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A Step-by-Step Argument for Causal Finitism

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I defend a new argument for causal finitism, the view that nothing can have an infinite causal history. I begin by defending a number of plausible metaphysical principles, after which I explore a host of novel variants of the Littlewood-Ross and Thomson’s Lamp paradoxes that violate such principles. I argue that causal finitism is the best solution to the paradoxes.

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  1. This rules out infinite regresses of causes. It’s important to note that there are different ways of fleshing out this definition. Here are a few: (i) there is no possible world in which there exists an infinitely descending chain of causal-power dependencies; (ii) nothing can be affected (directly or indirectly) by infinitely many causes (Pruss, 2018, p. 2); and (iii) there cannot be infinitely many things that causally impact one target state (Pruss, 2018, p. 25). All of these ways accurately capture the general thrust of CF.

  2. Discussion of the Littlewood-Ross paradox has (somewhat) stagnated, as the majority of substantive treatments of the paradox appear in the 1990s and early 2000s. Also, the recent proposals to which I’m referring are those contained in Pruss (2018), Koons (2014), and Huemer (2016) concerning CF and infinite intensive magnitudes, among others.

  3. Some of these terms merit definition. Roughly: (i) a process is a determinate sequence of actions on or within a system; (ii) a system is a bounded, specified area or domain; (iii) a state of a system is the condition or character of the system’s contents at a specified point (e.g. point in time or space) and along a specified axis (e.g. mass, shape, numbered contents, etc.); (iv) a step/sub-step of a process is a single, definite action or closely related (in terms of space, time, or execution) sequence of such actions which—when conjoined with more steps—forms a whole process; (v) identical means qualitatively identical and is always understood as identical along a specified axis (e.g. global condition of the respective systems, or else some specified property or properties). If the rough characterizations prove ultimately unhelpful, we can rest content with an intuitive grasp and obvious examples.

  4. Even if one thinks that symmetry can breed asymmetry—say, because one thinks this is entailed by indeterministic, stochastic quantum processes or libertarianly free actions—we can restrict the application of SP to deterministic processes. And this won’t affect the arguments I’ll give for CF, since the systems in which they operate are deterministic.

  5. Similar examples can be given in terms of spatially extended processes provided that space is continuous, as well as in terms of synchronous (at a single time) processes. More on this later.

  6. It’s worth noting that other authors have hinted at something like SP—though, few have explored its motivations and implications. An example of an author who hints at (something like) SP is Forrest (1999), who espouses a principle called the Impotence of Individuality, “the principle that if qualitatively identical processes result in products which are not qualitatively identical then the results are not determined by which process is which,” and he finds it plausible that such a principle is “a metaphysically necessary principle” (p. 445). This paper adds (what I take to be) much needed clarifications, motivations, and implications of SP.

  7. If one demurs at the specificity, we can modify the principle to a more general principle along the lines of: “Given an alternating process, identical sequences of individual alternations yield identical states (provided the process/system is explanatorily closed).” The argument for CF I develop can proceed (mutatis mutandis) with either formulation. (I stick with the specific formulation in the main text.) I am grateful to an anonymous referee for bringing the specificity of the principle—as well as this more general reformulation—to my attention.

  8. The paradox first appeared in Littlewood (1953, p. 26), though a description of the same paradox within Ross (1988, p. 46) captured philosophers’ attention first. Both Littlewood and Ross argue that the urn is empty at noon, with Allis and Koetsier (1991; 1995) and Earman and Norton (1996) arguing likewise. Holgate (1994) adds mathematical clarifications on the debate, while van Bendegem (1994) argues for the impossibility of the paradox in virtue of the contradictory states obtained (both empty and infinitely full). Other authors have contributed to the debate concerning the Littlewood-Ross paradox (or versions of it) not only concerning the urn’s end state but also the scenario’s metaphysical possibility. In addition to those already mentioned, see (inter alia) van Bendegem (1995; 2003), Forrest (1999), Byl (2000), Friedman (2002), Oppy (2006), Huemer (2016), Manchak and Roberts (2016), and Cook (2020).

  9. By ‘adding them’, I simply mean adding (i.e. actually performing) such interactions to an original state of the system (such as being off at 11:00).

  10. I’m thankful to an anonymous referee for bringing these approaches to my attention.

  11. Most others working on the paradox agree. Among others, these include Littlewood and Ross themselves as well as Huemer (2016), Oppy (2006), Cook (2020), Holgate (1994), and Earman and Norton (1996).

  12. Why photons? Because “[i]t is possible for two or more photons to share the same physical state, a condition that would not be possible for [say] electrons. To have a large cardinality of photons in a space–time such as ours would require that some photons be in the same place, and indeed in the same state” (Pruss 2006, p. 100). Note, though, that in this context we need not restrict ourselves to ‘space-times such as ours’. Nevertheless, this point should help assuage worries about co-location.

  13. See Sect. 6 for more details. Recall: these need only be logically possible.

  14. Recall that in motivating SP, the intuitive plausibility and explicability considerations didn’t hinge on total or global qualitative identity, in every single respect, of the respective systems. I was cautious to articulate the principle along some specified axis. (One might object: if there is some other axis, O*, along which the respective systems differ, might that end up providing an explanation for why the end states of the respective systems differ along the original axis, O, upon completion of the relevant processes? I don’t think so. This could only be the case if O is dependent on O*. But at least in the cases I have specified, the numbers on the balls are independent of the positions and trajectories. Changing the positions and trajectories does nothing to alter the numbers; only the mechanism can change the numbers.)

  15. What about RIP? I have addressed its relation to the ‘underdetermined end state’ worry at the end of Sect. 3.2. In short, RIP engenders paradox regardless of whether the end state in the original Thomson’s Lamp story is underdetermined—all RIP needs to get off the ground is that the lamp is either on or off at noon in the ‘actual’ world.

  16. A second reason (with which Huemer agrees) to restrict our focus to intensive magnitudes is that it seems possible for there to be infinite extensive magnitudes. For instance, absent strict finitist qualms, plausibly space could be infinite in extent, and plausibly there could be a universe with an infinite amount of mass or infinitely many electrons. Plausibly, moreover, for philosophers of a realist bent, it’s at least possible for there to be infinitely many abstracta (numbers, say).

  17. Note that as a zero is printed on ball n, the ball shrinks to the size of ball 10n. This stipulation ensures the absolute qualitative identity between the two urns’ contents. Though, absolute qualitative identity is technically not required; all we need is qualitative identity along some dimension or axis (with respect to numbered contents, say). That suffices for a violation of SP.

  18. I’m using ‘machine’ for simplicity, but keep in mind that I’m referring to any mechanism that performs the steps.

  19. For more on hypertime, see Hudson (2014) and Lebens and Goldschmidt (2017).

  20. Again, the ‘mechanisms’ could be point particles with the causal power to produce the (to-be-specified) effect, or whatever.

  21. We seem to have some reason to think (on the basis of quantum entanglement phenomena) that some kind of influence or action at a distance is not only possible but actual.

  22. We need not worry about the details of the signal—it suffices to note simply that the causal influence travels across space and time to cause a specific change in L, namely the addition (and/or subtraction) of labeled balls. Again: the balls are not essential to the story—particles could take their place, where a label on the particles could be represented by some special physical property or quantity.

  23. There will also be angelic variants on the persistence conditions. E.g., nothing destroys them throughout the task or at any point shortly thereafter; if an angel thinks of a number, the angel persists in thinking of that number unless caused (internally or externally) not to think that number; etc. Again, the argument only requires the logical possibility of angels (or something like angels, such as distinct thoughts in the divine intellect). As I use it, an angel is just a non-divine, non-physical, non-embodied, non-human mind.

  24. I don’t have any worked out account of what an area for angels would be. This is only a functional term that serves the role of the urn (i.e. the system). Maybe angels are (or would be) utterly non-spatial, in which case we could simply define the system as the totality of the number-thinking angels, or as the non-spatial realm in which they reside, or whatever.

  25. Or, rather, infinitely many angels each of which thinks of a natural number followed by an omega sequence of zeroes.

  26. See Erasmus and Luna (2020) and Pruss (2018, ch. 3) for more precise and extended discussions of the inference from the possibility of ungrounded chains (i.e. a chain, sequence, or series with (i) a strict total ordering (e.g. an ‘earlier than’ or ‘caused by’ relation) among its members and (ii) no ‘first’ member) to the possibility of paradoxes (viz. Benardete-type paradoxes involving Grim Reapers, Deafening Peals, etc.) analogous to the ones explored in my paper.

  27. See Koons (2014, esp. pp. 257–260) for an elaboration and defence of this inference.

  28. While some such synchronic versions employ continuous space, others don’t require it and instead employ strategies like infinite space plus instantaneous action at a distance (as in Sect. 4.3) or even infinite space plus an infinite past (with the speed of light as the speed limit to causal interaction, again as in Sect. 4.3).

  29. And the difference maker couldn’t come upon completion of the process, since that’s ‘too late’ in the order of explanation.

  30. Thanks to an anonymous referee for this perceptive point.

  31. More precisely, I have only argued that CF is the best solution to the paradoxes of this paper. I have here put the argument in deductive form for the sake of simplicity.

  32. I shall hereafter speak simply of ‘rejecting (a)’, but note that this includes merely rejecting (a) as applied to divergent infinities.

  33. For instance, (i) the seeming absence of any prevention of the construction’s arising given the possibility of infinite causal chains, (ii) patchwork principles, etc.

  34. Thanks to Joshua Rasmussen, Alexander Pruss, and two anonymous referees for their valuable insights and/or comments on earlier drafts.


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Schmid, J.C. A Step-by-Step Argument for Causal Finitism. Erkenn 88, 2097–2122 (2023).

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