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A paradox is a situation in which seemingly profound and rigorous reasoning leads to an impossible result -- or, in the case of reductio ad absurdum, an absurd result. Writers often encounter this in the guise of an overextended metaphor; metaphors are simplifications of reality, not full descriptions thereof, so you can only extend them so far before they crumble under their own weight. Paradoxes form a specific sub-class of the larger category of contradictions: two or more conclusions about a given subjective or physical aspect of reality that point in different and incompatible directions. Paradoxes represent a large red flag that tells us that we don’t understand something nearly as well as we think we do. In the case of reductio ad absurdum, the misunderstanding most often depends on our forgetting that there are limits or constraints to even the most logical statements (probably including this one!) and that we failed to account for them in framing our point of departure on this particular chain of logic.

A common problem I encounter in my scientific editing arises from an author’s assumption that a relationship between two variables is linear simply because they see what appears to be a single overall trend in a scatterplot (the pattern in a graph formed by the various combinations of values of the variables). This leads to an occasional paradox that is easily explained by taking a step back to examine the basis (the underlying assumptions) for inferring a single trend. In nature, most phenomena are bounded, and don’t follow a constant pattern ad infinitum. For example, trees can only grow to a certain maximum height before they topple over, and as they approach that height, their height growth slows because increasing amounts of energy must be devoted to building a thick stem and a widespread root system that keeps them from falling over when the wind blows*. Thus, the tree’s growth curve (a graph that shows its height at a given point in time) resembles something of a tilted and horizontally stretched S, referred to as a sigmoidal growth curve. There are many other bounded biological functions, and they have their roots in something profound: biological systems need to maintain homeostasis because homeostasis represents the state in which the system performs optimally. You can’t extrapolate much beyond the boundaries of that state without encountering contradictions.

* Trees are also limited by their ability to transport water up to the top of the tree, which is where the height growth occurs. (Trees grow from what’s called an apical meristem, not from the stump.) Water relations of trees involve many complex processes that we still don’t completely understood, though we’re probably about 95% of the way to a complete understanding.

Chemistry and physics also provide many paradoxes that lead us to important insights. For example, consider a chemical reaction in which ingredients 1 and 2 combine to produce a product. The rate of reaction between the two products is often constant, so that the amount of product increases at a constant rate. But eventually that rate will slow, even though we have complete faith in our observation that the reaction rate should be constant. This is a simple paradox because a little thought tells us that as we use up the ingredients, it becomes increasingly unlikely that the few remaining bits of ingredients 1 and 2 will encounter each other, thereby permitting a reaction to occur. This leads to the important realization that in any chemical reaction for which the inputs are limited, the reaction rate will slow down or decline abruptly to zero at some point as you run out of ingredients.

Physics provides more complex examples, such as the paradox of water. Most substances shrink more or less continuously as their temperature decreases, but water shows a radically different behavior: at some point, it instead begins to expand. (This is why water that enters cracks can shatter stone as it freezes: it begins to occupy more space than the space it originally filled, and having nowhere to go, forces the constraining materials around it to move apart to make room. For something relatively rigid like rock, the substance shatters. For something plastic (capable of stretching), like a metal water bottle full of water that has been left too long in the freezer to chill, the substance stretches, and sometimes the stretch is irreversible*.

* Yes, that happened to me. I now own a spherical-bottomed water bottle that can no longer stand up on its own.

To many, a paradox is something frustrating to be avoided, because it introduces undesired complexity into a situation we thought we understood. Worse, it may require us to rethink our understanding of a problem we’d considered until now to be “solved”. But that’s the wrong way to look at a paradox. Rather, the power of paradox is in how it leads us to re-examine a flawed understanding and sometimes learn something new and important. Consider a situation in which we have two conclusions, A and B. The two may be reached by entirely different chains of reasoning based on entirely different bodies of evidence (a process called “triangulation”), or B may be a logical consequence of A, arrived at through logic rather than experimentation. The important thing is that in both cases, they contradict each other, or seem to. Fortunately, we have a relatively limited number of possible explanations for the contradiction:

If A is right, B can be right, wrong, or partially right.
If A is wrong, B can be right, wrong, or partially right.
If A is partially right, B can be right, wrong, or partially right

This gives us nine possible explanations we must test to determine which combination explains the contradiction.

One of the things I love about working with really good scientists is that they don’t throw up their hands in despair when they encounter a paradox. Instead, they roll up their sleeves and set about the hard work required to narrow down the possible explanations. Arthur Conan Doyle famously had Sherlock Holmes state that “when you have eliminated the impossible, whatever remains, however improbable, must be the truth”. Hart’s corollary is that when you encounter a paradox, you should apply Holmesian logic to identify whether A, B, or both are incorrect, whether totally or partially, so that you can revise the offending assumption or propose a constraint that defines its boundaries. That way lie the big discoveries.

Note that this way of thinking isn’t limited to the sciences or philosophy. It’s also a powerful tool for thinking through problems with human interactions*, personal problems, plot problems in fiction, and other whole categories of thought problem. So the next time you encounter a seemingly paradoxical situation, allow yourself a moment to experience frustration, or even existential despair about the abnegation of your cherished notion. Then take a deep breath, muster your courage, and re-examine that notion to determine which aspects that led to it in the first place are right, wrong, or partially right. The resulting insights can be amazing.

* Psychologist George Miller described this as follows: "To understand what another person is saying, you must assume that it is true and try to imagine what it could be true of." Communication often fails when you (person A) and your partner in paradox (person B) have different understandings of just what it is you’re talking about. As the saying goes, “it ain’t always about what it’s about”.

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