The Impossibility Theorem

  • David Wick


Nothing takes place without a sufficient reason,” wrote the German philosopher and mathematician Gottfried Leibniz in the 18th century; “that is to say, nothing occurs for which one having sufficient knowledge might not give a reason... why it is as it is and not otherwise.” When 19th-century physicists invoked chance in their statistical theories of matter, it cast no cloud over the principle of sufficient reason; probability is useful, but causal determinism rules the universe. So it was most surprising when, in 1932, a young Hungarian mathematician announced a mathematical proof that determinism is dead.


Quantum Mechanic Sufficient Reason Impossibility Theorem Billiard Ball Modern Mathematical Logic 
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  1. I profited from Steven J. Heims’ John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death (1980), Norman Macrae’s John von Neumann (1992), and Wigner’s various memoirs.Google Scholar
  2. [p. 63]
    Heisenberg anecdote: see Macrae, p. 142.Google Scholar
  3. [p. 63]
    Arthur Wightman assured me that they did read von Neumann in Princeton in the 1940s.Google Scholar
  4. [p. 65]
    Von Neumann reaches the punchline on p. 325 of his book, von Neumann (1955).Google Scholar
  5. [p. 66]
    The example of an unstable dynamical system is called “Sinai’s billiard” after the Russian mathematical physicist Ya. Sinai; see Sinai (1976).Google Scholar
  6. [p. 67]
    See Jammer (1974) for a detailed history of the responses and criticisms to von Neumann’s theorem. Bell’s example is in his first paper “On the Problem of Hidden Variables in Quantum Theory,” Rev. Mod. Phys. 38 (1966), reprinted in Bell (1987), and his analysis is trenchant. But for the author, only rediscovering Bohm’s and Bub’s model of a quantum apparatus (see Chapter 18) finally caused the fog to lift.Google Scholar
  7. [p. 68a]
    Other “impossibility theorems” are those of Gleason , J. Math. & Mech. 6, 885 (1957);MathSciNetMATHGoogle Scholar
  8. [p. 68b]
    Jauch and Piron , Helv. Phys. Acta. 36, 827 (1963);MathSciNetMATHGoogle Scholar
  9. [p. 68c]
    Kochen and Specker, J. Math. & Mech. 17, 59 (1967), and there are many more. Although these authors proved interesting theorems about Hilbert spaces, they failed to attain their goal for the same reason as did von Neumann: ignoring the role of the apparatus in the alternative “hidden variable” theories. Statisticians, naturally, are the group that understands this point best; see Chapter 12, “Dice Games and Conspiracies,” and ChapterMathSciNetMATHGoogle Scholar

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© Birkhäuser Boston 1995

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  • David Wick

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