Tuesday, June 2, 2009

Fitness Landscapes and Kuhnian revolutions

Fitness Landscapes

A fitness landscape, in evolutionary studies, is a map of various combinations of characteristics along with their survival/reproductive fitness. Imagine that all the combinations could be mapped into two dimensions, the third, height, would represent fitness.

Figure 1: A sample fitness landscape of characteristics in two dimensions. (From Rosio Pavoris)

Although the notion began with Darwinian evolution in biology, it has been applied to all sorts of scenarios, from corporate management to computer programs. What I'm going to do here is apply it to scientific theories, although not quite in the manner of Scharnhorst et al.1

Basically, We'll start with each point on the map representing a complete theoretical description of how the universe works, or a subset of theories representing a complete scientific field. Any change to any theory, even just tweaking a constant, represents a slight movement to another point on the map. Obviously, the overall map represents a lot more than two dimensions, but for illustrative purposes I'm going to try to keep it to two, making it easier to visualize.

The third dimension, height, will represent how well, or closely, the total theoretical description represents actuality. Any actual representation of scientific theory will be considerably more complex than this, since we must allow for unknowns (waiting to be determined experimentally) and many theories don't actually fit into continuous dimensions but rather discrete. However, I believe I can communicate the essence with a continuous two dimensions.

Introducing Kuhn

Now, normal science may be compared to climbing the current hill. The hill itself may be compared to a Kuhnian paradigm: ...
According to Kuhn, the scientific paradigms preceding and succeeding a paradigm shift are so different that their theories are not comparable (incommensurable). The paradigm shift does not merely involve the revision or transformation of an individual theory, it changes the way terminology is defined, how the scientists in that field view their subject, and, perhaps most significantly, what questions are regarded as valid, and what rules are used to determine the truth of a particular theory. Kuhn observes that they are incommensurable — literally, lacking comparability, untranslatable. The new theories were not, as the scientists had previously thought, just extensions of old theories, but were radically new world views. Such incommensurability exists not just before and after a paradigm shift, but in the periods in between conflicting paradigms. It is simply not possible, according to Kuhn, to construct an impartial language that can be used to perform a neutral comparison between conflicting paradigms, because the very terms used are integral to the respective paradigms, and therefore have different connotations in each paradigm. The advocates of mutually exclusive paradigms are in an invidious position: "Though each may hope to convert the other to his way of seeing science and its problems, neither may hope to prove his case. The competition between paradigms is not the sort of battle that can be resolved by proof." (SSR, p. 148). (From Wiki)

Anybody familiar with science knows that the use of mathematics by science is heuristic: mathematical terms are "defined" for use in a scientific field because they work: the mathematical relationships model the real world. Thus, the assignment of "meaning" to mathematical terms is actually part of the structure of definition and theory: the paradigm. In pure mathematics, symbols have no real meaning relating to the real world, they are simply part of an internally consistent grammar.

The same is true for all terms used by science, in a less rigorous sense (sometimes). Which means that any specific point on our "map" represents a system of "meaning" for terms as well as a system of theories about how those terms relate (i.e. how they map to some mathematical system).

Thus, a jump from one hill to another can cleanly represent a Kuhnian paradigm shift. Paradigm shifts aren't necessarily as catastrophic as the examples he used, or the introduction of plate tectonics in the '70's. Just as hilltops can be far apart and separated by deep valleys or close together and separated by just a saddle (or anything in-between), a paradigm shift can be anything from a major redefinition of a scientific field to a minor semantic shift.


Now, one of the things that such paradigms include is a (loose) definition of parsimony. I've written on this subject before, but my current thrust here is somewhat different. The definition of Occam's "entities" is clearly part of the paradigm, and the distinction between a mechanism as one entity and a number of different events as many can make the difference between which theory (or theoretical system) is more "parsimonious". Indeed, parsimony is one of those terms that changes its meaning between paradigms, and thus can instigate great controversy.

To pick an extreme example, consider the divide between creationism and "evolutionism". For most scientists, evolution is a set of mechanisms, including the various causes of mutation. But to the creationists, each mutation represents a separate entity (at least the more "improbable" mutations). Conversely, to the scientist each separate act of "divine intervention" represent a separate entity, while religious fundamentalists consider God a single entity (in much the same way that scientists regard mutation). Thus, each consider their position the most "parsimonious".

Another Example: Redefining "Emotion"

Now, I want to go on to a more moderate example, something I mentioned in a recent discussion. The basic idea comes (to me) from Cancace Pert's Molecules of emotion: That what we call emotion is basically chemical, caused by various hormones in our bloodstream. Now, I'm in a quandary here, because the further development seems obvious (i.e. state of the art) to me: why don't we redefine "emotional state" as simply an n-dimensional vector where n is the total number of hormones floating around in the bloodstream. I took this directly from Pert's book back in '98 or so when I first read it. But after re-reading it, I don't actually see that suggestion made, so I don't know how obvious the idea turns out to be.

But obvious or not, such a redefinition represents a paradigm shift, if only in a small sense. Like any paradigm shift, it changes the meanings of certain terms (such as "emotion"), and redefines the universe of questions that rigorous researchers can ask.

New Areas for Research

If we define the "emotional state" of the individual as an n-dimensional vector where n is the number of hormones, and the intersection of the vector on each dimensional axis is the concentration of that hormone, then we can both put the whole issue of emotion into rigorous terms, and extend the study of emotion to animals without the risk of anthropomorphism. This opens up a number of regions for rigorous research, including mapping the classical "emotions" to precise study. Take a simple emotion such as "anger" for instance. Defining it in classical terms is fairly easy, but finding correlates in other mammalian species is much more difficult. But if we can map the expression of "anger" in humans to a specific territory in the n-dimensional space of possible emotions, we can study animals from fish to birds to apes looking for equivalent territories and the behavioral signs associated with them (if any). These can then be compared back to the "emotional" expressions the even Darwin noticed, looking for correlations.

Another fruitful subject for research is the individual variation in "mapping" the feeling of classical "emotions" to the hormone levels involved. These variations, in turn, can be analyzed for correlation to genetic differences, cultural differences, and differences in background, educational opportunities, etc.

The functional limits of "emotion" in this new paradigm also offer great research oppportunities. The settling time for changes to concentrations in the bloodstream can be measured in something like tens of seconds to minutes, probably sufficient for most emotional responses. However, mechanisms exist for much faster internal communication: the autonomic and limbic nervous systems. These systems seem to deliver a variety of chemicals that act as hormones in the body, but their delivery seems to be both faster and more precise than hormones dissolved in the general bloodstream.

Generally speaking, the autonomic system (or systems: there appear to be three discrete systems, the sympathetic, the parasympathetic, and the enteric nervous system nervous systems) deliver their messages throughout the body. This may all be a simple matter of "front-loading" parts of the body to respond more quickly to emotional reactions than could be accomplished by waiting for the bloodstream to distribute them. Or perhaps not: a fruitful area for research.

The limbic system is somewhat more controversial, but a least part of it probably (IMO) does the same thing for the brain, allowing thought processing to shift emotional modes at the same speed as other aspects of thought. However, it seems likely (to me) that parts of the limbic system are doing more than just "emotion". If hormones are being "front-loaded" to the brain to allow it to react faster to a general change in hormone levels, that would reasonably constitute "emotional" activity. But what if neurohormones are delivered selectively to only a few areas of the brain? This could be simply because only those parts need the "front-loading", but it could also be that this delivery is part of a more "calculating" process. A very fruitful area for research, IMO.

The generation of "emotions", that is the generation of alterations to the total, chemical, emotional state, is another fruitful area for research. It appears to be established that many parts of the body, including the immune system, are capable of releasing hormones with "emotional" content, and in any event, given that we probably haven't discovered all, or even the major fraction, of the hormones in play, any type of cell in the body could be generating its own "emotional" reaction to its current state, with the results being integrated into signal of the better known "emotional" hormones, perhaps in the pituitary. However, it's obvious that regular brain activity is also involved in generating emotions. How? Could we possibly create a network map of all the influences, both neural and hormonal?

Back to Fitness Landscapes

Enough of this example, let's return to the core subject for this post: fitness landscapes. In normal science we can see scientific activity (research, publication, money) clustering around some particular point of the map, which we may equate to a "hillside" in the fitness landscape. Research activity will correspond to "climbing the hill", that is developing a "fitter" overall theoretical structure by answering open questions and discovering currently unknown facts. However, this all takes place within the current paradigm, which we have equated to a single hill. A move to another hill is a long-distance jump, which will likely open up a considerable opportunity for new research, especially if the current "hill" has already been pretty much climbed.

The key question then, in regard to the example I've discussed, is how much incentive there is to make the "paradigm shift": to allow the redefinition and begin researching the questions it opens up. This, in turn, will depend on how much "hill climbing" there remains to be done on the current hill. There are limited amounts of money to pay for research, and efforts to do research in a new paradigm will engender opposition, even hostility and perhaps ridicule, from people ensconced in the halls of power of the current paradigm. Since that paradigm generally has to dismiss "emotions" as vaguely defined, at best, and thus not good objects for research, it comes to a simple question of whether to allocate money for answering questions that make sense in the new paradigm, but don't in the old, at least to the level of rigor needed to justify money.

It's an interesting question. I wonder how it'll turn out.

Links: (Not all of these are peer-reviewed, and not all are called out in the text.) (I've included only the link referenced in this leader.)

1. Scientific information in continuous characteristics spaces

2. Philosophy is to science, as ornithologists are to birds: 3. Science is a Dynamic Process

3. Modeling evolutionary landscapes: Mutational stability, topology, and superfunnels in sequence space


  1. You might like to check out my paper:

    John S. Wilkins (2008). The Adaptive Landscape of Science. Biology and Philosophy 23 (5):659-671.

  2. Thanks for your comment, John. I'd like to, but there seems to be a paywall.

    I generally try not to reference papers behind a paywall, because even if I can see them, my readers can't. I'd like to read it, but I can't even do that.

    But thanks for the reference.