Hooper’s Impossibility Theorem

The Sticky Business of Healthy Peanut Butter

A friend watched me spread peanut butter on my bread and suggested that I ditch my “mass market” peanut butter and instead use the “healthy” kind that separates and must be stirred each time. I sort of grunted and filed this away under the heading of “things to think about.”

Later, I realized that I don’t like “healthy” peanut butter. I don’t like being required to stir it. Plus, it is lumpy and it doesn’t spread as easily—part of it is too solid and part is too liquid—resulting in ripped bread, and I don’t like the taste as much. As a result, I stubbornly kept buying and eating my “unhealthy” peanut butter, knowing that I would inevitably be questioned about my choice of spread.

Recently I watched an advocate of the “healthy” kind wrestle with a jar of particularly recalcitrant peanut butter and end up throwing out the whole mess. I kept quiet.

This topic has piqued me for years. How did my friend know that one jar of peanut butter was healthier than another? Obviously, someone else who supposedly thoroughly researched the topic had told him. But how did that person get good data? And even if the “healthier” peanut butter is healthier, is it a lot better or just a little? Would someone who eats peanut butter three or four times a month even notice a difference? Would the “healthy” peanut butter prevent the cancer that the other type of peanut butter would have given me? And, further, health isn’t my only objective—I want to enjoy my eating experience. I want to enjoy my life, and I dislike the messy job of stirring peanut butter. I would happily trade a small amount of “health” for a bit of convenience and taste. Further, if the perceived cost of eating peanut butter increases, perhaps I’ll switch to unhealthier alternatives and be worse off.

Ultimately, I concluded that there’s not a single person on this Earth who could tell me how much healthier the old fashioned, “healthy” peanut butter would be for me. This led to Hooper’s Impossibility Theorem, fashioned after Kenneth Arrow’s Impossibility Theorem regarding voting in a democracy.

Hooper’s Impossibility Theorem: It is impossible to know a priori the precise effects of a specific dose of a certain substance on a particular person.

Two weeks ago, the Food and Drug Administration revoked its 1969 approval of artificial red food dye—specifically FD&C Red Dye No. 3—the coloring found in beverages, snacks, cereals, and candies. Interestingly, the FDA’s press release basically explained why the fear of red dye is overblown: the cancer that red dye gives male rats in laboratory experiments does not apply to humans and the laboratory experiments showed that male rats given lower doses—doses closer to normal human exposure—caused no cancer. This press release made me a little sorry for the FDA, forced into an action that isn’t supported by data.

How dangerous (or beneficial) is the consumption of red food dye for a given person? I claim that it’s impossible to know with any certainty and, in the absence of certainty, the FDA should leave the matter to individuals.

As a thought experiment, let’s consider an 11-year-old boy eating a bowl of cereal manufactured with FD&C Red Dye No. 3. This boy lives in Bandon, Oregon. He’s Caucasian, of normal weight, a better-than-average student, moderately active, a new entrant to the world of reading outside of school, an active video-game player, somewhat sloppy, and, ultimately, a lazy person without a clear career or life plan. Oh, in addition to cereal, he likes peanut butter and jelly sandwiches.

Problem 1: How could this food dye affect him?

FD&C Red Dye No. 3 could cause him to live a shorter or longer life, cause or prevent cancer, cause or prevent heart disease, cause or prevent diabetes, change the way his body absorbs vitamins, change his metabolic rate, change his mental state, etc. There are virtually an unlimited number of ways that this food dye could affect this boy and to know the full effect on him, we would need to measure all of them repeatedly.

Problem 2: How could we run an experiment?

We could have two groups of laboratory mice and give half of them red dye and measure the differences between the two groups over time. But how do we know the two groups of mice are otherwise identical? Were they genetically identical? Did they have the same diet? Were their gut microbiomes the same? Were their cages held at the same temperatures?

What dose should be given, how long should we run the study, and what should we measure? And does the very act of measurement change the mice?

And then, ultimately, we must ask whether the results we get are applicable to an 11-year-old boy. In medical parlance, mice aren’t a perfect model for humans.

We could put the red dye on human tissue samples and see what happens. But that doesn’t mimic what happens in the body because food is aggressively processed by the digestive system, starting with saliva in the mouth and a highly acidic environment in the stomach.

We could run an experiment with two groups of 11-year-old boys, giving half of them dye and half not. But can we adequately control such an experiment? Will any of the boys eat red dye outside of the experiment, perhaps at a friend’s birthday party? Will they be careful enough for us to be sure that some of them got the dye and others didn’t?

What dose should be given, how long should we run the study, and what should we measure? And does the very act of measurement—weekly blood pressure checks, for instance—change the boys?

And then, ultimately, we must ask whether the results we get are applicable to this particular 11-year-old boy. In medical parlance, one 11-year-old boy isn’t a perfect model for another 11-year-old boy. People have different genetics, environmental factors, gut microbiomes, fat consumption, physical activity levels, intelligence, emotional states, etc.

Instead of looking at groups of 11-year-old boys, we could feed this particular 11-year-old boy red dye and follow him closely, measuring everything we can think of. But we still have five problems. 1: Did the very act of measuring him repeatedly in a health care setting change him? 2: What would have happened to him without the dye? In other words, we need to address the counterfactual. 3: Even if we come to a conclusion after 30 years, is the result too late to be helpful? 4: Such a study will be expensive, especially if we study the hundreds of other food ingredients he consumes. Would the benefits even approach the costs? 5: If he eats hundreds of other things, how will we separate the effects of red dye from all the other ingredients?

This is not to say we know nothing. For instance, we do know that the smallpox vaccine is highly effective at preventing smallpox infections, with about 95% of people developing protection within 5 to 10 days of receiving a single dose. We know that folic acid, taken early in pregnancy, is effective at preventing birth defects.

But we don’t know which regimen of folic acid will avert a birth defect and which person won’t develop full protection from a smallpox infection.

Which comes back to Hooper’s Impossibility Theorem: It is impossible to know a priori the precise effects of a specific dose of a certain substance on a particular person.

Even if the FDA has a couple of large, well-controlled studies that show clear and substantial deleterious effects of red dye on people—which, by the way, it doesn’t—there may still be people who will benefit from red dye and, most importantly, there may be people who would prefer to continue consuming products containing red dye, even if they were aware of the deleterious effects. Or, stated differently, if the FDA has this evidence, it would be better to warn us to avoid products containing red dye and not ban red dye outright. That way we can make decisions that take into account our own particular circumstances, circumstances of which the FDA has zero information.

Some possible questions for skeptics

What about large doses of extremely dangerous substances, such as radiation or cyanide? While we can generally rule out long-term benefits, we can’t know with much clarity how quickly a particular person will suffer from a particular dose. Plus, these substances are present naturally and they are often used for other purposes, which means that getting rid of them isn’t practical. And so we fall back on general rules of thumb.

What about withholding clearly beneficial substances, such as water? For instance, what would happen to a person who has been denied water for 15 days? Well, they would die of dehydration. But we can’t know a priori how quickly they will die. One person might die after three days while another would last six. It is impossible to know beforehand.

Conclusion

Because we can’t know the exact benefit or harm anything will provide to someone, and enforcement is always problematic, authorities should never prohibit nor mandate the use of any substance, even mRNA COVID-19 vaccines. Recommendations, however, are a good idea, provided they are backed by solid data and analysis. With peanut butter, this is a sticky matter.

Hooper’s Impossibility Theorem explicitly states what doctors have known for years. Doctors tell patients to take Drug X and report back later. Doctors don’t tell patients that taking Drug X will cause them to lose 12 pounds or reduce their systolic blood pressure by 22 mm Hg. Doctors practice Hooper’s Impossibility Theorem; I simply spelled it out.