## Abstract

“**N**othing 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.

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### Notes

- 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 - [p. 63]Heisenberg anecdote: see Macrae, p. 142.Google Scholar
- [p. 63]Arthur Wightman assured me that they did read von Neumann in Princeton in the 1940s.Google Scholar
- [p. 65]Von Neumann reaches the punchline on p. 325 of his book, von Neumann (1955).Google Scholar
- [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
- [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 - [p. 68a]Other “impossibility theorems” are those of Gleason ,
*J. Math. & Mech.***6**, 885 (1957);MathSciNetMATHGoogle Scholar - [p. 68b]
- [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