Why is Predicting The Future Mathematically Impossible?

The success of classical physics, particularly Newton’s theory of gravity, induced the excitement of several scientists at the beginning of the nineteenth century. For instance, it led the famous French scientist Pierre Simon Marquis de Laplace to argue that the universe is completely deterministic. Laplace suggested that there should be a certain set of scientific laws that allow us to completely predict the state of the universe, past and future. For example, knowing the current state of the sun and the planets, allows us, according to Newton’s laws, to completely predict the next states of the Solar System at any other time. But Laplace went further to assume that there are broader and more general unification laws that allow us to predict everything at every time, including the human behaviour.

The British scientists Lord Rayleigh and Sir James Jeans published a paper called « LIII. Remarks upon the law of complete radiation . » in 1900, in which they suggested that a hot particle, or a system of particles, such as star, must radiate energy at an infinite rate. At that time, according to the laws we believed in, a hot¹ body ought to give off electromagnetic waves, such as light, equally at all frequencies. For example, a hot body should radiate waves with frequencies between one and two million million (one and 12 zeros after it) waves a second. Since the number of waves a second is unlimited, this would mean that the total energy radiated would be infinite. This result is evidently repugnant as it violates the very basic laws of physics.

In order to to avoid this this ridiculous result, the German scientist Max Planck, after few months, argued that light cannot be arbitrarily emitted but can only be emitted in packets of energies, which he called quanta, that is defined by his famous equation, see equation 1:

ρν𝑑ν = 8πν2𝑑ν 𝑐3 𝐸.

Implications for the scientific determinism were not realized until 1926, when another German scientist, Werner Heisenberg, published a paper « Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik »

(On the Perceptible Content of Quantum Theoretical Kinematics and Mechanics), in which he formulated his uncertainty principal. In order to determine the position and the velocity of a particle, one has to calculate their position and velocity precisely. One obvious way to do this is to shine light on the particle so that some of the electromagnetic wave can be scattered by the particle. However, in order to determine the precise position of such small particle, one should use smaller wavelengths. However, according to Planck’s theorem that I stated earlier in this article, one cannot emit light arbitrarily but at least a quantum. That would increase the particle’s energy hence accelerate it. This leads to the core definition of the Uncertainty Principal which states that the more accurate you determine the particle’s position, the less likely you calculate its velocity and vice versa. This can simply be described by Equation 2 where the product of the difference in position and the difference in momentum² is always less than half of the reduced Planck constant h.

Δx Δp ≥ ħ/2

This shatters the Laplacian dream of having a completely deterministic universe, hence predicting your future, given your initial state, is mathematically almost certainly unrealistic.

Glosses:

¹hot: a hot body or particle is a system that contains high levels of kinetic energy, they are moving at high speeds which leds to what we think of as heat

²momentum: the momentum is a physical value that is equivalent to the force required to bring the object to a stop in a unit length of time. It is the product of the mass and the velocity: p=mv

Written by Iheb Gafsi