

The Ontology of Mind: The Free Will Theorem “If the atoms never swerve so as to originate some new movement that will snap the bonds of fate, the everlasting sequence of cause and effect—what is the source of the free will possessed by living things throughout the earth?” - Titus Lucretius Carus, Roman philosopher and poet Previously in this series, I went through a mathematically intensive deductive argument that demonstrated (via reductio ad absurdum) that the mind was not mechanistic. In this installment I will be detailing what physics has to say about the prospects for determinism in the physical universe and the notion of free will. [Note: Before proceeding below the fold I highly recommend that the reader revisit the introductory entry to this series, which has almost been completely rewritten since it was first published] The Classical Understanding “We may regard the present state of the universe as the effect of its past and the cause of its future. 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.” - Pierre-Simon Laplace From a modern perspective, the great success of the Western scientific program of reductionism starts with Newton, who developed a theory of dynamics for explaining the basic interactions of solid objects and gravity. The mathematical machinery behind this theory, which was independently discovered by himself and Leibniz, was deterministic in its workings to such a degree that practically any physical setup that you could imagine under this theory had a unique time evolution. Moreover, Newtonian dynamics and the mathematical machinery behind it was so successful that it was subsequently expanded to include the phenomena of fluids and electromagnetism. The irresistible metaphor that emerged from all this was that of conceptualizing the universe in terms of one grand machine that is governed by a precise collection of rules which are revealed by physics in the form of mathematical equations. It is a metaphor famously articulated by Laplace in the quotation above. The consequences of this metaphor to traditional notions of free will were profound, namely that those notions had to be wrong and that the experiences on which they were based had to be illusory. For if the universe is one big machine, whose workings can be completely determined by a set of mathematical equations, then what room is there for free will? And if our experience of free will is illusory then what are we to make of those things that we believed to have been the source of our will as anything other than falsehoods built on illusions! The Dome “Since 2003, there have been many reactions to the dome. Some are amused to see that indeterminism arises in so simple an example in Newtonian physics. Others are indifferent. The response that surprised me, however, came from those who had a full grasp of the technical issues, but nonetheless experienced a powerful intuition that the dome somehow lies outside what is proper in Newtonian theory. It is not always easy to discern the grounding of such instinctive reactions.” - John Norton Amusingly enough, the long-standing lore of classical dynamics being completely deterministic was always wrong! John Norton, an American philosopher of science, discovered a counterexample just a few years ago that he is now calling “The Dome”. The dome is a radially symmetric surface with its shape defined by the equation given above; where r is the radial distance coordinate in the surface of the dome, h is the vertical distance below the apex at r=0 and g is the constant acceleration of free unit mass in the vertical gravitational field surrounding the surface. A point-like, unit mass slides frictionlessly over the surface. Initially, at time t=0, it is at rest exactly at the apex. As it turns out, classical mechanics doesn’t distinguish between an infinite family of possible outcomes for the dome; specifically, the unit mass may stay at the top of the dome indefinitely or it may slide down the dome at any given radial direction at any given moment in time (the details of the physics can be found in Norton’s paper here). The situation couldn’t be less deterministic. Modern Developments “It is safe to say that nobody understands quantum mechanics.” - Richard Feynman Although Norton’s Dome was only discovered a few years ago we’ve known that there are problem with this metaphor of the universe as machine for quite a while now. To the surprise of many, classical dynamics wasn’t the end of the story as it was eclipsed in the 20th century by the new physics of relativity and quantum mechanics (abbrev. QM), the latter considered to be the most fundamental theory of physics currently available. However, also to the great surprise of everyone, QM turned out to be an incomplete theory; in other words, one cannot use QM to completely predict the future state of the universe in terms of its elementary particles. Specifically, the mainstream interpretation of QM is that the universe undergoes periods of unitary development, which are completely deterministic, followed by a wave function collapse that is not deterministic. It was presciently observed by the Ancient Greek atomist Titus Lucretius that in order for there to be free will that the atoms that make up the universe must, every so often, exhibit a “slight swerve” in their motions. Perhaps such an indeterministic “slight swerve” can be found in QM with rigid scientific proof and finally put to rest the soulless phantasm of determinism, which William James famously observed as robbing from mankind that most precious hope. The Eternal Return Interestingly enough, we can give a philosophical proof for why such a task is impossible by considering as a thought experiment something similar to Nietzsche’s concept of an eternal recurrence. The thought experiment goes as follows: Suppose God creates a cyclic universe that periodically restarts itself at constant intervals, and that when it restarts itself it does so exactly in the same manner every time. Consequently, regardless of whether the actors in the first running of the universe had free will (or not) the actions of their counterparts in subsequent installments of the universe would be completely determined by what their antecedents did in the first running of the universe. And since the actors of such a cyclic universe have no way of knowing which installment of the universe they inhabit they cannot be able to demonstrate, via a rigid scientific argument, that the evolution of their universe is not completely determined as a function of the past. Of course, however unlikely it is entirely possible that this state of affairs could apply to us and, therefore, we too have no way of demonstrating via such an argument that the universe we inhabit is not a deterministic function of the past. The Free Will Theorem The stage is now set for a very recent and provocative result that comes to us from the famous English mathematician John Conway, inventor of the Game of Life, and his American colleague the logician Simon Kochen in what they are calling The Free Will Theorem (abbrev. FWT), which states roughly that if indeed we humans have free will, then elementary particles already have their own small share of this valuable commodity. The FWT makes use of three axioms that are given the names SPIN, TWIN, and MIN, which I will give below along with a technical statement of the main theorem (all of which are taken from the linked article). SPIN Axiom: Measurements of the squared (components of) spin of a spin 1 particle in three orthogonal directions always give the answers 1, 0, 1 in some order. TWIN Axiom: For twinned spin 1 particles, suppose experimenter A performs a triple experiment of measuring the squared spin component of particle a in three orthogonal directions x, y, z, while experimenter B measures the twinned particle b in one direction, w. Then if w happens to be in the same direction as one of x, y, z, experimenter B’s measurement will necessarily yield the same answer as the corresponding measurement by A. MIN Axiom: Assume that the experiments performed by A and B are space-like separated. Then experimenter B can freely choose any one of the 33 particular directions w, and a’s response is independent of this choice. Similarly and independently, A can freely choose any one of the 40 triples x, y, z, and b’s response is independent of that choice. The Free Will Theorem: The axioms SPIN, TWIN and MIN imply that the response of a spin 1 particle to a triple experiment is free—that is to say, is not a function of properties of that part of the universe that is earlier than this response with respect to any given inertial frame. [Therefore the twinned particles have exactly the same kind of freedom as the experimenters in MIN] [I recommend that the interested reader examine the proof of the FWT, which can also be found in the linked article, as it is easy to read and only requires a modicum of mathematical facility and doesn’t require any knowledge of physics] The axioms SPIN and TWIN are straightforward consequences of QM that can also be operationally verified independent of QM, which is just a fancy of saying that these axioms can be established by performing experiments in a suitably equipped laboratory without needing any knowledge of QM. Indeed these axioms have been verified in this way and are most definitely true. It is the axiom MIN that contains this assumption that an experimenter can make a “free” choice, which is to say a choice that is independent of the past history of the universe, with regards to the directions in which to measure the squared spin component of a particle. Conway points out that this assumption of independence is customary to make in empirical science otherwise it might be possible that Nature conspires against the scientist to only conduct those experiments that might produce misleading results. In other words, the cost of denying MIN is that we don’t have any propositional grounds on which to believe that the results of empirical science are reasonable. Before moving on to the last section of this entry, I would like to point out that there is a six part video lecture series that has been made available to the public here and consists of Conway lecturing on every aspect of the FWT in front of a live audience at Princeton. A purely nontechnical presentation of these ideas and their consequences can be had by watching the first and last lecture in this series of talks. Conclusions We conclude this entry with a series of inductive arguments in support of free will. Moreover, we shouldn’t balk at our use of inductive arguments toward this end for if we have free will then there should be no rigid scientific argument in favor of free will by our considerations of the eternal recurrence thought experiment. (1) If we have free will then that would explain why we seem to have free will. In contrast, determinism cannot explain why we seem to have free will. (2) If we have free will then the FWT gives us an explanation for why QM is incomplete. (following Conway) (3) If we have free will then the FWT tells us that the particles that make up our bodies have their own small share of this valuable commodity, which explains how we can have free will in the first place. (following Conway) (4) If we do not have free will then we do not have any propositional grounds on which to believe that the results of empirical science are reasonable. (following Conway) (5) There is currently no evidence for determinism in physics. Personally, I find these inductive arguments compelling and feel that they provide more than enough philosophical evidence for embracing some form of free will and rejecting determinism.



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