Can mathematics predict the future?
People have long imagined what the future holds,
whether that's long term or short term.
Some modern applications of big data crunching include
facebook predicting when a couple will enter a relationship, history looking
for patterns when violence will spike, or researchers choosing
the best March Madness bracket.
But what if past behavior is irrelevant for predictions?
The trajectory of a woman jumping off a cliff doesn't
depend on what she did an hour ealier, but rather, on
her intial position and speed and the forces of gravity, wind,
and air resistance. Mathematician Pierre-Simon Laplace asserted
that if we have full knowledge of everything at one instant of time, and
understand how things change as time progresses,
we could, at least theoretically, predict the future.
In other words, if you know what a system is like now and
know the rules for how it will change, then you know
what it will be like in the future.
Let's illustrate this with a simple mathematical model, the Game of Life.
Start with a grid where at a given moment each cell is
either alive (shaded black) or dead (shaded white). In the next generation,
each cell stays the same or changes according to the
following four rules:
1) Any live cell with fewer than two live neighbors dies,
as if caused by under-population. 2) Any live cell with two or three live neighbours
lives on to the next generation.
3) Any live cell with more than three live neighbours dies,
as if by overcrowding. And 4) Any dead cell with exactly three live neighbours
becomes a live cell, as if by reproduction.
These rules produce interesting dynamics such as
still lifes that don't change, oscillators, and spaceships that move across the grid.
Starting with just a few cells on, you can see that the
dynamics can be pretty crazy. The point is that knowing the state
of every cell in one generation and knowing the rules lets us
determine each cell's state at any time in the future.
Another example is called Langton's ant. Again, let each cell
be either black or white. An ant starts at one cell pointing
in a specified direction. At each step there are two possible moves:
1) If the ant starts at a white square, turn 90 degrees right, flip the color
of the square, and move forward one unit, or
2) If the ant starts at a black square, turn 90 degrees left, flip the color
of the square, and move forward one unit
Starting on a completely white grid, something fascinating happens.
The ant's progress seems somewhat random, but after about
10,000 moves, the ant starts moving along a "highway" forever.
In fact, it seems that no matter how the grid is initially colored,
a highway eventually appears. Nobody can prove this or
explain why it happens.
The Game of Life and Langton's Ant bring out a challenge
to predicting the future: we can do a lot of calculations to
see what happens after, say, one million steps, but
usually there is no theorem that immediately tells you what
that state will be.
In contrast to these scenarios that use time moving in a stepwise manner,
models for studying fluid flow, planetary motion, or the weather
use continuous time. Sometimes the equations involved, like for
studying how a planet moves around the sun, can be solved exactly, so we can predict where
any planet will be located with great accuracy.
But fluid flow is governed by the Navier-Stokes equations,
differential equations which are impossible to solve exactly,
but are critical to modeling weather, air flow which produces
flight, and of course water flow. Understanding solutions to these
equations is so important that the Clay Mathematics Institute is offering
a one million dollar prize for concrete progress.
Researchers can solve these equations approximately but the errors from the approximations grow
over time. You can see this when a double pendulum is released at nearly
identical initial states; the positions change radically over
time. This is why weather prediction is so difficult;
approximate solutions to equations and the growth of errors means it
is nearly impossible to accurately predict the weather after several days.
You can check this yourself by looking at the predicted highs and
lows in your area for one week later and compare it to the what
actually happens. Other factors, like geography, can magnify this instability.
Of course, if we have some control in our system,
like firing thrust out from a lander, we may be able to adjust
for any instability, and predict the kind of future
we want. Or not.