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									The Strong Free Will
John H. Conway and Simon Kochen

                    he two theories that revolutionized           fortunate because these theoretical notions have
                    physics in the twentieth century, rela-       led to much confusion. For instance, it is often said
                    tivity and quantum mechanics, are full        that the probabilities of events at one location can
                    of predictions that defy common sense.        be instantaneously changed by happenings at an-
                    Recently, we used three such para-            other space-like separated location, but whether
        doxical ideas to prove “The Free Will Theorem”            that is true or even meaningful is irrelevant to
        (strengthened here), which is the culmination of          our proof, which never refers to the notion of
        a series of theorems about quantum mechanics              probability.
        that began in the 1960s. It asserts, roughly, that if        For readers of the original version [1] of our
        indeed we humans have free will, then elementary          theorem, we note that we have strengthened it
        particles already have their own small share of           by replacing the axiom FIN together with the as-
        this valuable commodity. More precisely, if the
                                                                  sumption of the experimenters’ free choice and
        experimenter can freely choose the directions
                                                                  temporal causality by a single weaker axiom MIN.
        in which to orient his apparatus in a certain
                                                                  The earlier axiom FIN of [1], that there is a finite
        measurement, then the particle’s response (to
                                                                  upper bound to the speed with which informa-
        be pedantic—the universe’s response near the
                                                                  tion can be transmitted, has been objected to by
        particle) is not determined by the entire previous
        history of the universe.                                  several authors. Bassi and Ghirardi asked in [3]:
            Our argument combines the well-known conse-           what precisely is “information”, and do the “hits”
        quence of relativity theory, that the time order of       and “flashes” of GRW theories (discussed in the
        space-like separated events is not absolute, with         Appendix) count as information? Why cannot hits
        the EPR paradox discovered by Einstein, Podolsky,         be transmitted instantaneously, but not count as
        and Rosen in 1935, and the Kochen-Specker Para-           signals? These objections miss the point. The only
        dox of 1967 (See [2].) We follow Bohm in using a          information to which we applied FIN is the choice
        spin version of EPR and Peres in using his set of 33      made by the experimenter and the response of
        directions, rather than the original configuration         the particle, as signaled by the orientation of the
        used by Kochen and Specker. More contentiously,           apparatus and the spot on the screen. The speed
        the argument also involves the notion of free will,       of transmission of any other information is irrel-
        but we postpone further discussion of this to the         evant to our argument. The replacement of FIN
        last section of the article.                              by MIN has made this fact explicit. The theorem
            Note that our proof does not mention “probabil-       has been further strengthened by allowing the
        ities” or the “states” that determine them, which is      particles’ responses to depend on past half-spaces
                                                                  rather than just the past light cones of [1].
        John H. Conway is professor of mathematics at Princeton
        University. His email address is jhorcon@yahoo.com.
        Simon Kochen is professor of mathematics at Prince-       The Axioms
        ton University. His email address is kochen@math.         We now present and discuss the three axioms on
        princeton.edu.                                            which the theorem rests.

 226                                        Notices of the AMS                               Volume 56, Number 2
      quantum mechanics, but only two of its testable          the choice an experimenter makes is not a func-
      consequences, namely this axiom SPIN and the             tion of the past. We explicitly use only some very
      axiom TWIN of the next section.                          special cases of these assumptions in justifying
         It is true that these two axioms deal only with       our final axiom.
      idealized forms of experimentally verifiable pre-             The MIN Axiom: Assume that the experiments
      dictions, since they refer to exact orthogonal and       performed by A and B are space-like separated.
      parallel directions in space. However, as we have        Then experimenter B can freely choose any one of
      shown in [1], the theorem is robust, in that approx-     the 33 particular directions w , and a’s response
      imate forms of these axioms still lead to a similar      is independent of this choice. Similarly and inde-
      conclusion. At the same time, this shows that any        pendently, A can freely choose any one of the 40
      more accurate modifications of special relativity         triples x, y, z, and b’s response is independent of
      (such as general relativity) and of quantum theory       that choice.
      will not affect the conclusions of the theorem.               It is the experimenters’ free will that allows the
                                                               free and independent choices of x, y, z, and w . But
      (ii) The TWIN Axiom and the EPR Paradox
                                                               in one inertial frame—call it the “A-first” frame—
          One of the most curious facts about quantum
                                                               B’s experiment will only happen some time later
      mechanics was pointed out by Einstein, Podolsky,
                                                               than A’s, and so a’s response cannot, by temporal
      and Rosen in 1935. This says that even though the
                                                               causality, be affected by B’s later choice of w . In a
      results of certain remotely separated observations
                                                               B-first frame, the situation is reversed, justifying
      cannot be individually predicted ahead of time,
                                                               the final part of MIN. (We shall discuss the mean-
      they can be correlated.                                  ing of the term “independent” more fully in the
          In particular, it is possible to produce a pair of   Appendix.)
      “twinned” spin 1 particles (by putting them into
      the “singleton state” of total spin zero) that will
                                                               The (Strong) Free Will Theorem
      give the same answers to the above squared spin
                                                               Our theorem is a strengthened form of the original
      measurements in parallel directions. Our “TWIN”
                                                               version of [1]. Before stating it, we make our terms
      axiom is part of this assertion.
                                                               more precise. We use the words “properties”,
          The TWIN Axiom: For twinned spin 1 particles,
                                                               “events”, and “information” almost interchange-
      suppose experimenter A performs a triple exper-
                                                               ably: whether an event has happened is a property,
      iment of measuring the squared spin component
                                                               and whether a property obtains can be coded by
      of particle a in three orthogonal directions x, y, z,    an information-bit. The exact general meaning of
      while experimenter B measures the twinned par-           these terms, which may vary with some theory
      ticle b in one direction, w . Then if w happens to       that may be considered, is not important, since we
      be in the same direction as one of x, y, z, experi-      only use them in the specific context of our three
      menter B’s measurement will necessarily yield the        axioms.
      same answer as the corresponding measurement                To say that A’s choice of x, y, z is free means
      by A.                                                    more precisely that it is not determined by (i.e.,
          In fact we will restrict w to be one of the 33 di-   is not a function of) what has happened at earlier
      rections in the Peres configuration of the previous       times (in any inertial frame). Our theorem is the
      section, and x, y, z to be one of 40 particular or-      surprising consequence that particle a’s response
      thogonal triples, namely the 16 such triples of that     must be free in exactly the same sense, that it is
      configuration and the 24 further triples obtained         not a function of what has happened earlier (with
      by completing its remaining orthogonal pairs.            respect to any inertial frame).
      (iii) The MIN Axiom, Relativity, and Free Will           The Free Will Theorem. The axioms SPIN, TWIN
          One of the paradoxes introduced by relativity        and MIN imply that the response of a spin 1 parti-
      was the fact that temporal order depends on the          cle to a triple experiment is free—that is to say, is
      choice of inertial frame. If two events are space-       not a function of properties of that part of the uni-
      like separated, then they will appear in one time        verse that is earlier than this response with respect
      order with respect to some inertial frames, but          to any given inertial frame.
      in the reverse order with respect to others. The         Proof. We suppose to the contrary—this is the
      two events we use will be the above twinned spin         “functional hypothesis” of [1]—that particle a’s
      measurements.                                            response (i, j, k) to the triple experiment with
          It is usual tacitly to assume the temporal causal-   directions x, y, z is given by a function of proper-
      ity principle that the future cannot alter the past.     ties α, . . . that are earlier than this response with
      Its relativistic form is that an event cannot be influ-   respect to some inertial frame F. We write this as
      enced by what happens later in any given inertial              F
      frame. Another customarily tacit assumption is                θa (α) = one of (0, 1, 1), (1, 0, 1), (1, 1, 0)
      that experimenters are free to choose between            (in which only a typical one of the properties α is
      possible experiments. To be precise, we mean that        indicated).

228                                       Notices of the AMS                                Volume 56, Number 2
   Similarly we suppose that b’s response 0 or 1                We now define
for the direction w is given by a function                                 G         G
                                                                          θ0 (w ) = θb (x, y, z, w ; β0 ) ,
                θb (β)   = one of 0 or 1                     noting that since by MIN the response of b cannot
of properties β, . . . that are earlier with respect to      vary with x, y, z, θ0 is a function just of w .
a possibly different inertial frame G.                           Similarly there is a value α0 of α′ for which the
    (i) If either one of these functions, say θa , is        function
                                                                         F              F
influenced by some information that is free in the                       θ1 (x, y, z) = θa (x, y, z, w ; α0 )
above sense (i.e., not a function of A’s choice of           is defined for all 40 triples x, y, z, and it is al-
directions and events F-earlier than that choice),           so independent of w , which argument we have
then there must be an an earliest (“infimum”)                 therefore omitted.
F-time t0 after which all such information is avail-            But now by TWIN we have the equation
able to a. Since the non-free information is also                    F               G       G       G
available at t0 , all these information bits, free and              θ1 (x, y, z) = (θ0 (x), θ0 (y), θ0 (z)) .
non-free, must have a value 0 or 1 to enter as               However, since by SPIN the value of the left-hand
arguments in the function θa . So we regard a’s              side is one of (0, 1, 1), (1, 0, 1), (1, 1, 0), this shows
response as having started at t0 .                           that θ0 is a 101 function, which the Kochen-
    If indeed, there is any free bit that influences          Specker paradox shows does not exist. This com-
a, the universe has by definition taken a free                pletes the proof.
decision near a by time t0 , and we remove the
pedantry by ascribing this decision to particle a.           Locating the Response
(This is discussed more fully in the section “Free           We now provide a fuller discussion of some delicate
Will Versus Determinism”.)                                   points.
    (ii) From now on we can suppose that no such                (i) Since the observed spot on the screen is the
new information bits influence the particles’ re-             result of a cascade of slightly earlier events, it
sponses, and therefore that α and β are functions            is hard to define just when “the response” really
of the respective experimenters’ choices and of              starts. We shall now explain why one can regard
events earlier than those choices.                           a’s response (say) as having already started at any
    Now an α can be expected to vary with x, y, z            time after A’s choice when all the free information
and may or may not vary with w . However, whether            bits that influence it have become available to a.
the function varies with them or not, we can intro-             Let N(a) and N(b) be convex regions of space-
duce all of x, y, z, w as new arguments and rewrite          time that are just big enough to be “neighborhoods
θa as a new function (which for convenience we               of the respective experiments”, by which we mean
give the same name)                                          that they contain the chosen settings of the ap-
                    F                                        paratus and the appropriate particle’s responses.
                   θa (x, y, z, w ; α′ )             (⋆)
                                                             Our proof has shown that if the backward half-
of x, y, z, w and properties α independent of
                                                             space t < tF determined by a given F-time tF is
x, y, z, w .                                                 disjoint from N(a), then the available informa-
   To see this, replace any α that does depend               tion it contains is not enough to determine a’s
on x, y, z, w by the constant values α1 , . . . , α1320 it   response. On the other hand, if each of the two
takes for the 40 × 33 = 1320 particular quadru-              such half-spaces contains the respective neigh-
ples x, y, z, w we shall use. Alternatively, if each         borhood, then of course they already contain the
α is some function α(x, y, z, w ) of x, y, z, w , we         responses. By varying F and G, this suffices to lo-
may substitute these functions in (⋆) to obtain              cate the free decisions to the two neighborhoods,
information bits independent of x, y, z, w .                 which justifies our ascribing it to the particles
                               G                             themselves.
   Similarly, we can rewrite θb as a function
                   θb (x, y, z, w ; β′ )                        (ii) We remark that not all the information in the
                                                             G-backward half-space (say) need be available to
of x, y, z, w and properties β′ independent of                                                                   G
                                                             b, because MIN prevents particle b’s function θb
x, y, z, w .                                                 from using experimenter A’s choice of directions
   Now for the particular choice of w that B will            x, y, z. The underlying reason is of course, that rel-
make, there is a value β0 for β′ for which                   ativity allows us to view the situation from a B-first
                    G                                        frame, in which A’s choice is made only later than
                   θb (x, y, z, w ; β0 )
                                                             b’s response, so that A is still free to choose an
is defined. By the above independence of β′ from              arbitrary one of the 40 triples. However, this is our
w , the function θb (x, y, z, w ; β0 ) is defined with        only use of relativistic invariance—the argument
the same value β0 for all 33 values of w . (The fact         actually allows any information that does not re-
that MIN allows B to freely vary his choice of w             veal A’s choice to be transmitted superluminally,
makes this intuitively clear.)                               or even backwards in time.

February 2009                                          Notices of the AMS                                                 229
         (iii) Although we’ve precluded the possibility               Historically, this kind of correlation was a great
      that θb can vary with A’s choice of directions, it is        surprise, which many authors have tried to ex-
      conceivable that it might nevertheless vary with a’s         plain away by saying that one particle influences
      (future!) response. However, θb cannot be affected            the other. However, as we argue in detail in [1],
      by a’s response to an unknown triple chosen by               the correlation is relativistically invariant, unlike
      A, since the same information is conveyed by the             any such explanation. Our attitude is different:
      responses (0, 1, 1), to (x, y, z), (1, 0, 1) to (z, x, y),   following Newton’s famous dictum “Hypotheses
      and (1, 1, 0) to (y, z, x). For a similar reason θa     F    non fingo”, we attempt no explanation, but accept
      cannot use b’s response, since B’s experiment                the correlation as a fact of life.
      might be to investigate some orthogonal triple                  Some believe that the alternative to determin-
      u, v, w and discard the responses corresponding              ism is randomness, and go on to say that “allowing
      to u and v.                                                  randomness into the world does not really help
                                                                   in understanding free will.” However, this objec-
         (iv) It might be objected that free will itself
                                                                   tion does not apply to the free responses of the
      might in some sense be frame-dependent. Howev-
                                                                   particles that we have described. It may well be
      er, the only instance used in our proof is the choice        true that classically stochastic processes such as
      of directions, which, since it becomes manifest in           tossing a (true) coin do not help in explaining free
      the orientation of some macroscopic apparatus,               will, but, as we show in the Appendix and in §10.1
      must be the same as seen from arbitrary frames.              of [1], adding randomness also does not explain
         (v) Finally, we note that the new proof involves          the quantum mechanical effects described in our
      four inertial frames—A-first, B-first, F, and G.               theorem. It is precisely the “semi-free” nature of
      This number cannot be reduced without weak-                  twinned particles, and more generally of entan-
      ening our theorem, since we want it to apply to              glement, that shows that something very different
      arbitrary frames F and G, including for example              from classical stochasticism is at play here.
      those in which the two experiments are nearly                   Although the FWT suggests to us that determin-
      simultaneous.                                                ism is not a viable option, it nevertheless enables
                                                                   us to agree with Einstein that “God does not play
      Free Will Versus Determinism                                 dice with the Universe.” In the present state of
      We conclude with brief comments on some of the               knowledge, it is certainly beyond our capabilities
                                                                   to understand the connection between the free
      more philosophical consequences of the Free Will
                                                                   decisions of particles and humans, but the free
      Theorem (abbreviated to FWT).
                                                                   will of neither of these is accounted for by mere
         Some readers may object to our use of the
      term “free will” to describe the indeterminism of
                                                                      The tension between human free will and phys-
      particle responses. Our provocative ascription of
                                                                   ical determinism has a long history. Long ago,
      free will to elementary particles is deliberate, since
                                                                   Lucretius made his otherwise deterministic parti-
      our theorem asserts that if experimenters have a             cles “swerve” unpredictably to allow for free will.
      certain freedom, then particles have exactly the             It was largely the great success of deterministic
      same kind of freedom. Indeed, it is natural to               classical physics that led to the adoption of deter-
      suppose that this latter freedom is the ultimate             minism by so many philosophers and scientists,
      explanation of our own.                                      particularly those in fields remote from current
         The humans who choose x, y, z, and w may of               physics. (This remark also applies to “compati-
      course be replaced by a computer program con-                balism”, a now unnecessary attempt to allow for
      taining a pseudo-random number generator. If we              human free will in a deterministic world.)
      dismiss as ridiculous the idea that the particles               Although, as we show in [1], determinism may
      might be privy to this program, our proof would              formally be shown to be consistent, there is no
      remain valid. However, as we remark in [1], free             longer any evidence that supports it, in view of the
      will would still be needed to choose the random              fact that classical physics has been superseded by
      number generator, since a determined determinist             quantum mechanics, a non-deterministic theory.
      could maintain that this choice was fixed from the            The import of the free will theorem is that it is not
      dawn of time.                                                only current quantum theory, but the world itself
         We have supposed that the experimenters’                  that is non-deterministic, so that no future theory
      choices of directions from the Peres configuration            can return us to a clockwork universe.
      are totally free and independent. However, the
      freedom we have deduced for particles is more                Appendix. Can There Be a Mechanism for
      constrained, since it is restricted by the TWIN              Wave Function Collapse?
      axiom. We introduced the term “semi-free” in [1]             Granted our three axioms, the FWT shows that
      to indicate that it is really the pair of particles          nature itself is non-deterministic. It follows that
      that jointly makes a free decision.                          there can be no correct relativistic deterministic

230                                         Notices of the AMS                                Volume 56, Number 2
theory of nature. In particular, no relativistic ver-   choices of directions x, y, z, and w .1 There are
sion of a hidden variable theory such as Bohm’s         40 × 33 = 1320 possible fields in question. For
well-known theory [4] can exist.                        each such choice, we have a distribution X(FA , FB )
   Moreover, the FWT has the stronger impli-            of flashes, i.e., we have different distributions
cation that there can be no relativistic theory         X1 , X2 , . . . , X1320 . Let us be given “in advance”
that provides a mechanism for reduction. There          all such random sequences, with their different
are nonlinear extensions of quantum mechanics,          weightings as determined by the different fields.
which we shall call collectively GRW theories (after    Note that for this to be given, nature does not have
Ghirardi, Rimini, and Weber, see [5]) that attempt      to know in advance the actual free choices FA (i.e.,
to give such a mechanism. The original theories         x, y, z) and FB (i.e., w ) of the experimenters. Once
were not relativistic, but some newer versions          the choices are made, nature need only refer to
make that claim. We shall focus here on Tumul-          the relevant random sequence Xk in order to emit
ka’s theory rGRWf (see [6]), but our argument           the flashes in accord with rGRWf.
                                                           If we refer to the proof of the FWT, we can see
below applies, mutatis mutandis, to other rela-
                                                        that we are here simply treating the distributions
tivistic GRW theories. We disagree with Tumulka’s
                                                        X(FA , FB ) [= X(x, y, z, w )] in exactly the same way
claim in [7] that the FWT does not apply to rGRWf,
                                                        we treated any other information-bit α that de-
for reasons we now examine.
                                                        pended on x, y, z, w . There we substituted all the
   (i) As it is presented in [6], rGRWf is not a de-    values α1 , . . . , α1320 for α in the response function
terministic theory. It includes stochastic “flashes”     θa (x, y, z, w ; α). Thus, the functional hypothesis
that determine the particles’ responses. However,       does apply to rGRWf, as modified in this way by
in [1] we claim that adding randomness, or a            the recipe.
stochastic element, to a deterministic theory does         Tumulka [7] grants that if that is the case, then
not help:                                               rGRWf acquires some nasty properties: In some
   “To see why, let the stochastic element in a         frame Λ, “[the flash] fy will entail influences to
putatively relativistic GRW theory be a sequence        the past.” Actually, admitting that the function-
of random numbers (not all of which need be             al hypothesis applies to rGRWf has more dire
used by both particles). Although these might           consequences—it leads to a contradiction. For if,
only be generated as needed, it will plainly make       as we just showed, the functional hypothesis ap-
no difference to let them be given in advance.           plies to the flashes, and the first flashes determine
But then the behavior of the particles in such a        the particles’ responses, then it also applies to
theory would in fact be a function of the informa-      these responses, which by the FWT leads to a
tion available to them (including this stochastic       contradiction.
element).”                                                 (ii) Another possible objection is that in our
   Tumulka writes in [7] that this “recipe” does        statement of the MIN axiom, the assertion that a’s
not apply to rGRWf:                                     response is independent of B’s choice was insuf-
   “Since the random element in rGRWf is the set        ficiently precise. Our view is that the statement
of flashes, nature should, according to this recipe,     must be true whatever precise definition is given
make at the initial time the decision where-when        to the term “independent”, because in no inertial
flashes will occur, make this decision ‘available’       frame can the past appearance of a macroscopic
to every space-time location, and have the flash-        spot on a screen depend on a future free decision.
es just carry out the pre-determined plan. The             It is possible to give a more precise form of MIN
problem is that the distribution of the flashes          by replacing the phrase “particle b’s response is
depends on the external fields, and thus on the          independent of A’s choice” by “if a’s response is
free decision of the experimenters. In particular,      determined by B’s choice, then its value does not
                                                        vary with that choice.” However, we actually need
the correlation between the flashes in A and those
                                                        precision only in the presence of the functional
in B depends on both external fields. Thus, to
                                                        hypothesis, when it takes the mathematical form
let the randomness ‘be given in advance’ would                                                    F
                                                        that a’s putative response function θa cannot in
make a big difference indeed, as it would require
                                                        fact vary with B’s choice. To accept relativity but
nature to know in advance the decision of both
                                                        deny MIN is therefore to suppose that an exper-
experimenters, and would thus require the theory
                                                        imenter can freely make a choice that will alter
either to give up freedom or to allow influences to      the past, by changing the location on a screen of
the past.”                                              a spot that has already been observed.
   Thus, he denies that our “functional hypothe-
sis”, and so also the FWT, apply to rGRWf. However,     1
                                                         This unfortunately makes rGRWf non-predictive—it can
we can easily deal with the dependence of the           only find the flash distribution that “explains” either
distribution of flashes on the external fields FA         particle’s behavior when both experimenters’ fields are
and FB , which arise from the two experimenters’        given.

February 2009                                     Notices of the AMS                                               231
          Tumulka claims in [7] that since in the twinning      [3] A. Bassi and G. C. Ghirardi, The Conway-Kochen
      experiment the question of which one of the first              argument and relativistic GRW models, Found. Phys.
      flashes at A and B is earlier is frame-dependent, it           37(2) (2007), 169–185.
      follows that the determination of which flash influ-        [4] D. Bohm, Quantum Theory in terms of “hidden”
                                                                    variables, I, Phys. Rev. 85 (1952), 166–193.
      ences the other is also frame-dependent. However,
                                                                [5] G. C. Ghirardi, A. Rimini, and T. Weber, Unified dy-
      MIN does not deal with flashes or other occult
                                                                    namics for microscopic and macroscopic systems,
      events, but only with the particles’ responses as             Phys. Rev. D34 (1986), 470–491.
      indicated by macroscopic spots on a screen, and           [6] R. Tumulka, arXiv:0711.oo35v1 [math-ph] Octo-
      these are surely not frame-dependent.                         ber 31, 2007.
          In any case, we may avoid any such questions          [7]        , Comment on “The Free Will Theorem”, Found.
      about the term “independent” by modifying MIN                 Phys. 37 (2) (2007), 186–197.
      to prove a weaker version of the FWT, which nev-
      ertheless still yields a contradiction for relativistic
      GRW theories, as follows.                                    Authors’ Note: We thank Eileen Olszewski for
          MIN′ : In an A-first frame, B can freely choose any    typesetting the paper and Frank Swenton for the
      one of the 33 directions w , and a’s prior response       graphics.
      is independent of B’s choice. Similarly, in a B-first
      frame, A can independently freely choose any one
      of the 40 triples x, y, z, and b’s prior response is
      independent of A’s choice.
          To justify MIN′ note that a’s response, signaled
      by a spot on the screen, has already happened in
      an A-first frame, and cannot be altered by the later
      free choice of w by B; a similar remark applies to
      b’s response. In [7], Tumulka apparently accepts
      this justification for MIN′ in rGRWf: “. . . the first            Mathematicians rarely make it into
      flash fA does not depend on the field FB in a frame               the newspaper, much less into fashion
      in which the points of B are later than those of A.”            spreads. But on September 21, 2008, the
          This weakening of MIN allows us to prove a                  fashion section of the New York Times
      weaker form of the FWT:                                         Sunday magazine carried a picture of An-
          FWT′ : The axioms SPIN, TWIN, and MIN′ imply                nalisa Crannell, a professor of mathemat-
      that there exists an inertial frame such that the re-           ics at Franklin & Marshall College (and a
      sponse of a spin 1 particle to a triple experiment is           book reviewer in this issue of the Notices).
      not a function of properties of that part of the uni-           The magazine contacted Crannell saying
      verse that is earlier than the response with respect            that it was doing a photo portfolio about
      to this frame.                                                  academics and had chosen her because
          This result follows without change from our                 she had received the Haimo Award from
      present proof of the FWT by taking F to be an                   the Mathematical Association of America.
      A-first frame and G a B-first frame, and applying                 “By phone, I learned that the ‘photo port-
      MIN′ in place of MIN to eliminate θa ’s dependence              folio’ was really about fashion, which was
                     G                                                pretty funny because part of my Haimo
      on w and θb ’s dependence on x, y, z.
                                                                      Award speech included the price tags of
          We can now apply FWT′ to show that rGRWf’s
                                Λ                                     the clothes I usually wear (US$1 or less),”
      first flash function (fy of [4]), which determines
                                                                      she said.
      a’s response, cannot exist, by choosing Λ to be the
                                                                          A crew of eight spent three hours
      frame named in FWT′ .
                                                                      dressing Crannell and doing her hair and
          The Free Will Theorem thus shows that any
                                                                      makeup. One of them told Crannell he
      such theory, even if it involves a stochastic ele-
                                                                      loves mathematics and asked her some
      ment, must walk the fine line of predicting that for
                                                                      questions about fractals. The picture of
      certain interactions the wave function collapses to
                                                                      Crannell, which can be found on the Web,
      some eigenfunction of the Hamiltonian, without
                                                                      does not show the US$2,500 Gucci boots,
      being able to specify which eigenfunction this is.
                                                                      which Crannell called “pretty darned un-
      If such a theory exists, the authors have no idea
                                                                      comfortable”. “I was wearing my blue AMS
      what form it might take.
                                                                      ‘I love math’ bracelet that I’d picked up at
                                                                      MathFest, and the guy admired it so I gave
      References                                                      it to him,” Crannell recalled. “I didn’t get
      [1] J. Conway and S. Kochen, The Free Will Theorem,             to keep any of the clothes they brought,
          Found. Phys. 36 (2006), 1441–1473.
                                                                      but he got to keep my bracelet!”
      [2] S. Kochen and E. Specker, The problem of hidden
          variables in quantum mechanics, J. Math. Mech. 17
          (1967), 59–88.

232                                       Notices of the AMS                                Volume 56, Number 2

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