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Composition as pattern

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Abstract

I argue for patternism, a new answer to the question of when some objects compose a whole. None of the standard principles of composition comfortably capture our natural judgments, such as that my cat exists and my table exists, but there is nothing wholly composed of them. Patternism holds, very roughly, that some things compose a whole whenever together they form a “real pattern”. Plausibly we are inclined to acknowledge the existence of my cat and my table but not of their fusion, because the first two have a kind of internal organizational coherence that their putative fusion lacks. Kolmogorov complexity theory supplies the needed rigorous sense of “internal organizational coherence”.

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Notes

  1. van Inwagen (1990, p. 82).

  2. Though Dennett himself has foresworn developing this proposal as a full-blown ontology, there has been other such work on his behalf, most notably Ross (2000) and the subsequently developed Ladyman and Ross (2007). Haugeland (1993) is a critical response to Dennett’s paper. Independent philosophical work on patterns and ontology is in Johansson (1998, 2004).

  3. The standard Kolmogorov text is Li and Vitányi (2008). Incidentally, for what it’s worth: after working on this Kolmogorov approach to patterns independently, I was surprised on going back to Dennett’s paper to find he had just the same notion in mind.

  4. “Proof”: given any finite data string x (longer than one bit of course), pick some UTM and prefix it with a program that has x stored as a hardcoded value, and outputs that value on any input that begins with 1—including just 1. On any input that starts with 0, strip the initial 0 and pass the rest to the unmodified UTM. The result is a new UTM that compresses x to one bit.

    Dennett either did not notice, or chose not to address this problem.

  5. This is how I would gloss what Dennett calls the “scientific path to realism” in his “Real Patterns”, according to which for example “centers of gravity are real because they are (somehow) good abstract objects. They deserve to be taken seriously, learned about, used” (p. 29).

  6. Goodman (1954). Of course this particular example of a “natural” property is complicated by the fact that in order to explain what the green surface spectral reflectances have in common, it seems we need to refer to our own cognitive architecture.

  7. Li and Vitányi (2008, p. 112).

  8. Chalmers (2009), Hale and Wright (2009), Hirsch (2009), McDaniel (2009) and Thomasson (2009).

  9. See Correia and Schnieder (2014) for an overview of issues in grounding.

  10. In fact I would argue that being built to get by in the world is necessary for being an epistemic agent—which, if true, would help explain pragmatism’s prevalence among philosophers of cognitive science.

  11. This is from the probabilistic rather than algorithmic information theory perspective, of course, but see Grünwald and Vitányi (2004) for an overview of the close ties between Shannon information theory and Kolmogorov complexity.

    Incidentally, Friston et al. (2015) further argue that any agent worthy of the name must be seeking to minimize such surprisal (p. 5).

  12. See for example Solomonoff (1960) and Grünwald (2007). Baum (2004), like Hutter, insists that cognition is essentially a form of information compression.

  13. See https://www.quantamagazine.org/new-theory-cracks-open-the-black-box-of-deep-learning-20170921/.

  14. For reasons like evolutionary pressure I am open also to “logical depth” versions of complexity, which consider time constraints; see Bennett (1988).

  15. Compare Bas van Fraassen (1980) on “observable”, p. 17.

  16. By a “standard” UTM I mean what Li and Vitányi (2008) p. 103 call “additively optimal”; contrast the perverse UTM in Example 2.1.2 on p. 107.

  17. Thanks to Terrance Tomkow for this comparison.

  18. For issues with gunk see Sider (1993) or Zimmerman (1996). Ladyman and Ross (2007) reject the atomic-gunky dichotomy—see for example pp. 20 and 44—but whether their own view is best summarized as “composition all the way down” is contentious.

  19. This usage is meant to echo, but not be the same as, that of Simons (1987, pp. 44–45), where for the Fs to serve as a “base” in an atomless mereology means \(\forall x \exists y (Fy \wedge y < x)\) and \(\forall z ((Fz \supset (z< x \equiv z < y)) \supset x = y)\).

  20. For scattered objects, see Cartwright (1975). “Connected” should be read as the topologists’ “path connected”, but not as their “simply connected”—that is, I allow for regions with holes.

  21. I try to treat the word ‘plurality’ as a plural collective noun (what better candidate for one?).

  22. As with “base”, this usage is meant to echo, but not be the same as, that of Simons (1987, p. 357).

  23. See p. 290. I like to think of patterns as unifying and generalizing Simons’ subsequently developed notion of “integrity”.

  24. Each “bit” of information is basically a yes-no answer—in this case, one for each of the seven independent base properties assumed earlier. There would also be some computational overhead for header information like image size and such.

  25. Though it would not endorse the full “Doctrine of Arbitrary Undetached Parts” from van Inwagen (1981), since small enough subregions will not have enough of the pattern to be compressible.

  26. Let rr be the base plurality over region R, and let K(rr) be the Kolmogorov complexity of the configuration of plurality rr—that is, the length of the shortest input to a reference universal Turing machine U required to output that configuration. U is normally understood to take inputs encoding the Turing machine (or “program”) to emulate, and the input to that emulated machine, as (p, n). Let P(rr) be the shortest p such that U(p, n) returns the configuration of R, where the length of (p, n) is minimal (that is, \(K(rr) = |(p,n)|\)). This shortest such TM that most compresses the rr is thought by Kolmogorov complexity theorists to represent what I am calling the “pattern”; the n represents the noise. So as a first pass, we want to say that if for some region S containing R, \(P(ss)=P(rr)\), then the rr do not compose anything.

    There is a complication here, though, because one part of the regularity of a region can be its shape. Thus a perfectly spherical region embedded in an irregular clay lump will have not just the clay pattern, but also the comparatively regular sphere pattern. This added regularity will show up in the “pattern” part of the encoding for its configuration, and so the “same pattern” will not extend to the totality. The fix for this is to factor out the region specifications by looking at the conditional Kolmogorov complexity. Thus let K(R) be the Kolmogorov complexity of specifying just the region R, without its plurality and their properties. Then K(rr|R) is the complexity of the configuration conditional on its region specification, and the proper statement is this: if for some region S containing R, \(P(ss|S) = P(rr|R)\), then the rr do not compose anything.

  27. Unger (1980).

  28. van Inwagen (1990, pp. 216–219).

  29. Shorter, unless the two subroutines are frequently paired over a larger region, in which case you would gain savings. But in this case we plausibly have what I later call super-composition.

  30. Suppose S is a subregion of R. Let \(R {{\setminus}} S\) be the region difference of R minus S (note this resulting region might be disconnected!), let \(rr {\setminus} ss\) be the resulting plurality, and let \(K(rr {\setminus} ss)\) represent the Kolmogorov complexity of that plurality’s configuration. Then for the plurality of R to compose something, it must be the case that \(K(rr {\setminus} ss) + K(ss) > K(rr)\). In other words, including the information in S must make it easier to compress the rest of R.

  31. More formally, if rr is the plurality over the table-lump region, tt is the plurality over the table region, and cc the plurality over the clay region, it seems \(K(rr) = K(tt) + K(cc) + c\), where c is needed to perform the conjunction. Thus \(K(rr) > K(rr {\setminus} cc) + K(cc)\), in violation of the more formal version of POP-Min. In general POP-Min implies that no composite can be partitioned into subregions \(\{S_{i}\}\) such that \(\sum _i K(ss_i) + c = K(rr)\).

  32. Note that this notion of “anomalous boundary” does not appeal to internal cohesion—a diachronic notion, presumably, and a problematic one for composition anyway, as van Inwagen (1990) argued. Something like cohesion may explain why boundaries are anomalous—or vice-versa—but I do not assume so.

  33. Parts must be simpler than wholes as a simple consequence of POP: the best compression of the totality would get more for less by doing the putative whole without detailing the putative part separately. And wholes must be simpler than the sum of their parts or else we will have a non-trivial partition that violates the POP-Min formal inequality.

  34. As part of a pyramid, though, the pattern of “this block here, then that one here” extends further than any two, so when embedded within a pyramid two adjoining blocks are not maximal, and therefore do not themselves compose anything further. All blocks in one layer of a pyramid are likely to be maximal, though, since their best compression probably employs the more specific “blocks placed according to this square of locations” algorithm; the regularity of the square shape makes a more concise pattern that does not extend over the pyramid, and the best compression of the latter probably employs the former. If so, we have blocks composing layers, which in turn compose a pyramid.

  35. Whether an otherwise completely disordered pile built by careful deliberation counts as a composite will depend on the role intentions play in the diachronic account of patterns (I suspect it will be a significant role).

  36. In some moods, perhaps too beholden to my view, I confess I am not even willing to grant this: controlling for teleology, I want to insist that two similar-enough things in contact just do compose, even “pre-theoretically”. It reminds me of the case Hart (1991) makes that four grains are minimally sufficient for a heap—a case Williamson (1994) calls “astonishingly plausible” (p. 213).

  37. I am taking something like wide reflective equilibrium for granted as the methodology; see for example Daniels (1979).

  38. Four, if we count the requirement for synchronic composition that it must be over a connected region of space.

  39. van Inwagen (1990, p. 138).

  40. Schrödinger (1945).

  41. See Kitcher (1989) for unificationist explanation.

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Acknowledgements

Thanks to Karen Bennett, Daniel Dennett, Maureen Donnelly, John Keller, Dan Korman, David Kovacs, Kris McDaniel, Catherine Nolan, James Overton, Lewis Powell, Ken Regan, Barry Smith, Terrance Tomkow, Peter van Inwagen, Dean Zimmerman, the anonymous referees, and others.

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    Steve Petersen

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Petersen, S. Composition as pattern. Philos Stud 176, 1119–1139 (2019). https://doi.org/10.1007/s11098-018-1050-6

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