Friday, April 09, 2010

Essay: Orthogonal Complexity by Peter Grice

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. (MP3 Audio | RSS | iTunes)

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.

I began by implying that this is part of an cumulative case.  Personally, I regard the evidence for Christianity to be broad-ranging and convincing, and I encourage readers to explore this through other essays in this series.  I trust that my contribution has at least highlighted a major point of departure between rival explanations.  Is it legitimate for an anti-supernatural philosophical stance to reject out of hand whole swathes of potential evidence for Christianity?  It seems to me that this issue turns on the quality of complexity we are now discovering, which undermines the claim that telos in biology is illusory.

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,
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;


Ken Pulliam said...


Thanks for the essay. I agree with you that the universe appears to have some design. Whether this is real or apparent is the question. As you know, philosophers debate the teleological argument.

One thing that I find interesting is that if biology so clearly points to a designer, then why aren't more biologists theists? It would seem that since they have the best understanding of what is going on there, they would be forced to conclude design and yet most don't. I know this is not a formal argument because of the fallacy of ad populum but I don't think it can be ignored either.

Peter said...

Hi Ken,

Appreciate your comment, thanks.

I should think more biologists aren't theists because they are generally not philosophers, and conversely, philosophers might dispute the presence of telos in organisms, to the extent that they aren't savvy with biological advances. This might sound like smoke and mirrors, but I do think (along with Kuhn) that specialization can prop up a failing paradigm, since one might use the consensus of another department as a baseline, assuming it is firmly established (when in fact they could be making similar assumptions).

The previous two essays contribute to an explanation by flagging naturalistic interpretive bias. [Moreover if Christianity happens to be true it would seem there is a fundamental human motive to prejudice naturalistic interpretation, but that obviously can't be used in service of arguing for Christianity.]

So I tend to think those factors that might account for a mistaken consensus are reasonably glimpsed now, and would emerge in more detail should the consensus change.


Ken Pulliam said...


Thanks for the reply. I think its crucial to make a distinction between methodological naturalism (which science must practice) and metaphysical naturalism (which is solely the domain of philosophy).

While the unexplained in science may lead one to a philosophical conclusion that there must be a God, it needs, in my opinion, to be clearly stated that science as a methodology can neither prove nor disprove God.

Peter Grice said...


Certainly. That distinction has different construals, and is arguably not properly practiced, but I agree that rightly construed it is a useful distinction in principle. It would be a category mistake to seek or ascribe empirical proof to a supernatural being. However, in Christianity God is seen as the ultimate cause, orderer and sustainer of all the things with which empirical science has to do. The posited supra-natural domain is qualitatively distinct from but not causally unrelated to the natural domain of regulated, physical processes. Cast in that light, methodological science would actually be probing epiphenomenal effects, or contingent events, not ultimately closed causal systems.

But naturalism is nothing if not the dogma of closed causal systems. Its core assumption is the mechanistic paradigm, and as such it is inherently reductionist. Naturalism proper, or the worldview of metaphysical naturalism, is an extrapolation from empirical science. Not from its results per se, but from the epistemological program itself. However illegitimate it may be to prejudice that philosophical stance, and idealistic to keep them distinct, we do have a situation where the project is lauded as an epistemology for our time, yet surreptitiously reduces causality itself. The net result is a problematic admixture of ideology and method.

I notice you flagged the unexplained in science as being a potential platform to infer God, but that seems both speculative and tending towards a god-of-the-gaps approach. If you had in mind other aspects of reality, such as human experience, aspirations, and so on, where claims to ascertain the transcendent might be more legitimate, I'm not sure these would be outside the purview of science. Stephen Gould's "non-overlapping" scheme might have seemed like a good idea at the time, but the reality is science crossed over into philosophical territory long ago [and vice versa]. Assuming mechanism as it does, broadly speaking, reductionist implications are unavoidable, and these are integrated into new hypotheses.

For example, discovering that our moral sensibilities are subject to magnetic interference is one thing, but to interpret this as evidence that our thoughts are solely physical is quite another. Yet the latter is the "natural" way to construe things in the current milieu. Philosophers of mind do get involved in this area, quite legitimately I think. It is not philosophy or metaphysics that have no import here, but rather purely speculative metaphysic, and there is a difference.

Better I think to seek the unification of knowledge. Better to include the conceptual prowess of philosophers regarding the constructs deployed in biological science, rather than adopt a sort of non-philosophical naive empiricism.

Peter Grice said...

Back to my essay, though, I'm not sure that issues of consensus or what science is (or should be) is the primary concern. If we can locate the issue more fundamentally as "explanation," deploying all the best of our rational resources and all the datum of our experience, perhaps we will be in the most unprejudiced position. In my quest I'm going to run with what I think we can and can't know, and can and can't infer, about reality. Science is an enormously useful tool in service to this. The ideal human being (from which I'm very far) is a fully equipped scientist-philosopher-artisan-athlete-provider-steward-lover-theologian-humanitarian, or something along those lines. The unification or integration of human knowledge and experience may turn out to undermine or expunge one or more traditional categories — theology for instance — but it is still integration we should pursue, on the assumption that all things participate in relation to some unified reality. Classically speaking, we should pursue at least the True, if not also the Good and Beautiful.

I see orthogonal complexity, if acknowledged, to be a paradigm-determining concept. Not just one "conceptual merger," which could easily be coincidental (and therefore not a true conceptual merger, from a mind). Natural selective pressures can indeed result in an aggregate of coincidences, given enough time. However, when one finds apparent high-level "conceptual mergers" extending every which way, dimensionally, interposed and superimposed, and highly localized and congruent with each other, this is a different challenge. It is not just that it might propel complexity into stratospheric orders of magnitude, in turn straining to breaking point the available time-frame for this to have occurred naturally [though that would be a strong argument]. Rather, it is a direct challenge to the assumption of linear (and closed) causal processes in the first place. There is a definite sense that you "can't get there from here," and this is a prima facie rational inference from empirical data.

It awaits rigorous mathematical formulation (which I conceded in my essay), but that's all the more reason to pursue it. "Mathematical certainty" is a worthy goal, but when placed in a scientific context, assumption and inference is always involved, and "proof" becomes an elusive notion. Those who properly understand science are aware that scientific knowledge ought to be regarded as tentative. So I commend tentatively held inferences as the proper sense for one's beliefs, whatever they are, and complete or unified explanation as the ideal goal.


Ken Pulliam said...


You said: It would be a category mistake to seek or ascribe empirical proof to a supernatural being. However, in Christianity God is seen as the ultimate cause, orderer and sustainer of all the things with which empirical science has to do. The posited supra-natural domain is qualitatively distinct from but not causally unrelated to the natural domain of regulated, physical processes.

But your conclusion (or inference) is a theological or philosophical one--not a scientific one.

You also say: Naturalism proper, or the worldview of metaphysical naturalism, is an extrapolation from empirical science. Not from its results per se, but from the epistemological program itself.

Metaphysical naturalism is an inference drawn from the results of science but so is methaphysical supernaturalism. They both belong in the realm of philosophy not science.

The problem with replacing methodological naturalism with metodological supernaturalism in scientific experiments is the fact that supernatural activity cannot be observed. We have no tools for which to see it or examine it.

I have no problem with you drawing theological inferences from the scientific data but I can't see how you can call it anything but theology or philosophy.

Edward T. Babinski said...

Is the "Peacock Feather" the Newest Icon of I.D. & Creationism?

Peacocks in general are members of the pheasant family, see pic:
There are two species of brightly tailed pheasants assigned to the genus, Pavo, see pic: and a third species of brightly tailed pheasant assigned to its own genus, Afropavo, see pic: Therefore, out of the many pheasant species that have ever evolved on earth only three living species presently exist with brightly colored tails.

But that's how natural selection works. Only a few adaptions ever evolve to extremes, while the majority remain somewhere in the mean. Take the example of evolution of mammalian brains. There's large-brained mammals among dolphin, elephant and ape species. And there are fossilized species of upright apes with larger cranial capacities than modern apes. And there's a few homo species like Erectus and Neanderthal, and some other species of homo as well that have recently been discovered. In other words out of all large-brained mammalian social species there is only a single living species that developed writing and an impressive array of knowledge about the world that we can keep adding to, generation after generation, and thus manipulate the world to a greater degree than other large-brained social mammalian species.

Getting back to the peacock, some research has been done on the colors found in peacock feathers:;100/22/12576

As for the pattern of the colors, all animals exhibit coloration patterns, whether spots, lines or stripes. Patterns occur on fish scales, on mammalian skin and/or hair, and on feathers. Some patterns are more distinctive than others, and some less distinctive, some patterns cover more of the body's surface, some less of the body's surface. And patterns are subject to change AS SPECIES CONTINUE TO EVOLVE. One fish for example appears to have recently evolved "eye" spots on a feathery portion of its body, see links
something that the vast majority of other members of its fish species lack. Some butterfly and moth species have spots on their wings, some big cat species such as leapords have spots, and peacock feathers have spots.

Exactly how patterns evolve goes back to genetics. The peacock feather itself consists of a shaft and the "eye spot" is produced by the symmetrical growth of feathery tendrils out of opposite sides of the same central shaft. As you move upward along the shaft, slightly different gradients of chemicals are released producing a symmetrical pattern in the feather. Circular patterns are produced by gradients of chemicals released less generously, and then more generously, and then less generously as you move up the shaft. In fact circular bisymmetrical patterns are what one would expect from a central shaft producing tendrils out of both sides. Non-symmetrical patterns radiating off a central shaft would be less easily explicable, such as patterns resembling a letter Q, or E split down the middle by the same central shaft.

Also, peacocks fit in with what might be expected in a world of evolving species. Before any new traits can arise and be carried to the next generation a species must be able to survive. Hence, peacocks can fold up their tails, which lessens the danger of being noticed by predators or getting caught in the bushes. In addition, as with most species of bird where the males are colorful or fancy to attract the females, the females or peahens are much duller and more camouflaged than the males, in large part because they are the ones who guard the eggs and chicks.

Edward T. Babinski said...

Lastly, at what point are the creationists and I.D.ists going to throw in the towel? In the past they have claimed that there's no way for a human to evolve from an ape, no way for whales and dolphins to have evolved from land-based mammals, no way for bats to have evolved, no way to explain the evolution of the blood clotting mechanism, or the evolution of bacterial flagellum. And in each case the creationists continue to move the goal posts in retreat. Now what, the peacock feather is going to save creationism or I.D.?

How many creationists and I.D.ists know Micahel Behe, author of Darwin's Black Box, and senior member at the Discovery Institute has come out in favor of common ancestry of humans and apes based on the genomic evidence, including not just overall similarity but based on shared non-functioning genes (for creating vitamin C for example, that are still there but no longer functioning in both chimps and humans), and shared non-functioning genes from retroviruses that have inserted themselves into the same regions of human and chimpanzee genomes. Even young-earth creationist, Todd C. Wood, who teaches at Bryan College has admitted that creationists don't have a superior point of view when it comes to the genetic evidence for shared common ancestry of humans and apes. He composed an entire research paper on the subject, and also admitted:

"Evolution is not a theory in crisis. It is not teetering on the verge of collapse. It has not failed as a scientific explanation. There is evidence for evolution, gobs and gobs of it. It is not just speculation or a faith choice or an assumption or a religion. It is a productive framework for lots of biological research, and it has amazing explanatory power. There is no conspiracy to hide the truth about the failure of evolution. There has really been no failure of evolution as a scientific theory. It works, and it works well."

Wood also addressed the recent flurry of creationist hope/hype concerning the latest findings concerning Y chromosome differences between chimps and humans:

P.S., When and if the genomes of pheasants from around the world are known and compared with the genomes of the three living species of peacocks, we'll begin to get a better grasp on which living pheasants are the closest ancestors to peacocks, and also what pseudogenes and retroviral genes their ancestors shared, and the kinds of genomic changes that led to the colorful tails of peacocks. Also, there's already enough genomic evidence in from bird species around the world that has led evolutionists to agree that all perching birds species probably originally evolved in south east asia, i.e., not dispersing from Noah's Ark in the middle East, not at all.

Peter Grice said...

Hi Ken,

Thanks again very much for your perspectives on this.

Again I think the categories we use are valid in principle. I agree that supernaturalism is to science as naturalism is to science — they are both as metaphysics to physics. It is clear enough that one should not be prejudiced over the other, but equally clear that resolving toward one or the other should be a pursuit of any open-minded quest for knowledge about the way things are. If we can't go sufficiently far to make a determination either way that's fine, but we can't know that from the outset.

I'm not particularly fussed on what anything is labeled, so long as labels are used consistently (often they're not) and in service of unification rather than fragmentation.

For example I said "as metaphysics to physics" above. But did I mean to refer to nonphysical and physical aspects of a single reality (ontology), or did I have in mind the associated fields of inquiry (epistemology)? One can see intuitively that approaches to inquiry might be more varied and distinct than the domains themselves. Even to ascribe "domain" to existence seems a perspectival distinction that may not be respected by reality itself (as Whitehead bemoaned the "bifurcation of nature").

Human inquiry might involve distinct tools and methods, but one takes it that reality is not similarly compartmentalized into special discontinuous categories. Explanation should be parsimonious as far as reality permits.

"Explanation in philosophy and science is inherently unifying, subsuming a multiplicity of phenomena under classificatory unity. [However,] Proponents of monism... must attend to the risk of neglecting genuine data and truths resistant to a monistic explanatory scheme. What monism gains by unification of multiplicity in data may be
lost by neglect of genuine recalcitrant data. Explanatory unity may be a virtue, but it will be virtuous only if pertinent truths and data are not excluded for the sake of theoretical simplicity." - Moser & Yandell.

Which is why philosophy is connected to the scientific enterprise. Positing monism (closing causality) is a philosophical stance, but it's not entirely not-science. Similarly, positing a transcendent causal principle is not not-science. They are not scientific data, they are not scientific method, but they do proffer scientific hypotheses about physical systems, that can be tested (at least in principle).

As is true in a lot of areas of life, knowing where to draw a distinction (and conversely when to dissolve an artifical one) is a critical activity.

For instance: is there such a thing as a theological inference? Of course, if one means to say that it is an inference to God. But why isn't it termed by what it is an inference *from,* which is the pole more solidly established? And are there really species of inference, or is it rather unhelpful to qualify an inference since inferences remain what they are?

Don't forget that Christian theology is not theology in general, and I have not really had in mind the mere study of God, but the specific claims for ontological reality that flow from the Christian hypothesis in that area. Once again I affirm the category mistake of seeking to "prove" a nonphysical entity empirically/physically, however, my discussion is located in the area of causality and effects. Both are real in principle, and intimately connected, and insofar as the full scientific enterprise can legitimately infer causes, philosophy is already involved, and transcendent causality should not be foreclosed prematurely.

Peter Grice said...

Hi Edward,

Thanks very much for your comments, particularly the portions relating to peacocks and animal coloration, which I will interact with below. I don't here plan on debating anything/everything that might be relevant to the controversial claims of I.D./creationism: the extra material you supplied is being discussed (some would say ad nauseum and without sufficient clarity) on forums dedicated to the broad scope of the debate. At a glance though, one could concede most or all of your points and still affirm what I said (such as from within the stance of dynamical monism, or, as you pointed out yourself, Behe's acceptance of common descent).

On animal coloration, essentially, I agree with you.

The first set of two links you supplied were to the same 2003 paper (which I had read in the course of my research), which is itself referenced in the report I linked to above (ref. #10). I chose a basic summary for a general audience, but the range of relevant discoveries in this area of late is both broad and exciting.

In my discussion I affirmed the structural nature of the peacock's plumage (and the photonic crystals used to generate color), with its linear, developmental unfolding in various growth patterns. This is all reasonably well understood.

My argument was not at all that God somehow caretakes this process as it occurs, and I felt I was quite clear that the whole system can be physically reduced/reverse-engineered (which implies its linear development with the unfolding organism, though not step-wise gradual historical emergence).

The patterns are not merely 'circular,' involving as they do cardioud [r = sin 0/2] and ellipsoid [x2/a2 + y2/b2 = 1] curves, and considering the ogee curves of the 'T' feathers as well (extending from the top of each main feather). But whatever their complexity, it is constructive, so they may easily be reduced to equations (which can further be 'reduced' to linear genetic bit-encodings).

You said: "Non-symmetrical patterns radiating off a central shaft would be less easily explicable, such as patterns resembling a letter Q, or E split down the middle by the same central shaft."

Asymmetrical patterns in themselves would not be any less physically reductive [I take it you didn't mean to suggest anything conspicuous about resemblance to English letters as per this image ]. Same for asymmetrical bilateral patterns. [This study shows that bilateral asymmetry in insects is not as common as the theory of sexual selection would predict: ]. In fact, any conceivable pattern of animal coloration will be physically reductive because it is embedded in and emerges from DNA. It might be fruitful to inquire into the uncanny resemblance to external objects of some types of camouflage (although natural+sexual selection is confident in having explained this), but that too still reduces physically. If by "split down the middle" you meant to invoke the idea of digital/discontinuous structures (I don't think you did), that's a little closer to where I'm coming from.

In your second set of two links (to the same paper), this might be relevant if this Lionfish design is of a complex digital nature. Could you provide any reference either way on that? (I don't have current access and don't wish to pay $47).

None of what I've suggested rules out the use of peacock plumage for signaling and mating. Instead it calls into question its adequacy as a causal explanation.

Why does it do that? Because it is about the orthogonality involved conceptually in correlating many elements of one key feature (the pattern, with its complex photonic structures and differing coloration methods) with many elements of another (the physical discontinuous structures extended in space). The straightforward growth patterns are a part of this argument, not a challenge to it.

Peter Grice said...

"If we look at the speculum on a duck's wing, we might imagine that an artist had drawn his brush across some ten blank feathers, which overlap sideways - making white, bluey-green, and black lines - so that the stroke of the brush touched only the exposed part of each feather. The pattern is a single whole, superimposed on the individual feathers, so that the design on each, seen by itself, no longer appears symmetrical. We realize the astonishing nature of such a combined pattern only when we consider that it develops inside several or many feather sheaths completely separated from one another; and that in each individual feather it appears at an early stage while it is still tightly rolled up, the join pattern not being produced until these feathers are unfolded. What sort of unknown forces direct the construction work in the 'painting' of these feather germs?"

- Adolf Portmann, Animal Forms and Patterns

Post a Comment

Thanks for taking the time to comment. By posting your comment you are agreeing to the comment policy.

Blog Archive