THE CONWAY-KOCHEN ‘FREE WILL
THEOREM’

AND UNSCIENTIFIC DETERMINISM

David
Hodgson

** **

Determinism is the doctrine
that everything that happens is fixed (‘determined’) in advance. Broadly, there are two versions of
determinism, which can be asserted either independently or in combination. One has it that earlier circumstances and
the laws of nature uniquely determine later circumstances, and the other has it
that past present and future all exist tenselessly in a ‘block universe,’ so
that the passage of time and associated changes in the world are illusions or
at best merely apparent.

Both of these versions are associated with scientific
ideas, and determinism is often considered to be a doctrine supported or even
required by science.

The first version found classic expression in the writings
of the eighteenth-century French mathematician Pierre Laplace, who supposed
that the entire universe consisted of a few kinds of basic objects moving
through space in accordance with Newton’s laws of motion, which specify how
physical quantities associated with these objects uniquely determined how they
move. This general idea still has a
strong hold on many scientists and philosophers today, despite challenges to it
from twentieth century science, in particular quantum mechanics.

The second version is considered by many scientists and philosophers
to follow from relativity theory, which treats time and space as interdependent
dimensions in a reality of four or more dimensions, in which time does not pass
and every event in the past present and future has a location in the unchanging
space-time continuum.

A theorem recently propounded[1]
by Princeton mathematicians John Conway (who invented the famous Game of Life)
and Simon Kochen (one of the originators of the Kochen-Specker paradox of
quantum mechanics) supports a powerful challenge to the scientific credentials
of determinism, by showing that two cornerstones of contemporary science,
namely acceptance of the scientific method as a reliable way of finding out
about the world, and relativity theory’s exclusion of faster-than-light transmission
of information, together conflict with determinism, in both its versions. Belief in determinism may thus come to be
seen as notably *unscientific*.
This conclusion has previously had some support from a theorem devised
by John Bell and experiments undertaken by Alain Aspect, but in my
understanding the Conway-Kochen theorem supports it more strongly.

The theorem, which Conway and Kochen call the *free will
theorem*, has been reported briefly in *New Scientist[2]*
and has been the subject of considerable discussion on the internet, but
otherwise has had remarkably little publicity, despite what seems to me to be
its great importance. It seems hardly
to have been noticed by philosophers.
In this article, I will discuss the theorem in an informal way, with a
view to making its significance understandable by people who are not
mathematicians.

Conway
and Kochen make three assumptions, which they set out as axioms:

There
exist “particles of total spin 1” upon which one can perform an operation
called “measuring the square of the component of spin in a direction *w*” which always yields
one of the answers 0 or 1.

We shall write *w *→* **i* (*i* = 0 or 1) to indicate the result of this
operation. We call such measurements
for three mutually orthogonal directions *x*, *y*, *z* a *triple experiment for
the frame* (*x*, *y*, *z*).

**The
SPIN axiom:** A triple experiment for the frame (*x*, *y*, *z*) always yields the outcomes
1, 0, 1 in some order.

We can write this as: *x* → *j*, *y* → *k*, *z* →*l*, where *j*, *k*, *l* are
0 or 1 and *j *+ *k* + *l* = 2.

It is possible to produce two distantly separated spin
1 particles that are “twinned,” meaning that they give the same answers to
corresponding questions. A symmetrical form of the TWIN axiom would say that if
the same triple *x*, *y*, *z* were
measured for each particle, possibly in different orders, then the two
particles’ responses to the experiments in individual directions would be the
same. For instance, if measurements in the order *x*, *y*, *z* for one particle produced *x* → 1, *y* → 0, *z* → 1, then measurements in the
order *y*, *z*, *x* for
the second particle would produce *y* → 0, *z* → 1, *x* → 1. Although
we could use the symmetric form for the proof of the theorem, a truncated form is
all we need, and will make the argument clearer:

**The
TWIN axiom: ** For
twinned spin 1 particles, if the first experimenter A performs a triple
experiment for the frame (*x*, *y*, *z*), producing the result *x* → *j*, *y* → *k*, *z* → *l* while the second experimenter B measures a single spin in direction *w*, then if *w* is one of *x*, *y*, *z*, its result is that *w* → *j*, *k*, or *l, *respectively.

**The
FIN Axiom:** There is a finite upper bound to the speed
with which information can be effectively transmitted.

This is, of course, a well–known consequence of
relativity theory, the bound

being
the speed of light.

They then state their
theorem:

**The
Free Will Theorem** (assuming SPIN, TWIN, and
FIN)].

If the choice of
directions in which to perform spin 1 experiments is not a function of the
information accessible to the experimenters, then the responses of the
particles are equally not functions of the information accessible to them.

Before giving my informal explanation of the theorem, I
will say something about the axioms.

The first two axioms are well-supported conclusions of
quantum mechanics. Both of them follow
from the mathematics of quantum mechanics, and have experimental support in
that they have been extensively tested and never falsified.

Spin is a property of particles of matter dealt with by
quantum mechanics, of the same nature as polarisation of light; and according
to quantum mechanics, some particles of matter are particles of total spin 1,
having the properties set out in the SPIN axiom. The TWIN axiom deals with properties of pairs of such particles
that have been correlated in a particular way by interaction between them and
then have moved far apart in such a way as to preserve the correlation. In such a case the mathematics of quantum
mechanics indicates, and experiments have confirmed, that when experimenters
measure the spin of these particles the results are correlated in the way
stated by the TWIN axiom, even if the experiments have space-like separation –
that is, even if the experiments are performed at times and places such that no
signal travelling at light speed or less could pass between them in either
direction. Conway and Kochen in fact
postulate experiments performed on Earth and on Mars, separated by a distance
of five light minutes, at pre-arranged times.

As noted above, the FIN axiom is a consequence of
relativity theory, and it is widely accepted although, as Conway and Kochen
also note, it is ‘not experimentally verifiable directly’. They say it also follows from what they call
‘effective causality,’ that effects cannot precede their causes. This in turn reflects another consequence of
relativity theory, namely the equal status of inertial frames of reference, and
the fact that there are frames of reference according to which signals sent
back and forth at greater than the speed of light (if this were possible) could
arrive back to where they originated earlier than when they were sent.

When I
discuss the theorem, and in particular the implications of the first two
axioms, it will become apparent that there is some tension between these two
axioms and the third axiom. However,
the first two axioms are well supported, and I will argue (as do Conway and
Kochen) that this tension is not such as to justify the rejection of the third
axiom. In any event, rejection of the
third axiom (and thus of a central plank of relativity theory) would directly
undermine the basis for at least the second version of determinism, associated
with the block universe.

This
[requirement] is the free will axiom in its modified form. This, we claim, is
why one should really want ‘free will’ to be there. It is not the free will to
modify the present without affecting the past, but it is the freedom to choose
the initial state, regardless its past, to check what would happen in the
future.

Indeed, when Tumulka, in the quoted text, talks about
conspiracy, stating that conspiratorial theories appear to be unacceptable, it
was actually this modified form of free will that he had in mind. *But this
is not the free will that is assumed in the Conway-*

*Kochen
argument!*

One
cannot modify the present without assuming some modification of the past.
Indeed, the modification of the past that would be associated with a tiny
change in the

present
must have been quite complex, and almost certainly it affects particles whose

spin
one is about to measure.

[1]
‘The free will theorem’, (2006) *Found. Phys.
36 (10)*, 1441; arXiv:quant-ph/0604079v1.

[2]
‘Free will -
you only think you have it’, *New Scientist*, 6 May 2006, 8.

[3] ‘The free-will postulate in quantum mechanics’, arXiv:quant-ph/0701097v1.

[4] Most recently in ‘Partly free’, *Times Literary Supplement*, 5 July 2007, and ‘Making our own luck’, (2007) *Ratio
20*, 278.