THE CONWAY-KOCHEN ‘FREE WILL THEOREM’
AND UNSCIENTIFIC DETERMINISM
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 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 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-
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.
 ‘The free will theorem’, (2006) Found. Phys. 36 (10), 1441; arXiv:quant-ph/0604079v1.
 ‘Free will - you only think you have it’, New Scientist, 6 May 2006, 8.
 ‘The free-will postulate in quantum mechanics’, arXiv:quant-ph/0701097v1.
 Most recently in ‘Partly free’, Times Literary Supplement, 5 July 2007, and ‘Making our own luck’, (2007) Ratio 20, 278.