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Intrinsic local distances: a mixed solution to Weyl’s tile argument

Abstract

Weyl’s tile argument purports to show that there are no natural distance functions in atomistic space that approximate Euclidean geometry. I advance a response to this argument that relies on a new account of distance in atomistic space, called the mixed account, according to which local distances are primitive and other distances are derived from them. Under this account, atomistic space can approximate Euclidean space (and continuous space in general) very well. To motivate this account as a genuine solution to Weyl’s tile argument, I argue that this account is no less natural than the standard account of distance in continuous space. I also argue that the mixed account has distinctive advantages over Forrest’s (Synthese 103:327–354, 1995) account in response to Weyl’s tile argument, which can be considered as a restricted version of the mixed account.

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Notes

  1. My presentation of the argument follows Salmon (1980).

  2. I put “atoms” in quotes because it is not entirely clear what philosophical theory of spacetime we should explicate from Hogan (2012). More technically, the tested hypothesis implies that the geometry of spacetime is not commutative below the Planck level. Among other things, this means that unextended points do not exist because the coordinates of a point are necessarily commutative (e.g., in the (xy)-coordinate system, for any point (ab), \(ab-ba=0\)).

  3. Strictly speaking, it is more natural to think that the distance between a and b is the length of a shortest path from a to b minus one. For example, while the length of the side AB is four in Fig. 1, it’s more natural to think that the distance between A and B is three. However, for the sake of generalization in later discussions, it’s better to use Distance.

  4. Another intuitive option is to assume that two atoms are adjacent iff their representing tiles are horizontally or vertically adjacent. Under this option, the diagonal BC is represented by the zigzag region along the diagonal direction (Fig. 3). But this option has the same problem: the ratio of the diagonal to the side is about 2:1 rather than \(\sqrt{2}:1\).

  5. Here I am using “dimension” in an informal (and hopefully intuitive) way that every region of N-dimensional atomistic space is also N-dimensional. In other words, dimensionality is an intrinsic property of an atom. But we can have alternative definitions of dimension in atomistic space, which will be briefly discussed in Sect. 6.

  6. Note that this example does not solve Weyl’s tile argument: even though the sides and the diagonal of the square region satisfy the Pythagorean theorem, the distances along other directions don’t.

  7. In Fritz’s formalism, atomistic space is modeled by an infinite graph composed of \(\mathbb {Z}^d\)-translates of a certain finite pattern—call each of those translates a “cell.” For example, in the hexagonal tile space, each cell contains just one vertex and six edges. According to Fritz, a cell must contain a very large number of edges in order for the metric of the graph to approximate Euclidean geometry closely at the large scale. This means that, if there is an atomistic space represented by a tile space that approximates Euclidean space very well at the large scale, the repeated pattern must be very complicated. I thank Fritz for clarifying the gist of Fritz (2013) in personal correspondence.

  8. See McDaniel (2007) for more discussion of the view. McDaniel argued that the intrinsic account is true in some possible worlds, and in such worlds, atomistic space can approximate Euclidean distance.

  9. In a general context, I use “point” to simply refer to an ultimate part of an arbitrary space.

  10. Here, “connected” is used in the sense that a path \(a_1,\ldots ,a_k\) can be connected with a path \(a_k,\ldots ,a_n\) to form a single path \(a_1,\ldots ,a_n\) (\(1\le k\le n\)).

  11. A semimetric is a generalized distance function that does not satisfy triangle inequality. Under the intrinsic account, it is hard to see why a space cannot have a semimetric.

  12. This condition is violated in some approaches to discrete spacetime, such as that of Crouse and Skufca (2018). According to Crouse and Skufca, a particle can jump in any direction as long as the minimal length of a step is a constant number \(\chi \). This allows every point in continuous space to be a potential position of a particle. So it may be more natural to consider their approach to be about a discrete dynamics rather than a discrete spacetime.

  13. The construction of distance from proto-distance is closely related to the definition of geodesic distance in a weighted graph in graph theory, and to the construction of metric from semi-metric or quasi-metric (for example, see Harary 1969; Paluszyński and Stempak 2009).

    In more general settings, especially for continuous space, it is standard to define the distance between two points as the infimum of the lengths of paths between them, since a shortest path between them may not exist. However, this definition coincides with my definition in the case of atomistic space due to the requirement that for any atom a and any real number r, there are only finitely many atoms x with \(\mathbf{d} (a,x)< r\).

  14. The mixed account can accommodate curved space as well. I will not go into details here, but one can refer to Forrest (1995, pp. 334–340), in which Forrest explained how an atomistic model can approximate curved space once we have a model that approximates Euclidean space.

  15. For instance, in two-dimensional Euclidean space (or any flat two-dimensional Riemannian manifold), the length of a tangent vector expressed by \((\frac{dx}{dt}, \frac{dy}{dt})\) is \(\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}\).

  16. More formally, consider a path in two-dimensional Euclidean space. Let g be a metric tensor and T range over tangent vectors along a path. Then the length of that path is \(\int \sqrt{g(T,T)}dt\).

  17. For example, Weatherson (2006) argued that we should define duplicates in terms of fundamental properties and relations in a way that weeds out neighborhood-dependent aspects. Bricker (1993) suggested that local metrics are distances in infinitesimal neighborhoods of points.

  18. Van Bendegem (1987, 1995) also proposed solutions to Weyl’s tile argument. I consider his later proposal as a restricted version of Forrest’s account. We can have a one-to-one correspondence between points (a technical notion) in Bendegem’s model and atoms in Forrest’s model that preserves distance. But Forrest’s account allows models that are incompatible with Bendegem’s account.

  19. I change some of Forrest’s terminology to align with mine. He calls atomistic space “discrete space” and atoms “points.”

  20. For the proof, see Forrest (1995, pp. 344–346).

  21. The parameter \(m=10^{30}\) is a number given by Forrest to ensure the model to approximate Euclidean geometry at the large scale (Forrest 1995, p. 333).

  22. For example, see ’t Hooft (2016).

  23. As shown in Appendix A, in order for the atomistic model to approximate Euclidean space, the longest primitive distance needs to be about as large as the shortest primitive distance divided by the permitted distortion (as expressed by “\(M>3r/\delta \)” in the Appendix).

  24. Suppose Euclidean mixed model is more than two-dimensional locally, then there are more than three atoms in a local neighborhood equidistant from each other. But their distances just are the Euclidean distances among their representative pairs of integers. Thus there are more then three pairs of integers that are equidistant from each other on the Euclidean plane. But this is known to be impossible. Thus, Euclidean mixed model is no more than two-dimensional locally. Moreover, it is clear that Euclidean mixed model is not one-dimensional locally, so it is exactly two-dimensional.

  25. Forrest needs the definition of dimensionality to be relative to the scale because he wants to recover some sense in which space is three (or four) dimensional.

  26. This definition is analogous to the definition of the dimension of a manifold (i.e., a continuous space). One may try to translate this definition into a more intrinsic form such as this:

    Dimension\(\dagger \). A space is N-dimensional iff N is the least number that there are at most \(N+1\) atoms that bear the same primitive distance to each other.

    The problem with Dimension\(\dagger \) is that it leads to counterintuitive results. For instance, if no two pairs of atoms in the same local neighborhood have the same primitive distance, then Dimension\(\dagger \) would imply that the space is one-dimensional. But when such a space is not embeddable into one-dimensional continuous space, it is intuitively not one-dimensional.

  27. Here, the notion of approximation is cast in a different way from Forrest’s (1995). Forrest showed that his model approximates Euclidean space in the sense that we can map Euclidean space into his model such that the distances are approximately preserved. Here, it is the other way around: a model approximates Euclidean space in the sense that we can map this model into Euclidean space that preserves distances approximately. I do not consider either interpretation of approximation to be better than the other, but I work with this one because I feel it a bit more natural.

  28. Here’s a proof for the simple case in which two circles in question have the same radius, which is adequate for our purpose. Let two circles be \(x_1=r\cos \theta _1\), \(y_1=r\sin \theta _1\), \(x_2=r\cos \theta _2+n\), \(y_2=r\sin \theta _2.\) Then, \((x_1-x_2)^2+(y_1-y_2)^2=n^2-2r^2\cos (\theta _1-\theta _2)-2nr(\cos \theta _1-\cos \theta _2)+2r^2\le n^2+2r^2+4nr+2r^2=(n+2r)^2.\) That is, for two circles with the same size, the largest distance between two points on them is equal to the distance between their centers plus their radii.

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Correspondence to Lu Chen.

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I thank Philip Bricker and Jeffrey Russell for very helpful guidance, feedback, and discussions. I thank the audience at my talks based on this paper in Metaphysical Mayhem at Rutgers University in 2018, and in Philosophy of Logic, Mathematics, and Physics Graduate Conference at the University of Western Ontario in 2019. Among the audience, I especially thank Cian Dorr for his helpful feedback. I’d also like to thank a referee of Synthese for pressing me on the application of my account to relativistic settings, which helps clarify the relevance of the account.

Appendix A

Appendix A

Now I shall turn to how well space approximates Euclidean space under the mixed account. Under this account, an atomistic space can be represented by a set of points with a shortest path metric that assigns some pairs of points real-valued distances (bounded by a finite number) and derives other distances as their least sums.

We will understand “approximation” in terms of “almost isometry.” Let e(pq) be the Euclidean distance between two points pq in Euclidean space. Let \(\epsilon , r\) be two positive numbers. A metric space X with a metric d is \(\epsilon \)-isometric to Euclidean space E with regard to r iff there is a map f from X to E such that (1) for \(x,y\in X\), we have

$$\begin{aligned} 1-\epsilon \le \frac{e(f(x),f(y))}{d(x, y)}\le 1+\epsilon \end{aligned}$$

(the smallest \(\epsilon \) such that f satisfies this condition is called the distortion of f);Footnote 27 (2) for every \(p\in E\), there is a \(x\in X\) such that \(e(p,f(x))\le r\). In other words, the embedded points cover E reasonably well so that there are no obvious “clusters” and “holes.”

Theorem A.1

For any \(\epsilon \) and r, there is a set of points with a shortest path metric (with distances being bounded by a finite number) that is \(\epsilon \)-isometric to Euclidean space with regard to r.

Proof

For brevity, I will resort to the following abbreviations when applicable. Given an embedding f of a metric space into Euclidean space, for any points xy in the space, let \(\Vert xy\Vert _f=e(f(x),f(y))\) (the subscript “f” is omitted if it is clear which embedding we refer to). Also, for any points pq in Euclidean space, let \(\Vert pq\Vert =e(p,q).\)

Let G be an embedding of an infinite set X to Euclidean space E such that there is an r such that for any \(p\in E\), we can find an \(x\in X\) with \(e(p,G(x))<r\). (For example, if G maps members of X to Euclidean points represented by pairs of integers, then r in question is at least \(\sqrt{2}/2\).) We will construct a metric over X such that the resulting metric space is \(\epsilon \)-isometric to Euclidean space under G, where \(\epsilon \) is a small number we choose.

M is a real-number parameter that will play an important role in assigning weights and in determining the distortion of the intended embedding. For any \(x,y\in X\), if \(\Vert xy\Vert > M\), we can find a sequence of points \(p_1,p_2,\ldots p_n\) in E such that \(p_0=G(x)\), \(p_n=G(y)\), \(\Vert p_0p_1\Vert =\Vert p_1p_2\Vert =\cdots =\Vert p_{n-2}p_{n-1}\Vert =M\) and \(\Vert p_{n-1}p_n\Vert <M\). Let \(N=\Vert p_{n-1}p_n\Vert \). Consider \(p_i,p_{i+1}\), where \(i=1,\ldots ,n-2\). We can find \(x_i,x_{i+1}\in X\) such that \(e(G(x_i),p_i)<r\) and \(e(G(x_{i+1}),p_{i+1})<r\). We know that the largest distance between points on two circles is equal to the distance between their centers plus their radii.Footnote 28 Thus, \(\Vert x_ix_{i+1}\Vert <M+2r\). Now, for any two \(a,b\in X\), if \(\Vert ab\Vert <M+2r\), then let them be connected by an edge with the weight \(d(a,b)=\Vert ab\Vert \); otherwise, ab are not connected by an edge. Then, \(M\le d(x_i,x_{i+1})<M+2r\). Moreover, it’s easy to see that \(M\le d(x,x_1)\le M+r\) and \(N\le d(x_{n-1},y)\le N+r\). It follows that \(d(x,y)\le d(x,x_1)+d(x_1,x_2)+\cdots +d(x_{n-1},y)< n\cdot (M+2r)+(N+r)\). Furthermore, if \(x, x_1, \ldots x_n, y\) is a shortest path, then \(d(x,y)=d(x,x_1)+d(x_1,x_2)+\cdots +d(x_{n-1},y)=\Vert xx_1\Vert +\cdots +\Vert x_{n-1}y\Vert \ge \Vert xy\Vert \). Thus, we have:

$$\begin{aligned} 1\le \frac{d(x,y)}{\Vert xy\Vert }<\frac{n\cdot (M+2r)+(N+r)}{nM+N}=1+\frac{(2n+1)r}{nM+N} \end{aligned}$$

The distortion \(\displaystyle \delta =\frac{d(x,y)}{\Vert xy\Vert }-1<\frac{(2n+1)r}{nM+N}<\frac{(2n+1)r}{nM}<\frac{3r}{M}\). Then, for any small positive number \(\epsilon \), we can make \(\delta <\epsilon \) by letting \(M=3r/\epsilon \). (Note that if we are only concerned with distances that involve a large n, we only need M to be \(2r/\epsilon \).) This completes the case for any \(x,y\in X\) with \(\Vert xy\Vert > M\). If \(\Vert xy\Vert \le M\), then we have \(d(x,y)=\Vert xy\Vert \), in which case there is no distortion. Therefore, we have found a metric space, in which all distances are bounded by \(3r/\epsilon +2r\), that is \(\epsilon \)-isometric to Euclidean space at any scale. \(\square \)

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Chen, L. Intrinsic local distances: a mixed solution to Weyl’s tile argument. Synthese 198, 7533–7552 (2021). https://doi.org/10.1007/s11229-020-02531-4

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Keywords

  • Weyl’s tile argument
  • Atomistic space
  • Discrete space
  • Intrinsic distance
  • Path-dependent distance
  • Locality
  • Metric tensor