The free will theorem of John H. Conway and

Simon B. Kochen Simon Bernhard Kochen (; born 14 August 1934) is a Canadian mathematician, working in the fields of model theory, number theory and quantum mechanics. Education and career Kochen was born in Antwerp, Belgium, and escaped the Nazis with his family ...

states that if we have a

free will Free will is generally understood as the capacity or ability of people to (a) choice, choose between different possible courses of Action (philosophy), action, (b) exercise control over their actions in a way that is necessary for moral respon ...

in the sense that our choices are not a function of the past, then, under specific assumptions drawn from

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

and relativity, so must some

elementary particles In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model presently recognizes seventeen distinct particles—twelve fermions and five bosons. As a con ...

. That is, if human experimenters possess a form of free will—defined as the ability to make choices not entirely determined by prior events—then certain elementary particles must also exhibit a corresponding form of indeterminacy. The theorem argues that

stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stoc ...

do not satisfy this definition of "

freedom Freedom is the power or right to speak, act, and change as one wants without hindrance or restraint. Freedom is often associated with liberty and autonomy in the sense of "giving oneself one's own laws". In one definition, something is "free" i ...

," because random values can, in principle, be pre-determined or embedded in the past (for example, sampled from a pre-existing

table Table may refer to: * Table (database), how the table data arrangement is used within the databases * Table (furniture), a piece of furniture with a flat surface and one or more legs * Table (information), a data arrangement with rows and column ...

). Consequently, the theorem implies that no

physical theory Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict natural phenomena. This is in contrast to experimental physics, which uses experi ...

relying solely on a combination of deterministic laws and pre-existing randomness can fully account for the observed outcomes of quantum measurements. Conway and Kochen's paper was published in ''

Foundations of Physics ''Foundations of Physics'' is a monthly journal "devoted to the conceptual bases and fundamental theories of modern physics and cosmology, emphasizing the logical, methodological, and philosophical premises of modern physical theories and procedur ...

'' in 2006. In 2009, the authors published a stronger version of the theorem in the ''

Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume was published in 1953. Each issue of the magazine ...

''. Later, in 2017, Kochen elaborated some details.

Axioms

The proof of the theorem as originally formulated relies on three axioms, which Conway and Kochen call "fin", "spin", and "twin". The spin and twin axioms can be verified experimentally. # Fin: There is a maximal

speed In kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a non-negative scalar quantity. Intro ...

for propagation of

information Information is an Abstraction, abstract concept that refers to something which has the power Communication, to inform. At the most fundamental level, it pertains to the Interpretation (philosophy), interpretation (perhaps Interpretation (log ...

(not necessarily the

speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...

). This assumption rests upon causality. # Spin: The squared spin component of certain elementary particles of spin one, taken in three orthogonal directions, will be a permutation of # Twin: It is possible to "entangle" two elementary particles and separate them by a significant distance, so that they have the same squared spin results if measured in parallel directions. This is a consequence of

quantum entanglement Quantum entanglement is the phenomenon where the quantum state of each Subatomic particle, particle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic o ...

, but full entanglement is not necessary for the twin axiom to hold (entanglement is sufficient but not necessary). It isn't clear why the caveat of a maximal speed of transmission isn't "necessarily the speed of light". That's because there can only be one universal speed, or the powerfully proven

Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...

s cannot be even approximately true. And the speed of light has to be that universally determined speed; that's because it is the ratio between the two electromagnetic forces the electrical and the magnetic otherwise the laws of nature (in particular, the reality that gives our minds the ability to think) would be impossible. In their later 2009 paper, "The Strong Free Will Theorem", Conway and Kochen replace the Fin axiom by a weaker one called Min, thereby strengthening the theorem. The Min axiom asserts only that two experimenters separated in a space-like way can make choices of measurements independently of each other. In particular, it is not postulated that the speed of transfer of ''all'' information is subject to an upper limit, but only of the particular information about choices of measurements. In 2017, Kochen argued that Min could be replaced by Lin experimentally testable Lorentz covariance.

The theorem

The free will theorem states: That is an "outcome open" theorem: Since the theorem applies to any arbitrary physical theory consistent with the axioms, it would not even be possible to place the information into the universe's past in an ad hoc way. The argument proceeds from the

Kochen–Specker theorem In quantum mechanics, the Kochen–Specker (KS) theorem, also known as the Bell–KS theorem, is a "no-go" theorem proved by John S. Bell in 1966 and by Simon B. Kochen and Ernst Specker in 1967. It places certain constraints on the pe ...

, which shows that the result of any individual measurement of spin was not fixed independently of the choice of measurements. As stated by Cator and Landsman regarding hidden-variable theories: "There has been a similar tension between the idea that the hidden variables (in the pertinent causal past) should on the one hand include all ontological information relevant to the experiment, but on the other hand should leave the experimenters free to choose any settings they like." Rephrasing one more time: The experimenters themselves would be bound by the same restraining hidden variables, so the premise of the experimenters making an arbitrary choice would be false, and hence a contradiction. So either there is free will, or the phrasing of the axiom is inadequate. But the phrasing of the axiom appears to be as representative as any.

Reception

According to Cator and Landsman, Conway and Kochen prove that "determinism is incompatible with a number of ''a priori'' desirable assumptions". Cator and Landsman compare the Min assumption to the locality assumption in

Bell's theorem Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measuremen ...

and conclude in the strong free will theorem's favor that it "uses fewer assumptions than Bell’s 1964 theorem, as no appeal to probability theory is made". The philosopher David Hodgson supports this theorem as showing quite conclusively that "science does not support determinism": that quantum mechanics proves that particles do indeed behave in a way that is not a function of the past. Critics, however, argue that the theorem applies only to deterministic, and not even to stochastic, models. Sheldon Goldstein, Daniel V. Tausk, Roderich Tumulka, and Nino Zanghì (2010)
What Does the Free Will Theorem Actually Prove?
''Notices of the AMS'', December, 1451–1453.

Notes

References

*
Introduction to the Free Will Theorem
videos of six lectures given by J. H. Conway, Mar. 2009. * {{cite book , last= Wüthrich , first= Christian , editor1-first= Claus , editor1-last= Beisbart , editor2-first= Stephan , editor2-last= Hartmann , title= Probabilities in Physics , chapter= Can the World Be Shown to Be Indeterministic After All? , date= September 2011 , publisher= Oxford University Press , isbn= 978-0199577439 , doi= 10.1093/acprof:oso/9780199577439.003.0014 , pages= 365–390 , chapter-url= http://philsci-archive.pitt.edu/8437/ , url= http://philsci-archive.pitt.edu/8437/1/WuthrichChristian2010PhilSci_IndeterministicWorld.pdf Physics theorems Free will John Horton Conway

Axioms

The theorem

Reception

See also

Notes

References