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How can a line segment with extension be composed of extensionless points?

From Aristotle to Borel, and Beyond

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Abstract

We provide a new interpretation of Zeno’s Paradox of Measure that begins by giving a substantive account, drawn from Aristotle’s text, of the fact that points lack magnitude. The main elements of this account are (1) the Axiom of Archimedes which states that there are no infinitesimal magnitudes, and (2) the principle that all assignments of magnitude, or lack thereof, must be grounded in the magnitude of line segments, the primary objects to which the notion of linear magnitude applies. Armed with this account, we are ineluctably driven to introduce a highly constructive notion of (outer) measure based exclusively on the total magnitude of potentially infinite collections of line segments. The Paradox of Measure then consists in the proof that every finite or potentially infinite collection of points lacks magnitude with respect to this notion of measure. We observe that the Paradox of Measure, thus understood, troubled analysts into the 1880’s, despite their knowledge that the linear continuum is uncountable. The Paradox was ultimately resolved by Borel in his thesis of 1893, as a corollary to his celebrated result that every countable open cover of a closed line segment has a finite sub-cover, a result he later called the “First Fundamental Theorem of Measure Theory.” This achievement of Borel has not been sufficiently appreciated. We conclude with a metamathematical analysis of the resolution of the paradox made possible by recent results in reverse mathematics.

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Notes

  1. See Skyrms (1983). We will adopt this label throughout the paper. It has also been called “Zeno’s metrical paradox” (see Grünbaum (1952)), “Zeno’s metrical paradox of extension” (see Friedman (2012)), and the “Paradox of Infinite Divisibility” (see Huggett (2019)).

  2. Aristotle, On Generation and Corruption, 316a15-30, trans. Joachim in Barnes (1984).

     (Greek text from Rashed (2005)).

    The argument comes in the context of Aristotle’s refutation of Democritus’ theory of atomic magnitudes. For a detailed treatment of Aristotle’s overall strategy in this chapter, see Sedley (2004). Although Aristotle does not explicitly attribute this paradox to Zeno, Simplicius, who had access to Zeno’s work, does attribute it to him in his commentary On Aristotle’s Physics, 139.20-24 (Simplicius, 2011).

  3. See Lando and Scott (2019).

  4. Thus, we restrict our project to interpreting, and resolving, the paradox understood as a mathematical antinomy.

  5. When we say a point p lies on a line segment \({\mathsf {I}}\), we require that p is not an endpoint of \({\mathsf {I}}\), in other words, p is interior to \({\mathsf {I}}\).

  6. There is no direct evidence from Eudoxus for this attribution, but there is little disagreement over the inductive evidence in its favor. Archimedes explicitly relies on Eudoxus for the ‘method of exhaustion’ in On the Sphere and Cylinder. On which topic, see Archimedes’ précis to Dositheus: “For, though these properties also were naturally inherent in the figures all along, yet they were in fact unknown to all the many able geometers who lived before Eudoxus...” (Heath, 2002, p. 2). Also, given that the method of exhaustion is essentially an alternative formulation of Elements V, Definition 4, and a crucial component of Elements X, Proposition 1, there can be no doubt that Archimedes is not the originator. Moreover, since Eudoxus spent time at Plato’s Academy, it would not be implausible to suppose that his ideas reached Aristotle through the teachings of the Academy. Wilbur Knorr, for example, suggests that Eudoxus was active in Plato’s Academy at roughly the same time as Aristotle (see Knorr (1990, p. 318)).

  7. See Stolz (1883). Although his title for this axiom is something of a misnomer, we retain it.

  8. Indeed, all commentators of whom we are aware.

  9. See, for example, Physics 215b12-20, trans. Hardie and Gaye in Barnes (1984). Henceforth, and unless otherwise noted, translations of Aristotle’s Physics are from Hardie and Gaye, occasionally modified for terminological consistency: “Now there is no ratio in which the void is exceeded by body, as there is no ratio of 0 to a number. For if 4 exceeds 3 by 1, and 2 by more than 1, and 1 by still more than it exceeds 2, still there is no ratio by which it exceeds 0; for that which exceeds must be divisible into the excess and that which is exceeded, so that 4 will be what it exceeds 0 by and 0. For this reason, too, a line does not exceed a point-unless it is composed of points.”

     (Greek text from Ross (1936)).

    Aristotle’s reason for claiming that there is no ratio between any whole number () and 0 (), and by analogy that there is no ratio between a line () and a point (), is a decisive invocation of the Archimedean proscription of infinitesimal elements. There is no n-th multiple of such that exhausts any m. This arithmetical application of the method of exhaustion pointedly recalls the iterated on which the Paradox of Measure turns. Heath rightly connects Aristotle’s reasoning in this passage to Definition IV in Book V of Euclid’s Elements (cf. footnote 8 above), which is the Axiom of Archimedes in another guise (see Heath (1970, pp. 116–117)). Cf. Heath (1981, pp. 384–385) and Elements Book X, Proposition I. For additional evidence, see footnote 16 below.

  10. That is, we refrain from appealing to additivity properties of measures, or the failure thereof, in our interpretation of the Paradox or its resolution.

  11. Borel’s resolution of the Paradox does not flow from his development of measure theory, but from his Finiteness Theorem, that predated this development and served as a prolegomenon thereto, as we discuss at length in Sect. 3.4. The Finiteness Theorem first appeared in his doctoral thesis, Sur quelques points de la théorie des fonctions, submitted in 1893, defended in 1894, and published a year later as Borel (1895).

  12. We retain the name “Covering Principle” for this generalization of the earlier Principle from points to point-sets.

  13. Indeed, given \(p_j\), we could construct \(S_j\) with midpoint \(p_j\) and length \(1/2^{n+j}\).

  14. From a contemporary perspective, it is virtually irresistible to identify a process with an algorithm, and its stage-wise execution with computations via this algorithm on numerals representing successive positive integers, thereby identifying the notion of potentially infinite set with the notion of computably enumerable set. For our purposes, we need not make any such identification.

  15. We offer one explication of the notion of potential infinity that we regard as particularly fruitful in understanding the Paradox of Measure. We are of course aware that this notion has received extensive treatment in the literature. See, for example, Lear’s authoritative account in Lear (1980).

  16. We imagine 0 to the left of 1 along the unit segment \({\mathsf {I}}\).

  17. The argument here is implicit in Aristotle’s observation that the sum of a geometric series is finite which can be found in Physics, 206b4-12: “In a way the infinite by addition is the same thing as the infinite by division. In a finite magnitude, the infinite by addition comes about in a way inverse to that of the other. For just as we see division going on ad infinitum, so we see addition being made in the same proportion to what is already marked off. For if we take a determinate part of a finite magnitude and add another part determined by the same ratio (not taking in the same amount of the original whole), we shall not traverse the given magnitude. But if we increase the ratio of the part, so as always to take in the same amount, we shall traverse the magnitude; for every finite magnitude is exhausted by means of any determinate quantity however small.”

    The connection between this passage and the Axiom of Archimedes has been noted by commentators. See Heath (1921, pp. 342–343), Ross (1936, p. 556), Hussey (1983, p. 84). Indeed, this passage lends considerable support to our approach, insofar as it emphasizes the role of the Axiom of Archimedes in explicating the fact that points lack magnitude.

  18. One may view this principle as yet another manifestation of the Axiom of Archimedes.

  19. As did analysts of the 1880’s. See Sect. 3.3.

  20. Of course, we are only interested in its application to finite and potentially infinite collections of points, and to line segments, insofar as they might legitimately be regarded as collections of points. The notion of an arbitrary collection of points only developed through the work of nineteenth-century mathematicians, and is thus not germane to our interpretation of the Paradox of Measure, nor even to its resolution.

  21. It is “ancient” for deploying potentially infinite, rather than countably infinite, covers.

  22. See Harnack (1885). We introduce Harnack’s notion below in Definition 5.

  23. Note that we admit the case that \(c=0\), the magnitude of a “degenerate” line segment.

  24. We would like to thank Jeremy Avigad for pointing out that there is a significant gap between the mathematical resources necessary to articulate the definition of ancient-measure in the case that \(c=0\) and \(c>0\). In particular, the requirement expressed in Definition (1.1) is trivial to verify in the case that \(c=0\). Thus, in our argument for Claim 3, we only needed to provide a substantive verification of the requirement expressed in Definition (1.2). On the other hand, it is obscure that the requirement expressed in Definition (1.1), in the case \(c>0\), could have been grasped by the ancients, insofar as its verification in a particular case would involve refuting the existence of a potentially infinite cover of total length less than some strictly positive c. It is exactly such verification, in application to potentially infinite covers of a non-degenerate line segment, that would be required to resolve the Paradox as we understand it. Thus, from our point of view, this represents a fundamental conceptual obstacle to the resolution of the Paradox in antiquity.

  25. The forthcoming resolution of the Paradox will provide further, indeed compelling, justification for identifying the magnitude of a collection of points with its ancient-measure.

  26. The astute reader will have noticed that the potential infinity of intervals \(\Xi _2\) deployed in our proof that the bisection point process lacks magnitude, already witnesses that the ancient-measure of the bisection point process, and by extension, any potential infinity of points, has ancient-measure at most 1/2, which is already paradoxical on our interpretation. We would like to thank Henry Towsner for this observation.

  27. Borel (1895).

  28. See Stolz (1884), Cantor (1884), and Hawkins (2001, pp. 61–66).

  29. Harnack uses the term “Inhalt” for the notion defined here (see Harnack (1885)). It is now generally referred to as outer-content in texts on analysis (see, for example, Bressoud (2008)) to distinguish it from related notions that were introduced during the development of the theory of measure and integration. Since we will make no use of these other notions, we retain the simplicity of Harnack’s terminology.

  30. The proof of Proposition 2 is essentially the same as that of Proposition 1.

  31. Chapter 4 of Hawkins (2001) presents a riveting account of this struggle.

  32. See Cantor (1883).

  33. Cantor attached considerable significance to this result. See, for example, his letter to Mittag-Leffler of November 26, 1883, Meschkowski and Nilson (1991, p. 151).

  34. Indeed, Hankel purported to prove that every nowhere-dense collection of points has content 0 (see Hawkins (2001, p. 167)). Hankel’s “result” had been anticipated by Dirichlet in 1829. Alas, this simple connection between topological notions and magnitude proved to be illusory. Smith, and then Volterra, constructed nowhere-dense sets with content greater than 0 (see Hawkins (2001, p. 169)).

  35. See Oxtoby (1996, p. 10).

  36. It is worth remarking that the modern-measure of X is identical to the Lebesgue measure of X, for every set X that is Lebesgue measurable. (A collection of points X is Lebesgue measurable if and only if for every \(n>0\) there is a closed set C and an open set O such that \(C\subseteq X\subseteq O\) and the difference between the modern-measure of O and the modern-measure of C is less than 1/n.) It is also worth remarking that every collection of points that is relevant to our discussion of the Paradox of Measure, that is, line segments and potentially infinite collections, is Lebesgue measurable. From our point of view, the existence of non-Lebesgue-measurable sets is a twentieth-century curiosity that has no direct relevance to Zeno’s Paradox of Measure or its resolution.

  37. See Harnack (1885).

  38. See Ferreirós (1999), Section V.1.

  39. Hawkins (2001, p. 172). Harnack’s argument for this conclusion is essentially the same as that given in the proof of Theorem 1, except that he had no need for the care we have taken to observe that the ever shrinking collections of covering segments for a potentially infinite collection of points can themselves be constructed to be potentially infinite.

  40. Hawkins (2001, p. 172).

  41. See Bressoud (2008, p. 63).

  42. The following quotation from Cantor’s letter to Paul Tannery dated October 5, 1888, Meschkowski and Nilson (1991, pp. 323–325), trans. A. Newstead, Newstead (2001), suggests the intriguing possibility that Cantor may have anticipated the Finiteness Theorem of Borel discussed below.

    You are right to point out that, I so to speak, renew the Pythagorean view, insofar as I teach that the geometrical continuum is a real compound of separate points, geometrical individuals, just as a forest is composed out of trees, but because the Pythagoreans understood the continuum as a sum of points, [a view] which is powerless against the demonstrations of Zeno of Elea, I take the continuum to be a point set (ensemble of points) of a more definite, precisely specified nature. My grasp of the geometrical (and temporal) continuum is one which harmoniously combines the advantages of the Aristotelian view with what is true in the Pythagorean way of understanding, so that there will be no Zeno waiting for me who will demonstrate any kind of contradiction whatsoever in my most well-considered concept of the continuum.

    As mentioned earlier, Cantor’s construction of a set of content zero equipollent to the linear continuum made it clear that the resolution of the Paradox of Measure did not lie in the cardinality of an interval. But his result, Proposition 3 above, tells us immediately that if one could establish that the measure of a line segment is equal to its content, then the Paradox would be overcome. We have no direct evidence that Cantor actually recognized this finiteness result, though he made implicit use of the compactness of Euclidean n-space in arguments advanced in Cantor (1884); see Hawkins (2001, p. 62). Nonetheless, it is exactly this deep property of the linear continuum that allows for a harmonious combination of “the advantages of the Aristotelian view with what is true in the Pythagorean way of understanding.”

  43. See Hawkins (2001, pp. 97–106) for a detailed description of the problem and Borel’s contribution to its resolution. It is worthy of note that Poincaré himself was one of the rapporteurs for Borel’s thesis.

  44. Borel (1895, p. 51) cited in Hawkins (2001, pp. 101), fn. 9.

  45. The result, and its generalizations, are often referred to as the Heine-Borel Theorem, though it is widely recognized that this is a misnomer, since Heine neither stated nor proved any such result.

  46. The appellation “Le premiere théorème fondamental” first appears in the second edition of Leçons sur la Théorie des Fonctions in a lengthy note to the first edition and is reprinted in Borel (1950, p. 223). In this note, Borel explains the significance of the result in establishing that his approach to assigning a measure to (what we now call) the Borel sets is well-defined.

  47. See Sect. 3.5, Corollary 1.

  48. See Borel (1898, p. 42). We adopt the formulation for countably infinite covers, since this is all that is required for the resolution of the Paradox of Measure. Borel’s error in claiming the stronger result created some confusion, even among mathematicians of the stature of Lebesgue, who studied Borel (1895), and applied the Theorem in his 1901 thesis in an argument that required the result for uncountable covers. When Lebesgue realized that Borel had only established the result for countable covers, he gave a proof, published in 1904, for the case of arbitrary open covers. As it happens, Pierre Cousin had proved a version of the two-dimensional case of Borel’s Finiteness Theorem for arbitrary open covers in 1895! Additional proofs of the Theorem were given by Schoenflies in 1900 and Young in 1902. It is Schoenflies who first, mistakenly, attributed the result to Heine, based on the similarity of methods of his own proof with Heine’s proof of the significant result that a continuous function on a closed interval is uniformly continuous (a proof Heine apparently pirated from Dirichlet without attribution). Cf. Lebesgue (1904) and Andre et al. (2013).

  49. See Andre et al. (2013) and Hildebrandt (1926) for detailed discussions of the history and mathematics of Borel’s Finiteness Theorem.

  50. See Cantor (1872) and Dedekind (1996).

  51. For example, Salmon (1980, p. 35):

    We should begin by noting that, although the calculus was developed in the seventeenth century, its foundations were beset with very serious logical difficulties until the nineteenth century – when Cauchy clarified such fundamental concepts as functions, limits, convergence of sequences and series, the derivative, and the integral; and when his successors Dedekind, Weierstrass, et al., provided a satisfactory analysis of the real number system and its connections with the calculus. I am firmly convinced that Zeno’s various paradoxes constituted insuperable difficulties for the calculus in its pre-nineteenth-century form, but that the nineteenth-century achievements regarding the foundations of the calculus provide means which go far toward the resolution of Zeno’s paradoxes [of motion].

  52. Aristotle, Physics, 227a11-12.

    Commentators recognize this passage as central to understanding Aristotle‘s conception of the linear continuum. See, for example, White (1988).

  53. See Dedekind (1996), Sect. 3. Stein (1990) emphasizes the similarity between Aristotle’s and Dedekind’s formulations.

  54. See Propp (2013) for a taxonomy of the logical relations among several continuity properties of the linear continuum, among them the Dedekind Cut Property, Nested Interval Completeness, and Order Completeness, also known as the Least Upper-Bound Principle. See Sinkevich (2015) for a history of the use of the Nested Interval Completeness Property from antiquity to the late nineteenth century. Note that our convention that, unless stated otherwise, line segments are understood to include their endpoints, remains in force in our statement of this property.

  55. The proof we present is essentially the same as that given by Borel in Borel (1898, pp. 42–43). The reader may observe a similarity between this proof and that of the König Infinity Lemma: the statement that a binary tree with infinitely many levels has an infinite path. Both involve an iterated application of the infinite pigeonhole principle - the statement that if you sort infinitely many objects into two pigeonholes, at least one of the holes will contain infinitely many objects - followed by inference of the existence of a sequence that witnesses the choice of an infinite hole at each stage. See Stillwell (2013, p. 75) for discussion of this point, and Sect. 4 below for an examination of the deeper connection between these results.

  56. Insofar our interpretation of the the Paradox involves a sound mathematical argument that makes use only of notions entirely intelligible to mathematicians of the ancient world, the reader may legitimately wonder whether the resolution we propose is similarly accessible to such thinkers. It is reasonably clear to us that the sequence \(\{{\mathsf {J}}_1,{\mathsf {J}}_2,\ldots \}\) constructed in the proof of Borel’s Finiteness Theorem is not potentially infinite. Indeed, this may be the only point in the argument that lies beyond the grasp of ancient mathematicians. We reflect on this point briefly in Sect. 4.

  57. Borel (1895).

  58. Quine (1976, p. 9). The well-known ancient dictum is that philosophy begins in wonder (Plato, Theaetetus 155d; Aristotle, Metaphysics 982b)–which is to say it begins with a feeling of puzzlement () from which we recoil and flee () in search of understanding (). We believe that the history of thought about the Paradox of Measure is a testament to this philosophical trajectory, but we also acknowledge that, from the ancient point of view, what we call ‘understanding’ might yet constitute another encounter with . It is interesting to note that our advance in understanding about the Paradox has, over the centuries, proceeded at times accidentally, owing much of its progress to tangential inquiries in the footnotes and to the practically oriented applications of mathematical thinking to real-world problems (what Aristotle would call and respectively).

  59. See Grünbaum (1952), Grünbaum (1967), Skyrms (1983), Holden (2004), Friedman (2012), and Huggett (2019) for some recent endorsements of this view.

  60. See Borel (1895). This alternative proof of the uncountability of the linear continuum is well-known to students of analysis. Cf. Oxtoby (1996, pp. 1–4). Oxtoby presents this alternative “measure-theoretic” proof and contrasts it with a formulation of Cantor’s original “topological” proof (see Cantor (1996)) that makes use of only the Nested Interval Completeness Property. We compare these two arguments from a meta-mathematical point of view in the following subsection.

  61. Hirschfeldt (2014, pp. 3–5) emphasizes the interest of Reverse Mathematics in this connection.

  62. Simpson (2009, pp. xxiii–xiv). This work remains the standard reference for Reverse Mathematics, though the field has developed rapidly in the little more than a decade since its publication.

  63. See Simpson (2009), Chapter II, for details. Some sense of the strength of \(\mathsf {RCA}_{\mathsf {0}}\) can be gleaned from the fact that its minimum \(\omega \)-model consists of the recursive sets of natural numbers.

  64. See Simpson (2009, pp. 76–77).

  65. See especially footnote 55.

  66. See Simpson (2009, pp. 127–130) for discussion and a proof of Theorem 5 which was announced in Friedman (1974).

  67. See Simpson (2009, p. 31), for a proof of Theorem 6.

  68. See footnote 51. It is worth remarking that though \(\mathsf {WKL}_0 (=\mathsf {RCA}_0+\mathsf {WKL})\) is far richer than \(\mathsf {RCA}_0\) from a mathematical point of view, it is nonetheless still comparatively weak from a metamathematical point of view—it is a conservative extension of primitive recursive arithmetic with respect to \(\Pi ^0_2\) sentences. This finititistic reduction is of great significance from the point of view of partial realizations of Hilbert’s Program, cf. Simpson (1988) and Simpson (2009, pp. 377–378).

  69. That is, \(\mathrm {WWKL}\) asserts that if there is a fixed positive lower bound on the density of the levels of a binary tree T, then T has an infinite path. Cf. Simpson (2009, p. 393).

  70. Part 1 of the theorem follows immediately from Theorem 1 of Yu and Simpson (1990, p. 175) and Theorem 3.3 of Brown (2002, p. 196), while parts 2 and 3 are established in Yu and Simpson (1990, p. 172). Denis Hirschfeldt (private communication) informs us that a further reversal of \(\mathrm {WWKL}\) of interest in connection with the Paradox of Measure may be obtained via methods developed by Brown et al. (2002) and Dorais et al. (2016). Namely, let \(\mathbf {MNZ}\) be a formalization of the statement that the modern-measure not zero; \(\mathsf {RCA}_{\mathsf {0}}\vdash \mathbf{WWKL} \Longleftrightarrow \mathbf{MNZ} \).

  71. Chen (2021, p. 4442).

  72. Note that if a set X of cardinality less than \({\mathfrak {c}}\), the cardinality of \({\mathbb {R}}\), has positive modern-measure, then X is not Lebesgue-measurable. This follows from the fact that every Lebesgue-measurable set contains a closed set of positive modern-measure. But every set of positive modern-measure is uncountable, and every uncountable closed set has cardinality the continuum, by the Cantor-Bendixson Theorem.

  73. Jech (2002, pp. 529–537). Indeed, let \(\kappa \) be the least cardinal such that there is a collection of points of cardinality \(\kappa \) that has positive modern-measure, and let \(\lambda \) be the least cardinal such that there is a family of cardinality \(\lambda \) of collections of points, each of modern-measure 0, whose union has positive modern-measure. It is consistent with \(\mathsf {ZFC}\) that \(\lambda<\kappa <{\mathfrak {c}}\). This suggests that the Paradox of Measure, even conceived in terms of cardinality of point-sets of positive measure, is a separate issue from questions of additivity.

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Acknowledgements

We would like to thank Jeremy Avigad, William Ewald, Denis Hirschfeldt, Steven Lindell, Susan Sauvé Meyer, Phil Nelson, Mary-Angela Papalaskari, Brian Skyrms, Stephen Simpson, Henry Towsner, and two anonymous referees for valuable comments on earlier drafts of this paper.

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  1. University of Central Florida, Orlando, USA

    Brian Reese

  2. University of North Carolina at Chapel Hill, Chapel Hill, USA

    Michael Vazquez

  3. University of Pennsylvania, Philadelphia, USA

    Scott Weinstein

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Proof of Proposition 3

Proof of Proposition 3

Proof

Let X satisfy the hypothesis of the Proposition and suppose that X is dense in [0, 1]. We must show that

  1. 1.

    for every finite collection of intervals \(\Xi \) covering X, \({\tau (\Xi )}>1\).

  2. 2.

    for every positive integer n, there is a finite collection of intervals \(\Xi \) such that \(\Xi \) covers X and \({\tau (\Xi )}<1+n^{-1}\).

(2): Fix n, and let \(\Xi =\{(-(3n)^{-1}, 1+(3n)^{-1})\}\). \(\Xi \) covers X and \({\tau (\Xi )}=1+2\cdot (3n)^{-1}<1+n^{-1}\).

(1): Suppose that \(\Xi =\{(a_1,b_1),\ldots ,(a_k,b_k)\}\) is a finite collection of open intervals covering X. We may suppose, without loss of generality, that the intervals are ordered in such a way that \(a_1<a_2<\ldots<a_k<1\) and \(b_1<b_2<\ldots <b_k\), for if that were impossible, one interval would be a subinterval of another and could thus be removed from \(\Xi \) and the remaining collection of intervals would cover X and be of smaller total length. To conclude the proof it suffices to show that

  1. (1.1)

    \(a_1\le 0\),

  2. (1.2)

    \(1\le b_k\), and

  3. (1.3)

    for every \(1\le j<k\), \(a_{j+1}\le b_j\).

(1.1): Suppose to the contrary that \(0< a_1\). Since X is dense in [0, 1], it follows that for some \(p\in X\), \(0<p<a_1\). Since \(a_j>a_1\) for all \(j>1\), it follows that p is not contained in any interval in \(\Xi \). This contradicts the hypothesis that \(\Xi \) covers X.

(1.2): The argument here is virtually identical to the that for (1.1).

(1.3): The argument again is very similar to that for (1.1). Suppose to the contrary that that for some \(1\le j<k\), \(b_j< a_{j+1}\). Since X is dense in [0, 1], it follows that there is a \(p\in X\) such that \(b_j< p<a_{j+1}\). Therefore, contrary to hypothesis, \(\Xi \) fails to cover X. \(\square \)

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Reese, B., Vazquez, M. & Weinstein, S. How can a line segment with extension be composed of extensionless points?. Synthese 200, 85 (2022). https://doi.org/10.1007/s11229-022-03538-9

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