Orthogonal Complexity

by Peter Grice

Something resembling Christianity must be true, in my view, due to a pervasive phenomenon I'd like to call orthogonal complexity. It is distinct from two related concepts, irreducible complexity and specified complexity, as elaborated below.

All three concepts fall under the general category of teleology. Telos is a mode of explanation described by Aristotle,1 where a physical object or system has a purpose that exists in prior causal relation to its features of form and function. In other words, its traits serve the interests of a goal.

For example, we understand that a steak knife is for cutting steak. Its own teleological ‘end’ helps to explain both why the knife exists (to function for cutting steak) and why it has particular features (such as its serrated edge and proper balance when held by a human hand). Although a steak knife could be fully measured and described scientifically without invoking its known purpose, this would be a reduced rather than complete explanation.2

Given the inability of steak knives to intend and manufacture themselves, the clear implication is that they are artefacts of beings with sufficient intelligence and creative power. While this is not disputed for steak knives, it certainly is controversial when it comes to human beings and other biological systems, for obvious reasons.

Yet it seems all too easy to dismiss contemporary discussion about this as “merely an updated form” of William Paley’s argument3 – whatever that might mean in detail. It is precisely the detail that matters, since the design argument is not unsound. Rather, its application is disputed. Our knowledge of biological complexity has come a long way in the past 200 years, making it more applicable than ever to the question of telos in the organic world.

Irreducible complexity4 is the notion that all constituent parts are necessary for a given biological system to maintain its function relative to the organism.5 Specified complexity6 refers to systems that are both specified, as with a single letter of the alphabet, and complex, as with a string of letters. If verified, either of these concepts would show that any stepwise, trial-and-error meandering of naturalistic evolution has in fact been transcended by intelligence.

What I mean by orthogonal complexity7 is the confluence of multiple linear pathways of development, in a coordinated way, resulting in an emergent structure or pattern of different dimensionality. This pattern, such as the impressive fan of “eyes” in a peacock’s train, would be characterised as epiphenomenal, complex, specified and also digital in terms of traversing discontinuous structures (as with pixels on a computer screen). The feat must be accomplished via advanced calculations and conceptual mergers far beyond the capacity of undirected, linear processes to procure. While strictly reducible to physical constituent parts, the presence of an effect is real. It dissipates rather than participates in a physical reduction, so in that sense it is also irreducible.

Imagine an exquisite tapestry – its ornate, intricate design the trademark of a particular family of artisans, along with the knowledge of precise over-and-under weavings for its reproduction. Reflect for a moment on the necessity of the craftsman to the process.8 One could attempt to explain this away by unraveling the weave, one strand at a time, to show the tapestry composed entirely of linear threads. Yet this is inadequate as a full explanation, since it excludes genuine data – the telos of the arrangement.

Tapestries exhibit orthogonal complexity in the way their vertical ‘warp’ threads interlace with horizontal ‘woof’ threads. There is further orthogonality at each point of virtual intersection, with its calculation to reference the superimposed design. The canvas is an assemblage of linear threads and not a continuous flat surface, and therein lies the challenge.

So it is with a peacock’s tail, only here the physical “canvas” comprises myriad linear filaments of different scales, in fractal-like configuration, fixed in precise positions in space to facilitate the overall arrangement. Just as a pile of threads would seem a poor choice on which to paint a masterpiece, so is the peacock’s splay of feathers entirely nonconductive to a two-dimensional picture. Yet it is plain to see one superimposed.9

In the case of the rounded “eye” of a single feather, this involves a requisite colour abruptly starting, continuing and stopping along a given barb or barbule – all at precise locations and specified lengths that only make sense within the overall scheme. Adjacent elements of the design are juxtaposed on adjacent digits, with empty space in between.

The mappings involved are analogous to mathematical transformations between lower and higher dimensions. The colours themselves are effects of complex 3D microscopic structures known as photonic crystals,10 introducing yet another complex transformation. In fact, the whole panoply unfolds from a linear encoding of information inside DNA.

If this boggles the mind of human beings,11 one has to be suspicious that it all ensues straightforwardly once the humble peahen conspires with nature to simulate a master weaver. We are asked to believe that the mating preferences of peahens largely account for this phenomenally complex feat, despite the disputed nature of any evidence for this.12 Even the brightest human minds could not produce such a masterpiece without indulging in mimicry.

Multiple interposed levels of orthogonal complexity cry out for adequate explanation. Just as a relatively simple tapestry necessitates a weaver, so it would seem that nature’s orthogonality requires transcendent, intelligent, creative causal agency.



  1. Aristotle assigns telos the role of “Final Cause,” from his doctrine of the Four Causes expounded in his text Metaphysics.
  2. Hence the pejorative sense of the term reductionism. Within the full range of data present to human understanding, whole categories exist that seem to fall outside the bounds of what science alone is capable of analyzing.
  3. Paley’s design argument, from his 1802 work Natural Theology, takes this form: if we were to chance upon a wristwatch on some remote ground, we would realise its obvious purpose in measuring time, and infer from this that it had been designed. By analogy, it seems rational to make the same kind of inference from the apparent purposiveness of biological systems, to an intelligent cause.
  4. A concept first put forward by Michael Behe in his bestselling Darwin’s Black Box (1996).
  5. My wording here is significant, since critics have suggested that some parts or substructures of a proposed irreducibly complex system have been co-opted from other contexts, yet this appears to sidestep the claim, which is about the particular system’s function in its present context.
  6. Championed by William Dembski in The Design Inference (1998).
  7. In proposing my own concept I don’t mean to imply that it isn’t subsumed by the work of Behe, Dembski and others, or that it is rigorously formulated elsewhere (I am not a complexity theorist). Nonetheless I trust that my humble observation will provoke the reader to reflect on candidates for orthogonal complexity and their adequate explanation.
  8. While afterwards it may be reproduced mechanistically, as with a Jacquard Loom, this wouldn’t have been possible without the initial involvement of an intelligent agent.
  9. While there is orthogonality in the diverging and converging growth process, the more interesting and sophisticated orthogonality is in the superimposition of the familiar 2D design on to the underlying structure.
  10. See for instance, http://www.nnin.org/doc/2007nninREUSmyth.pdf
  11. Little wonder Charles Darwin wrote to a colleague, “Trifling particulars of structure often make me very uncomfortable. The sight of a feather in a peacock's tail, whenever I gaze at it, makes me sick.”
  12. Takahashi et al., Peahens do not prefer peacocks with more elaborate trains; http://bit.ly/aK3BzL