Consider the wheel again. If you are a Stone Age historical thinker
called on to predict the future in a comprehensive report for your chief
tribal planner, you must project the invention of the wheel or you will miss
pretty much all of the action. Now, if you can prophesy the invention of
the wheel, you already know what a wheel looks like, and thus you already
know how to build a wheel, so you are already on your way. The
Black Swan needs to be predicted!
But there is a weaker form of this law of iterated knowledge. It can be
phrased as follows: to understand the future to the point of being able to
predict it, you need to incorporate elements from this future itself. If you
know about the discovery you are about to make in the future, then you
have almost made it. Assume that you are a special scholar in Medieval
University's Forecasting Department specializing in the projection of future
history (for our purposes, the remote twentieth century). You would
need to hit upon the inventions of the steam machine, electricity, the
atomic bomb, and the Internet, as well as the institution of the airplane
onboard massage and that strange activity called the business meeting, in
which well-fed, but sedentary, men voluntarily restrict their blood circulation
with an expensive device called a necktie.
This incapacity is not trivial. The mere knowledge that something
has been invented often leads to a series of inventions of a similar nature,
even though not a single detail of this invention has been disseminated—
there is no need to find the spies and hang them publicly. In mathematics,
once a proof of an arcane theorem has been announced, we frequently
witness the proliferation of similar proofs coming out of nowhere, with
occasional accusations of leakage and plagiarism. There may be no plagiarism:
the information that the solution exists is itself a big piece of the
solution.
By the same logic, we are not easily able to conceive of future inventions
(if we were, they would have already been invented). On the day
when we are able to foresee inventions we will be living in a state where
everything conceivable has been invented. Our own condition brings to
mind the apocryphal story from 1899 when the head of the U.S. patent ofHOW
TO LOOK FOR B I R D POOP 1 73
fice resigned because he deemed that there was nothing left to discover—
except that on that day the resignation would be justified.*
Popper was not the first to go after the limits to our knowledge. In Germany,
in the late nineteenth century, Emil du Bois-Reymond claimed that
ignoramus et ignorabimus—we are ignorant and will remain so. Somehow
his ideas went into oblivion. But not before causing a réaction: the
mathematician David Hilbert set to defy him by drawing a list of problems
that mathematicians would need to solve over the next century.
Even du Bois-Reymond was wrong. We are not even good at understanding
the unknowable. Consider the statements we make about things
that we will never come to know—we confidently underestimate what
knowledge we may acquire in the future. Auguste Comte, the founder of
the school of positivism, which is (unfairly) accused of aiming at the scientization
of everything in sight, declared that mankind would forever remain
ignorant of the chemical composition of the fixed stars. But, as
Charles Sanders Peirce reported, "The ink was scarcely dry upon the
printed page before the spectroscope was discovered and that which
he had deemed absolutely unknowable was well on the way of getting
ascertained." Ironically, Comte's other projections, concerning what we
would come to learn about the workings of society, were grossly—and
dangerously—overstated. He assumed that society was like a clock that
would yield its secrets to us.
I'll summarize my argument here: Prediction requires knowing about
technologies that will be discovered in the future. But that very knowledge
would almost automatically allow us to start developing those technologies
right away. Ergo, we do not know what we will know.
Some might say that the argument, as phrased, seems obvious, that we
always think that we have reached definitive knowledge but don't notice
that those past societies we laugh at also thought the same way. My argument
is trivial, so why don't we take it into account? The answer lies in a
pathology of human nature. Remember the psychological discussions on
asymmetries in the perception of skills in the previous chapter? We see
flaws in others and not in ourselves. Once again we seem to be wonderful
at self-deceit machines.
* Such claims are not uncommon. For instance the physicist Albert Michelson imagined,
toward the end of the nineteenth century, that what was left for us to discover
in the sciences of nature was no more than fine-tuning our precisions by a few decimal
places.
1 7 4 W E J U S T C A N ' T P R E D I CT
Monsieur le professeur Henri Poincaré. Somehow they stopped making this kind of
thinker. Courtesy of Université Nancy-2.
THE NTH BILLIARD BALL
Henri Poincaré, in spite of his fame, is regularly considered to be an undervalued
scientific thinker, given that it took close to a century for some
of his ideas to be appreciated. He was perhaps the last great thinking
mathematician (or possibly the reverse, a mathematical thinker). Every
time I see a T-shirt bearing the picture of the modern icon Albert Einstein,
I cannot help thinking of Poincaré—Einstein is worthy of our reverence,
but he has displaced many others. There is so little room in our consciousness;
it is winner-take-all up there.
Third Republic-Style Decorum
Again, Poincaré is in a class by himself. I recall my father recommending
Poincaré's essays, not just for their scientific content, but for the quality of
his French prose. The grand master wrote these wonders as serialized articles
and composed them like extemporaneous speeches. As in every masterpiece,
you see a mixture of repetitions, digressions, everything a "me
too" editor with a prepackaged mind would condemn—but these make
his text even more readable owing to an iron consistency of thought.
Poincaré became a prolific essayist in his thirties. He seemed in a hurry
and died prematurely, at fifty-eight; he was in such a rush that he did not
bother correcting typos and grammatical errors in his text, even after spotting
them, since he found doing so a gross misuse of his time. They no
HOW TO LOOK FOR B I R D POOP 1 75
longer make geniuses like that—or they no longer let them write in their
own way.
Poincaré's reputation as a thinker waned rapidly after his death. His
idea that concerns us took almost a century to resurface, but in another
form. It was indeed a great mistake that I did not carefully read his essays
as a child, for in his magisterial La science et l'hypothèse, I discovered
later, he angrily disparages the use of the bell curve.
I will repeat that Poincaré was the true kind of philosopher of science:
his philosophizing came from his witnessing the limits of the subject itself,
which is what true philosophy is all about. I love to tick off French literary
intellectuals by naming Poincaré as my favorite French philosopher.
"Him a philosophe? What do you mean, monsieur?" It is always frustrating
to explain to people that the thinkers they put on the pedestals, such
as Henri Bergson or Jean-Paul Sartre, are largely the result of fashion production
and can't come close to Poincaré in terms of sheer influence that
will continue for centuries to come. In fact, there is a scandal of prediction
going on here, since it is the French Ministry of National Education that
decides who is a philosopher and which philosophers need to be studied.
I am looking at Poincaré's picture. He was a bearded, portly and imposing,
well-educated patrician gentleman of the French Third Republic,
a man who lived and breathed general science, looked deep into his subject,
and had an astonishing breadth of knowledge. He was part of the
class of mandarins that gained respectability in the late nineteenth century:
upper middle class, powerful, but not exceedingly rich. His father
was a doctor and professor of medicine, his uncle was a prominent scientist
and administrator, and his cousin Raymond became a president of the
republic of France. These were the days when the grandchildren of businessmen
and wealthy landowners headed for the intellectual professions.
However, I can hardly imagine him on a T-shirt, or sticking out his
tongue like in that famous picture of Einstein. There is something nonplayful
about him, a Third Republic style of dignity.
In his day, Poincaré was thought to be the king of mathematics and science,
except of course by a few narrow-minded mathematicians like
Charles Hermite who considered him too intuitive, too intellectual, or too
"hand-waving." When mathematicians say "hand-waving," disparagingly,
about someone's work, it means that the person has: a) insight,
b) realism, c) something to say, and it means that d) he is right because
that's what critics say when they can't find anything more negative. A nod
from Poincaré made or broke a career. Many claim that Poincaré figured
1 7 6 WE J U S T C A N ' T P R E D I CT
out relativity before Einstein—and that Einstein got the idea from him—
but that he did not make a big deal out of it. These claims are naturally
made by the French, but there seems to be some validation from Einstein's
friend and biographer Abraham Pais. Poincaré was too aristocratic in both
background and demeanor to complain about the ownership of a result.
Poincaré is central to this chapter because he lived in an age when we
had made extremely rapid intellectual progress in the fields of prediction—
think of celestial mechanics. The scientific revolution made us feel that we
were in possession of tools that would allow us to grasp the future. Uncertainty
was gone. The universe was like a clock and, by studying the movements
of the pieces, we could project into the future. It was only a matter
of writing down the right models and having the engineers do the calculations.
The future was a mere extension of our technological certainties.
The Three Body Problem
Poincaré was the first known big-gun mathematician to understand and
explain that there are fundamental limits to our equations. He introduced
nonlinearities, small effects that can lead to severe consequences, an idea
that later became popular, perhaps a bit too popular, as chaos theory.
What's so poisonous about this popularity? Because Poincaré's entire
point is about the limits that nonlinearities put on forecasting; they are not
an invitation to use mathematical techniques to make extended forecasts.
Mathematics can show us its own limits rather clearly.
There is (as usual) an element of the unexpected in this story. Poincaré
initially responded to a competition organized by the mathematician
G?sta Mittag-Leffer to celebrate the sixtieth birthday of King Oscar of
Sweden. Poincaré's memoir, which was about the stability of the solar system,
won the prize that was then the highest scientific honor (as these were
the happy days before the Nobel Prize). A problem arose, however, when
a mathematical editor checking the memoir before publication realized
that there was a calculation error, and that, after consideration, it led to
the opposite conclusion—unpredictability, or, more technically, nonintegrability.
The memoir was discreetly pulled and reissued about a year later.
Poincaré's reasoning was simple: as you project into the future you
may need an increasing amount of precision about the dynamics of the
process that you are modeling, since your error rate grows very rapidly.
The problem is that near precision is not possible since the degradation of
your forecast compounds abruptly—you would eventually need to figure
HOW T O LOOK FOR B I R D POOP 1 77
FIGURE 2: PRECISION AND FORECASTING
One of the readers of a draft of this book, David Cowan, gracefully drew this picture
of scattering, which shows how, at the second bounce, variations in the initial conditions
can lead to extremely divergent results. As the initial imprecision in the angle is
multiplied, every additional bounce will be further magnified. This causes a severe
multiplicative effect where the error grows out disproportionately.
out the past with infinite precision. Poincaré showed this in a very simple
case, famously known as the "three body problem." If you have only two
planets in a solar-style system, with nothing else affecting their course,
then you may be able to indefinitely predict the behavior of these planets,
no sweat. But add a third body, say a comet, ever so small, between the
planets. Initially the third body will cause no drift, no impact; later, with
time, its effects on the two other bodies may become explosive. Small differences
in where this tiny body is located will eventually dictate the future
of the behemoth planets.
Explosive forecasting difficulty comes from complicating the mechanics,
ever so slightly. Our world, unfortunately, is far more complicated
than the three body problem; it contains far more than three objects. We
are dealing with what is now called a dynamical system—and the world,
we will see, is a little too much of a dynamical system.
Think of the difficulty in forecasting in terms of branches growing out
of a tree; at every fork we have a multiplication of new branches. To see
how our intuitions about these nonlinear multiplicative effects are rather
weak, consider this story about the chessboard. The inventor of the chessboard
requested the following compensation: one grain of ricè for the first
1 7 8 WE J U S T C A N ' T P R E D I CT
square, two for the second, four for the third, eight, then sixteen, and so on,
doubling every time, sixty-four times. The king granted this request, thinking
that the inventor was asking for a pittance—but he soon realized that he
was outsmarted. The amount of rice exceeded all possible grain reserves!
