How to Transform Gold into Mud: The Inverse Philosopher’s Stone

Let us take the same example as before, using as the function the square root (the exact inverse of squaring, which is concave, but much less concave than the square function is convex).

The square root of the expected (average) payoff is then √(1+2+3+4+5+6 divided by 6), equals √3.5, here 1.87. The function of the average equals 1.87.

But the average of the function is as follows. Take the square root of every payoff, (√1+√2+√3+√4+√5+√6), divided by 6, that is, the average square root payoff, and you can see that the average of the function equals 1.80.

The difference is called the “negative convexity bias” (or, if you are a stickler, “concavity bias”). The hidden harm of fragility is that you need to be much, much better than random in your prediction and knowing where you are going, just to offset the negative effect.

Let me summarize the argument: if you have favorable asymmetries, or positive convexity, options being a special case, then in the long run you will do reasonably well, outperforming the average in the presence of uncertainty. The more uncertainty, the more role for optionality to kick in, and the more you will outperform. This property is very central to life.

1 The method does not require a good model for risk measurement. Take a ruler. You know it is wrong. It will not be able to measure the height of the child. But it can certainly tell you if he is growing. In fact the error you get about the rate of growth of the child is much, much smaller than the error you would get measuring his height. The same with a scale: no matter how defective, it will almost always be able to tell you if you are gaining weight, so stop blaming it.

Convexity is about acceleration. The remarkable thing about measuring convexity effects to detect model errors is that even if the model used for the computation is wrong, it can tell you if an entity is fragile and by how much it is fragile. As with the defective scale, we are only looking for second-order effects.

2 I am simplifying a bit. There may be a few degrees’ variation around 70 at which the grandmother might be better off than just at 70, but I skip this nuance here. In fact younger humans are antifragile to thermal variations, up to a point, benefiting from some variability, then losing such antifragility with age (or disuse, as I suspect that thermal comfort ages people and makes them fragile).

3 I remind the reader that this section is technical and can be skipped.

4 The grandmother does better at 70 degrees Fahrenheit than at an average of 70 degrees with one hour at 0, another at 140 degrees. The more dispersion around the average, the more harm for her. Let us see the counterintuitive effect in terms of x and function of x, f(x). Let us write the health of the grandmother as f(x), with x the temperature. We have a function of the average temperature, f{(0 + 140)/2}, showing the grandmother in excellent shape. But {f(o) + f(140)}/2 leaves us with a dead grandmother at f(0) and a dead grandmother at f(140), for an “average” of a dead grandmother. We can see an explanation of the statement that the properties of f(x) and those of x become divorced from each other when f(x) is nonlinear. The average of f(x) is different from f(average of x).

BOOK VI

Via Negativa

R ecall that we had no name for the color blue but managed rather well without it—we stayed for a long part of our history culturally, not biologically, color blind. And before the composition of Chapter 1, we did not have a name for antifragility, yet systems have relied on it effectively in the absence of human intervention. There are many things without words, matters that we know and can act on but cannot describe directly, cannot capture in human language or within the narrow human concepts that are available to us. Almost anything around us of significance is hard to grasp linguistically—and in fact the more powerful, the more incomplete our linguistic grasp.

But if we cannot express what something is exactly, we can say something about what it is not—the indirect rather than the direct expression. The “apophatic” focuses on what cannot be said directly in words, from the Greek apophasis (saying no, or mentioning without mentioning). The method began as an avoidance of direct description, leading to a focus on negative description, what is called in Latin via negativa, the negative way, after theological traditions, particularly in the Eastern Orthodox Church. Via negativa does not try to express what God is—leave that to the primitive brand of contemporary thinkers and philosophasters with scientistic tendencies. It just lists what God is not and proceeds by the process of elimination. The idea is mostly associated with the mystical theologian Pseudo-Dionysos the Areopagite. He was some obscure Near Easterner by the name of Dionysos who wrote powerful mystical treatises and was for a long time confused with Dionysos the Areopagite, a judge in Athens who was converted by the preaching of Paul the Apostle. Hence the qualifier of “Pseudo” added to his name.

Neoplatonists were followers of Plato’s ideas; they focused mainly on Plato’s forms, those abstract objects that had a distinct existence on their own. Pseudo-Dionysos was the disciple of Proclus the Neoplatonist (himself the student of Syrianus, another Syrian Neoplatonist). Proclus was known to repeat the metaphor that statues are carved by subtraction. I have often read a more recent version of the idea, with the following apocryphal pun. Michelangelo was asked by the pope about the secret of his genius, particularly how he carved the statue of David, largely considered the masterpiece of all masterpieces. His answer was: “It’s simple. I just remove everything that is not David.”

The reader might thus recognize the logic behind the barbell. Remember from the logic of the barbell that it is necessary to first remove fragilities.