Abstract
An infinite lottery machine is used as a foil for testing the reach of inductive inference, since inferences concerning it require novel extensions of probability. Its use is defensible if there is some sense in which the lottery is physically possible, even if exotic physics is needed. I argue that exotic physics is needed and describe several proposals that fail and at least one that succeeds well enough.
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20 June 2017
An erratum to this article has been published.
Notes
Here I will restrict myself to objective approaches to inductive inference. Analogous problems arise if we ask what should be our subjective belief in each outcome.
If we accelerate the jumping in a supertask, so infinitely many jumps are completed in finite time, the flea will have jumped “off to infinity,” behaving like what is called a “space evader” in Manchak and Roberts (2016, §1.4).
They are uniform in the sense that any two angular segments of the same size host rational angles that can be mapped one-one onto each other; and the entire set of rational angles is mapped back to itself if we shift all the angles by adding any fixed rational angle to them.
I learned of this argument from the ever-inventive Jim Brown.
These conditions may not be necessary. Paraconsistent logics support deductive reasoning in domains with inconsistent descriptions.
To keep the selection conservative, I set aside Newtonian indeterministic systems such as the dome or the infinite mass-spring-mass-spring-… system, described in (Norton 2008: Norton 1999, §1.1), even though van Strien (2014) has shown that dome-like indeterminism had already been found in Newtonian physics in the nineteenth century.
For an introduction to the extensive literature on supertasks, see Manchak and Roberts (2016).
Davies (2001) describes a supertask computer able to compute Turing non-computable functions through its use of unlimited miniaturization.
This supertask has a venerable history that includes a celebrated von Neumann anecdote. See Halmos (1973, pp. 386–87).
The fastest way to arrive at (5) is to generate the random walk from a simplified half-step random walk, in which the walker can only move up a half step (“+1/2”) or down a half-step(“-1/2”) with probability 1/2 in each case. The result of two half steps is +1, −1 or 0, conforming with the probabilities of (4). In N steps = 2N half-steps, the probability of m half steps up and 2N–m half steps down is just \( \frac{\left(2 N\right)!}{m!\left(2 N- m\right)!}\frac{1}{2^{2 N}} \). This results in a net motion of m – (2N–m) = 2(m-N) half steps up, which corresponds to the walker moving to cell n = (m-N). This last probability becomes equation (5) with the substitutions m = N + n and 2N–m = 2N–(n + N) = N–n.
That is, for any m, n, \( {}_{N\to \infty}^{\lim}\frac{P\left( m, N\right)}{P\left( n, N\right)}=\frac{\left( N+ n\right)!\left( N- n\right)!}{\left( N+ m\right)!\left( N- m\right)!}=1. \)
To avoid issues of failure of convergence in the positions of the balls in the past time limit, it is convenient to replace the urn and balls with an infinite panel of numbered switches. Removing a ball corresponds to setting its switch from “in” to “out.”
The probability that none of the gods in the collective remove the lowest numbered ball from its urn is \( \frac{N}{N+1}\times \frac{N+1}{N+2}\times \cdots \times \frac{2 N-1}{2 N}=\frac{N}{2 N}=\frac{1}{2} \).
For a concern that quantum theory likely precludes such unnormalizable states, see Earman, Additivity requirements in classical and quantum probability, unpublished.
This scheme can also be implemented (but with the same two problems) as an infinite superposition of the discrete energy eigenstates of a particle in an infinitely deep potential well; or a harmonic oscillator.
A more modern calculation (Johnson 2007, p. 34), using the Schrödinger wave for the electron orbitals, affirms that the position expectation for the nth s state <r > grows with n 2.
The complication of the example, as a referee has reminded me, is to find a precise enough stipulation of what constitutes a galaxy and the count of the stars it contains. An easier course is to divide space into large volumes of equal size and assign an “H” or “T” according to whether the net charge of the volume is even or odd, when measured in units of electron charge.
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Acknowledgements
I thank John Earman, Casper Storm Hansen, Bryan Roberts, David Snoke, Teddy Seidenfeld and Porter Williams for helpful discussion.
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The original version of this article was revised: Formulae in footnotes 11, 12 and in Section 11.3 Nonmeasurable outcomes, 3rd sentence, paragraph 5, were incorrectly presented.
An erratum to this article is available at https://doi.org/10.1007/s13194-017-0175-3.
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Norton, J.D. How to build an infinite lottery machine. Euro Jnl Phil Sci 8, 71–95 (2018). https://doi.org/10.1007/s13194-017-0174-4
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DOI: https://doi.org/10.1007/s13194-017-0174-4