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.

 

The Axioms and the Theorem

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.

 

Explanation of the Theorem

Conway and Kochen commence their proof of the theorem by relying on a version of the Kochen-Specker paradox, which shows on the basis of the SPIN axiom that the result of measurement of spin in every possible direction cannot exist prior to and independently of measurement.  That is, given that measurements of the squares of components of spin of spin 1 particles in three orthogonal directions (directions at right angles to each other) always gives the results 1, 0 and 1 in some order, and given that such measurements may be made for any such combination of directions, it cannot be the case that the state of a particle and/or the universe is such that every one of these results is fixed in advance of and independently of measurement.  It is quite easily shown that, if it were the case that these results were fixed in advance of and independently of measurement for some directions, this could not be the case for at least one direction:  measurement for that direction would have to give the result 1 or 0, depending on measurement events in relation to other directions; that is, depending on which other directions were measured and/or the results of those measurements and/or factors in the measurement process affecting those results.

          This of itself does not refute determinism in the behaviour of the particles.  For instance, it could still be the case that, in relation to any combination of directions that can be measured, the outcome is fixed in advance for that combination of directions, and so will certainly give that result if that combination is measured.  More generally, it could be the case that the result of measurement for one or more directions is determined in part by measurement events in relation to other directions.

          What the theorem shows is that in the case of twinned particles, as postulated by the TWIN axiom, the FIN axiom means that determinism cannot be saved in this way.  A direction that would have to give the result 1 if one set of measurement events occurs for other directions and 0 if another set of measurement events occurs for other directions could be the single direction measured by the second experimenter B; and the FIN axiom means that neither B nor the twinned particle measured by B could have the information what measurement events occur in measurements by the first experimenter A.  Thus it must be open that the result of measurement by B could be either 1 or 0; but which one it is could not be determined in part by measurement events that occur in measurements by experimenter A, as would be required if determinism were to be saved in the way suggested.

          The bottom line is that, if determinism and the SPIN and FIN axioms are all maintained, there would have to be directions such that, if measurements were made for those directions by experimenter A and experimenter B, the TWIN axiom would be contradicted.  The only way that both determinism and all three axioms can be maintained together would be to postulate that the experimenters are somehow prevented from measuring for these directions:  that is, as Conway and Kochen put it, that the experimenters do not have free will to measure for these directions.

          This is my non-mathematical explanation of the theorem; and it also suggests some tension between the first two axioms and the FIN axiom, as referred to earlier.  It might seem that the same problem arises, even if determinism is rejected.  If the outcomes of all possible measurements are not fixed in advance, beyond being subject to the constraints of the SPIN and TWIN axioms, then it might seem that communication between the locations of experimenter A and experimenter B is required if the correlations required by the TWIN axiom are to be achieved, thereby contradicting the FIN axiom.

          The resolution of this tension is to be found in the non-locality of unmeasured quantum systems, in which there are correlations between spatially separated parts of the systems.

A simple illustration of this is given by a single particle system, for which there may (according to the mathematics of quantum mechanics) be a 0.5 probability that the particle will be found at location X and a 0.5 probability that it will be found at a distant location Y.  Then, if it is found at X, according to the mathematics of quantum mechanics the probability of it being found at Y is zero, not because of any passage of information from X to Y but simply because the total probability cannot be other than 1.  If in such a case there is a measurement at Y, which has space-like separation from the measurement at X, the particle will not be found at Y.  According to some frames reference, it will be the measurement at X that occurs first and establishes that the particle is at X and is therefore not at Y; and according to other frames of reference, it will be the measurement at Y that occurs first and establishes that the particle is not at Y and is therefore at X.  According to relativity theory and the FIN axiom, both points of view have equal validity, and there is no fact of the matter whether it was the measurement at X or the measurement at Y that was “truly” causative.  There is nothing in quantum mechanics, or in the SPIN or TWIN axioms, that contradicts this.

          In the same way, the mathematics of quantum mechanics gives interdependent probabilities for the results of spin measurements by experimenter A and experimenter B, and the outcome of relevant measurements alters these probabilities in a way that has to be correlated as required by the TWIN axiom.  The process is not a causative one requiring the conveying of information from the location of experimenter A to the location of experimenter B or vice versa, but rather a logical one depending on the necessity that probabilities of alternatives always add up to 1.  In that way there can be a reconciliation of the TWIN axiom and the FIN axiom; but this reconciliation is not available if the outcome of any of the relevant measurements is fixed in advance, rather than being a matter of probability only.

 

Implications

Although the authors called their theorem the free will theorem, it does not actually either depend on any assumption of free will as generally understood, or directly support any conclusion about free will.  Its conclusion (that the outcomes of measurements are not fixed in advance) follows if it is the case both that experimenters can measure spin in any direction (at least in the sense that there is no causal link with the states of the particles to be measured that limits the directions that can be measured), and that (in accordance with the axioms) laws of nature correctly specify features of the outcome in the event that any such measurement is made.

          This conclusion follows in a straightforward way in relation to the first version of determinism I identified at the outset, namely that earlier circumstances and the laws of nature uniquely determine later circumstances.  The theorem shows that, on the basis of the axioms, if it is the case that experimenters can measure spin in any direction, then prior to measurement being made existing circumstances and the laws of nature can do no more than determine probabilities for various outcomes.  Accordingly a deterministic version of quantum mechanics would require either that one or more of the axioms must be rejected or else that experimenters are somehow prevented from making those measurements that would contradict the laws of quantum mechanics.  Since the causal antecedents of the experimenters (or of whatever it is that determines which measurements are made) may be effectively independent of the causal antecedents of the particles being measured, such a limitation on what measurements can be made would require not merely determinism but a thoroughgoing conspiracy of nature.  As Conway and Kochen point out, this in turn would undermine the scientific method, because it would mean that scientists cannot have access to random samples but rather are sometimes prevented by a conspiracy of nature from making measurements that if made would refute a hypothesis being tested.

          In relation to the other version of determinism, the block universe, the position may seem less clear.  It could be said that the block universe theory does not involve any particular view as to how earlier states are causally linked to later states, and that the circumstance that outcomes would have been different if earlier circumstances had been different, that is, if experimenters had made different measurements, is entirely unremarkable.  The falsity of determinism only follows if there is nothing preventing experimenters from making measurements in any direction; and it may be said that, on the block universe view, experimenters cannot make any measurements other than the ones they actually make.  The problem with this is that the block universe view then cannot provide any explanation, or any causal story, as to why experimenters never make those measurements which, on the basis of the three axioms, would falsify well-established laws of quantum mechanics if their outcomes were fixed in advance.  As pointed out above, the causal antecedents of whatever determines which measurements are made and those of the particles being measured may be effectively independent of each other; so even on the block universe view, a conspiracy of nature would be required in order to provide an explanation or a causal story as to why experimenters never make these measurements.

 

‘t Hooft’s Response

A response to the Conway-Kochen theorem has been published[3] by Nobel laureate Gerard ‘t Hooft, who has for some time been developing a deterministic version of quantum mechanics.

          He accepts that the ‘model of the world’ provided by a physical theory must give ‘credible scenarios for a universe for any choice of initial conditions.’  He claims that this requirement can be satisfied by a deterministic theory, despite the Conway-Kochen theorem, on the following basis:

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.

I do not think this response is adequate, in that it fails to recognize that the causal antecedents of whatever determines which measurements are made, and those of the particles being measured, may be effectively independent of each other.  For example, the measurements to be made by experimenter A and experimenter B could be determined by a signal sent from some location far way and equidistant from both of them; and there would be no opportunity for that signal to affect the particles being measured until it arrived with experimenter A and experimenter B immediately before the measurements are made.  So ‘t Hooft’s argument does not avoid the necessity for a highly artificial and improbable conspiracy of nature, which would undermine the scientific method.

 

What About Free Will?

Recognition that the Conway-Kochen theorem makes it unscientific to accept determinism would not directly support free will.  Quantum mechanics treats the indeterminism involved in the results of such measurements as being random within probability parameters specified by the mathematics; and as has often been pointed out, randomness is inimical to free will rather than supportive of it.

          However, as I have argued in various publications,[4] there are other reasons for believing in free will associated with rational conscious decisions and actions; and refutation of determinism supports an argument that the physical world is not closed to influences from rational conscious processes, and thus that there is ‘room’ for free will.  Much recent philosophical debate concerning free will has concentrated on the question whether free will is compatible with determinism.  The substantial refutation of determinism supported by this theorem may have the beneficial effect of encouraging philosophers to give more attention to other questions concerning free will, in particular the relevance and role of conscious mental processes in our decisions and actions.

……………………………………………………………………………..

Return to Home

……………………………………………………………………………..



[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.