The Paradox Paradox

By denying scientific principles, one may maintain any paradox.
Galileo Galilei

Paradox: [Latin paradoxum, from Greek paradoxon from neuter sing. of paradoxos, conflicting with expectation, para-, beyond; see para–¹, + doxa, opinion (from dokein, to think; see dek-).]

(noun)

1. A seemingly contradictory statement that may nonetheless be true: the paradox that standing is more tiring than walking.
2. One exhibiting inexplicable or contradictory aspects: “The silence of midnight, to speak truly, though apparently a paradox, rung in my ears” (Mary Shelley)
3. An assertion that is essentially self-contradictory, though based on a valid deduction from acceptable premises.
4. A statement contrary to received opinion.

This morning I ran across an article discussing the "paradox" that obesity seems to play a protective role in heart disease. We seem to be presented with a flood of paradoxes relating to health and nutrition - and indeed said paradoxes present equal confusion to (too) many scientists. Let's talk a bit about what a paradox really is, and then I'll show why the Galileo quote was right on the money. To say it another way, any scientist who cries "paradox" is being fundamentally unscientific. You'd never get them to admit it (because they probably don't believe it), but their use of paradox is in the sense of the 4th definition above, rather than indicating a true logical paradox. And we all know how well science and opinion mix.

Most paradoxes are only superficially paradoxical, and can be resolved on deeper inspection. Real paradoxes are rare. Consider this example from the Wikipedia entry on "paradox":

... consider a situation in which a father and his son are driving down the road. The car collides with a tree and the father is killed. The boy is rushed to the nearest hospital where he is prepared for emergency surgery. On entering the surgery suite, the surgeon says, "I can't operate on this boy. He's my son."

Sounds paradoxical, right? But the issue is simply a bad assumption: since most surgeons are men, one erroneously extrapolates that ALL surgeons are men. Obviously the surgeon must be the boy's mother. This is a common source of claimed paradoxes in science: extrapolating something that is believed at some level (e.g. obesity causes heart disease) to a statement of absolute truth.

Let's consider mathematics, starting with simple Boolean logic. The point of logic is to reason deductively about the truth of a statement, given the truth of other statements. A paradox would imply you could get different answers depending on how you worked through the problem, i.e. two different sets of steps valid within the rules of logic would give different answers. If such paradoxes did exist, they clearly render logic useless, since you could never consistently prove something true. The dictionary definition of "paradox" admits a subtly different situation, which is a statement like "I am a liar". The rules of logic can neither prove nor disprove this statement. But this more an artifact of language and technical aspects of formal mathematical systems as opposed to the sort of "scientific paradox" claimed by the authors of the heart disease/obesity paper.

Generalizing the case of logic to all math leads to the same conclusion. A mathematical system which admits true paradoxes is pointless. A true paradox would indicate inconsistency in the rules and assumptions used to build the system. Problems labeled "paradoxical" in math are really counter-intuitive, like the Banach-Tarski Paradox, where one can prove that there is a way of dividing up a 3-dimensional ball, moving the pieces around without stretching them, and reassembling to get two balls of the same size as the original. Sounds pretty paradoxical, right? But it's really just counter-intuitive: the size of the set of points in the one ball (called the cardinality) is actually the same as the size of the set of points in the two balls. The size of a set is different than it's measure (which in this case would be the volume). The result that we can double the volume of a set of points without changing the cardinality of that set violates our intuition, but is consistent within the mathematical definitions of measure and cardinality (this is roughly equivalent to realizing that that size of the set of even integers is the same as the size of the set of all integers: they're both infinite).

Can we ever have a true scientific paradox? Mathematical truth is purely conceptual, and can thus be "absolute". We define the axioms and rules and mentally manipulate these to prove or disprove other statements. Science is messier. Nothing is absolute in science, because all scientific theories must be supported by observational evidence from the real world. Our observations are limited by various practical considerations. Our data is never 100% accurate, we can never be sure we've observed all of the relevant variables, etc. So our belief in a scientific hypothesis is always conditioned on the evidence which itself is subject to limitations of our ability to observe and collect information. Scientific belief thus exists in a continuum between absolute truth and falsehood, and is always conditioned on the available evidence. As new evidence is obtained, we update our beliefs accordingly toward greater or less truth as indicated by the new evidence.

So you can never have a scientific paradox. Scientific honesty demands that observation of evidence contradicting a hypothesis causes you to lower your belief in that hypothesis. A paradox requires two statements which can be shown to be contradictory yet simultaneous true. But neither evidence nor hypotheses carry absolute truth, and our beliefs in either are always conditioned on the other. The scientifically relevant method evaluates belief of hypotheses conditioned on evidence.

Science as most often practiced, using frequentist statistics, evaluates belief in data assuming the truth of a hypothesis, so it's no wonder scientists spend so much time confused about "paradoxes". Take a hypothesis and data that appears to contradict that hypothesis. Then try to test the hypothesis quantifying your belief in the data presuming truth of that hypothesis. When the number comes back low, you basically have two choices: come up with a reason why the data is "wrong" (e.g. a mistake in experimental design, broken instrument, drunken graduate student), or realize that your hypothesis (again, whose truth was assumed as part of the analysis) is possibly not true. If you believe your data AND are 100% convinced of the hypothesis (which begs the question of why you did the experiment in the first place), you'll think you've got a paradox. The only real paradox is that people get paid to make this fundamental error in inference - over and over and over . . .

Our friends who observed the apparently paradoxical protective effect of obesity in heart disease patients have fallen into this trap. The right thing to do upon observing this effect is to update belief in the hypothesis that obesity causes heart disease. The new evidence lowers our belief in that hypothesis, and simultaneously signals that we should evaluate competing hypotheses in the light of all of the available evidence. Indeed, if one were to do a proper analysis of the evidence, it would be clear that no more supports the hypothesis that obesity causes heart disease any more than it does the hypothesis that heart disease causes obesity. Not all heart disease patients are obese, and not all obese people suffer from heart disease. Further, there's no strong metabolic evidence indicating the arrow of causality.

The smart thing to do in such situations is to start looking at hypotheses where a third culprit is the underlying cause of the observed associated effects. So what might cause both obesity and heart disease, or in some people one but not the other?

A growing body of evidence links poor blood sugar control to heart attack risk (see this recent study, for instance). The body maintains blood glucose in a narrow range, because both too little or too much are dangerous. Too little and the brain starves. Too much and you overwhelm the systems which repair the damage caused by sugar, in particular that to the arterial lining. You cannot excrete excess blood glucose like you can excess water or salt (at least not without severely damaging the kidneys). So your options are to either store it, or turn it into something else. The muscles and liver have a limited capacity for storage of glucose. Once they're full, the liver, as directed by insulin, will turn the rest into fat, and your fat tissue, again as directed by insulin, will store that fat.

At least that's how it's supposed to work. Insulin is a hormone, and hormones activate genes to manufacture proteins. The response to a hormonal stimulus is thus partially determined by genetics. Your genes will determine, for instance, the relative expression of lipoprotein lipase and hormone sensitive lipase in response to insulin levels. This in turn governs the ability to take fat from the blood and store it, or release that fat from fat cells to be used as energy. Similarly one guesses that insulin sensitivity of muscle and liver tissue has some genetic basis, and these may further be altered by disease, nutrition, etc. (overconsuption of either alcohol or fructose will make the liver insulin resistant, thus impeding its ability to store glucose, transform it to fat, or scale back manufacture of glucose from protein).

In the framework of this hypothesis, a person with greater propensity towards fat storage has a potential advantage when it comes to heart disease, as it provides another "sink" for excess blood glucose. A perpetually skinny person may be at a disadvantage. If your fat cells don't respond to insulin signals, then the fat has nowhere to go and stacks up in your blood as "triglycerides". If your liver and/or muscle don't properly respond to insulin, glucose begins to build up in the blood. Neither situation is likely good for the development of heart disease, and in reality both seem to occur simultaneously for susceptible individuals.

The news blurb doesn't state whether blood glucose or triglycerides tested, and the publisher of Journal of the American College of Cardiology doesn't provide free access to the publication. Perhaps a reader with access can post a comment as to whether blood glucose was tested and the results. Regardless, it is the unwillingness or inability of the authors to consider alternative hypotheses which leads them to cry "Paradox!" in such a public manner. Such individuals are clearly mired in irrational dogma and/or trying to drum up extra funding. From a broader view, any hypothesis (like diet-heart) which embraces paradoxes (like the "French paradox") are probably junk science. Treat them accordingly lest you extinguish your own spark of reason.