the probability of having your [health insurance] policy torn up given a massively expensive condition is pushing 50%. One in two.

Assuming for the moment that that is correct, there’s really only one word to properly describe individual health insurance: a scam.

And shocking as it may be, I see no reason to doubt the article’s analysis. In a prepared statement, the CEO of Assurant Health recently informed Congress that

Rescission is rare. It affects less than one-half of one percent of people we cover.

This sounds pretty minuscule at first. However, it’s only for a small minority of very sick people that the insurance company has any financial incentive to cancel the policy — they are happy to take your money so long as you’re healthy.

As the article describes, no more than 5% of a insurer’s health policies cost the company enough to give it any financial incentive to tear up the policy. Furthermore, it’s only the sickest, most expensive 1% of the company’s policies (those which pay out more than $35000 per year) that it really has a strong incentive to rid itself of. If only these 1% of cases are targeted for rescission, then given that you are suffering a major medical problem costing more than $35000 per year, the chance of the insurance company booting you out is 50%.

Now, individual health insurance contracts certainly could be written to protect the insurance buyer from being defrauded by the insurer in this way. But they aren’t. The core problem is that health insurance policies are by their nature hideously complicated beasts (compared to say, fire or life insurance policies, which are far simpler). Before you sign one, you’d be well advised to hire a doctor, a contract lawyer, and a mathematician review it; the insurance company most assuredly has hired such people to help it write its health policies.

But in practice, no one goes to that much trouble and expense to evaluate a health policy. Consequentially, health insurance contracts are written entirely to protect the interests of the insurer, not the individual consumer.

A lot of regulation aimed at the health insurance industry is misguided and does more harm than good for the people who want to buy health insurance. The issue of rescission, however, is one case where regulation is badly needed.

The problem is that the measured length of an island’s coastline depends on the scale at which you measure it. Using a shorter ruler to measure the coastline captures smaller nooks, and so the measured length of the coast increases as the length of the ruler used decreases. Wikipedia provides an excellent visual presentation of the problem. Consequentially, it’s more or less meaningless to state that the coast of Iceland is 4970 km long without stating the length of the ruler used to produce that number. Wolfram really ought to know better.

Sam considers whether the most unlikely possible event is not presently occurring. By ‘the most unlikely possible event’, Sam intends to designate whatever was, at the immediately preceding instant, t – 1, the possible event most unlikely to occur at the next instant, t, which is the moment at which her deliberation occurs. Sam understands the proposition in question. Solely in virtue of that understanding, it seems to her—i.e. she intuits—that the proposition is true, though not necessarily so. On the basis of this intuition, she believes that the most unlikely possible event is not presently occurring. Her belief is true.

To be a bit more precise, Sam starts out believing “X will not occur” with probability 1-ε. By hypothesis, she has some justification for believing so. After drawing on her powers of intuition, she believes “X will not occur” with probability 1. Turri claims that this tiny increase in certainty is justified.

Turri never explains why he builds this argument using the “most unlikely” event instead of an unlikely event, but he does admit that ε must be small in absolute terms for his argument to work. He arbitrarily proposes an upper bound of 10^-18 for ε.

I maintain that Sam’s intuition has committed a rounding error. Under the circumstances Turri sets up, there is abundant reason to mistrust human intuition. One need travel no farther than the nearest casino to observe humans attempting to do probabilistic reasoning by intuition. Human intuition doesn’t just commit tiny rounding errors, it commits big errors — errors of much greater magnitude than 10^-18 — and commits them quite frequently. Given this extremely spotty record, an optimist might hope that Turri would proceed to demonstrate that in this particular case intuition guides us true.

If [a philosopher] reproached Sam, “Pardon me, Miss, but you of course meant that [the unlikely event] is almost certainly not occurring,” she could rightly respond, “Oh, come on! It’s not occurring, and we all know it.”

That’s Turri’s argument.

Actually, there’s a little bit more to Turri’s argument, but it’s really nothing more than semantic smoke and mirrors. He worries extensively about the definitions of knowledge and assertion. Personally, I think that one can do epistemology quite well without ever using those words. But for the sake of argument, I’ll accept whatever definitions Turri happens to prefer. My question is this: are these concepts (however you’ve chosen to define them) suitable for addressing delicate probabilistic reasoning and small variations in degrees of certainty?

If the answer is No, then there simply isn’t any difference between gaining an ε of knowledge and gaining zero knowledge. There’s no longer a meaningful distinction between knowing/asserting that X is true with probability 1 vs. probability 1-ε.

If Turri wants to answer Yes then I want to see him explicitly wed those concepts to the axioms of probability and redo his argument in a correspondingly rigorous fashion.

A study by the Centre for Science in the Public Interest showed that soft drinks were the single biggest contributor to calories in the American diet…

This sentence conveys no information whatsoever. An example will serve to illustrate the problem. Suppose that every day I drink two non-diet sodas (about 100 kcal each) and eat 1800 kcal of stew. Thus the “single biggest contributor to calories” is stew, at 90% of my total energy intake. Now, instead of just stating that I eat “stew”, let’s break it down into components. Suppose my stew contains beef, rice, goat, barley, peas, clams, mushrooms, noodles, tomatoes, and tofu, with each of these 10 components constituting 10% of the energy of the stew. Then my diet consists of 180 kcal of each of those items and 200 kcal of soda, making soda the “single biggest contributor to calories” in my diet.

All of this mess can be avoided by simply stating that 10% of my energy intake comes from soda, which is ultimately the relevant number, and the number which The Economist should have provided.

Lastly, I’d like to highlight a comment by Jacob (not me), who points out that Swoopo stands to make a lot of money by dishonestly bidding up its own auctions.

Hat tip to Greg Mankiw.

Hat tip to The Quantum Ponitiff for finding this hilarious video (or should that be a mitre tip?).

Police in Orlando, Florida claim “a typical prostitute that’s HIV-positive could potentially infect more than 18,000 people a year.”

Taking into account HIV transmission probabilities, Classically Liberal estimates that SuperProsti would have to serve at least “9 million customers [per year]. Disney World only manages 27 million.”

It is true that the results of pure mathematics do follow from whatever axioms one starts with a priori of any empirical observations. Indeed, empirical observations are quite irrelevant to pure math (except in an inspirational role). However, mathematics by itself cannot tell us anything about the physical world.

A concrete example will help illuminate the relationship between math and the physical sciences. Let’s consider the theory of General Relativity. GR is a mathematical model, and there are a couple of different ways to look at a mathematical model.

A pure mathematician might just be interested in the abstract “mathematical formalism” of GR. This pure mathematician could study the Einstein field equations, et cetera, irrespective of their physical significance. Indeed, an alien mathematician in an alternate universe with completely different physics could study the same mathematical formalism and derive the same results, even though the theory and results would have no physical significance in that alternate universe.

However, there’s more to a mathematical model than just an abstract formalism. A mathematical model additionally makes the empirical claim that the behavior of the real world is analogous, in a specified manner, to the behavior of its abstract formalism. This linkage between the formalism and the real world is sometimes called “the interpretation” (especially in cases where there is controversy about just how to link things up, such as quantum mechanics).

To return to the example of GR, you can derive from its mathematical formalism (in a completely abstract, a priori manner) the existence of gravitational waves. However, this most certainly does not prove that gravitational waves exist in the real world. It might turn out that there are no gravitational waves and the real world doesn’t behave in the way the model says it should. It might turn out that the model is wrong.

In light of the empirical fact that the predictions of GR have thus far agreed with experimental tests, physicists believe with fairly high certainty that GR does accurately model the behavior of the real world. We have not yet experimentally observed gravitational waves, but you’d be hard-pressed to find a physicist willing to bet money against their existence. No one, however, thinks that we’ve proved, a priori, that gravitational waves exist. If scientists did think their existence had been proved a priori, we wouldn’t bother going to the great trouble and expense of attempting to find empirical evidence for them.

In some cases, the correspondence between a mathematical model and the real world is so obvious and uncontroversial that no one bothers to explicitly lay out the formalism’s interpretation or the empirical evidence for its correctness. Consider a basket with 2 apples in it. Now toss in 2 more apples. Examine the basket, and you will find (surprise!) 4 apples. However, you cannot prove a priori that there will be 4 apples in the basket. It is an empirical question, albeit a trivial one, whether baskets of apples (which are physical things) behave in the same manner as the non-negative integers under addition (which is an abstract logical construct).

This distinction might seem hopelessly pedantic at first, but you can easily go astray by ignoring it. For example, many people naively expect photons to behave in the same manner as integers under addition, but they don’t. “Number of photons” is not a conserved quantity in the way that “number of apples” is; photons can be created/destroyed, one photon can be split into two, et cetera. Richard Feynman tells an interesting story about trying, and failing, to explain to his father how photons can do these things. I strongly suspect his father was implicitly assuming that all “particles” — a rather misleading term — must behave like apples and integers under addition. However, once you clearly understand the distinction between abstract math and empirical questions, it becomes clear that there is no a priori reason to believe that photons behave like apples or integers.

Without empirical evidence, math can’t tell us anything about the physical world. Scientists have never, ever proven anything about the real world a priori, and never will.

• Several groups have found different ways to use the Casimir effect to levitate objects. The New Scientist has a nice summary; specific papers can be found on arXiv.org.
• Mike Dunford, The Questionable Authority, gives us an almost poetic description of the formation of the moon. A beautiful example of how science opens our eyes to the wonders of the universe.
• If you know of any internet dating sites with female tyrannosaurs who like to dress up as Maxwellian demons, please send a link to T-Rex (of Dinosaur Comics). Otherwise, he’ll probably just take out all that sexual frustration by stomping more humans.
• Laurie Snell explains why, contrary to what you may have heard, Oscar winners do not live longer. Amusingly, the reader is given a homework problem at the end: using analogously bad statistics, develop a simple model which shows that “breaking your hip increases your life expectancy”.

That’s all for now. Maybe later I’ll add some sciency posts of my own.

Hat tip to MarkCC for finding this.

This problem is the bane of anyone wanting to build tiny swimming robots. But now, as reported by PhysOrg, a team of scientists has devised an ingenious way to turn this bane into a boon by having a micro-robot “treadmill” across the surface. The robot would roll across the fluid surface in the same way that a tank rolls across solid ground; what is marvelous is that this works precisely because the viscous forces are so large on this length scale that the fluid surface is, in effect, solid. The analysis shows that this method of locomotion loses very little energy to friction.

This idea seems obvious in retrospect, but there are a lot of smart people who didn’t think of it first. In any case, it’s a really beautiful solution.