Download KB Paper2 Free Will Theorem

The Free Will Theorem
Kushal Byatnal
Summary
For a long time, physicists have been in pursuit of the elusive Theory of Everything
(TOE). If found, the TOE would allow us to describe and predict the actions of everything in the
Universe. Right now, the biggest hurdles lie in combining two very mysterious fields. One is
general relativity – the theory developed by Einstein that describes objects in the macroscopic
world such as planets, stars, galaxies, and black holes. It proved Newton’s theories incorrect and
allows us to describe the nature of the very fabric of space-time. The other is quantum
mechanics – the theory that explains how the microscopic world works. It accurately predicts
the actions of electrons, muons, atoms, etc. Combining these two may lead to the TOE, but it
has proved impossible so far.
One of the biggest hurdles lies within quantum mechanics, which brings us to the main
topic of this article: free will vs. determinism. Many philosophers have taken sides while others
have tried to combine their views on both fields. Descartes, for example, believed in a system
known as “disconnected determinism” which claimed that all material objects (such as bodies,
the environment) were all controlled by determinism. However, the human mind itself was
under the principles of free will.
Titus Lucretius Carus was the first, however, to suggest that science could be used as a
support for the ideology behind free will. Lucretius’s main issue was finding the source for
human free will. He settled on a description very similar to that of the free will theorem
(developed nearly 2000 years after his time). He claimed that the slight wavering of atoms along
a path at random location and time intervals proved that the particles themselves have free will,
leading a logical argument that supported free will governing all living things.
In our age of modern technology, Lucretius’s definition has been further refined through
experimentally derived results leading to three axioms. The SPIN axiom reveals that the spin of a
particle (which must be measured as either 1 or 0) must follow a 1-0-1 orientation along three
directions. However, since these directions are subject to change, it can be seen that the spin
along a certain direction is not defined until it is actually observed (a statement that goes
against the views behind determinism). The TWIN axiom describes that two “entangled”
particles will give the same measurement for spin along the same direction. This occurs even if
the measurements are both taken simultaneously from far distances apart. It places a restriction
on free-will by ensuring that it applies to a pair of particles (semi-free will). The final axiom, FIN,
states that nothing – not even information – can travel faster than the speed of light. If this were
not the case, then an observer could determine the outcome of a choice before the choice was
observed. Many paradoxical situations can arise and so the FIN axiom further supports free will
by stating that one entangled particle cannot change the outcome of another (going against the
deterministic view that the past can affect the present).
When combined together, the three axioms give rise to the free will theorem. The free will
theorem states that if humans have free will, then the particles which operate under the
mysterious laws of quantum mechanics also have free will. However, it is important to note that
the theorem does not prove free will as it is also impossible to disprove determinism. Free will
gives rise to many philosophical implications – perhaps the most important is that our Universe
does not operate on randomness, but rather order very much different from determinism.
I. Theory of Everything
“However, if we discover a complete theory, it should in time be understandable by
everyone, not just by a few scientists. Then we shall all, philosophers, scientists and just ordinary
people, be able to take part in the discussion of the question of why it is that we and the universe
exist. If we find the answer to that, it would be the ultimate triumph of human reason -- for then
we should know the mind of God.” – Stephen Hawking
The Theory of Everything (TOE) – one simple idea that would reveal all the secrets of the
Universe. On the one hand, you have quantum theory. The world of small, mysterious, and
unexplored particles and weak, strong, and electromagnetic forces. Imagine a world where
things pop into – and out of – existence without warning. A world where the location of objects
isn’t described as exact spots, but as probabilities and possibilities. A world in which human
intuition hopelessly breaks down.
“Those who are not shocked when they first come across quantum theory cannot
possibly have understood it.” – Niels Bohr
On the other hand, there exists general relativity. Used to describe the world of the
macroscopic, it is very well understood in comparison to quantum theory. It’s used to describe
black holes (to an extent – they’re still incredibly mysterious and our laws of gravity break down
when applied to them), galaxies, stars, and the force of gravity.
“General relativity is the cornerstone of cosmology and astrophysics. It has also provided
the conceptual basis for string theory and other attempts to unify all the forces of nature in
terms of geometrical structures.” - Paul Davies
On their own, general relativity and quantum theory predict the behavior of stars and
atoms (respectively) to incredible accuracy. Together, they would reveal a system allowing
physicists to predict the nature of virtually everything in the universe – this is the infamous
Theory of Everything. Many physicists, including Einstein, have attempted to unify the two,
using various techniques such as String Theory, but all have failed in pursuit of this elusive
equation. But why is it so important to find the TOE?
“An intellect which at a certain moment would know all forces that set nature in motion,
and all positions of all items of which nature is composed, if this intellect were also vast enough
to submit these data to analysis, it would embrace in a single formula the movements of the
greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would
be uncertain and the future just like the past would be present before its eyes.”
— Essai philosophique sur les probabilités, Introduction (1814)
As Stephen Hawking stated, the TOE will allow us to understand the mind of God –
which brings us to the discussion of free will and quantum theory. The free will theorem states
that if living beings have free will, then elementary particles themselves also exhibit free will.
This is partially because quantum theory is so mysterious and complicated that free will of the
particles becomes a valid explanation for explaining their behavior. Having used the TOE to set
up the background, the remainder of this paper will explain the history and ideology behind the
free will theorem.
II. Free Will vs. Determinism
The debate between free will and determinism stretches far back into the history of
mankind. Determinism has deep religious roots and argues that that the fate of the Universe –
the formation of the Andromeda galaxy, your first kiss, 9/11, Watergate – was all foreseen and
predicted before the Universe even existed. On the other side, free will states that every action
is not based on the something in the past. That is, every occurrence is an independent action,
unaffected by any outside influence. While interesting, it is a debate that is at the same time
never ending. After all, determinism can never be disproved – ie. it can never be shown that
there isn’t someone writing the fate of the Universe in the “Big Book” above.
Gottfried Leibniz – the famous mathematician who developed calculus independent of
Newton – had views known as the “Principle of Sufficient Reason”. It’s a law of logic that states
everything must have a cause or reason. While not mentioned directly, the principle goes
against the idea of free will simply because it requires some external cause for an action to
occur. Since this external cause must have taken place before the action itself, it implies that the
action is based on something in the past – a direct contrary to the idea of free will.
Pierre-Simon de Laplace was another famous French mathematician and astronomer
who attempted to develop his own system for resolving the debate between free will and
determinism. Believing in the rationality of the Universe, he proclaimed an idea very similar to
that of the Theory of Everything that supports determinism. Laplace claimed that a combination
of scientific formulas could be used to understand and describe the actions of everything in the
Universe, both past and future. In theory, this seems like a very valid idea – however, the entire
ideology breaks down when applied to the quantum scale. As mentioned in the previous
section, human intuition completely breaks down when applied to the quantum scale. Formulas
help us predict what we believe will happen, but they don’t allow us to understand what exactly
happens since they defy our human instincts.
In the middle, you have those philosophers who believe that the Universe was not so
black and white after all – the grays were represented with a mix of both determinism and free
will. Thomas Hobbes believed in such a system, accurately named “compatibilism”. According to
Hobbes, both determinism and free will can exist together because they are complements, not
substitutes. He says that the will is determinate, but man has free will in whether or not he
wants to follow it. In other words, man is free to follow his will but he is not free to will.
Descartes was another well-known philosopher who shared the same beliefs with
Hobbes. In Descartes’s case however, he believed in something termed “disconnected
determinism”. According to Descartes, the world is split into two – mind and matter. All matter
– that is, things that have physical space such as the human body, environment, etc. – operate
according to determinism. The significant break comes with the mind, which is governed by the
properties of free will. This is because “The will is by its nature so free that it can never be
constrained.”
It may be surprising to note that the philosopher who will lead us into the discussion of
the free will theorem and its relation to quantum theory is also the oldest philosopher on the
list. Titus Lucretius Carus was a Roman poet and philosopher who lived from 99 – 55 BCE.
Lucretius believed strongly in human free will, but he had to address the issue of where exactly
the source of this free will was located. In his poem, De Rerum Natura, Lucretius presented the
themes of atomism to argue that the Universe operates according to a set of physical principles
and not by the divine actions of the traditional Roman deities. In particular, he states a property
that hints towards the free will theorem. He says “When atoms are traveling straight down
through empty space by their own weight, at quite indeterminate times and places, they swerve
ever so little from their course, just so much that you can call it a change of direction.” By
making this claim, Lucretius lends support to his views on free will by describing a source for the
free will – that is, since the atoms that make up humans have free will, humans themselves can
adopt those characteristics and demonstrate free will.
III. The Three Axioms
The entire premise of the Free Will Theorem rests upon three core axioms – SPIN, FIN,
and MIN.
The SPIN axiom involves measuring the spin of a particle, resulting in either a 1 or 0.
Additionally, the SPIN axiom is operationally defined, meaning that it has been proved a
multitude of times. What it states is that if the spin of a particle is measured along three
orthogonal directions, it will result in a combined measurement of 1-0-1. That is, two of the
directions will reveal a spin of 1 while the other will have a spin of 0. There are three main rules
of the SPIN axiom that can be derived. Since three orthogonal directions must have only one 0,
perpendicular directions cannot both be zero. The second rule states that opposite directions
(ie. extending the axis line the other way) must be the same spin. The final rule intuitively claims
that three directions cannot all be 1.
However, the most important result of the SPIN axiom is that it implies the quantity
being measured cannot exist before the actual measurement is taken. Specifically, imagine that
two out of the three orthogonal directions are selected. The measurement of the third direction
cannot exist until the spin along the other two has been determined (since all three directions
must follow the 1-0-1 pattern, as proved experimentally). A shocking revelation has been made
– the spin of a particle along a direction is not determined until it is observed. This is almost an
exact statement for the support of free will. Since the spin of the particle is not pre-determined,
it means that it cannot be based on past events, ruling out the possibility of any determinism.
The TWIN axiom supposes that there are two researchers working with two different
particles (A & B). There is a crucial detail to this axiom – the two particles have been entangled
in what is known as the “singleton state”. At this stage, the particles are separated by a large
distance. Now if the spin is measured instantaneously along the same direction for both
particles, it’s been experimentally proven that the value will be the same every time. That is,
two entangled particles will give the same spin along a direction even if the answer is not known
before. The two particles somehow “communicate” and match the spin to be the same. This
occurs even if the spin of the two particles is measured at the same time – the information
seems to travel instantaneously from one particle to the other.
The TWIN axiom also gives rise to the EPR paradox. The EPR paradox attempts to
address a flaw in the theory of quantum mechanics – specifically, Heisenberg’s uncertainty
principle. The uncertainty principle states that both the momentum and position of a particle
cannot be known at the same time. For years, the explanation was thought to be that any
attempt at observing one would disturb the other (perhaps by introducing or removing energy
from the system). However, the TWIN axiom shows that measuring the spin of A will change the
spin of B even if there was no direct disturbance involved in measuring B. The TWIN axiom also
places a restriction on free will (semi-free) by constraining the impacts of free will to a pair of
entangled particles.
The final axiom is the FIN axiom. The FIN axiom relates the theory of special relativity
(which states that the universal speed limit is the speed of light) to the idea of free will. Take a
scenario where special relativity didn’t apply and it was possible for information to travel faster
than the speed of light. This means that a situation may arise where an observer knows the
outcome of a choice before seeing the choice being made. The observer can then take an action
to change the choice (since free will is assumed to apply), resulting in a paradox. In order to
make logical sense of the situation, the FIN axiom is used to support the idea of free will and
prevent situations such as the one above from arising. Put a different way, the FIN axiom
ensures that the result of A cannot affect the result of B and thereby maintains free will (since
outcomes are no longer based on what occurs in past situations).
When combined together, the axioms of SPIN, FIN, and TWIN can uphold the free will
theorem. The spin of any particle is based on free will because the three axioms can ensure that
outcomes from the past don’t affect any present results.
IV. Philosophical Implications
At first glance, it seems like a completely ridiculous idea. That if humans had free will,
then an explanation for the actions of quantum mechanics is that particles themselves somehow
have free will. And yet, it’s important to keep in mind that the free will theorem does not prove
that humans have free will – it merely supposes what would happen if they did. Thus, it is not a
proof for free will and not evidence against determinism (after all, it is impossible to prove that
determinism isn’t the answer).
If nothing else, the free will theorem proves that Einstein was correct in saying “God
does not play dice with the Universe.” While the theorem may not explain bizarre phenomena
(such as quantum entanglement), it does show that if objects were to operate under free will,
then their decisions are not based on randomness. In the end, the free will theorem may have
been all speculation but it has led us one step closer to finding the Theory of Everything – one
step closer to understanding the mind of God.
References:
1. Free Will and Determinism in Science and Philosophy. Perf.
John Conway. Princeton University, 2009. Recording
2. The Paradox of Kochen and Specker. Perf. John Conway.
Princeton University, 2009. Recording
3. The Paradoxes of Relativity. Perf. John Conway. Princeton
University, 2009. Recording
4. Quantum Mechanics and the Paradoxes of Entanglement.
Perf. John Conway. Princeton University, 2009. Recording
5. Proof of the Free Will Theorem. Perf. John Conway.
Princeton University, 2009. Recording
6. The Theorem’s Implications for Science and Philosophy.
Perf. John Conway. Princeton University, 2009. Recording