasing plan that ignores mathematical principles of competitive bidding. The new plan began with an apparently good idea: Medicare has started buying durable medical equipment through a competitive bidding process rather than through fixed prices influenced by industry lobbying. In theory, this could save the government a big chunk of the $14 billion it has spent on such equipment in a year. But in...

Human longevity is largely a modern phenomenon. (p. 10)