Abstract
In this paper, I present a puzzle involving special relativity and the random selection of real numbers. In a manner to be specified, darts thrown later hit reals further into a fixed well-ordering than darts thrown earlier. Special relativity is then invoked to create a puzzle. I consider four ways of responding to this puzzle which, I suggest, fail. I then propose a resolution to the puzzle, which relies on the distinction between the potential infinite and the actual infinite. I suggest that certain structures, such as a well-ordering of the reals, or the natural numbers, are examples of the potential infinite, whereas infinite integers in a nonstandard model of arithmetic are examples of the actual infinite.
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Notes
I discuss potential problems with this notion of probability below.
Chris Freiling (1986, p. 190) presents “a simple ‘philosophical’ proof of the negation of Cantor’s continuum hypothesis (CH).” He imagines that two darts are sequentially thrown at [0, 1]. A crucial assumption that Freiling (pp. 192, 199) makes is that “…the real number line does not really know which dart was thrown first or second…” and again “[t]he real number line cannot tell the order of the darts.” Freiling argues: [0, 1] cannot tell the order of the darts, along with other assumptions, proves not CH. In this paper, I argue: if CH, then [0, 1] can tell the order of the darts. And so we can respond to Freiling’s argument by defending CH and accepting that [0, 1] can tell the order of the darts. Of course, I go on to create a puzzle if [0, 1] can tell the order of the darts. But I still suggest that [0, 1] can tell the order of the darts.
To make the example work in terms of the relative motion, it may be necessary to employ two ‘dartboards’, that is, two copies of [0, 1], and have one dart thrown at each. The conclusion (puzzle) still follows.
‘Random’ means that any single positive integer has the same chance of selection as any other, from which it follows that the chance of selecting a fixed positive integer is either 0 or infinitesimal.
Due to finite additivity and the fact that each singleton’s chance of selection is either 0 or infinitesimal, as noted in footnote 4.
In reply to the worry that events of probability 0 can occur, it may also be helpful to think of repeated trials. That is, imagine running the experiment (involving two darts throws and special relativity) many times, e.g., 100. One player must win a minimum of 50 trials (in the case where each player wins 50 trials). It would be odd to suggest that, in repeated trials, an event of probability 0 occurs half of the time.
To my knowledge, Norton (2004, pp. 1147–1148) comes closest in the literature to endorsing the position that darts get larger through time. Norton does not couch his discussion in temporal terms. Instead he uses the terms “direct property” and “inverse property,” but the point seems very much the same.
I believe that recognizing this correct conception of infinite number dissolves many paradoxes of the infinite, not only the puzzle presented in this paper. For example, consider Thomson’s Lamp (Thomson 1954). If a lamp button is pressed infinitely many times, and if “infinitely many” is an infinite integer in a nonstandard model of arithmetic, then there is no paradox. Any infinite integer is either even or odd. If the button was pressed an even number of times, then the lamp is in its starting state. If the button was pressed an odd number of times, then the lamp is in the opposite state from its starting state. But why, it may be asked, can’t we ask about button presses of the structure ω, that is, a super-task as presently conceived? Simply put, because ω is only potentially infinite—it is never complete and actual; it is impossible to complete a task of structure ω. See Gwiazda’s (2012a) “A Proof of the Impossibility of Completing Infinitely Many Tasks” for an argument to this conclusion.
References
Brown, J. R. (2004). Peeking into Plato’s Heaven. Philosophy of Science, 71, 1126–1138.
Freiling, C. (1986). Axioms of symmetry: Throwing darts at the real number line. The Journal of Symbolic Logic, 51, 190–200.
Gwiazda, J. (2006). The train paradox. Philosophia, 34, 437–438.
Gwiazda, J. (2012a). A proof of the impossibility of completing infinitely many tasks. Pacific Philosophical Quarterly, 93, 1–7.
Gwiazda, J. (2012b). On infinite number and distance. Constructivist Foundations, 7, 126–130.
Hallett, M. (1988). Cantorian set theory and limitation of size. Oxford: Clarendon Press.
Moore, A. W. (2001). The infinite (2nd ed.). London: Routledge.
Norton, J. (2004). On thought experiments: Is there more to the argument? Philosophy of Science, 71, 1139–1151.
Thomson, J. F. (1954). Tasks and super-tasks. Analysis, 15, 1–13.
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Gwiazda, J. Throwing Darts, Time, and the Infinite. Erkenn 78, 971–975 (2013). https://doi.org/10.1007/s10670-012-9371-x
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DOI: https://doi.org/10.1007/s10670-012-9371-x