Science Generates Limit Paradoxes


The sciences occasionally generate discoveries that undermine their own assumptions. Two such discoveries are characterized here: the discovery of apophenia by cognitive psychology and the discovery that physical systems cannot be locally bounded within quantum theory. It is shown that such discoveries have a common structure and that this common structure is an instance of Priest’s well-known Inclosure Schema. This demonstrates that science itself is dialetheic: it generates limit paradoxes. How science proceeds despite this fact is briefly discussed, as is the connection between our results and the realism-antirealism debate. We conclude by suggesting a position of epistemic modesty.

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Fig. 1


  1. 1.

    One could argue that the Tractatus does not fit the kind of “ladder paradoxes” we explore here because one can, in fact, jettison the ladder of the Tractatus, moving beyond its propositions to the correct understanding of the world (Wittgenstein himself appears to have thought so). However, Priest has argued (2002, pp. 191–192) that the Tractarian ladder cannot be used and then jettisoned, so Wittgenstein is saddled with paradox. Priest puts it thus: “… far from being the rungs of a real ladder that one can ascend, [the propositions of the Tractatus] are the like the rungs of a holographic ladder that will not support any weight put on them …” (p. 191).

  2. 2.

    “Apophenia” has precedence over Shermer’s (2008) term “patternicity” and is more general than “pareidolia.”

  3. 3.

    For example, standard statistical meta-analysis techniques applied to multiple experimental studies regularly yield positive evidence for ESP-like phenomena (e.g. Radin 2006; Tressoldi 2011); meta-analysis similarly yields positive evidence for retrocausation (Mossbridge et al. 2012). However, no level of statistical significance beyond which such phenomena would be deemed to have been demonstrated has been widely agreed upon within the relevant research communities, and positive claims about either are typically met with suspicion or dismissal (e.g. Miller 2011).

  4. 4.

    It is not often noted how important the assumption that some patterns are meaningless or “random” is for either science or common sense. Finding significance in every turn of a leaf can be overwhelming. While a full discussion of this issue is beyond the present scope, we return to it briefly in the section that follows.

  5. 5.

    Specifically, we are faced with a Pyrrhonic regress: an unmeetable demand for a further criterion that identifies “real” patterns. As is well known, such a regress is undefeatable by evidence. As Russell (1921) puts it,

    There is no logical impossibility in the hypothesis that the world sprang into being five minutes ago, exactly as it then was, with a population that “remembered” a wholly unreal past. There is no logically necessary connection between events at different times; therefore nothing that is happening now or will happen in the future can disprove the hypothesis that the world began five minutes ago.

    pp. 159–160

    Pyrrhonic scepticism is, however, also unsustainable: It is impossible to spend any length of time believing that one knows nothing. To get through our day, we have to at least behave as if we know things. But even this is impossibly arduous for our minds, and we wind up implicitly adopting the position that we actually know things. We then, following Chisholm (1982), turn to a burden-of-proof argument, saddling the sceptic with the impossible task of proving that the world did begin just five minutes ago.

  6. 6.

    It is worth noting that confabulation of motives for actions can also be intersubjective, as the effectiveness of crowd psychology and subliminal advertising demonstrate.

  7. 7.

    What counts as an “observer” in quantum theory has, in our opinion, received less attention than it deserves. Many theorists use the word “observer” merely to indicate a logically-possible point of view. There is, in particular, no requirement that “observers” be “like us” in any significant way. “Observer independent” is therefore considerably stronger than “intersubjective” if the latter term is used in a way that invokes or assumes any degree of psychological or experiential similarity. See Fields (2012a) for a discussion.

  8. 8.

    Many scientists also insist that the possibility of doing experiments commits them to free will (e.g. Gisin 2012). This assumption creates yet another “ladder” phenomenon in any science that both claims applicability to scientists themselves and assumes some form of deterministic causation. This is particularly evident in quantum theory, which in standard formulations claims both universality (and hence applicability to physicists as components of the physical universe) and the unitary time evolution given by the Schrödinger equation. As discussed in more detail below, unitary time evolution entangles observer with observed. A physicist that is entangled with her apparatus may appear to herself to be manipulating its state and hence may appear to herself to be exercising free will, but to a third party who observes them interacting, neither she nor her apparatus can be even assigned a determinate individual state. That experimenter and apparatus are interacting and hence entangled is itself, moreover, not an observer-independent fact; yet a fourth observer could describe the world in a way that renders them mutually isolated and hence non-interacting (e.g. Zanardi 2001). Hence if unitary evolution is taken seriously, even the question of whether an experiment has been performed, let alone the question of whether it has been performed by an agent executing free will, does not have an objective, i.e. observer-independent, answer. We return to this point in Sect. 6.

  9. 9.

    An enormous literature attempts to show that the predictions of quantum theory can be made consistent with the existence of an objectively classical world, i.e. a world that is classical independently of any or all observers, and hence with the existence of objective, observer-independent boundaries around objectively-existing objects. While we cannot exhaustively review that literature here, our position is that all conceptually successful attempts to establish objective classicality are inconsistent with quantum theory and hence have long empirical odds against them, while weaker positions produce “emergent” classicality only by assuming it. The simplest gambit, of course, is to claim that quantum theory is false or “in need of modification”; this is the tactic of all “objective collapse” theories (a recent example is that of Weinberg 2012). As no experimental evidence supports the idea of an objective collapse, i.e. of an objectively non-unitary time evolution for any physical system, these theories remain largely curiousities (relevant experimental evidence as well as additional conceptual considerations are discussed in Schlosshauer 2006; Jordan 2010). Considerably more common is the claim that “collapse” and hence classicality are only apparent; this is the claim of standard decoherence theory (e.g. Zurek 1998; Schlosshauer 2007). Merely apparent, subjective and hence observer-dependent classicality is fully consistent with quantum theory, but provides no sense in which boundaries between systems—including boundaries between observers and their apparatus—can be ontological. It is often claimed, however, that decoherence provides a mechanism via which classicality can be objectively apparent, i.e. equally apparent to all observers. This claim, primarily advanced under the rubric of “quantum Darwinism” (e.g. Blume-Kohout and Zurek 2006; Zurek 2009) is inconsistent with the observer-relativity of entanglement (Zanardi 2001; Zanardi et al. 2004; Harshman and Ranade 2011) as has been argued (Dugić and Jeknić 2006; Dugić and Jeknić-Dugić 2008; Fields 2014). Either formulation of decoherence, moreover, assumes both a fixed system-environment boundary and a classical-statistics (i.e. “heat bath”) description of the environment, either of which render any claim to “explain” classicality circular (Fields 2014; Kastner 2014). Assuming that the system of interest is quantum and its environment is classical is also, it should be noted, assuming that the laws of physics change from quantum to classical at the system-environment boundary, an assumption that makes the calculations simpler, but denies the universality of quantum theory and has neither empirical nor theoretical justification. Purely ontological approaches fare no better. Lam and Esfeld (2012), for example, claim that quantum theory can be supported by an ontic structural realist ontology in which well-defined “quantum objects” are held together by essential, and hence observer-independent, relational properties such as entanglement, an approach that is clearly inconsistent with entanglement being observer-relative.

  10. 10.

    This is not to deny that many quantum theorists are idealists, but merely to emphasize that they needn’t be idealists about those aspects of reality that quantum theory treats as objective.

  11. 11.

    The semantic relationship between \(Kp\) and \(\kappa p\) is clearly complicated (e.g. Koons 2013; Pollock 1987, 1995). One might insist that \(Kp\) implies \(\kappa p\), and deny the converse. The former inference could be based on the notion that p being only defeasibly known does not entail that it is actually able to be defeated. However, if one interprets “defeasibly known” as “defeatable in the actual world” or as “merely believed (even with high confidence)” then \(Kp\) cannot imply \(\kappa p\) since \(Kp\) implies that p is true and hence not defeatable in the actual world. Still further, one might insist that the central notion of knowledge, at least in science, is defeasible knowledge: knowledge relative to assumptions (e.g. Moses and Shoham 1993). Except for footnote 13 below, we are going to ignore the complex epistemological and metaphysical issues surrounding the relation between \(Kp\) and \(\kappa p\), as addressing them is not required to demonstrate that the Ladder Schema is an instance of the Inclosure Schema.

  12. 12.

    To more faithfully capture how A works in real scientific revolutions, we would need to add, at least, a time index, thus making A a 3-ary function of a time, a proposition p, and a set s of propositions. The time would be the pivot time, before which we would assert that we know the propositions in the set s (these constitute the alleged knowledge of the “old” paradigm), but after which we would assert that we know the new proposition p as well as the fact that p ushers in a revolution by rendering false the propositions of the old paradigm in s. However, since we assume here that scientific truths are timeless, we are going to ignore this complexity. Also, in scientific revolutions, A sometimes renders the propositions of the old paradigm false in the particular sense that they become approximations of propositions in the new paradigm. Relativity and quantum theory, for example, rendered Newtonian physics false, but it is still a “good enough” approximation to provide predictions that are accurate to within the relevant measurement resolutions for medium-sized objects moving at moderate velocities, and so is treated as “true for all practical purposes” in that circumscribed domain. We ignore this complication, too.

  13. 13.

    The claim that \(\omega\) is known is the tip of some interesting epistemology. The most important point for us is that A is part of a knowledge engine. The engine works something like this. At step 1, we have \(A(\kappa \omega , \kappa I_{\omega }) = \kappa (\lnot I_{\omega })\). At step 2, the very falsity of \(I_{\omega }\), through a process only partially understood, elevates the defeasible knowledge of both \(\omega\) and \(\lnot I_{\omega }\) to genuine knowledge, giving us \(A(K \omega , \kappa I_{\omega }) = K(\lnot I_{\omega })\). Indeed, demonstrating that the propositions in \(I_{\omega }\) are false is sometimes regarded as a kind of aesthetic evidence that \(\omega\) is true. Such aesthetic considerations are particularly significant in scientific revolutions in which the most dramatic confirmatory findings—the discoveries of the structure and replications mechanism of DNA, for example, or the demonstration of quantum entanglement across kilometer distances—come decades after the revolutionary theory is proposed or even accepted. This is part of what we mean by saying that scientific revolutions uncover (produce?) more truth. On the notion that defeasible knowledge aspires to genuine knowledge, see Williamson (2000), especially the introductory chapter.

  14. 14.

    From the Introduction, in the case of the observer effect \(\omega\) is “observing can strongly affect the system being observed” and in the case of number theory \(\omega\) is Gödel’s First Incompleteness Theorem. Both of these, though quite important, have limited application. The same is true for the scientific revolutions we discussed above.

  15. 15.

    This is a good place to point out the relation between our two paradoxes. Basically, we are saying that apophenia running amok is an inability on science’s part to draw a boundary between real patterns and merely “psychological” ones.

  16. 16.

    Formally, of course, if \(I_{\omega }\) simply engulfs \(\sigma\), we get paradox because \(\omega \in \sigma\), and hence is both true and false. But the case were interested in is when \(I_{\omega }\) expands to \(\Psi\) for semantic reasons.

  17. 17.

    See, e.g. and the Wikipedia entry

  18. 18.

    One might ask at this point: what about emergent properties? Surely one can say, for example, that solidity is a real emergent property, and solid objects have real boundaries—try kicking one!—even if these boundaries are somehow dependent on emergence. If one carefully avoids the idea that “emergence” is a physical process, e.g. following Butterfield (2011) and defining emergence in terms of relationships between observables, this is all fine. As in the cases of “collapse” or “decoherence,” the problems arise when one imagines that emergence is physical, and hence must postulate that physical laws change abruptly at some emergent boundary.

  19. 19.

    Classical, axiomatic set theory dodges this contradiction by insisting that the “set” of all sets is not itself a set, but is rather a “class” to which Cantor’s Theorem does not apply.

  20. 20.

    Two other of the many examples of science cum philosophy are Kelly 2008 (a philosopher does science) and, going the other way, Conway and Kochen 2006 (scientists do philosophy).

  21. 21.

    Our case of closure has a property common to many other cases of closure: “[it is] established by reflecting on the conceptual practice in question; [in] a polemical context, this can appear as an ad hominem argument” (Priest 2002, p. 4).

  22. 22.

    There is another way to get this conclusion. If \(K(\lnot \Psi )\) is in \(\Psi\) (via closure, as we’ve shown), then \(\Psi\) contains its own refutation (since \(K(\lnot \Psi )\) implies \(\lnot \Psi\)). We have then that science contains its own refutation. Science emerges as one big Liar Paradox (provided we construe science as one big proposition, as we can easily do). But as Priest has shown (1994, 2002), the Liar can be rendered as an inclosure paradox, giving us both transcendence and closure (though the latter is now redundant).

  23. 23.

    A “Boltzmann brain” is a statistical fluctuation with the apparent perceptions, thoughts, and memories of a human or other sentient being. With plausible assumptions, standard inflationary cosmology predicts that virtually all observers are Boltzmann brains in otherwise empty universes, not evolved systems in universes with material structure (Bousso and Freivogel 2007).

  24. 24.

    Note that this position almost diametrically opposes the constructive empiricism of van Fraassen (1980, 2001), which is realist about “observables” but anti-realist or at least agnostic about theoretical entities.

  25. 25.

    But many do not, e.g. the religious are rarely dissuaded that an image of some deity or revered person does not bespeak a communicating god, and to this day, some believe the “face” in the Cydonia region of Mars was built by space aliens, and that NASA’s denial of this is a conspiratorial cover-up.


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Dietrich, E., Fields, C. Science Generates Limit Paradoxes. Axiomathes 25, 409–432 (2015).

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  • Apophenia
  • Dialetheism
  • Psychology
  • Quantum measurement
  • Realism
  • Scientific knowledge